Diffusion in Channeled Structures

Diffusion in Channeled Structures Benoit Palmieri

Doctor of Philosophy

Chemistry Department

McGill University Montr´eal, Qu´ebec May 2, 2007

A thesis submitted to McGill University in fulfillment of the requirements of the degree of Doctor of Philosophy c °Benoit Palmieri, 2007

CONTRIBUTIONS OF AUTHORS

I derived all the expressions presented in all manuscripts, and wrote all the computer programs. All of this with the help and supervision of David Ronis. The only exceptions are the hypernetted chain calculation presented in Ref. [112] and the general thermodynamic work bound relation developed in Ref. [137] which are the work of David Ronis alone.

I, David Ronis, hereby give copyright clearance for the inclusions of all papers (Refs. [53, 112, 125, 137, 120]) for which I am the co-author with Benoit Palmieri and which are presented in this dissertation.

David Ronis

date:

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REMERCIEMENTS

Je commence par mes parents, Paulette Tessier et Philippe Palmieri. Je vous remercie de m’avoir encourag´e dans tous les projets que j’ai men´es a` terme, mais surtout dans les autres. Pour votre soutien continu dans mes e´ tudes et ailleurs, merci. J’en aurai encore besoin. Je vous aime tous les deux sinc`erement. Pour m’avoir beaucoup aid´e, pour avoir collabor´e et construit les projets de recherche qui sont pr´esent´es ici, merci a` mon superviseur David Ronis. Merci de m’avoir appris beaucoup plus que ce qui est contenu dans cette th`ese. Je remercie aussi Juan Gallego pour son impressionnante efficacit´e et pour tous les probl`emes informatiques qu’il a r´egl´es pour moi. Pour leur support financier, je remercie le CRSNG et le FQRNT. Merci a` tous mes amis, mais plus sp´ecialement, merci a` Fred Tremblay, Karl Therrien et Ben Dinello. Ben, ton m´epris pour les e´ tudiants est plus drˆole aujourd’hui. Karl, ta curiosit´e rafraˆıchissante va me manquer. Fred, je t’ai d´ej`a tout dit vieux. Prends ces remerciements comme une grosse e´ treinte que je ne te donnerai pas. Je veux aussi remercier tous les membres du groupe que j’ai cˆotoy´es pendant mon s´ejour, mˆeme toi Tobie. Kyunil Rah pour tous les caf´es que je lui dois, pour son e´ coute int´eress´ee et pour eˆ tre une des meilleures personnes que j’ai connue. Annie Dorris et Alison Palmer avec qui j’ai v´ecu les meilleurs moments de mon passage ici; c¸a aurait e´ t´e insipide sans vous. Caroline Leblanc, je te remercie pour ta patience bruyante. Merci pour les efforts que tu mets dans notre petite famille rapi´ec´ee. Finalement, merci a` Vincent Palmieri, mon petit garc¸on, mon grand complice. Merci de m’amuser, d’ˆetre beau, profond´ement bon et de rire d’aussi bon coeur. Merci d’avoir balafr´e ma vie jusque l`a immacul´ee, de m’avoir rendu plus aimable, plus tendre. Je suis une meilleure personne depuis que tu es l`a. Je t’aime comme un dingue.

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ABSTRACT

The theory of Ronis and Vertenstein [J. Chem. Phys. 85, 1628, (1986)] is used to calculate the permeability of Xenon in Theta-1 and of Argon in α-quartz, both crystalline sodalites containing large, one-dimensional channels in the first case and narrow interconnected channels in the second. The simulated dynamics of a small part of the crystal atoms exactly reproduce those of the full crystal by the means of a generalized Langevin classical equation of motion. An approximate expression for the potential of mean force inside the crystal is derived. The Theta-1 energy landscape is smooth with small energy barriers while the α-quartz has large energy barriers to diffusion. The permeability is reported for both systems and compared in detail with that obtained from transition state theory. The role of the lattice vibrations is also investigated. For Xenon in Theta-1, transition state theory does not properly describe the diffusion process and the lattice vibrations do not play a large role. For Argon in α-quartz, transition state theory is more appropriate but there, the lattice vibrations cannot be neglected. For systems where the lattice vibrations play a role, the quantum mechanical corrections to the diffusion are computed. The diffusion is studied using the path integral formalism. Forward-Backward path integrals are combined and, using the MSR [Phys. Rev. A., 8, 423 (1973)] formalism, are transformed to a set of generalized Langevin equations that reduce to the classical equations of motion at high temperatures. The quantum mechanical treatment of the lattice vibrations results in a decreased permeability. The quantum corrections to the potential of mean force are computed from an approximate density matrix. A modification to the original Feynman-Kleinert variational method[Phys. Rev. A., 34, 5080 (1986)] to calculate quantum mechanical partition functions is suggested. This modification, which uses a time-dependent reference potential is shown to improve the calculation of density matrices. Finally, the Jarzynski equality’s connection with thermodynamics is considered.

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The work upper bound that results from the equality is shown to be an upper bound to the thermodynamic work upper bound and the free energy derived from the equality is not the same as what appears in nonequilibrium thermodynamics.

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´ E ´ ABREG

La m´ethode d´evelop´ee par Ronis et Vertenstein [J. Chem. Phys. 85, 1628, (1986)] est utilis´ee pour calculer la perm´eabilit´e du X´enon a` l’int´erieur du z´eolite Theta-1 et de l’Argon a` l’int´erieur d’un cristal d’α-quartz. Ces deux sodalites contiennent des canaux qui sont larges et unidimensionnels dans le premier cas et e´ troits et interconnect´es dans le deuxi`eme. La dynamique d’une petite partie des atomes du cristal est explicitement simul´ee. Cette dynamique est d´ecrite a` partir d’´equations de Langevin g´en´eralis´ees qui reproduisent l’effet du reste du cristal. L’´energie libre du gaz absorb´e a` l’int´erieur du cristal est approxim´ee. Le profil e´ nerg´etique a` l’int´erieur du z´eolite Theta-1 est presque plat et contient des barri`eres e´ nerg´etiques peu e´ lev´ees. Celui a` l’int´erieur du quartz contient de larges barri`eres a` la diffusion. La perm´eabilit´e des deux syst`emes est rapport´ee et compar´ee en d´etail avec celle obtenue a` partir de la th´eorie dite des e´ tats de transitions. Le rˆole qu’ont les modes de vibrations du cristal sur la diffusion est aussi e´ tudi´e. Pour le X´enon a` l’int´erieur du z´eolite Theta-1, la th´eorie des e´ tats de transitions ne d´ecrit pas ad´equatement la diffusion du gaz et les vibrations du cristal ne jouent pas un grand rˆole. Pour l’argon dans le quartz, la th´eorie des e´ tats de transitions est plus appropri´ee et les vibrations du cristal ne peuvent eˆ tre n´eglig´ees. Pour les syst`emes o`u les vibrations du cristal jouent un rˆole, les premi`eres corrections quantiques sont calcul´ees. Dans ce cas, la diffusion est e´ tudi´ee a` partir de la formulation des int´egrales de chemins. Les int´egrales de chemins sont combin´ees et, en utilisant la th´eorie d´evelopp´ee par Martin, Siggia et Rose [Phys. Rev. A., 8, 423 (1973)], r´eduites a` un syst`eme d’´equations de Langevin g´en´eralis´ees qui est e´ quivalent aux e´ quations de mouvements classique lorsque la temp´erature est e´ lev´ee. Le traitement quantique des modes de vibration a pour effet de r´eduire la perm´eabilit´e calcul´ee. Les corrections quantiques de l’´energie libre sont calcul´ees a` partir d’une matrice de densit´e approximative. Une modification a` la m´ethode variationelle sugg´er´ee par Feynman

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et Kleinert [Phys. Rev. A., 34, 5080 (1986)] pour le calcul de fonctions de partition quantiques est propos´ee. Cette modification, o`u un potentiel de r´ef´erence, fonction du temps, est utilis´e, am´eliore le calcul approximatif de la matrice de densit´e. Finalement, l’´egalit´e de Jarzynski et ses connections avec les lois de la thermodynamique sont e´ tudi´ees. L’´egalit´e de Jarzynski impose une borne sup´erieure sur le travail et il est d´emontr´e que celle-ci est sup´erieure a` la borne thermodynamique. Aussi l’´energie libre qui apparaˆıt dans l’´egalit´e est diff´erente de celle qui peut eˆ tre conventionnellement obtenue a` partir des lois de la thermodynamique hors-´equilibre.

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TABLE OF CONTENTS CONTRIBUTIONS OF AUTHORS . . . . . . . . . . . . . . . . . . . . . . . . .

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REMERCIEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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´ E´ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABREG

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LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2

1.3 1.4 1.5 1.6 2

Absorption studies in Channeled structures . . . . . . Earlier diffusion studies . . . . . . . . . . . . . . . . 1.2.1 Experimental diffusion studies . . . . . . . . 1.2.2 Earlier theoretical diffusion studies . . . . . . The role of lattice vibrations . . . . . . . . . . . . . Quantum mechanical corrections . . . . . . . . . . . Improving the approximate density matrix evaluation Other Problems . . . . . . . . . . . . . . . . . . . .

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1 6 9 9 12 18 23 27 29

c Diffusion in channeled structures: Xenon in a crystalline sodalite [53](°American Physical Society, 2003) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 2.2

2.3

2.4

2.5 2.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Microscopic Expressions for the Permeability . . . . . . 2.2.2 Correlation Function Expression for D(z) . . . . . . . . 2.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Effective Forces and Force Correlation Functions . . . . 2.3.2 Differential Equations . . . . . . . . . . . . . . . . . . . 2.3.3 Potential of Mean Force Approximation . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Specification of the System and Potentials . . . . . . . . 2.4.2 Simulation Results and Permeability of Xenon in Theta-1 Transition State Theory . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . viii

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33 37 37 38 39 40 40 47 51 55 55 64 71 77

2.7 2.8 3

Appendix: Equations of motion . . . . . . . . . . . . . . . . . . . . . Appendix: Energy cost of a local displacement in an elastic medium .

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Diffusion in channeled structures II: Systems with large energy barric ers [112](°American Chemical Society, 2005) . . . . . . . . . . . . . . .

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3.1 3.2 3.3

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Diffusion in Channeled structures III: Quantum corrections induced by the c lattice vibrations. [120](°American Physical Society, 2007) . . . . . . . . 112 4.1 4.2 4.3

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4.5

4.6 4.7 4.8 4.9 5

Introduction . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . 3.3.1 Specification of the system 3.3.2 Permeabilities . . . . . . . Transition state theory . . . . . . . Comparison with experiment . . . Discussion . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 The Time-Correlation Function Form of the Permeability . . . . . . . 116 Evaluation of anti-commutator correlation functions using Path integrals 120 4.3.1 The density matrix . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.2 Forward-Backward path integrals and connection with MSR . . 126 4.3.3 The permeability revisited . . . . . . . . . . . . . . . . . . . . 134 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.4.1 Potential of mean force . . . . . . . . . . . . . . . . . . . . . 138 4.4.2 Noise terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.5.1 Potential of mean force for Neon in α-quartz . . . . . . . . . . 151 4.5.2 Guest-Free correlations . . . . . . . . . . . . . . . . . . . . . 157 4.5.3 Diffusion and the permeability . . . . . . . . . . . . . . . . . 159 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Appendix: Anti-commutator for Target-Target correlations in the absence of the guest . . . . . . . . . . . . . . . . . . . . . . . . . . 168 ˆ g , t), J(r ˆ 0 )}/2i in Appendix: The Anti-commutator correlation h{J(r g the semi-classical limit . . . . . . . . . . . . . . . . . . . . . . . . 170 Appendix: An alternate approximation for W (xg ) . . . . . . . . . . . 171

Effective classical partition functions with an improved time-dependent c reference potential. [125](°American Physical Society, 2006) . . . . . . . 175 5.1 5.2

Introduction and Theory . . . . . . . . . . . . . . . . . . . . . . . . . 175 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 181

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Other problems: The Jarzynski equality: Connections to thermodynamics and c the Second Law. [137](°American Physical Society, 2007) . . . . . . . . 191 6.1 6.2 6.3 6.4 6.5 6.6

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Introduction . . . . . . . . . . . . . . . . . . . One-dimensional expanding ideal gas . . . . . Work bounds . . . . . . . . . . . . . . . . . . The Jarzynski relation and response theory . . . Discussion . . . . . . . . . . . . . . . . . . . . Appendix: non-equilibrium distribution function

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191 195 204 211 221 225

Conclusions and Ideas for Continued Research . . . . . . . . . . . . . . . . . 228 7.1 7.2

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Suggestions for Continued Research . . . . . . . . . . . . . . . . . . 230

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 A

Copyright information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

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Table

LIST OF TABLES

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2–1 Silicate Force Constants [101] . . . . . . . . . . . . . . . . . . . . . . . .

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2–2 Parameters for zeolite silicon and oxygen . . . . . . . . . . . . . . . . . .

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2–3 Parameters for the noble gas atoms . . . . . . . . . . . . . . . . . . . . .

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2–4 Lennard-Jones parameters for the gas-channel interactions (T = 300 K). .

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2–5 The diffusion coefficient in different planes . . . . . . . . . . . . . . . . .

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3–1 D0 and P 0 as a function of temperature. PT0 ST is the number obtained from transition state theory. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3–2 Comparison with experimental diffusion coefficients for the two potential models, (a) and (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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Figure

LIST OF FIGURES

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1–1 For this system, the net flux, j, through the dividing interface, which is the crystalline channeled structure, is forced to be perpendicular to the interface by the impermeable walls. The flux obeys the continuity equation, Eq. (1.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1–2 The external sinusoidal potential, U (x) in Eq. (1.10), is shown (top panel) together with two velocity correlation function (lower panel) obtained for trajectories where the particle is released at a potential maximum (A) or a potential minimum (B). In the inset, the time integral of the velocity correlation function is reported for both starting points. . . . .

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2–1 The exact density of states (full line) for Theta-1 obtained in a Brillouin zone calculation is compared with the approximate density of states (dashed line) that is generated using our representation of the memory function, Eq. (2.41). The force constants are specified in Sec. 2.4.1. . .

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2–2 In this figure, the gray atom is a bath atom while the black ones are target atoms. The motion of the second target atom is illustrated. For potentials that include only stretching and bending energies, the bath atom does not feel the motion since the angle θ remains unchanged. . .

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2–3 The force acting on xenon in Theta-1 during the aging is shown as the noisy curves. The curve showing large fluctuations is obtained at 300K, the other at 3K. The straight lines are the approximate values for the ˚ 8.07333A, ˚ mean force at 300 K and 3K. The guest is at (6.49633A, ˚ The system and the potential are defined in Sec. 2.4.1 . . . 2.36156A).

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2–4 The target zone. The oxygen atoms are in red and the silicon atoms are in blue. A minimum (A) and a maximum (B) W (z) plane are shown. A Xenon atom in a binding pocket is shown as a green ball. The z-axis (channel axis) is the vertical axis. . . . . . . . . . . . . . . . . . . . .

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2–5 Constant potential of mean force surfaces for Xenon in Theta-1 (2 unit cells along the channel axis are shown) at 300 K. The surface energy is indicated in the corner of each sub-figure. The absolute minimum is at -6.94kB T . Steepest descent reaction coordinates are shown in red (path 1) and blue (path 2). . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2–6 The potential of mean force along path 1 (dashed line) and path 2 (full line). The energy is plotted as a function of z (and as a function of the distance d along the path in the inset). The activation energy for path 1 is 1.95kB T and, for path 2, 2.15kB T or 2.06kB T (depending on the starting point) at 300 K. . . . . . . . . . . . . . . . . . . . . . . . . . .

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2–7 The plane potential of mean force for the middle unit cell of the channel is represented by the full curve. The dashed curve shows the value of the minimum in every plane. . . . . . . . . . . . . . . . . . . . . . . . . .

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˚ is extrapolated from the plateau 2–8 (D(z)/n∞ )eW (z)/kB T for z = 2.519A value of the full line. The dashed line is the uncorrected result. This plane is a maximum energy plane, with respect to W (z) . . . . . . . .

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˚ is extrapolated from the plateau 2–9 (D(z)/n∞ )eW (z)/kB T for z = 0.944625A value of the full line. The dashed line is the uncorrected result. This plane is a minimum energy plane with respect to W (z) . . . . . . . . . R 2–10 The Boltzmann factor e−W (r)/kB T and the factor dt hvz (t)vz ir e−W (r)/kB T in the maximum (A) and minimum (B) W (z) plane. The z-axis has arbitrary units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2–11 The space-dependent diffusion coefficient D(z)/n∞ (solid line) and (D(z)/n∞ )eW (z)/kB T (dashed line). . . . . . . . . . . . . . . . . . . .

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2–12 The steepest descent path 1 (A), and the steepest descent path 2 (B) (in red) are compared with the conditional averages of the trajectories started at the appropriate saddle point at 300 K (blue) and 3 K (green). The surface is at -5.3kB T . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2–13 The x and y components of the steepest descent path 1 (dashed line) and ˚ the average trajectory (full line). The starting point is at (6.9295A, ˚ 3.27A). ˚ The average x-component has to follow the reaction 7.8445A, coordinate because it lies in the reflection plane. The error bars indicate the standard deviation associated with the average. . . . . . . . . . . . R 2–14 The Boltzmann factor e−W (r)/kB T and the factor dt hvz (t)vz ir e−W (r)/kB T in saddle plane 1 (A) and saddle plane 2 (B). The z-axis has arbitrary units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–1 The exact density of states (full line) for α-quartz obtained from a Brillouin zone calculation is compared with the approximate density of states (dashed line) that is reproduced from Eq. (3.7)). . . . . . . . . . . . . .

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3–2 Constant potential of mean force surfaces for argon in α-quartz at 300 K. The surface energy is indicated in the corner of each part. Two symmetry equivalent low energy reaction coordinates are shown as path 1 (red) and 2 (blue). The kink is the position of a local minimum along the path. Also shown is a high energy reaction coordinate (green) connecting different channels in different unit cells (the saddle point is indicated by an “X”). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3–3 The potential of mean force along the reaction coordinate (Path 1 and 2 in Fig. 3–2) is shown as the full line. The dashed line shows the potential of mean force for a reaction coordinate that connects different channels, cf. Fig. 3–2. The saddle point at “X” is indicated in Fig. 3–2. The rate constants used in transition state theory, cf. Eq. (3.10), are also indicated at the two saddle points 1 and 2. . . . . . . . . . . . . . . . .

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3–4 Plane average potential of mean force. The temperatures are, starting with the bottom line, T = 100K, 300K and 800K. . . . . . . . . . . . . . . .

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3–5 The intrinsic permeability, P 0 , shows Arrhenius behavior in the temperature range,100-800K. The top line shows ln (P 0 /n∞ eβWBulk ) while the bottom line shows ln (PT0 ST /n∞ eβWBulk ). . . . . . . . . . . . . . . . . .

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3–6 The non-ideal bulk correction factor appearing in Eq. (3.17) for a LennardJones fluid with Argon-like parameters [27] as a function of pressure, p, using the HNC approximation and the compressibility relation for the pressure (σ is the Lennard-Jones size-parameter). Note that the HNC solution becomes unphysical for pressures beyond those shown in the figure at 100K, possibly signaling a phase transition. . . . . . . . . . . 101 Rt 3–7 The correlation function hv(t)vi and the integral 0 ds hv(s)vi are shown for an argon atom starting at a saddle point in an adiabatic, flexible and frozen lattice simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 104 3–8 The distribution of positions along z of the 2000 trajectories as a function of time. The left figure is for the adiabatic case while the right figure is for the unfrozen lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3–9 The steepest descent path (blue) is compared against the average trajectories (black) as obtained from our simulations started at the high energy saddle point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 R∞ 3–10 The factor 0 dt hvz (t1 )vz ir e−β(W (r)−W (z)) in the maximum W (z) plane ˚ × 3.46A ˚ is shown . . . . . . . . . . . . . . . 111 where an area of 2.757A 4–1 The exact density of states (full line) for α-quartz is compared with the one obtained from our approximate form for the memory function (dashed line), cf. Eq. (4.89). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 xiv

4–2 The two contours, C1 and C2 described in the text, are illustrated in the complex ω plane. Here, the absolute value of the real part of ω is bounded by Λ and the absolute value of the imaginary part is bounded by ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4–3 The target zone used in all simulations is shown in the box. Red and blue atoms are oxygen and silicon, respectively. The silicon atom whose correlations are reported in Fig. 4–8 appears in green. The positions labeled A and B are associated with the potential of mean force results of Fig. 4–6 and the guest correlations shown in Fig. 4–9, respectively. The locations of some of the binding sites are shown as small white spheres. In this figure, the z-axis is normal to the page. . . . . . . . . . 150 4–4 Potential of mean force energy surface is shown for Neon inside α-quartz. The surface is drawn for W (rg ) = 3.5kB T at 300K. . . . . . . . . . . . 152 4–5 Plane average potential of mean force is shown for Neon inside α-quartz at 300K. Our semi-classical approximation is compared against the classical expression. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 ˚ 0.0A, ˚ 0.0A) ˚ 4–6 The potential of mean force of Neon inside α-quartz at the point (−0.2305A, is shown as a function of temperature. The two sets of points are obtained from approximations similar to those in Refs. [68, 69, 70]. The points are obtained using reference frequencies that include curvature corrections at either the closest local minimum (circles) or global minimum (squares). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4–7 The potential of mean force of Neon inside α-quartz close to a global minimum is shown as a function of temperature. The set of points are obtained from approximations similar to those in Refs. [68, 69, 70]. . . 156 ˆ

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ˆ

ˆ

X1 } V1 } 4–8 The anti-commutator correlations, h {X1 (t), i and h {V1 (t), i, are shown 2 2 for T = 300K (the left) and T = 30K (right) for the silicon atom in green in Fig. 4–3. In each panel, we show the “exact” anti-commutator correlation, our simulated result and the classical correlations. . . . . . 158

4–9 The simulated velocity-velocity time correlation function for a trajectory ˚ −0.93A, ˚ 0.0A) ˚ is where the guest is initially at xg = (−0.2305A, reported for our semi-classical treatment and for the completely classical case at 300K. In the inset, we compare the time integral of the same correlation function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4–10 The simulated velocity-velocity time correlation function for a trajectory ˚ −0.93A, ˚ 0.0A) ˚ is where the guest is initially at xg = (−0.2305A, reported for our semi-classical treatment and for the completely classical case at 30K. In the inset, we compare the time integral of the same correlation function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 xv

4–11 The anti-commutator velocity-velocity correlation shown in Fig. 4–9, obtained within our semi-classical formalism, is compared with the velocity-velocity Kubo average. The region where the two curves differ the most is shown in the inset. . . . . . . . . . . . . . . . . . . . . . . 162 4–12 The Boltzmann factor, e−βW (rg ) in the maximum energy plane is compared Rt with 0 dt1 hvG,z (t1 )vG,z ixg (t=0)=rg e−βW (rg ) at 300K. The second panel has units of m2 /s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4–13 The quantum space-dependent diffusion coefficient, D(z), obtained from the plateau value of the solid curves for the maximum energy plane, z = 0 is compared with its classical counterpart. The dashed curve represents D(z, t) but where the denominator has been set to one in Eq. (4.93). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5–1 The free energy A = − log (Z)/β in the two approximate cases is compared with the “exact” answer. . . . . . . . . . . . . . . . . . . . . 182 5–2 β(Wef f (x) − A) for g = 0.4 and β = 10. Only x < 0 is shown because βWef f (x) = βWef f (−x). In the inset, βWef f (x) is shown for the region where our method differ most compared to the original case. . . 183 5–3 The quantum mechanical distribution function, ρ(x) for g = 0.4 and β = 10.185 5–4 The density matrix, ρ(xa , xb ) for g = 0.4 and β = 10. The figure on the left is our calculation while that on the right is the “exact” answer. . . . 186 5–5 The difference |ρ − ρexact | for g = 0.4 and β = 10. The left figure is our calculation while the right figure is the time-independent reference potential of Kleinert. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5–6 The classical path, xcl (τ ), defined by our reference potential and the original reference potential are compared with the “exact” average path linking the points (0.9, −0.9). Here, WKB(mean) is the average of the two energetically equivalent WKB paths (see text). In the inset, x¯(τ ) is compared for both reference potentials. . . . . . . . . . . . . . . . . . 188 5–7 The fluctuations around the classical paths linking xa = 0.9 and xb = −0.9 for g = 0.4 and β = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6–1 The initial equilibrium distribution function at t = 0 is shown on the left and the non-equilibrium distribution function at t = 4 is shown on the right. Here, Li = 10, V = 0.5 and Ti = 1 and the moving piston is on the right. The contours are drawn when f (x, u) = 0.005, 0.01, 0.015, 0.02, 0.025, 0.03 and 0.035. . . . . . . . . . . . . . . . . . . . . . . . 198

xvi

6–2 The local thermodynamics quantities defined by Eqs. (6.8)–(6.11) are shown as a function of time from t = 0 to t = 4 for an expansion where Li = 10 and V = 0.5. At t = 0, all quantities are uniform for 0 < x < Li and zero elsewhere. The curves are equally spaced at time intervals of 4/9. Larger values of t have larger non-uniform regions. . . 200 6–3 The fine-grained distribution function (on the left) is compared with the time-averaged distribution function that is later used to calculate the coarse-grained entropy according to Eq. (6.17b). Here, t = 1000 and τ = 100. The contours are drawn when f (x, u) = 0.005, 0.01, 0.015, 0.02, 0.025, 0.03 and 0.035. . . . . . . . . . . . . . . . . . . . . . . . 202 6–4 The average work W is obtained from Eq. (6.30) with T = 1, Li = 10 and V t = 10. This is compared with the Jarzynski bound, which is independent of the piston velocity. . . . . . . . . . . . . . . . . . . . . 209 6–5 The real free energy difference ∆A between the two equilibrium states is compared with ∆AJ . The conditions of the expansion are the same as in Fig. 6–4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

xvii

CHAPTER 1 Introduction The understanding of the motion of guest atoms or molecules inside porous crystals has been at the heart of many studies in the past decades. Porous crystals are usually characterized by the presence of hollow channels of varying sizes that extend throughout the crystal and that facilitate the absorption of the guest component. Because, for many guest molecules, the diffusion of guest components in and through these solid structures is facilitated by the channels, the diffusion process occurs on a fast enough time-scale such that it can be probed experimentally [1, 2]. The most famous family of porous crystal are called “Zeolites”, which means “stone that boils”. These crystals are made of Silicon, Aluminum and Oxygen atoms. There are many types of zeolitic channels; some contain large cages and smaller bottleneck regions, while others are roughly cylindrical everywhere, and, for many zeolites, the channels are interconnected. Each aluminum atom that appears in a zeolitic structure results in a net negative charge in the structure which is counterbalanced by a positive ion (often N a+ , K + or Ca++ ) such that the net charge of the crystal vanishes. Zeolites are far from being the only type of channeled structures. For example, a well-known crystal, quartz, contains narrow channels that allow small guest molecules to permeate, albeit quite slowly, into them. In general, channeled structures do not have to be crystalline. Cellular lipid bilayers containing ion channels is an example of a type of non-crystalline channeled structure. There are many reasons why channeled structures are particularly interesting to study. Channeled structures are used as catalysts in chemical reactions. In particular, zeolites have two properties which makes them suitable catalysis. First, they are cation exchangers which makes it possible to introduce a large variety of cationic species with different

1

2 catalytic properties (in acid or metal catalyzed reactions) inside the pore system [3]. Second, the zeolites pore dimension are often of the same size as many simple molecules and, hence, they possess molecular sieving properties (selective adsorption of molecules). Further, this shape and size selectivity also plays a role at the level of transition state of chemical reactions [3]. More practically, channeled structures are often used as catalysis in the fuel industry (more precisely, in the isomerization and hydrocracking of hydrocarbons and long chain alkanes) [4, 3] and in gas phase separation processes [2, 1]. They can also be used in water and wastewater treatment and have been shown too be effective to remove N H4+ and other ions [5]. Because of these applications, it is important to understand how the guest molecule moves in and out of these channeled structures. Theoretically, this problem is complicated by the fact that the guest is moving in a very inhomogeneous system. As a consequence, the standard diffusion equation [cf. Eq. (1.8) below] cannot be used to describe the motion of the guest inside the material and no unique diffusion constant [6, 7] can be defined. Moreover, from a microscopic point of view, if the energy barriers inside the structure are large, the guest will cross these barriers very infrequently. On the other hand, it is clear that these barrier crossing events are at the heart of the diffusion process, and one therefore faces the problem of sampling these rare events. Also, the crystal lattice, whose structures determines the energy landscape of the guest, is itself moving and the lattice vibrations generally participate in the guest motion. Practically speaking, propagating many lattice atoms in a computer simulation can have considerable problems. In this research, we carried out a thorough theoretical study of the microscopic dynamics of chosen guests inside typical crystalline channeled structures. The goal of this work is as follows. First, we develop a formalism, that relies on computer simulations, and that allows us to fully characterize the guest microscopic dynamics. This general formalism will allow us to quantify the role of the various players participating in the guest diffusion process. For example, we will be able to assess the effects of crystal vibration

3 0

z

Impermeable wall

µ+

µ−

j

Channeled Crystal

Impermeable wall

Figure 1–1: For this system, the net flux, j, through the dividing interface, which is the crystalline channeled structure, is forced to be perpendicular to the interface by the impermeable walls. The flux obeys the continuity equation, Eq. (1.1). and relaxation on the guest diffusion and we will be able to quantify what kind of guest trajectories, if any, dominate the behavior of the guest inside the structure. This microscopic information, which will largely be presented in terms of time-correlation functions, will then be combined with a theory developed by Vertenstein and Ronis [6] to make connection with macroscopic observables, here, the macroscopic observable is the net flux of guest component passing through the solid. In this work, we consider a steady state process where the channeled structure separates two regions of bulk guest component having different chemical potential. An idealization of the system’s geometry is illustrated in Fig. 1–1. The net flux through the interface, j, obeys the phenomenological continuity equation, known as Darcy’s law, j = P 0 [µ(z = −d) − µ(z = +d)],

(1.1)

where µ(z = ±d) is the chemical potential in the ∓ bulk phases and P 0 is the intrinsic permeability. The difference in chemical potential is taken at both interfaces separating the crystal from the bulk guest component. Note that in this setup, the flux is forced along a specific direction, normal to the channeled crystalline interface, by the presence of impermeable walls. The macroscopic transport is therefore completely characterized by

4 the permeability, P 0 , which is one of the key quantities that will be evaluated for various systems here. The formalism that will be used to calculate the permeability relies on the evaluation of various microscopic quantities that characterize the behavior of the guest inside the crystal. These quantities can be separated into two classes. The first contains timeindependent objects; it is related to the probability of finding the guest at some position inside the lattice. These probabilities are determined from the potential of mean force of the guest. The potential of mean force of the guest is an effective potential where the crystal degrees of freedom have been averaged out. This can also be viewed as follows. If the guest is frozen at a point in a vibrating lattice, it will feel a force whose fluctuations are caused by the lattice vibrations. The average of this force, the “mean force”, is derivable from the gradient of an effective potential: the potential of mean force. If the lattice atoms are assumed to be frozen, the potential of mean force reduces to the usual interaction potential between the guest and the crystal. In thermodynamics terms, the potential of mean force is the Helmholtz free energy change of the system when the guest is taken from vacuum to a particular position inside the crystal. The second class of microscopic quantities contain dynamical information. Here, this dynamical information will mainly be obtained from specific guest time-correlation functions that define a space-dependent diffusion coefficient (see Ref. [6]). Another approach that is often used in diffusion in channeled system studies is to treat the diffusion process as a series of hops where each hop is characterized by a hopping rate constant. This later rate constant can also be used to characterize the dynamics. For systems where the energy barriers to the guest diffusion are intermediate or large, the dominant contribution to the permeability will come from the free energy landscape (the potential of mean force) of the guest in the crystal, the static quantities, because, as will be shown below, this contribution of the energy landscape to the permeability appears exponentiated in the theory (as Boltzmann factors). More precisely, the permeability can ultimately be written in an Arrhenius

5 form, P 0 = Ae−∆E

† /k T B

(1.2)

where the activation energy, ∆E † is determined from the free energy landscape and the prefactor, A is obtained from the dynamics (kB is Boltzmann’s constant and T the temperature). Therefore, for intermediate and large barrier systems, very accurate potential models are mandatory for the permeability to be reported precisely. The importance of the accuracy of the potential in these kind of studies is not contested. Here, we use simple potential models and we spend most of our time studying the dynamical part; the type of motion inside the material (for example, diffusive or ballistic motion) will manifest itself in the prefactor. Because this dynamical part is the most difficult to quantify, numerous approximations have been used to study the guest dynamics in channeled structures. We will review these approximations below and, within the same potential model, quantitatively compare them. Specifically, we use a general simulation procedure, that focuses on the barrier regions and is unaffected by the sampling of rare events problem; this will allow us to check the validity of these various approximations. Further, it is in the guest dynamics, and not in the energy landscape, that most of the interesting physical phenomena happen. More precisely, barrier recrossings, energy exchange with the crystal and the decorrelation of guest velocity are some examples of phenomena that contribute solely to the pre-exponential factor. Of these, we will focus on the energy exchange between the guest and the crystal that here, happens through the lattice vibrations. We will also compare the time-scale of this energy exchange with the one that naturally arises from the randomization of the guest velocities. The goal here is to be able to assess how the dynamics of the guest, coupled to that of the lattice, modifies the pre-exponential factors of the permeability. For now, we start by reviewing the previous efforts that have been made to understand the behavior of guests in channeled structures

6 and later we show how our work proposes a more general and more powerful formalism to quantify and understand the guest diffusion in channeled structures. 1.1 Absorption studies in Channeled structures Prior to the studies of guest dynamics inside channeled structures, the thermodynamics of intracrystalline sorption was investigated. This was done in general terms by Hill [8] in a theoretical study which made connections with many experiments from which the heat of adsorption could be extracted. Many of these early adsorption experiments in zeolites where carried by Barrer and co-workers who is recognized as one of the fathers of zeolite chemistry. For example, in Refs. [9, 10], Barrer and Stuart obtained the heat of sorption of Argon and Nitrogen in faujasite type crystals in the temperature range 173 to 273K. There, the heat of sorption was obtained from pressure measurements combined with the Claussius-Clapeyron equation [11]. µ ln

P2 P1

¶

∆H = R

µ

1 1 − T2 T1

¶ ,

(1.3)

where P2 (P1 ) is the pressure at the temperature T2 (T1 ) , R is the Gas constant and ∆H = Ha − Hg is the heat of sorption defined in terms of Ha and Hg which are, respectively, the heat in the adsorbed phase and in the gas phase. In these experiments, the effects on the heat of sorption of the type of counter ions inside the crystal as well as the amount of guest adsorbed were quantitatively investigated. The temperature dependence of ∆H was also investigated for various systems. This was immediately followed by the first attempts to model the interaction potential between the adsorbed guest and the crystal framework. The goal there was to compare calculated heat of sorption using various potential models with the experimentally obtained results. This was done by Barrer et al. in Refs. [9, 10, 12] and by Mayorga and Peterson in Ref. [13]. In these earlier studies, the interaction potential between the guest and the

7 crystal, U , was chosen to be the 6-12 Lennard-Jones potential, X µ Agj Bgj ¶ U= − 6 + 12 , rgj rgj j

(1.4)

where j is summed over all crystal atoms. In this equation, rgj is the separation between the absorbed guest and the j th crystal atoms and Agj and Bgj are parameters than can be calculated using various approximate methods like the London or Kirkwood-Muller formulas [14, 15, 16]. Practically, the sum over j can be truncated for such a potential because it decays to zero quickly for large separations. Sometimes, a guest polarizability term, Up , was added to the interaction potential, α Up = − E 2 , 2

(1.5)

where α is the polarizability of the guest and E is the magnitude of the electric field produced from the crystal framework atoms partial charges. In Ref. [12], it was shown that neither of the two approximations (London and Kirkwood-Muller) is always better than the other to accurately reproduce the experimental heat of sorption for different guestcrystal systems. In fact, in all cases, both methods resulted in calculated heat of sorption that were within factors of 2 with the experimental value. As we discuss in detail in Chapters 2 and 3, this potential model strongly depends on the partial charges that are assigned to the silicon, oxygen and aluminum atoms in the crystal and, consequently, on the atoms effective sizes. The values of the potential parameters within the Lennard-Jones model are still controversial and, moreover, the model itself is not perfect. More precisely, the model is not accurate for very small separations, where the electron clouds of the two atoms overlap. Nevertheless, the potential model makes a prediction as to where the guest molecules are more likely to be found in the structures. The energy landscape of xenon in twenty different zeolites within the Lennard-Jones model only (i.e. no polarization terms) is reported in Ref. [17].

8 More sophisticated computational procedures were more recently combined with molecular dynamics [18] and Monte Carlo [19] techniques to calculate heat of sorption and Henry’s constant, Kh , defined by, cg = Kh cs ,

(1.6)

where cg and cs are, respectively, the equilibrium concentrations of guest component in the gas and in the adsorbed phases. These studies were again carried using the Lennard-Jones model potential. The quantities that are calculated in these computer experiments are equilibrium observables. Still, molecular dynamics and Monte Carlo contains information on the guest dynamics and, in the next section, we review how these techniques have been used to study the motion of the guest. In all the numerical studies referred to in this section, the crystal framework was assumed rigid. In other words, the computation was carried out with the crystal atoms fixed at their equilibrium position (in the absence of the guest), which was obtained from x-ray diffraction patterns. We conclude by pointing out that more recent experimental studies were carried for the adsorption of alkanes in various zeolites [20, 21]. This is relevant because, after all, the most interesting applications of zeolites occur with more complex molecules, i.e. alkanes and hydrocarbons. In Ref. [20], many experimental techniques (IR spectroscopy, calorimetry and gravimetry) were used to not only measure the heat of sorption, but also to characterize the ordering and the orientation of the adsorbed alkanes at equilibrium for various loadings in zeolites Ferrierite and Theta-1. In Ref. [21], low pressure adsorption of branched alkanes in zeolites Beta, ZSM-5, ZSM-22, zeolite Y and Mordenite was characterized using tracer and perturbation chromatographic techniques in the temperature range 473 − 648K. All of the above experimental or numerical studies focused on equilibrium properties of the adsorbed guest molecule. As such, these studies provided no information on the barrier regions of the free energy landscape, but only on the minimum energy regions.

9 1.2 Earlier diffusion studies The previous section briefly described how the energetics of an adsorbed guest component inside various channeled structures could be quantified. These studies, which were both experimental and theoretical, were naturally followed by others that focused on the dynamics of the guest inside the porous crystal. In other words, now that we know how much energy is released or has to be given when the guest is adsorbed, the following question emerges: How fast does the guest moves in and out of the crystal? 1.2.1 Experimental diffusion studies A summary of three experimental techniques that are often used to measure the guest self-diffusion inside channeled structures is now presented. The first of these techniques is called Pulsed-field gradient NMR [22, 23, 24, 25] This technique is a direct measure of the guest mobility in the crystal. Here, a magnetic field gradient pulse is applied after the two radio frequency pulses in a conventional spin-echo NMR technique. This produces a signal which is exponentially attenuated by the factor h|r(t) − r(0)|2 i where t is the time between the two magnetic field-gradient pulses and r(t) the position of a tagged particle at time t. Also note that h...i is an ensemble average. Using this information, the Einstein relation [26, 27], ­

® |r(t) − r(0)|2 = 6Dt,

(1.7)

which is valid for long times only, can be used to extract the diffusion coefficient, D. In this technique, the guest molecules inside the crystal are assumed to obey a diffusion equation, ∂c(x, t) = D∇2 c(x, t), ∂t

(1.8)

where c(x, t) is the concentration profile of the guest inside the crystal (this is called Fick’s Second Law [2]). For an example of this technique, see Ref. [22] where the self-diffusion of Xenon, CO2 and CO in various zeolitic frameworks are measured at different temperatures (in this study, the guest molecules were chosen because of the availability of C13

10 and Xe129 NMR techniques). As pointed out in Ref. [22], the diffusion coefficient that is measured here is some kind of average diffusion coefficient. By that, we mean hat the measured diffusion coefficient comes from the behavior of a large number of guest molecules adsorbed at various sites inside the macroscopic channeled structure. In particular, if the diffusion is anisotropic, D will be an average of the diffusion coefficients along the three orthogonal axis. This point will further be clarified when the early theoretical studies in the guest diffusion are described. In another technique, called Frequency Response, a dose of guest in the gas phase is brought into adsorption equilibrium with the sorbent [28, 29, 30]. A small amplitude periodic modulation of the gas-phase equilibrium volume is applied which results in a subsequent equilibrium pressure modulation. The pressure modulations are then analyzed in the context of Fick’s second law [2], Eq. (1.8). For a review of the frequency response technique, see Ref. [29]. In Ref. [28], frequency response is used to calculate the diffusion of CO2 in Theta-1 and Silicalite-1 as well as the diffusion of alkanes in larger pore zeolites Beta and NaX. Here, the experiments were performed around room temperature. The obtained experimental diffusion for CO2 in Silicalite-1 was in agreement with the results of Ref. [22] obtained with pulsed-field gradient NMR techniques. The dependence of the diffusion coefficient upon loading was also studied here by increasing the external gas pressure. For CO2 in Theta-1, the diffusion coefficient was roughly constant for the pressure range considered. This experimental technique also measures an averaged selfdiffusion coefficient. We conclude this subsection by describing a final type of experiment that is often used in the diffusion in channeled structures problems and that is particularly relevant here because the experimental setup is very similar to hypothetical setup described in Fig. 1–1. This technique is called Rutherford Backscattering Spectroscopy (RBS) [31]. In this setup, pressurized gas is exposed to a large single crystal whose contact surface has been polished. Here, the crystal is thick enough such that the gas never reaches the other

11 boundary. This setup is identical to what we described in Fig. 1–1, but with no gas on the left hand-side chamber and a thicker crystalline interface. The concentration profiles inside the crystal are obtained as a function of time using RBS and, from them, a diffusion coefficient is extracted assuming Eq. (1.8). Note that here, the concentration only depends on z and t (it is therefore somehow averaged in the x-y plane) where z represents the depth inside the crystal. In RBS, a 1–3 MeV 4 He+ ion beam is directed at the surface of the sample into which the guest component is diffusing. The incoming ions eventually backscatter and their final energies are detected. This final energy carries all the necessary information to draw a concentration profile because it depends on the type of atom with which the backscattering collision occurred and the depth at which the collision took place. More precisely, as the ions penetrate the sample, they lose energy through inelastic processes, from which the depth can be obtained. For example, when the channeled material is a silicate, the ions are backscattered from collisions with Si/O atoms or the guest particle, each resulting in a different energy loss for the incoming ions that comes from the difference in the masses of O, Si and the guest molecule. The proportion of collisions that occurred with the various species in the system at various depths is obtained by binning the number of backscattered ions with respect to their energies. From this, the concentration of the guest as a function of depth is obtained. The technique is capable of accurately reporting the concentration profile up to depths of the order of 100nm. It is hard to apply this technique with zeolites because many of them are rarely obtained as large single crystals. On the other hand, RBS has been used extensively for silica glass systems and quartz [32, 33, 34, 35]. In Refs. [32, 33], the diffusion of Argon was studied, as a function of temperature inside quartz and silica glass, respectively, while Refs. [34, 35] focused on lattice silicon self-diffusion in quartz. One of the main advantages of these techniques is that, because concentration depths profiles are extracted from it, one can characterize the diffusion well inside the lattice, where surface effects are negligible.

12 1.2.2 Earlier theoretical diffusion studies As stated above, the diffusion coefficient is related to the mean square displacement through the Einstein relation, Eq. (1.7). An equivalent expression for the self-diffusion coefficient is the Green-Kubo relation, 1 D= 3

Z

∞

dt hv(t) · v(0)i ,

(1.9)

0

where v(t) is the velocity a guest tagged particle and t is the time. Clearly, both diffusion coefficient relations, Eqs. (1.7) and (1.9), can easily be implemented within molecular dynamics simulations. In fact, this has been done extensively in the theory of real gases and simple liquids [27]. Many earlier theoretical studies assumed that the guest motion inside channeled structures can be described in terms of diffusion coefficients obtained from the Green-Kubo or Einstein relation combined with molecular dynamics computer simulations [36, 37, 38, 39, 40, 41, 42, 43]. In these simulation studies, the crystal lattice was assumed to be rigid [36, 38, 41, 42, 43] or flexible [37, 39, 40]. When the lattice vibrations were included [37, 39, 40], the motion of 576 crystal atoms was simulated. One of the conclusions that these studies reported was the fact that the lattice vibrations did not seem to modify the computed diffusion coefficient for the systems studied, which contained intermediate to large channels. On the other hand, there is a major problem with the Green-Kubo or Einstein relations, Eqs. (1.9) and (1.7), when they are used to study the diffusion inside channeled structures. This problem is related to the fact that the guest evolves in a very non-uniform media. This means that, even at infinite dilution, the guest will evolve according to the interaction potential with the crystal atoms which is clearly non-uniform (it contains hills and valleys). Hence, the guest inside the crystal is non-uniformly distributed. This is a completely different picture compared to what is observed in real fluids where all particles are uniformly distributed (any liquid is translationally and rotationally invariant).

13 For inhomogeneous systems, like crystalline channeled structures, the guest diffusion cannot be described by Eqs. (1.9) or (1.7). This can be seen in the following simple toy model. Consider a one dimensional particle evolving in an external potential as follows, x¨(t) = −

dU (x(t)) − x(t) ˙ + η(t), dt

(1.10)

where x(t) is the position of the particle at time t, x(t) ˙ = v(t) is a time derivative, U (x) is an external sinusoidal potential and η(t) is a random force that keeps the temperature, as defined for a canonical ensemble of these particles, constant. In Fig. 1–2, the external potential is shown together with the hv(t)v(0)i correlation function that is obtained when all trajectories are started at a barrier top (A) or at a minimum (B) and ensemble averaged. Clearly, the resulting time-correlation functions for different starting positions result in concomitantly different correlation functions, which will predict different values for D, should the Green-Kubo relation in its simplest form, cf. Eq. (1.9) be used. This illustrates why at the microscopic level, for inhomogeneous systems, there is no unique diffusion coefficient. Again, this is a completely different picture compared to real gas or liquid diffusion, for which the Green-Kubo relation in its usual form, Eq. (1.9), is appropriate and has been correctly used for decades [26, 27]. If a time-averaging procedure was used instead of the ensemble average, the velocity correlation function would still decay to zero, but the diffusion coefficient obtained from this procedure would be an average of the various diffusion coefficient as obtained from ensemble averages with fixed initial positions. Of course, this time-averaging procedure would make sense for simulation times that are long enough such that the particle samples all of phase space. Clearly, in such an approach, the minimum energy regions will be exponentially favored compared to the maximum regions. Something which can be problematic since it is well known that it is the dynamics near the maximum that governs the motion across a barrier. The diffusion in inhomogeneous system therefore as to be reformulated and cannot be described in terms of the usual Green-Kubo relation. This was done by Vertenstein and

14

Figure 1–2: The external sinusoidal potential, U (x) in Eq. (1.10), is shown (top panel) together with two velocity correlation function (lower panel) obtained for trajectories where the particle is released at a potential maximum (A) or a potential minimum (B). In the inset, the time integral of the velocity correlation function is reported for both starting points.

15 Ronis in Refs. [6, 7] where a space-dependent diffusion coefficient naturally arises. As will be shown later, this space dependent diffusion coefficient formalism makes a connection with the macroscopic observable, the permeability. As briefly discussed above, another approach which is common to many theoretical studies treats the guest diffusion inside channeled structures as a hopping process between neighboring binding sites [44, 45, 46]. Here, the hops are characterized by a hoping rate constant that can be calculated from transition state theory (TST) [47, 48]. The transition theory picture is simple and requires very little microscopic information about the system. In fact, the hopping rate constant, k, is obtained as follows, k=κ

kB T qη† qγ† e−β∆U h qx qy qz

(1.11)

where kB is Boltzmann’s constant, T is the temperature, h is Planck’s constant, β = 1/kB T , ∆U is the energy difference between the starting minimum and the activated state, the q’s are harmonic oscillator partition functions and κ is a transmission coefficient between 0 and 1. The partition functions with subscripts x, y, z are obtained from the eigenvalues of the potential energy curvature matrix at the starting minimum while qη† and qΓ† are obtained from the 2 positive eigenvalues of the curvature matrix at the activated state (remember that the activated state contains 2 positive eigenvalues and one negative, associated with a saddle point in energy landscape). Consider the guest at a saddle point; the negative eigenvalue eigenvector can be used to define a steepest descent path (reaction coordinate) that reaches the minima on either side of the activated state. In TST, this reaction coordinate is assumed to dominate the guest motion inside the structure. In TST, only the minima and saddle points in the potential energy landscape have to be analyzed to calculate k. A little more work has to be done if barrier recrossings are allowed. This shows up in the evaluation κ, the transmission coefficient [44]. In Refs. [44, 45, 46], a detailed study of TST applied for various molecules moving inside the zeolite NaY is presented. These studies included the flexibility of the lattice by

16 keeping a limited number of crystal atoms. Although TST has many practical advantages (it requires small numerical efforts), it relies on many assumptions. One of them is very restrictive and will be analyzed in detail in this work, namely that the motion of the guest is well described by reaction coordinates. There are some problems with this approximation. First, as we show in Chapters 2 and 3, the motion of the guest is sometimes very poorly described in terms of reaction coordinates. This is especially manifest for low barrier, soft potential systems. Second, for a very complicated energy landscape, it is sometimes not easy to find all the reaction coordinates (reaction pathways). Omitting one, if its barrier is comparable to the others, would certainly produce important errors in the calculated permeability. One of the successes of the formalism that will be used here is that it does not rely on the reaction coordinate assumption and that it is capable of testing it. The above techniques, with their advantages and disadvantages, give information about the microscopic dynamics of the guest. It is therefore necessary to complement them to make a connection with the macroscopic observable, the permeability. In TST, this is easily done by assuming a hopping model throughout the crystal. Since the crystal is periodic, this is very simple to implement. One has to identify reaction coordinates in the primitive unit cell only and then count the number of hops necessary to travel through the whole crystalline structure. This requires a careful analysis of the energy landscape inside the primitive unit cell. Again it is crucial that all reaction coordinates be included in the hopping model. The technique that we will use here, uses the generalized diffusion equation, ∂n(r, t) = ∇r · ∂t

Z

↔

dr0 L (r|r0 ) · ∇r0 βµ(r0 , t),

(1.12)

where n(r, t) is the number density, µ(r, t) is the chemical potential and where the Onsager ↔

coefficient, L (r|r0 ) is defined by, ↔

Z

∞

0

L (r|r ) ≡ 0

dt hJ† (r, t)J† (r0 )i,

(1.13)

17 where J† (r, t) is the irreversible part of the current in the sense of projection operators. This non-local diffusion equation is non-local in space and is fully consistent with linear response theory for inhomogeneous systems. The non-locality of this equation is not surprising. For example, the chemical potential in a small region around a barrier top will certainly affect the rate of change in local density at the neighboring binding sites. Hence, these two equation are more appropriate to describe the diffusion in an inhomogeneous medium. Starting from this generalized diffusion equation, Vertenstein and Ronis [6] obtained the following expression for the intrinsic permeability, P 0 , for a system where the net flow along the z axis, 1 1 = P0 β

Z

d

dz −d

1 , D(z)

(1.14)

where, as before, 2d is the thickness of the material. In this equation, the dynamics are contained in a space-dependent diffusion coefficient, D(z), which is written as, 1 D(z) ≡ A

Z

Z

∞

dt 0

Z drk

­ ® dr0 J†z (r0 , t)J†z (r) ,

(1.15)

where J†z is the z-component of the irreversible part of the current and A is the area of the crystalline medium. The fact that only the z component of the current vector fields are considered is a direct consequence of the experimental setup which forces the flow along the z axis (cf. Fig. 1–1). This space-dependent diffusion coefficient contains the mi­ ® croscopic dynamical information. In fact, the above correlations, J†z (r, t)J†z (r0 ) , could easily be calculated in standard MD simulations, although one must be careful in defining what is meant by the “irreversible” part of the current. It is important to point out that this formalism is strongly related to Smoluchowski processes [49]. In a one dimensional Smoluchowski process, a Brownian particle evolves under the influence of an external potential (W (z)) and the flux is written as j(z) = D0 e−βW (z) ∇βµ(z).

(1.16)

18 where D0 is space-independent. This postulate guarantees Fick’s law to be obeyed and forces the equilibrium distribution to be in the Boltzmann form. For steady states, Eq. (1.16) is easily solved, and Eq. (1.14) is recovered if D(z) = D0 e−βW (z) .

(1.17)

For multidimensional systems, it is not clear that the diffusion process will obey Smoluchowski everywhere. In this work, we will show that it does at least for the barrier tops regions, which are the more important contribution to the permeability, and we will verify this hypothesis numerically in Chapters 2 and 3. This formalism has many advantages over transition state theory because it contains the full microscopic dynamics. Precisely, combining Eqs. (1.14) and (1.15) clearly shows ­ ® that the dynamical information contained in J†z (r0 , t)J†z (r) for all r and r0 inside the crystal contributes to the permeability. In practice, for crystalline channeled structures, the periodicity of the lattice is used to greatly reduce these integration regions to the primitive unit cell. Moreover, in this approach, no reaction coordinates picture is assumed. This has the further advantage of avoiding the identification of these reaction coordinates, ­ ® which is not always so simple. Also note that the above J†z (r0 , t)J†z (r) correlations are expected to decay relatively quickly so that, in practice, short time simulations are required to compute D(z). Because of these advantages, this formalism is the one we chose to use and we will use it to test the underlying assumption of TST. The details of the formalism are already in the literature [50, 6, 7] and are overviewed in Chapter 2. Note that this formalism includes TST and, in the event that the reaction coordinate picture is valid, the space dependent diffusion coefficient, Eq. (1.15), would be dominated from the reaction coordinate contribution. 1.3 The role of lattice vibrations In the preceding description of earlier work, we pointed out that only some of them explicitly included the lattice vibrations in the calculation [37, 39, 40, 44, 45, 46]. As

19 stated above, in Refs. [37, 39, 40], the equations of motion for 576 crystal atoms where simulated and the interactomic potential between the crystal atoms was assumed to be harmonic. In Refs. [44, 45, 46], where the motion of the guest is analyzed within TST, an anharmonic potential was used to describe the crystal atoms bonded interactions. All the above methods included a small number of target atoms in their MD calculations. Clearly, this is more realistic than the rigid lattice model simulations [36, 38, 41, 42, 43], where no energy exchange between the guest and the lattice is possible and where the lattice atoms positions are assumed to remain unperturbed by the presence of the guest, but this is still problematic. First, it is well known that traditional MD conserves energy. Hence, consider the following situation. A trajectory is started with the guest molecule at an activated state (a saddle point in energy landscape) which is high in energy. The guest picks up a large amount of kinetic energy going down the barrier until it collides with some crystal atom to which some of this energy is transferred. In MD, this energy stays in the system and it can eventually come back to the guest. This situation is artificial. In reality, the energy transferred to the lattice will be dissipated through the infinite crystal. Of course, the infinite crystal atoms will transfer energy to the guest, but in a physical system, this energy transferred is around the thermal energy, kB T . For the situation described above energy larger than kB T can come back to the guest. This problem can be avoided by simulating a very large number of crystal atoms, but this leads to numerical problems. Further, in any realistic simulation that incorporates the crystal degrees of freedom, it is desirable to use a simulation procedure that reproduces the vibrational density of states of the crystal. For infinite harmonic crystal systems, the vibrational density of states is exactly obtained from Brillouin zone sum techniques [51, 52]. For a finite harmonic crystal, the density of states is obtained by binning the eigenvalues of the force constant matrix of the crystal. We have shown [53] that, the finite crystal vibrational density of states is equivalent to the exact result when approximately ten thousand atoms are included. Of course,

20 finite crystals miss the longer wavelength vibrations and, consequently, fail to properly describe the low frequency regions of the vibrational density of states. This can be important because the diffusing guest typically evolves on a time-scale comparable to that of the lower frequency vibrations. It is naturally expected that the low frequency vibrations interacts more strongly with the guest. The first attempt to overcome that problem without having to simulate a large number of crystal atoms by introducing dissipation (and fluctuation) in the crystal dynamics was done by Kopelevich and Chang [54]. In their work, they assumed that the guest interacts with a finite part of the crystal only (called the target) and was completely invisible to the rest of the crystal, the bath. This separation, that we will use extensively here, is justified because the guest motion is slow (the decorrelation time of the guest is much shorter than the time it takes for a typical guest to leave the target zone). This is often enough to guarantee that the guest remains in the target zone during the simulated time. For this model, Kopelevich and Chang averaged out all crystal degrees of freedom, a procedure first developed by Deutch and Silbey [55] for completely harmonic systems, and derived a generalized Langevin equation (GLE) for the guest degrees of freedom, Z

t

mg ¨r(t) = −∇Uef f (r(t)) −

ds η(r(t), r(s); t − s)˙r(s) + F† (r(t); t),

(1.18)

0

where r(t) is the guest position at time t, Uef f (r(t)) is an effective potential that differs from U (r(t)) because the crystal degrees of freedom have been averaged out, η(r(t), r(s); t− s) is a so-called memory function that describes the energy dissipated to the lattice and F† (r(t); t) is a random force that accounts for the energy fluctuations of the guest that comes from the interaction with the dynamic lattice. The vibrational modes of the entire crystal are fully contained in the memory function and, there, this memory function, which depends on past and present position of the guest, is approximated ad hoc as η(r(t), r(s); t − s) = Γe−|t−s|/τ ,

(1.19)

21 where τ is a correlation time to be determined. This model will certainly give dissipation and damping, but it is rather simple. In fact, for harmonic systems, it is well known that the decay of the memory function comes from the phase randomization of the different vibrational modes in the crystal. This model cannot possibly capture the details of the dynamics of the large crystal into one parameter, τ . Also, the dependence over the past history of the guest on the amplitude of the memory function was dropped altogether. As we will show, these two approximations are unnecessary, and this simple dissipation model is unlikely to properly describe the vibrational density of states of the full crystal. In this work, we do not eliminate all crystal degrees of freedom, rather we keep the target atoms explicitly in the simulation procedure. Hence our equations of motion describe the motion of both the guest and the target. The guest equation of motion is particularly simple while, in the harmonic crystal approximation, the target equations of motion transform to a GLE after the bath is averaged out [55], ∂U (rG , rα1 , . . . , rαNTarget ) dpG (t) = − dt ∂rG (1.20a) and ∂U (rG , rα1 , . . . , rαNTarget ) dpα = − dt ∂rα +eiLt hFα iBath + F†α (t) NTarget Z t X hF†α (t − t1 )F†γ iBath − dt1 · pγ (t1 ), m k T γ B 0 γ=1 (1.20b) where U (rG , rα1 , . . . , rαNTarget ) is the interaction between the guest and the target atoms, Fα is the net force on the target atom α that comes from other crystal atoms, F†α is colored noise, h. . .iBath is a projection operator that averages out the bath degrees of freedom and eiLt (where iL is the usual Liouville operator [56]) is an operator which propagates any

22 function of the system’s mechanical variables to time t. This procedure has many advantages compared with the one of Kopelevich and Chang [54]. First, the memory function, hF†α (t − t1 )F†γ iBath /kB T does not depend on the guest position here. Also, because all crystal interactions are harmonic, it can be calculated exactly at all times (the details are given in Chapter 2), no matter where the guest is. Had it depended on the guest position, we would have to recalculate it every time the guest moves, something which can drastically increase the numerical effort. Note that here and in Ref. [54], the memory function is proportional to the average over the bath degrees of freedom of the random force, F†α (t). This is known as the Einstein-Nyquist theorem and it relates the dissipative energy loss with the variance of the random force which accounts from energy transfer from the lattice to the guest. This relation was made explicit in Eq. (1.20b), but a similar one holds for Eq. (1.18). All of the simulation results that will be presented here rely on this model, a harmonic crystal that is divided into two parts, a target and a bath space. The target atoms interact anharmonically with the guest. The interaction between the guest and the bath is not direct and happens through the target atoms. This allows us to calculate the memory function exactly and to reproduce accurately the vibrational density of states of the crystal without having to simulate a large number of crystal atoms. Of course, the price to pay here is that we still need to simulate some of the crystal atoms motion (remember that Kopelevich et. al. [54] eliminated all of them). It is within this simulation procedure that we will calculate the desired guest time-correlation functions, D(z) and ultimately, the permeability, P 0 [cf., Eqs. (1.14) and (1.15)]. In Chapters 2, 3 and 4, we will apply this formalism to noble gases diffusion in various channeled structures, but the simulation procedure, Eqs. (1.20a) and (1.20b), could be easily generalized for molecules. This section is concluded with a brief remark about the conclusions that were obtained from earlier work about the role that the lattice vibrations play in the diffusion process. In most of these studies [54, 40, 37, 39], the lattice vibrations had very little to no effect on

23 the diffusion. The systems under study there were typical zeolites (ZSM-23, Mordenite, Silicalite) containing large channels and the diffusing molecules were rather small (Argon, Methane, Benzene). Here, we will use our more general procedure to assess the role of the crystal vibrations in two typical systems that are very different. In the first case, we will look at Xenon diffusing inside Theta-1 (a silicate with medium size pores and small energy barriers to diffusion) and second, we will look at Argon in α-quartz (a silicate with very narrow channels and high energy barriers to diffusion). As will be shown in Chapters 2 and 3, the flexibility of the lattice can play an important role in the guest diffusion when the interaction with the lattice is strong enough. 1.4 Quantum mechanical corrections As stated above, the role of the lattice vibrations on the guest diffusion will be studied for two different systems in Chapters 2 and 3. As the reader will see in the following chapters, for one of these systems, the lattice vibrations play a major role in the diffusion process. This system is Argon diffusing inside α-quartz. Briefly, for this system, the guest atom is always in the steep repulsive region of the interaction potential with one or many lattice atoms. For such systems, a rigid lattice model leads to extremely large barriers, which are drastically reduced when the lattice is allowed to relax to its new equilibrium position. Clearly, this process happens through lattice flexibility and vibrations. Because the guest is very close to some lattice atoms, and because the channels are very narrow, the vibrating framework seems to slow down the diffusion of the guest. These conclusions will be analyzed in details below. Naturally, for systems where the lattice vibrations participate significantly in the diffusion process, another question arises: Should we treat the lattice degrees of freedom quantum mechanically? In fact, it is well known that only a fraction of the crystal vibrational modes are strongly excited thermally at room temperature [48]. These modes behave classically. The rest are in a regime where quantum mechanical effects, like zeropoint motion, must be taken into account. Specifically, the dimensionless parameter that

24 controls the extent of the quantum mechanical corrections to the lattice vibrations is β¯ hω where ω is the frequency of a vibrational mode. At high temperatures, when β¯hω ¿ 1 for all modes, the crystal behaves classically. On the other hand, when β¯ hω À 1 for many vibrational modes, quantum mechanical corrections start to appear. For typical crystal, at room temperature, there are about 25 percent of the modes that behave classically. The rest should really be treated quantum mechanically. Therefore, in order to describe the lattice vibrations as realistically as possible, we should reformulate the theory of the permeability in the language of quantum mechanics. Hence, following our fully classical treatment we study the quantum mechanical corrections to the guest diffusion and to the permeability. The role of quantum mechanics for these types of systems has never been addressed before. Without going into to much detail, when the diffusion problem is cast in the language of quantum mechanics, the space-dependent diffusion coefficient and the permeability can still be obtained in terms of correlation functions, but these cannot simply be evaluated from the standard simulation of classical trajectories. In fact, as shown in Chapter 4, the space-dependent diffusion coefficient is more generally expressed in terms of correlation functions of operators that are defined in the guest sub-space. More precisely, the general ˆ is written as, quantum time-correlation function between two operators, Aˆ and B, h i ˆ Bi ˆ = Tr A(t) ˆ Be ˆ −β Hˆ /Q, hA(t)

(1.21)

ˆ is the Hamiltonian operator, Q the system’s partition function and Tr is a quantum where H mechanical trace. Note that, at this level, the Hamiltonian contains everything (the guest, the target and the bath). The goal here is to develop a simple formalism to accurately calculate such correlations and later use these correlations to reformulate the permeability. Here, the Heisenberg representation, where the time-dependence is carried by the operaˆ h ˆ −iHt/¯ ˆ h ˆ ≡ eiHt/¯ tors, will be used such that A(t) Ae .

25 Because it has a very intuitive classical limit, these propagators will be approximately evaluated within the path integral formalism [57]. Path integrals are often very difficult to evaluate exactly, but here the harmonicity of the bath allows us to perform all bath path integrals analytically. The path integrals will further simplify if we treat the guest semiclassically. Note that most experimental studies of diffusion in channeled structures are carried at room temperature or higher. For lower temperatures, the diffusion slows down considerably which makes it hard to measure experimentally. For moderate temperatures, the guest can be considered to be classical. We use this fact to evaluate the path integrals semi-classically where only the crystal degrees of freedom are completely quantum mechanical. According to Eq. (1.21), there are three propagators to be evaluated. The two ˆ

ˆ

real-time propagators associated with the factors e−iHt/¯h and eiHt/¯h that are respectively ˆ

called the Forward and Backward propagators. The other is associated with e−β H and it defines the density matrix. This last propagator is often called the complex-time propagaˆ in Eq. (1.21), depend solely on the guest, tor. The fact that the two operators, Aˆ and B considerably simplifies the evaluation of the path integrals. First, the complex time propagator is calculated assuming that the guest is roughly constant during the time interval β¯h (this argument is identical with the one of Feynman and Hibbs when they evaluate partition functions [58]). From the complex-time propagator, a guest potential of mean force can be obtained as in the classical case, by averaging out the quantum mechanical lattice crystal. This will be described in Chapter 4 and will be shown to reduce to its classical counterpart at high temperatures. Second, the treatment of the real-time propagators is based on the Martin-Siggia-Rose formalism that maps stochastic processes to path integrals. As shown in Chapter 4, the evaluation of the Forward-Backward propagators reduces to the numerical evaluation of guest time-correlation functions from the simulation of stochastic differential equations that are similar to the classical GLE, Eq. (1.20b), but with modified noise terms and modified initial conditions determined from the complex-time propagator. In fact, these new corrections to the classical stochastic equations account for the quantum

26 nature of the lattice and they vanish in the high-temperature limit. This modified set of GLE will then be used to simulate the correlation functions that are necessary to calculate D(z) and P 0 . Other semi-classical formulations have been studied in the past for simpler systems. In particular, semi-classical Langevin equations were obtained by Schmid et al. [59, 60] and Kleinert and Shabanov [61] for a particle coupled to a harmonic bath. In these two approaches, it was assumed that, at time zero, the bath is decoupled from the rest of the system, an approximation that is, as we show in Chapter 4, unnecessary and incorrect for our type of system, even in the classical limit. On the other hand, as will be shown below, our formalism also leads to a set of semi-classical Langevin equations. In another semi-classical formalism developed by Makri et al. [62, 63, 64, 65], both realtime propagators are individually evaluated within the semi-classical WKB [57] approximation and the complex-time propagator (or density matrix) is obtained by discretizing the path integral and evaluating it using Monte-Carlo integration techniques [66]. The complex-time propagators could also have been obtained using various approximate methods. For example, many authors came up with ad hoc approximate forms for the density matrix [67, 68, 69, 70] that are exact for harmonic systems and that reproduce the classical propagators at high temperatures. These procedures are much more tractable than dealing with the full path integrals. In these methods the guest itself contains some quantum mechanical corrections and these techniques will be compared with our formalism, where the guest quantum corrections are obtained as if the guest was a free-particle in a external uniform potential. This comparison will show that, at room temperature, the guest itself behaves almost classically. In conclusions, the effect of the semi-classical treatment of the diffusion process is that it generally slightly increases the potential of mean force of the guest throughout the lattice. This can be easily explained as follows. Within our approximation, the quantum

27 corrections to the potential of mean-force come from the lattice vibrations. As temperature increases, the potential of mean-force increases accordingly. When the vibrations are quantum mechanical, there is always residual zero point motion, even at low temperatures, which increases the potential of mean force. This results in a lower adsorption probability. Also, for the system studied (neon in α-quartz), the guest correlation time is shorter compared to the classical case. These two effects combine to reduce the permeability by a factor of about 25%. 1.5 Improving the approximate density matrix evaluation As we discussed above, the accuracy of the semi-classical formalism relies, in part, on an accurate evaluation of the density matrix which is often written as ρ(x, x0 ) in coordinate space. Only for a very limited number of models can this propagator be exactly evaluated in the path integral formalism. In the previous section, we cited a number of approximate methods that could be used to evaluate the density matrix. There is another approximate technique (due to Feynman and Kleinert [71]), which was not cited above, that gives a very good estimate of exact propagator in simple systems. We did not cite this method in the last section because it is inappropriate for diverging interaction potentials (like the Lennard-Jones potential, Eq. (1.4),) hence we could not apply it to calculate the free energy landscape of guest inside zeolites. On the other hand, we have found a way of improving the original Feynman-Kleinert method. This improved method is described in Chapter 5. Briefly, the original Feynman Kleinert method was first proposed as a tool to approximately calculate quantum mechanical partition functions. For any type of potential, the partition function, Q, is obtained from, Z Q=

1 dx ρ(x, x) = √ 2πβ

Z dx e−βWef f (x) ,

(1.22)

28 where, following Feynman and Kleinert, an effective potential has been defined, Wef f (x). Using an exactly solvable reference harmonic potential, with oscillator frequency ω, Feynman and Kleinert reported an approximate expression for Wef f (x) for which the true effective potential is a lower bound for any ω. This was used with a variational principle to obtain an equation that determines the oscillator frequency which gives the best estimate to the effective potential. The original Feynman-Kleinert method was tested numerically in Refs. [71, 72] and it gives very accurate effective potentials even for very low temperatures. Kleinert later extended the original formalism to calculate excited state energies [73] and density matrices [74]. In particular, one of the reference harmonic potentials used by Kleinert in Ref. [74] is, V0 (x(τ )) =

mω 2 (x(τ ) − x¯)2 , 2

(1.23)

where τ is the time (remember that all x(τ ), except the end points, are integrated in the path integral formalism) and where ω and x¯ are to be varied to give the best estimate to Wef f (x). As shown in Chapter 5, our modification to the method is the use of the following reference potential, V0 (x(τ )) =

mω 2 (x(τ ) − x¯(τ ))2 , 2

(1.24)

where we now allow x¯(τ ) to depend on time. This is justified by the following argument. Any possible path that connects the two end points contribute to the path integral. In particular, for ρ(x, x), the paths start and end at x. The overall contribution of a particular path is weighed by,

µ ¶ Z 1 β¯h mx(τ ˙ )2 exp − dτ + V (x(τ )) , h ¯ 0 2

(1.25)

where the integral is called the Euclidean action and where V (x(τ )) is the true potential. For low temperature, β¯h is large and the paths that dominate are the ones that spend most of their time close to an energy minimum. Therefore, for two specified end points, the paths can sample large spatial regions. The advantage of using a reference potential with

29 a moving minimum is that it can reproduce the average paths of the true potential with good accuracy, which results in a more accurate Wef f (x). In fact, we show in Chapter 5 that the variational principle of Feynman and Kleinert can be applied here and completely determines x¯(τ ), although the numerical effort to obtain Wef f (x) increases. We have tested our method for the one-dimensional double well potential and compared it against the original method. The improvement that this give is small for the effective potential, but becomes important when the formalism is used to obtain the off-diagonal elements of the approximate density matrix (in the language of path integrals, this means that the two end points differ). This work, which is presented in Chapter 5, is the result of some attempts we have made to improve the calculation of the quantum correction to the guest diffusion in channeled structures. Again, it cannot be applied it this context because of the nature of Lennard-Jones potential. 1.6 Other Problems The last part of the thesis is completely unrelated with the diffusion problem, but pertains to an important fundamental relation of statistical mechanics that was recently derived; the Jarzynski equality [75, 76, 77]. The equality is written as follows, ­

® Zb (β) eβW = ≡ e−β∆AJ , Za (β)

(1.26)

where W is the work done by the system, β is defined in terms of the initial temperature and Za/b are equilibrium canonical partition functions calculated for different work parameter (the work is done by varying a parameter in the Hamiltonian from a to b) at the initial temperature. The above ratio of partition functions can be written, as usual, in terms of the exponential of a free energy change. This free energy change is called ∆AJ here. We will show below that this relation does not depend on the rate of change of the work parameter contrary to the average work which depends strongly on how the work is done. Moreover, the Jarzynski equality is valid even if the process drives the system far from equilibrium.

30 This equality has been experimentally verified by Liphardt et al. [78] by stretching single RNA molecules in solution. The equality has often been the subject of some controversy since its original publication. In particular, Cohen and Mauzerall [79, 80] raised the objection that the temperature that appears in the Jarzynski equality is in general irrelevant to the actual temperature of the system after the work process. Moreover, when the process carries the system far from equilibrium, one can generally define a temperature only some time after the process, when the system as returned to local equilibrium. This statement is correct, but it is easy to show that the equality is indeed valid. Here, the derivation of the Jarzynski equality will be summarized because, it is the best way to show that it is correct. This will be done in the context of classical mechanics, although the equality has been extended for quantum system by Mukamel [81] and for classical stochastic systems by Crooks [82]. This proof of the equality was proposed by Jarzynski in Ref. [77] in answer to the criticism raised by Cohen and Mauzerall. In general, for Hamiltonian systems, there is no heat exchange because the Hamiltonian includes everything. Therefore, the process is adiabatic and the work is given by, Z

t

W =−

ds 0

dH(Γ, λ(t)) = H(Γ0 , a) − H(Γt , b) dt

(1.27)

where Γt represents the phase vector at time t, λ(t) is the work parameter (λ(0) = a and λ(t) = b). The duration of the work process is t. At t = 0, the system is at equilibrium. Because the system’s dynamics obeys the laws of classical mechanics, the position and velocities of all particles are completely determined from their initial values. Therefore, the average of eβW is formally written as, Z 1 Za (β) Z 1 = Za (β)

­ βW ® = e

dΓ0 −βH(Γ0 ,a)+βW e hN dΓ0 −βH(Γt ,b) e . hN

(1.28)

31 This last line is very close to the ratio of partition functions that appears in Eq. (1.26). In fact, if we could replace dΓ0 by dΓt in the above integrals, one would obtain the Jarzynski equality. This can in fact be done because, by Liouville’s theorem [56], µ det

∂Γ0 ∂Γt

¶ = 1.

(1.29)

The Jarzynski equality has been proved quite generally. Notice that this proof clearly shows why the initial temperature must be used in the factor eβW ; this allows the H(Γ0 , a) terms to cancel. The Jarzynski equality has been the subject of a large number of studies in the context of specific models. To give a few examples, Marathe and Dhar [83] tested the Jarzynski equality for spin systems, Oostenbrink and van Gunsteren [84] studied processes that result in no free-energy change; like redistribution of charges, creation and annihilation of neutral particles and conformational changes in molecules. Jarzynski [76] tested it for a harmonic oscillator system where the vibrational frequency is increased. Further, the equality has been extensively tested for the one dimensional expanding ideal gas model by Press´e and Silbey [85], Lua and Grosberg [86] and by Bena, Van den Broeck and Kawai [87]. Also note that the equality is now recognized as a tool for calculating free-energies [84, 88] In the last part of this thesis, we accept Eq. (1.26) as exact, and we rather focus on its connection with thermodynamics; i.e., how is ∆AJ related, if at all, to the free energy changes encountered in non-equilibrium thermodynamics. In fact, the Jarzynski equality is often combined with the Gibbs-Bogoliubov-Jensen-Peierls inequality to give an upper bound to the work, hW i ≤ −∆AJ ,

(1.30)

which is strongly reminiscent of the second law work bound for isothermal systems at constant volume. The questions that we investigate in Chapter 6 are the following. First, how is the free-energy change that appears in the Jarzynski equality related to the actual

32 free-energy change of the system during the work process? Second, how does the work bound obtained from the Jarzynski equality compare with the standard thermodynamic work bounds which is derived from the First and Second Law. This is an important question because it is sometimes believed that the Jarzynski equality is a proof of the Second law of thermodynamics (for example, see Refs. [87, 89]). These questions will first be examined in the context of the ideal expanding gas model, for which the Jarzynski equality is unambiguously satisfied [86, 87]. The work bound will then be analyzed within a general thermodynamic argument which shows that, even if the Jarzynski equality is correct, the work bound that it implies is, in general, an upper bound to thermodynamic upper bound. In Chapter 6, we show that this thermodynamic analysis is in complete agreement with the one dimensional ideal gas model and the harmonic oscillator model used by Jarzynski [76] to study the work bounds. The Jarzynski equality will then be analyzed in the context of response theory [90] which will be used to derive the conditions for when the two work bounds differ.

CHAPTER 2 c Diffusion in channeled structures: Xenon in a crystalline sodalite [53](°American Physical Society, 2003) 2.1 Introduction Understanding the diffusion of a guest component inside channeled structures (like membrane channels, zeolites and many silicates) has been a problem of interest for many years. Crystalline channeled structures have many applications in gas phase separation and are also widely used as catalysts in chemical reactions [2, 4]. The diffusion of one or more guest components inside the crystal plays an important role in any of these applications. In this work, we develop a systematic approach that allows us to understand the diffusion process microscopically and calculate the macroscopic permeability of channeled structures to a guest component. Specifically, we will apply the theory to the diffusion of noble gases through Theta-1, a high silicate zeolite that shows a remarkable selectivity in catalysis applications [4] and could be a good candidate to study single-file diffusion [91, 92]. Early theoretical calculations on channeled structures focused mainly on the heat of sorption [12, 13]. These calculations were performed using a model potential for the guest-crystal interactions on a rigid lattice. Next, the diffusion within channel structures was studied through molecular dynamics (MD) simulation [93, 38, 43], where a diffusion coefficient was calculated from Einstein or Green-Kubo relations, cf. Eqs. (2.1) and (2.2) below. For high internal potential energy barriers, where barrier crossing events are rare, a common way to proceed was to determine a hopping rate constant using transition state theory [44, 45, 46]. In homogeneous systems, the diffusion coefficient, D, can be obtained using an Einstein relation, h|r(t) − r(0)|2 i D = lim , t→∞ 6t 33

(2.1)

34 or equivalently by a Green-Kubo relation, 1 D= 3

Z

∞

dt hv(t) · v(0)i,

(2.2)

0

where r is the position of the guest, v is its velocity and h. . .i denotes an equilibrium average. These last two equations are valid only for an uniform system, and imply a diffusion equation for the guest component of the form, ∂n(r, t) = D∇2 n(r, t), ∂t

(2.3)

where n(r, t) is the number density of the guest. In the second approach, transition state theory or one of its modified versions (cf. Refs. [44, 45, 46]) is used in a hopping model to calculate the hopping rate constants. Of course, transition state theory makes several assumptions, the key ones being that the motion closely follows the reaction coordinate and that there are no re-crossings. This paper proposes an alternative and more general method that can also be used to verify the validity of transition state theory. For bounded systems with large potential gradients, a more correct starting point is the generalized diffusion equation, ∂n(r, t) = ∇r · ∂t

Z

↔

dr0 L (r|r0 ) · ∇r0 βµ(r0 , t),

(2.4)

where µ(r, t) is the chemical potential and where the generalized Onsager diffusion coefficient, ↔

Z

∞

0

L (r|r ) ≡

dt hJ† (r, t)J† (r0 )i,

(2.5)

0

leads to a space-dependent, non-local, diffusion coefficient generalizing Eq. (2.3). In this last equation, J† is the dissipative (random) part of the diffusion current defined by a projection operator [94, 95].

35 In all the approaches just mentioned, one still needs to make contact with what is measured experimentally; e.g., the net flux, j, of material passing through the channeled material. In steady-state, this typically obeys the macroscopic phenomenological constitutive relation j = P (µ+ − µ− )z=0 ,

(2.6)

where P is the permeability and µ± are the chemical potentials in the ± phases. The net flow is assumed to lie along the z-axis and z = 0 is the mid-plane inside the channeled structure. The simple diffusive or hopping models easily yield expressions for the permeability. The calculation based on the generalized diffusion equation, Eq. (2.4), is more involved, was considered in Ref. [6], and will be used here. Note that this theory does not assume a priori any reaction coordinate that dominates the dynamics of the guest. Previous molecular dynamics simulations were performed using rigid [93, 38, 43] or flexible [44, 45, 46, 96] lattices. The motion of the lattice in Refs. [44, 45, 46, 96] is, again, simulated with molecular dynamics. In this work, the motion of the lattice is described by a generalized Langevin equation (GLE) that mimics the effect of the infinite crystal. This approach is taken for many reasons. First, with the GLE, it is possible to reproduce the vibrational density of states of the infinite crystal with high accuracy. Second, the presence of the guest will inject energy in the crystal lattice. The use of GLE allows the dissipation of that energy in a physically consistent way. In conventional MD, this extra kinetic energy would stay in the system and could later on affect the guest dynamics. As was pointed out by Kopelevich and Chang [54], there are also more subtle problems associated with classical lattice models with periodic boundary conditions; specifically, artificial feed-back mechanism can lead to highly exaggerated sorbate transport rates in MD simulations. Of course, for the same number of degrees of freedom, the GLE is more numerically expensive than conventional MD. If the vibrational spectrum of the infinite crystal is to be

36 reproduced in MD simulations for typical zeolites, the motion of about 104 atoms needs to be simulated. Therefore, the flexibility of the lattice is often neglected in such problems. As shown by Kopelevich and Chang [54], neglecting the flexibility of the lattice does not lead to large errors for small guest in large channels structures. However, including the flexibility is mandatory for system where the size of the guest is smaller or equal to the pore size or if the guest has to pass through small bottlenecks. In other words, one needs to include the flexibility if it changes significantly the guest available volume inside the crystal. The importance of our approach relies in its generality. It has been shown that the use of Eqs. (2.1)–(2.3) is inconsistent for inhomogeneous systems [7]. We will show here that the transition state theory approach is not valid for system with low energy barriers (in agreement with the prediction made in Ref. [54]) where the flexibility of the lattice can usually be neglected. When the energy barriers are large, transition state theory is expected to give a more accurate result, but in that case, the flexibility of the lattice usually plays an important role. The method proposed here is general since it includes the flexibility of the lattice self-consistently through a GLE and it’s applicability is independent of the magnitude of the energy barriers. The paper is divided as follows. In Sec. 2.2, a summary of the theory leading to a projection-operator, correlation-function expression for the macroscopic permeability is presented and we show how to approximately reexpress time correlation functions containing projected dynamics in terms of those associated with Newtonian equations of motion. The evaluation of the permeability requires a space-dependent Onsager diffusion coefficient which is obtained in terms of equilibrium time correlation functions that are computed by simulating generalized Langevin equations of motion for the guest and harmonic lattice atoms presented in Sec. 2.2, an approach first discussed by Deutch and Silbey [55]. In Sec. 2.3, we show how the memory functions and random noise terms that appear in the Langevin equations of motion for the crystal atoms can be calculated and we

37 demonstrate that the vibrational density of states of the full Theta-1 crystal is reproduced. We also give an approximate way of calculating the potential of mean force for the guest inside the crystal and test it against the numerical simulations. Section 2.4 presents the details of the molecular model and gives results for the correlation functions and finally for the permeability for Xenon in Theta-1. A detailed comparison with transition state theory is made in Sec. 2.5. We summarize and make some concluding remarks in Sec. 2.6. 2.2 Theory 2.2.1 Microscopic Expressions for the Permeability The diffusion of the guest component inside the channeled structure is governed by the generalized diffusion equation, Eq. (2.4). The system will have many potential barriers and a nontrivial energy landscape. Microscopic expressions for the permeability starting from the generalized diffusion equation for such systems were obtained by Ronis and Vertenstein [6]. Here, we simply state their result. The macroscopic permeability of the channeled material is given by 1 1 = P β

Z

·

+∞

dz −∞

where, 1 D(z) ≡ A

Z

¸ 1 1 − , D(z) D+ Θ(z) + D− Θ(−z) Z

∞

dt 0

(2.7)

Z drk

dr0 hJ†z (r0 , t)J†z (r)i.

(2.8)

In these last equations, D(z) is a space-dependent Onsager diffusion coefficient, D± is the bulk chemical potential in the ± phases, J†z is the z-component of the irreversible part of the current and A is the area of the crystalline medium. The integral over rk (x, y) is a consequence of the fact that the net current through the interface is along the z axis. Note that this result is first order in membrane excess quantity and this choice of D(z) makes the higher order corrections smaller.

38 The expression for the permeability, Eq. (2.7), was derived on the basis of Eq. (2.6). The chemical potentials appearing in Eq. (2.6) are the bulk chemical potentials of the two regions extrapolated to the z = 0 plane. It is more convenient to rewrite Eq. (2.6) in terms of the chemical potential at the two outer surfaces of the channeled medium. Since j is constant in the bulk in steady-states, we rewrite Eq. (2.6) as j = P 0 [µ(d) − µ(−d)],

(2.9)

where 2d is the thickness of the interface and 1 1 = P0 β

Z

d

dz −d

1 , D(z)

(2.10)

is a permeability intrinsic to the material. 2.2.2 Correlation Function Expression for D(z) We already have a correlation function expression for the space-dependent Onsager diffusion coefficient D(z) in Eq. (2.8). Unfortunately, this equation cannot be used directly to compute D(z) because it involves the random part of the current. A common practice is to set J† = J, but, as was shown in Ref. [7], this is only valid in special cases, and in general, it is not valid in systems that are spatially inhomogeneous. A correlation function expression for D(z) that approximately includes the effects of the projection operator on the time dependence of the memory function was obtained by Ronis and Vertenstein [6] in terms of unprojected time correlation functions. Their final expression for D(z) is R∞ n∞ 0 dt hvG,z (t)vG,z iz e−βW (z) R∞ D(z) = , (2.11) 1 + 0 dt hβF (z(t))vG,z iz where n∞ is the number density in the bulk, vG,z is the z-component of the guest velocity, W (z) and F (z) are, respectively, the plane average potential of mean force and the mean force. Also, h...iz denotes an equilibrium conditional average for trajectories whose initial z coordinate is z. Infinite dilution was also assumed deriving Eq. (2.11). The correlations that appear in this expression will be evaluated by the means of numerical simulations of

39 the particle dynamics below. Given the correlation functions that appear on the right-hand side of Eq. (2.11), the calculation of the permeability is trivial. The equations of motion that will be used in the numerical simulations are described next. 2.2.3 Equations of Motion In this section, the equations of motion for the diffusing particle (hereafter referred to the “guest”) and the rest of the silicate atoms are described. For practical purposes, the motion of a relatively small number of crystal atoms must be simulated. The atoms in this part of the channel will be referred to as the target atoms and the rest of the crystal is called the bath. One of the main goals of this work is to preserve the effects of the crystalline bath on the motion of the target and guest atoms. In order to do this, we will use a projection operator approach introduced by Deutch and Silbey [55] in their derivation of the Langevin equation of motion for a particle in a harmonic lattice. This approach was subsequently used by Tully in his work on gas-surface interactions [97] and by Adelman, Diebold and Mou [98] in their work on gas-solid energy exchange processes. By assuming that the guest does not directly interact with the bath and that the crystal is fully harmonic, the equations of motion of the reduced (guest and target) system are ∂U (rG , rα1 , . . . , rαNTarget ) dpG (t) =− dt ∂rG

(2.12)

and ∂U (rG , rα1 , . . . , rαNTarget ) dpα = − dt ∂rα +eiLt hFα iBath + F†α (t) NTarget Z t X hF†α (t − t1 )F†γ iBath · pγ (t1 ), − dt1 mγ kB T 0 γ=1 (2.13) where F†α (t) ≡ eiLBath t (1 − P )Fα ,

(2.14)

40 is the contribution to the force on the α’th target atom at time t exerted by the bath in the presence of the frozen target atoms. In the last equations, the classical Liouville operator (iL = iLT arget + iLBath ) has been introduced as well as another projection operator. The projection operator, h...iBath is a normalized average over the bath degrees of freedom. Note, that the projection operator no longer appears in the time dependence of F † (t), and moreover, as was shown in Ref. [55], will evolve independently of the guest-target motion; as such, Eq. (2.13) is a generalized Langevin equation. F†α (t) is a colored noise and is considered in more detail in the next section. The last term on the right hand side of Eq. (2.13) is the expected friction term where the memory comes in through a force-force time correlation function. Note that it is possible to further project out the equations of motion for the target atoms if we linearize the guest-target interaction with respect to the target coordinates. This is basically the assumption of Deutch and Silbey, cf. Ref. [55], and this approach was taken by Kopelevich and Chang [54]. This approximation is not valid when the size of the guest is comparable or smaller than the pore sizes. Finally, note that the assumption that the guest does not interact directly with the bath can be relaxed if we can linearize the guest-bath forces in the bath degrees of freedom; this modifies Eq. (2.13) slightly, and in particular, makes the memory function depend on the instantaneous position of the guest at time t1 . In the next section, we show how the force-force correlation function can be calculated, and put everything together in order to perform the simulations. 2.3 Implementation 2.3.1 Effective Forces and Force Correlation Functions In this section, we reexamine Eq. (2.13) and show how the various terms that appear can be calculated. The separation of crystal atoms into target and bath subspaces allows

41 us to block the force constant matrix K as follows    KT T KT B  K= , KBT KBB

(2.15)

where KT T is the 3NTarget × 3NTarget matrix linking atoms in the target subspace and KBB is a 3NBath × 3NBath matrix linking atoms in the bath subspace only. The two rectangular matrices KBT and KT B connect the bath and target atoms. The first term of Eq. (2.13) can be written in matrix notation as eiLt hFT iBath = eiLt h−KT T rT − KT B rB iBath ,

(2.16)

where FT , rT and rB are, respectively, 3NTarget , 3NTarget and 3NBath column vectors. The Gaussian averages are performed to give eiLt hFT iBath = −Keff rT (t),

(2.17)

Keff ≡ KT T − KT B K−1 BB KBT ,

(2.18)

where

is an effective force constant matrix governing the harmonic motion of the target atoms in the presence of the bath. We now derive an expression for the force correlation function. Recall that F†T (t) is the force on the target exerted by the bath when the target atoms are frozen. In that case, the dynamics of the bath atoms are governed by, 0 d2 rB (t) 0 (t), = −KBB rB 2 dt

(2.19)

0 rB (t) ≡ rB (t) + K−1 BB KBT rT .

(2.20)

MB where

In the last equations, MB is the diagonal matrix containing the masses of the bath atoms.

42 The 3NBath eigenvectors, uB,i , of the matrix KBB are determined by, ˜ BB uB,i = ω 2 uB,i , K i

(2.21)

˜ XY ≡ M−1/2 KXY M−1/2 , with X, Y = T or B, K X Y

(2.22)

where

and where ωi2 is the eigenvalue associated with the i’th eigenvector. The shifted displace0 (t), can be expanded in terms of the mass scaled eigenvectors u˜B,i = ment vector, rB −1/2

MB

uB,i as 0 rB (t)

=

3N Bath X

µ u˜B,i

i=1

¶ bi ai cos(ωi t) + sin(ωi t) , ωi

(2.23)

where the ai and bi are related to initial positions and velocities, respectively, of the bath atoms and are Gaussian distributed. The random force, Eq. (2.14), is 0 F†T (t) = −KT B rB (t),

(2.24)

in matrix notation, and this can be rewritten in terms of the mass-scaled eigenvectors as F†T (t)

= −KT B

3N Bath X

µ u0B,i

i=1

¶ bi ai cos(ωi t) + sin(ωi t) . ωi

(2.25)

Since the ai and bi in the last equation are Gaussian distributed, Eq. (2.25) shows that F†T (t) is a Gaussian colored noise. Moreover, the random force-force correlation function is given by hF†T (t)(F†T )T iBath , and from Eq. (2.25), is easily written as

−1/2

hF†T (t)(F†T )T iBath = kB T KT B MB

3N Bath X i=1

uB,i uTB,i

cos(ωi t) −1/2 MB KBT , ωi2

(2.26)

where we have expressed the u0B,i in terms of the original eigenvectors uB,i , and where ha2i i = kB T /ωi2 .

43 As it turns out, the vibrational density of states of the infinite crystal is reproduced when the bath contains O(104 ) atoms. Therefore, Eq. (2.26) is not particularly convenient. In other words, the required eigenanalysis may be numerically too demanding. Brute Force The last section gives us a way of calculating the force-force correlation function in the time domain. By performing a Laplace transform on Eq. (2.26) and using the fact that, X

˜ BB , uB,i ωi2 uTB,i = K

(2.27)

i

we obtain, hF†T (s)(F†T )T i =

1 1/2 ˜ s ˜ BT M1/2 , MT KT B K T 2 ˜ ˜ β KBB (s + KBB )

(2.28)

where s is the Laplace transform variable. This last form of the force-force correlation function does not require an eigenanalysis. Instead, it requires the inversion of a large matrix. Inversion of matrices require less numerical effort than a full eigenanalysis, especially when the matrices involved are sparse. Also, the Laplace representation will be more convenient to use in the simulations. The inversion of a matrix of rank N requires O(N 2 ) computer memory and a simple estimate shows that our computation cannot be done on most common computers. One way out of this problem is to make an approximation about the nature of the forces within the crystal. From now on, we assume that the crystal atoms interact with their nearest neighbors through stretching interactions and with their second nearest neighbors through bending interactions, and that these are the only interactions present. Hence, the force constant matrix will be massively sparse and this allows us to perform the inversion in Eq. (2.28) even if KBB is large. This approach approximates the effect of the infinite bath using a large, but finite part of the crystal that reproduces the vibrational density of state accurately. We refer to this approach as the “brute force” method.

44 Brillouin Zones and Defects In this section, we will demonstrate how the force-force correlation function can be calculated in an exact way. This approach uses ideas first introduced by Maradudin in his study of defects in solids [52]. We rewrite Eq. (2.28) as hF†T (s)(F†T )T i = [Λ(0) − Λ(s)]/s,

(2.29)

1/2 ˜ 1/2 ˜ Λ(s) ≡ β −1 MT K T B G(s)KBT MT ,

(2.30)

˜ BB ]−1 . G(s) ≡ [s2 + K

(2.31)

where

with

As before, the problem with the last expression lies in the inversion of a large matrix. The ˜ −1 , where K ˜ is the mass-scaled force constant function G0 (s) defined as G0 (s) ≡ [s2 + K] matrix for the full crystal, can be obtained exactly by using a Fourier representation and then integrating over the first Brillouin zone of the crystal. We assume that G0 (s) is known ˜ in the following way: and obtain Λ(s) in terms of it. To proceed, we reblock K   ˜ ˜ ˜  KT T KT B1 KT B2    ˜ = K , ˜ ˜ ˜ K K K B T B B B B  1 1 1 1 2    ˜ ˜ ˜ KB2 T KB2 B1 KB2 B2

(2.32)

where we have split the bath into two parts: the primary bath subspace, B1 , refers to bath atoms that couple directly to the target (i.e. they have a target atom as their nearest or second nearest-neighbor); the secondary bath subspace, B2 , contains atoms that are not directly coupled with the target atoms (clearly, B2 is much larger than the other subspaces). ˜TB = K ˜ B T = 0. Note that K 2 2

45 ˜ BB = K ˜ −∆ ˜ where, We write K 

 ˜ ˜  KT T KT B1 0    ˜ ≡ K ∆ 0   ˜ B1 T 0 ,   0 0 0

and rewrite G(s) as ˜ − ∆] ˜ −1 G(s) = [s2 + K ˜ −1 G0 (s). = [1 − G0 (s)∆]

(2.33)

We separate G0 (s) into blocks as 



 g11 g12  G0 (s) =  , g21 g22

(2.34)

where the 1 subspace contains the target and the bath primary zone and 2 refers to the bath secondary atoms. In this representation, 



˜ 0 δK  ˜ ≡ ∆  , 0 0 where



(2.35)



˜TT K ˜TB K 1  ˜ ≡ δK  . ˜ KB1 T 0

(2.36)

˜ −1 G0 (s), we find that By evaluating [1 − G0 (s)∆] 

 ˜ −1  (1 − g11 δ K) g11 G(s) =  ˜ − g11 δ K) ˜ −1 g11 + g21 g21 δ K(1

˜ −1 g12 (1 − g11 δ K) ˜ − g11 δ K) ˜ −1 g12 + g22 g21 δ K(1

  . (2.37)

There are still multiplications of large matrices in the last expression, but notice that ˜ −1 , contains matrices in the 11 space. These the only inverse that we need, (1 + g11 δ K)

46 matrices are relatively small and the inversion is much more manageable. Moreover, by ˜ T B and K ˜ BT are nonzero only in the 11 subspace, only the 11 block of G(s) noting that K is needed in order to compute Λ(s), which thus becomes Λ(s) =

1 1/2 ˜ 1 ˜ BT M1/2 , MT KT B −1 K T ˜ β g11 (s) − δ K

(2.38)

using Eqs. (2.30) and (2.37). This expression is more convenient than Eq. (2.28) because it involves small matrices. Everything that we have done in this subsection is exact. It is very simple to work with Eq. (2.38) provided that we have calculated g11 (s) beforehand. The periodicity of the lattice can be used to obtain G0 (s), and hence, g11 (s), in terms of integrations over the Brillouin zone. Since these methods are standard (see, e.g., Ref. [51]), we simply state the result; i.e., ·

[G0 (s)]α,β i,j Z =

0

1 = ˜ s2 + K

¸α,β i,j

dk ik·(Ri −Rj ) X ²αp (k)[²βp (k)]T e . (2π)3 s2 + ωp2 (k) p (2.39)

where the indices i and j indicate which unit cell the atoms lie in, and where α and β denote the atoms within the unit cell and the Cartesian components of the displacements. The prime on the integral sign restricts the integration to the first Brillouin zone of the crystal. Also, ωp2 (k) and ²p (k) are, respectively, the p’th eigenvalues and eigenvectors of ˜ the matrix K(k) defined by ˜ α,β (k) ≡ K

X

˜ α,β , e−ik·R K R

(2.40)

R

where R is the lattice vector connecting the respective unit cells of atoms α and β. This method requires an eigenanalysis of a matrix of rank 3N0 , for every wavenumber (k), where N0 is the number of atoms in the unit cell. On the other hand, the numerical evaluation of the Fourier transform has to be done carefully such that an accurate result

47 is obtained. In particular, the sampling of wavevectors has to be on a scale finer than 2π/|Ri − Rj |, which is a problem when large separations are needed. Thus, we have two ways of calculating the force-force correlation function. The first, is a brute force way in the sense that we make the bath as large as we can (the upper bound is determined by the amount of computer memory we can use) and perform the matrix inversion using a sparse subroutine. The other approach is to use the theory of defects together with a Brillouin zone calculation of G0 (s). This approach is exact on paper, but the numerical integration prescribed by Eq. (2.39) introduces inaccuracies. Another approach would be to approximate the k-dependence of ωp (k) and do the integrals exactly. We tried all three approaches and they give comparable results. We decided to use the brute force method as it is free of the above problem. 2.3.2 Differential Equations The Langevin equation derived in Sec. 2.2.3 is not convenient for numerical use. First, while we are able to calculate the force-force correlation function in time or frequency, we do not have a simple analytic representation for this function. All we have are inefficient ways to obtain the function at a discrete collection of points. Second, the Langevin equation is a stochastic colored-noise integro-differential equation. In this section, we drop the integral term in Eq. (2.13) at the expense of introducing extra dynamical fields, and in order to do this, we introduce an analytic approximation to the memory-functions. In frequency space, the force-force correlation function is described by Eq. (2.28). We approximate the Laplace transform of the memory function matrix as βhF†T (s)F†T T i ≈

s A + Bs + Cs2

(2.41)

where A, B and C are 3NTarget ×3NTarget matrices. Analytical expressions for A and C can be obtained from the s → 0 and s → ∞ limit of Eq. (2.28). After examining several different schemes for obtaining B, each giving roughly equivalent results, we decided to obtain the A and the B matrices from a linear least square fit while the C matrix was obtained from

48 the asymptotic relations. Note that our approximation for the memory functions captures the decay and the oscillatory behavior of the memory function. As shown in Ref. [97], the vibrational density of states g(ω), can be expressed in terms of the memory function as ˜ g(ω) = Tr(Re[C(iω)]), where

à ˜ C(s) ≡

˜ eff K s+ + βM−1/2 hF†T (s)F†T T iM−1/2 s

(2.42) !−1 ,

(2.43)

In Fig. 2–1, we compare the approximate vibrational density of states with the exact result calculated using Brillouin zone sums. The agreement is excellent. Note that the matrix hF†T (s)F†T T i does not have a rank equal to 3NTarget . This is expected since every atom in the target space does not interact directly with the bath. In fact, for the harmonic interactions considered here, only target atoms which have a bath atom as their nearest or second-nearest neighbors can interact with the bath, and only these have nonzero random forces. In reality, the rank of the matrix is even smaller (e.g. as indicated by extra zero eigenvalues). This implies that there are extra motions of the target atoms that do not couple to the bath. An example of such a motion is illustrated in Fig. 2–2. Therefore, henceforth, we work in a reduced space (were A, B and C are non-singular) determined by the number independent target motions that couple to the bath. With our expression, Eq. (2.41), for the force-force correlation function, we can replace the noise and friction term in the Langevin equation, Eq. (2.13), by an extra dynamical field, −˙y(t). The equations of motion for the guest and target atoms are now written as, dpG (t) dU (rG (t), RT (t)) =− , dt drG

(2.44)

dPT (t) dU (rG (t), RT (t)) = −Keff RT (t) − − y˙ (t) dt dRT

(2.45)

49

Figure 2–1: The exact density of states (full line) for Theta-1 obtained in a Brillouin zone calculation is compared with the approximate density of states (dashed line) that is generated using our representation of the memory function, Eq. (2.41). The force constants are specified in Sec. 2.4.1.

50

Figure 2–2: In this figure, the gray atom is a bath atom while the black ones are target atoms. The motion of the second target atom is illustrated. For potentials that include only stretching and bending energies, the bath atom does not feel the motion since the angle θ remains unchanged.

51 and

µ

¶ d2 d C 2 + B + A y(t) = η(t) + M−1 PT (t). dt dt

(2.46)

The extra dynamical field, y(t) is a generalized Ornstein-Uhlenbeck process [49] with random initial conditions that satisfy hyyT i = kB T A−1 ,

(2.47)

h˙yyT i = hyy˙ T i = 0,

(2.48)

h˙yy˙ T i = kB T C−1 ,

(2.49)

and

and where the white noise variable η(t) satisfies hη(t)η(t0 )T i = kB T Bδ(t − t0 ).

(2.50)

In Appendix 2.7 we show that this new set of equations of motion for the target atoms is equivalent to Eq. (2.13), as long as the memory function can be written as in Eq. (2.41). 2.3.3 Potential of Mean Force Approximation At this point, we have everything that we need to perform simulations of the guest motion inside the channel used to calculate the correlation functions appearing in the diffusion coefficient [cf. Eq. (2.11)]. The only quantity that is still missing is the plane potential of mean force. In this section, we derive an approximation for the potential of mean force W (rG ) for the guest in the channeled structure. The mean force, F(rG ), can be obtained from the following potential of mean force: ·Z

¸ −β( 21 RT T Keff RT +U (rG ,RT ))

W (rG ) = −kB T ln

dRT e ·Z

= −kB T ln

¸ −βU 0

dRT e

,

(2.51)

52 where the definition of U 0 is obvious and where RT is now the displacement of the target atoms from their equilibrium positions in the absence of the guest. The interaction potential between the target and the guest, which is still unspecified, will not have a simple linear or quadratic form. Therefore, in general, the integral appearing in the last equation cannot be done analytically. Nonetheless, given the stiffness of the lattice, we can find a good approximation for W (rG ). (0)

We rewrite the target displacement vector as RT = RT + δRT and Taylor expand the (0)

(0)

interaction potential about RT . For the following choice for RT , (0)

(0)

Keff RT = F(rG , RT );

(2.52)

i.e., the position where the net force on the target atoms vanishes, the potential of mean force can be rewritten as 1 (0)T (0) (0) W (rG ) = U (rG , RT ) + RT Keff RT 2 µZ ¶ − β2 δRT D(rG )δRT T dδRT e −kB T ln −kB T ln he−βδU i, (2.53) where, (0)

D(rG ) ≡ Keff + R h(...)i =

∂ 2 U (rG , RT ) , ∂R2T β

(2.54)

T

dδRT e− 2 δRT D(rG )δRT (. . .) , R β T dδRT e− 2 δRT D(rG )δRT

(2.55)

and (0)

1 ∂ 3 U (rG , RT ) δU = × R3T + . . . , 3 3! ∂RT

(2.56)

where the × in the last equation implies that the multidimensional matrix product is taken appropriately. The first integral is just another Gaussian integral while the ln he−βδU i can be expanded in cumulants (see Ref. [99]). By neglecting terms that do not contribute to

53 the mean force, we can write the full expression for the potential of mean force as, 1 (0)T (0) (0) W (rG ) = U (rG , RT ) + RT Keff RT 2 kB T + ln[det(K−1 eff D(rG ))] 2 ∞ X hh(−βδU )j ii −kB T , j! j=1 (2.57) where hh. . .ii are cumulant averages and where the potential has been shifted by constants so as to vanish when the guest is noninteracting. At low temperatures, the first temperature correction to the potential of mean force will be linear in T , and the cumulants give higher order temperature corrections. In this work, we only keep the linear temperature dependence and drop the remaining terms; this turns out to give an excellent approximation at room temperature for our system. In Fig. 2– 3, we compare the numerically simulated force on a frozen guest with that obtained from Eq. (2.57). This section will be concluded with a brief remark. In the simulations, before releasing the guest, the lattice must be aged such that the target atoms have enough time to shift their equilibrium positions to ones that, on average, minimize the free energy of the system. Another scenario may be that, during the aging, the target atoms undergo a uniform collective translation that would put the guest at a minimum. Clearly, this should not happen. In order to prevent such a collective motion, we tethered some of the edge atoms of the bath (specifically, those atoms that were not fully coordinated). In Appendix 2.8, by using continuum elastic theory, we show that the tethering of boundary atoms does what we want for a 3-dimensional system, namely it makes a uniform translation of the target atoms impossible without an energy cost. On the other hand, this simple calculation shows that for 1- and 2-dimensional systems, the translation of a small portion inside the crystal

54

Figure 2–3: The force acting on xenon in Theta-1 during the aging is shown as the noisy curves. The curve showing large fluctuations is obtained at 300K, the other at 3K. The straight lines are the approximate values for the mean force at 300 K and 3K. The guest is ˚ 8.07333A, ˚ 2.36156A). ˚ The system and the potential are defined in Sec. 2.4.1 at (6.49633A,

55 Table 2–1: Silicate Force Constants [101] Motion Force Constant ˚ −2 Si-O stretch 5.0×10−18 J A O-Si-O bend 1.35×10−18 J rad−2 Si-O-Si bend 0.31×10−18 J rad−2 costs no energy even though the edges of the crystal are tethered, and is another manifestation of the well known Mermin-Wagner instability in low dimensional solids [100]. 2.4 Results 2.4.1 Specification of the System and Potentials In this section, we briefly describe the system that we will be working with. In particular, we specify the harmonic force constants and the form of the guest-target interaction potential. For practical purposes, we chose a sodalite having disconnected, one dimensional, channels. This will allow us to calculate plane averages using a single channel. The zeolite we chose is Theta-1 (TON). This system is a high silicate zeolite. We therefore assume that it has no Al atoms and thus, has the further advantage of not having any counter ions. Theta-1 contains two ten-membered oxygen-ring channels per unit cell. The target space that we used contains 5 unit cells along z and embeds the channel out to a ˚ from the channel axis. The target zone contains 210 atoms (140 oxygens radius of 6.5A and 70 silicons) and is electrically neutral. The crystallographic unit cell for Theta-1 is cubic and is described in Ref. [102]. The full unit cell contains 72 atoms. Notice that there is a reflection plane in x through the middle of the unit cell. The target zone is depicted in Fig. 2–4. The harmonic force constants were obtained from Ref. [101] and are summarized in Table 2–1.

56

Figure 2–4: The target zone. The oxygen atoms are in red and the silicon atoms are in blue. A minimum (A) and a maximum (B) W (z) plane are shown. A Xenon atom in a binding pocket is shown as a green ball. The z-axis (channel axis) is the vertical axis.

57 We assume that the potential energy of xenon inside a sodalite is well described by a Lennard-Jones term plus an induced dipole-electric field interaction; i.e., U (rG , r1 , . . . , rNTarget ) = "µ ¶12 µ ¶6 # NTarget X σi,G σi,G − 4²i,G r ri,G i,G i=1 αG − E · E, 2 where

NTarget

E≡

X i=1

qi ri,G , 3 4π²0 ri,G

(2.58)

(2.59)

is the electric field felt by the noble gas atom due to the partial charges on the crystal atoms. In the last two equations, ri,G ≡ |ri − rG |, ²i,G and σi,G are the Lennard-Jones parameters related to the guest-Si or guest-O interactions, ²0 is the permittivity of vacuum, qi is the partial charge on the i’th target atom, and αG is the polarizability of the guest. There seems to be consensus for the calculated values of the partial charges in silicates in the quantum mechanical literature (see, e.g., Refs. [103, 104]); namely, q0 = −1.2e and qSi = 2.4e, where e is the electron charge. For the values of the Lennard-Jones parameters, we did not find good agreement in the literature. A common way to proceed is to write the Lennard-Jones potential between molecule i and j as, φij = −

Aij Bij + 12 . 6 ri,j ri,j

(2.60)

If we have a way to calculate Aij and if we know the interatomic equilibrium separation eq ri,j = 21/6 σi,j , we can determine ²i,j . The equilibrium separation will be taken as the sum

of the radius of the atoms involved, while Aij is commonly determined by the London formula (cf. Ref. [14]), Ei Ej 3 , Aij = αi αj 2 Ei + Ej

(2.61)

58 Table 2–2: Parameters for zeolite silicon and oxygen Ref. [12, 13]

Atom O Si [105] O [17] O This Work O Si

q(e) -2 +4 -0.15, -0.20 0 -1.2 +2.4

˚ Radius(A) 1.40 0.42 1.52 1.52 1.08 0.53

˚ 3) α(A 1.65 0.02 1.25, 1.40 0.85 1.65 0.02

3

E(eV) χ( cm ) × 106 mol 13.55 12.58 166.73 1.00 N/A 10.0, 9.9 N/A N/A 3.887 39.855 -

or the Kirkwood-Muller formula (cf. Refs. [15, 16]), µ 2

Aij = 6mc αi αj

αj αi + χj χi

¶−1 ,

(2.62)

where αi is the polarizability of atom i and χi is its magnetic susceptibility. Table 2–2 contains a summary of what has been used in the literature to calculate the Lennard-Jones parameters. None of these studies used accurate partial charges for the silicon and the oxygen. In Refs. [12, 13]1 , a fully ionic structure is assumed. In Ref. [105]2 , the oxygen partial charge is introduced solely to balance the charge of the counter-ions, while in Ref. [17]3 , the partial charges are neglected. The atomic polarizability for the channel atoms determined in Refs. [12, 13] seems reasonable. The values for the polarizabilities for the ionic and neutral atoms are αO = 0.802, αO−2 = 3.88, αSi = 5.38 and ˚ 3 (cf. Ref. [106]). We expect values that are between these limiting cases αSi+4 = 0.0165A for silicate atoms and the values reported in Refs. [12, 13] are in that range. The other

1

In these references, the values for all the parameters are determined from the fully ionized atoms except for α which is determined more accurately from refractivity experiments (cf. Ref. [12]). 2

This comes from work on zeolite NaX and NaY. When two values are reported, it refers to the two zeolite types respectively. The charge on the oxygens is there to neutralize the charge carried by the counter-ions. Also, it is assumed that the Si/Al atoms do not contribute to the potential. 3

Here, the Si/Al atoms are also neglected as well as the partial charges.

59 Table 2–3: Parameters for the noble gas atoms Atom Ne Ar Xe

˚ [108] Radius(A) 1.560 1.900 2.224

˚ 3 ) [106] E(eV) [106] α(A 0.3956 21.56460 1.6411 15.75962 4.0440 12.12980

Table 2–4: Lennard-Jones parameters for the gas-channel interactions (T = 300 K). Gas Atom Ne Ar Xe

˚ σSi−X (A) ²Si−X /kB T 1.8622 0.0385 2.1622 0.0512 2.4537 0.0500

˚ σO−X (A) ²O−X /kB T 2.3522 0.1841 2.6522 0.3440 2.9437 0.4378

parameters in Refs. [12, 13] are those of a fully ionic crystal (the radius, the ionization potential and the magnetic susceptibility). On the other hand, Ref. [105] uses reasonable values for the polarizability while Ref. [17] uses the polarizability of neutral oxygen. In addition, Refs. [105, 17], use the same oxygen radius which is bigger than that of O−2 . This radius is the van der Waals radius of oxygen given by Bondi [107]. Because of the lack of agreement in these approaches, we decided to use our own parameters using the accurate partial charges values and interpolating the needed parameters (E, α, . . . ) from the CRC reference values (cf. Ref. [106]) of the neutral and ionized atoms. We will use the London formula with the polarizabilities of Refs. [12, 13] and we will interpolate the ionization potentials for Si+2.4 and O−1.2 using data in the literature [106]; the parameters thus obtained are summarized in Table 2–2. By using the London formula, Eq. (2.61), and the data in Tables 2–3 and 2–2 one obtains the Lennard-Jones parameters for the noble gas-zeolite atoms interactions listed in Table 2–4 Henceforth, we consider the case of xenon diffusing inside Theta-1. The potential of mean force inside the channel can be calculated using Eq. (2.57) and some constant potential of mean force surfaces are shown in Fig. 2–5. There are broad binding regions staggered on either side of the channel. The binding pockets are almost flat energetically; a closer examination shows that there are three binding sites in each pocket; one is exactly in the middle of the cell (in x) and the other two are symmetrically placed on either side.

60

Figure 2–5: Constant potential of mean force surfaces for Xenon in Theta-1 (2 unit cells along the channel axis are shown) at 300 K. The surface energy is indicated in the corner of each sub-figure. The absolute minimum is at -6.94kB T . Steepest descent reaction coordinates are shown in red (path 1) and blue (path 2).

61

Figure 2–6: The potential of mean force along path 1 (dashed line) and path 2 (full line). The energy is plotted as a function of z (and as a function of the distance d along the path in the inset). The activation energy for path 1 is 1.95kB T and, for path 2, 2.15kB T or 2.06kB T (depending on the starting point) at 300 K.

62 The barrier for motion between the central binding site and either of the ones to its side is very small, about 0.1kB T at 300 K; hence, there will not be any specific contributions to the permeability from the saddle points on these paths, and we have omitted them from the figures for the sake of clarity. The figures show that it is easier for the xenon atom to move between binding sites on the same side of the channel (the energy barrier is lower). The reaction coordinate linking two minimum energy sites are also shown in these figures. Notice that one of these paths (path 1 in Fig. 2–5) links binding sites that lie on the same side of the channel. The other (path 2 in Fig. 2–5) bridges binding sites that are on opposite sides. It turns out that path 1 has a lower activation energy than path 2. The free energy (potential of mean force) is plotted against the z-component along path 1 and path 2 in Fig. 2–6. The potential of mean force, W (r), can be used to calculate the plane potential of mean force, W (z), as −βW (z)

e

1 = ACell

Z dxdy e−βW (r) ,

(2.63)

Unit Cell

where ACell is the area of the unit cell perpendicular to z and the integration is restricted to the unit cell (note that each unit cell contains two channels). The resulting plane potential of mean force along the channel axis is shown in Fig. 2–7. This figure also shows the minimum potential of mean force in each plane. The enthalpy of sorption, ∆H, of xenon in Theta-1 can be estimated from Ref. [12] as 3

kB T 1X ∆H = Wmin + + h ¯ ωi , 2 2 i=1

(2.64)

where Wmin is the minimum in the potential of mean-force and where the last term is a sum over the zero point energy of vibration of the guest at the absolute minimum (this assumes that the potential of mean force near the minimum is almost harmonic). Assuming that this last term is small, we obtain a heat of sorption of ∆H ≈ 6.4kB T at 300K. Experimental measurements for xenon absorbed in mordenite gives ∆H = 14.1kB T [12] and for xenon

63

Figure 2–7: The plane potential of mean force for the middle unit cell of the channel is represented by the full curve. The dashed curve shows the value of the minimum in every plane.

64 absorbed in zeolite Na-Y, ∆H = 7.21kB T [18] at 300 K. Since the sorption occurs in different systems, we do not expect our number to agree. On the other hand, this confirms that our model potential does give heats of sorption that are the right order of magnitude. We did not find any experimental data for xenon absorbed in Theta-1. However, Ref. [17] calculated, using a rigid lattice and no polarization, a value for the activation energy of xenon in Theta-1, Eact = 1.24kB T at 300K. This number can be compared with the path’s activation energies of Fig. 2–6 which gives Wact = 1.95, 2.06 or 2.15kB T at 300K depending on the path. Also, K¨arger et al. [22] obtained an activation energy studying the self-diffusivity of xenon in silicalite ( a 3-d interconnected 10-oxygens ring channel silicate ) assuming that the temperature dependence of the self-diffusivity is well-described by, D = Do e−βEact .

(2.65)

Their number was, Eact = 2.0kB T at 300K. In our case, this should be compared with the barrier in W (z) which is 1.43kB T . Note that in Sec. 2.2.2 we assumed that the potential of mean-force were defined relative to their value in the adjacent bulk phases. If an experiment is carried where Theta-1 separates two solutions, the potential of mean force has to be shifted by the configurational Helmholtz free energy of the guest in the bulk, WBulk . 2.4.2 Simulation Results and Permeability of Xenon in Theta-1 Before presenting the results of the simulation, there are still a few remarks that must be made. First, the simulation will have to perform many matrix–column vector multiplications. These matrices, Keff , A, B and C, are all sparse to some extent. In order to reduce the computation time, we set the elements smaller than some threshold in these matrices to zero, and then use sparse matrix routines to perform the multiplications (specifically, we used the NIST sparse subroutine package [109]). The threshold is chosen such that the effect on the vibrational density of states of the crystal is negligible. Because the induced dipole/electric field interaction in the potential is long-range, we added a static bath background correction potential, obtained by the means of Ewald sums (see, e.g.,

65 Ref. [51]), in the simulations. Finally, the simulations were performed by integrating the set of differential equations, cf. Eqs. (2.44)–(2.46) using a second order stochastic RungeKutta integrator [110]. The aging time was 4.096 × 10−12 s and the simulation length was 8.192×10−12 s or 12.288×10−12 s. The time step used was 5.0×10−16 s. We calculated the correlations for every initial starting points by averaging over 2000 independent trajectories and performed this numerical work on a Beowulf cluster consisting of 16 processors. The space dependent diffusion coefficient D(z)/n∞ is obtained from the plateau value of Rt R n∞ 0 dt1 Unit Cell drk hvG,z (t1 )vG,z ir e−βW (r) D(z, t) = , Rt R Acell + 0 dt1 Unit Cell drk hβF (z(t1 ))vG,z ir e−β[W (r)−W (z)]

(2.66)

where W (r) is the potential of mean force at a point, W (z) is the plane potential of mean force defined in Eq. (2.63) and Acell is the xy-area of the unit cell. Each plane integration was performed using a grid that contains between 25 and 48 points, chosen in such a way that the potential of mean force in that plane and the plane average potential of mean force are accurately reproduced. The correlation functions were obtained from the numerical simulations and space group symmetries pertaining to a single channel were used to reduce the numerical effort (by four). Values between the grid points were interpolated using a bicubic spline and these were used to numerically perform the plane integration. The quantity D(z, t)eβW (z) /n∞ is shown in Figs. 2–8 and 2–9 for z = 2.519 (maximum W (z) ˚ (minimum W (z) plane). plane), and 0.944625A Figures 2–8 and 2–9 also illustrate the effect that the correction term in the denominator in Eq. (2.66) has on the integral of the velocity correlation function. In fact, neglecting that correction is equivalent to neglecting the † on the current fields J in Eq. (2.5) which has been shown to be incorrect [7], even if the naive Green-Kubo integral converges. Also, note that the correction factor in the maximum W (z) plane lowers the average of the velocity correlation function integral (see Fig. 2–8) while it raises the average of the velocity

66

˚ is extrapolated from the plateau value Figure 2–8: (D(z)/n∞ )eW (z)/kB T for z = 2.519A of the full line. The dashed line is the uncorrected result. This plane is a maximum energy plane, with respect to W (z)

67

˚ is extrapolated from the plateau Figure 2–9: (D(z)/n∞ )eW (z)/kB T for z = 0.944625A value of the full line. The dashed line is the uncorrected result. This plane is a minimum energy plane with respect to W (z)

68 Table 2–5: The diffusion coefficient in different planes ˚ Name z(A) minimum W (z) plane 0.944625 saddle plane path 2 1.48 intermediate plane 1 1.57438 intermediate plane 2 2.36156 maximum W (z) plane 2.519 saddle plane path 1 3.27

D(z) n∞

× 108 38.11 16.73 14.83 7.38 7.67 25.22

m2 s

e−βW (z) 24.05 13.09 10.85 5.75 5.74 18.03

D(z)eβW (z) n∞

× 108 1.5844 1.2782 1.3669 1.2847 1.3349 1.3988

m2 s

correlation function integral in the minimum W (z) plane. The effect of the correction in the minimum energy plane is in agreement with the prediction made in Ref. [7]. The dynamics can change the relative contributions to D(z) within a given plane over what would be expected simply on the basis of the Boltzmann weight (e.g., as in a Smoluchowski approach). This is illustrated in Fig. 2–10. It is clear from these figures that the dynamics can drastically affect the shape of the various contributions within a given plane. Regions with low potentials often have less correlated dynamics, while those with high potentials will have more coherent motion; what contributes to D(z) is a compromise between the Boltzmann weight and the coherence of the motion. Table 2–5 lists the calculated space-dependent diffusion coefficient D(z)/n∞ for several planes. We also show these results graphically in Fig. 2–11 for one unit cell along z. In Sec. 2.2.1, we assumed that the quantity D(z)eβW (z) is constant near the barrier tops. This quantity is shown as the dashed line in Fig. 2–11; clearly the assumption is valid. If the line is fitted to a constant, we find that (D(z)/n∞ )eβW (z) = (1.37 ± 0.10) × 10−8 m2 s−1 . Note that the results of this section were all obtained from simulations using sparse matrices; we have checked that the results are not significantly different when we use the full matrices. The permeability P 0 as defined in Sec. 2.2.1 can now be calculated. As is clear from Eqs. (2.10) and (2.11) the intrinsic permeability will be inversely proportional to the thickness of the material and independent of the area, as is expected from a resistor network analogy. By calculating the permeability of a single channel in a single unit cell,

69

R Figure 2–10: The Boltzmann factor e−W (r)/kB T and the factor dt hvz (t)vz ir e−W (r)/kB T in the maximum (A) and minimum (B) W (z) plane. The z-axis has arbitrary units.

70

Figure 2–11: The space-dependent diffusion coefficient D(z)/n∞ (solid line) and (D(z)/n∞ )eW (z)/kB T (dashed line).

71 Pchannel , it is straightforward to obtain the macroscopic permeability. For Theta-1 we find that P0 = 3.035 × 1013 s/(mkg), βW Bulk n∞ e where we have included the explicit correction associated with the free energy of the guest in the adjacent phases, WBulk , since the potentials used here have their zero defined relative to vacuum. The diffusion of xenon in Theta-1 has not yet been studied experimentally. However, as mentioned above, K¨arger et al. [22] examined the self-diffusion of xenon in silicalite. The high-temperature limit self-diffusion coefficient that they obtain with Eq. (2.65) is Do = (0.9±0.2)×10−8 m2 /s. This can be compared with our value, i.e., (D(z)/n∞ )eβW (z) = (1.37 ± 0.10) × 10−8 m2 s−1 , which, given the differences between the two systems and the quantities measured or calculated, is in reasonable agreement with the experimental value. In the next section, we discuss transition state theory within the context of the current approach. 2.5 Transition State Theory Another approach that one could have used in order to get the permeability of the system is transition state theory (see, e.g., Ref. [47]). This theory treats the motion of the guest between two neighboring binding sites as an activated hopping process, where the kinetics are described by hopping rate constants that are fully determined by motion near the steepest descent path linking two binding sites. We found two types of saddle points (transition states) for our potential of mean force. Their corresponding steepest descent path are shown in Fig. 2–5. The assumptions behind transition state theory are that there is an equilibrium between the reactants and the transition state and that there are no recrossings. Also, transition state theory will be accurate only if the average motion of the guest inside the crystal follows the reaction coordinates associated with each transition states. This last assumption can be verified in the following way.

72 If we start an ensemble of trajectories at the saddle point, we should see that, on the average, the guest moves to one of the 2 binding sites following the prescribed path. We started trajectories at the two saddle points (each ensemble contained 2000 members) and we averaged the trajectories conditional on which binding sites they end up in. The results are shown as 3-d plots in Fig. 2–12, and clearly show that the average paths are qualita-

Figure 2–12: The steepest descent path 1 (A), and the steepest descent path 2 (B) (in red) are compared with the conditional averages of the trajectories started at the appropriate saddle point at 300 K (blue) and 3 K (green). The surface is at -5.3kB T tively different than the steepest descent path. Moreover, while many of the trajectories that are started at a saddle point end up in the nearest neighbor minima, a significant fraction of trajectories also end up further away on the timescale of the velocity correlation function; for cases shown in Figs. 2–12 A and B, only 66.5% and 66.7%, at 300 K (blue curve), of the population is accounted for by those that end up in the nearest neighbor sites, respectively. In either example shown in Figs. 2–12 A and B, the short-time behavior of the average trajectories that do end up in the minima predicted by the minimum energy path is very different from the behavior expected from the steepest descent path. For the minimum energy path linking binding sites on the same side of the channel, the potential energy in the saddle plane suggests that trajectories that are directed away from the center along y will be backscattered towards the center of the channel (which

73 corresponds to the middle of the x − y plane in Fig. 2–12). For the other saddle plane, it is now the trajectories that are initially aimed away from the center in x that will be backscattered towards the center. The xy-components of the average trajectories for path 1 are plotted in Fig. 2–13 against their z-component to emphasize the differences with the steepest descent path and illustrate the last comments. The error bars in these figures show that the steepest descent path is within the standard deviation associated with the average of the trajectories, but it is clear that the steepest descent paths alone, do not accurately describe the dynamics. Another way to verify the validity of transition state theory is to look at the parameter R

dt hvz (t)vz ir e−W (r)/kB T in the transition planes, as shown in Fig. 2–14. For transition

state theory to be right, the plane averages in Eq. (2.66) must be dominated by the value R of dt hvz (t)vz ir e−W (r)/kB T along the reaction coordinates. Hence, we expect a spike at the saddle point in the saddle planes. It is clear from these figures (the saddle point is indicated by an ”X” in each plane) that the transition state contributes to the plane average, but that the rest cannot be neglected; Fig. 2–14 shows that the saddle point gives the largest contribution while in saddle plane 2, the saddle point is not even the point with the largest contribution. We conclude that for this particular temperature (300 K), transition state theory does not properly describe the average motion of xenon in Theta-1. Our potential, which is very flat, is probably one of the reasons for this breakdown of transition state theory. If the flatness of the potential is the main reason why transition state theory breaks down for our system, it is interesting to investigate the effects of reducing the temperature on the dynamics. The conditional average trajectories starting at the two saddle points at 3 K are shown in Figs. 2–12 A and B as the green curve. These curves represent 86.8% and 98.6% of the populations for trajectories started at the saddle points 1 and 2, respectively. As expected, at least for path 1, the trajectories follow the steepest descent paths more closely at lower temperature.

74

Figure 2–13: The x and y components of the steepest descent path 1 (dashed line) and ˚ 7.8445A, ˚ 3.27A). ˚ The the average trajectory (full line). The starting point is at (6.9295A, average x-component has to follow the reaction coordinate because it lies in the reflection plane. The error bars indicate the standard deviation associated with the average.

75

R Figure 2–14: The Boltzmann factor e−W (r)/kB T and the factor dt hvz (t)vz ir e−W (r)/kB T in saddle plane 1 (A) and saddle plane 2 (B). The z-axis has arbitrary units.

76 For the path 2 saddle point, the average trajectory still does not follow the steepest path as well as it did for path 1 at low temperature. This happens for two reasons. First, in the region where path 2 merges into path 1, the steepest descent (path 2) bends sharply and the guest jumps out of steepest descent region, using the kinetic energy it has picked up in moving down the barrier. Second, the binding pocket has its two absolute minima on either side of the x-reflection plane in the unit cell, and the barrier separating them is extremely small compared to the kinetic energy picked up down the barrier; hence, the forces aren’t large enough to keep the guest localized near the ends of the steepest descent curve. In a hopping model that incorporates hops along paths 1 and 2 and assumes fast equilibrium between the three binding sites in each of the low energy pockets, the steadystate flux is given by J =−

2ρc Keq (k1 K⊥ + 2k2 ) (µ+ − µ− ) (2K⊥ + 1)N n∞ kB T

(2.67)

where ki is the rate constant associated with path i, N , is the number of binding pockets in one channel (assumed large), ρc is the number of channels per unit area, and the equilibrium constants Keq and K⊥ govern the equilibrium between the bulk and the first binding pocket or that between the three binding sites within a pocket, respectively. Finally, we have used the linear approximation, δµ± ∼ kB T δn± /n∞ . Transition state theory makes an unambiguous prediction for the hopping rate constants, k1 and k2 [48]. By assuming that Keq can be obtained from a Langmuir adsorption model where bulk atoms are absorbed onto a surface (the first binding planes in the crystal), we find that 2k1 K⊥ Keq =2 2K⊥ + 1

Ã

2π(kB T )3 (1) (1) mG Kη Kζ

!1/2 ‡

e−β(W1 −Wbulk )

(2.68)

77 and 4k2 Keq =4 2K⊥ + 1

Ã

2π(kB T )3 (2)

(2)

mG Kη Kζ

!1/2 ‡

e−β(W2 −Wbulk ) .

(2.69)

(i)

where Wi‡ and the Kη,ζ ’s are the energy and vibrational force constants for motion transverse to the steepest descent path at the i’th saddle point, respectively. In writing Eqs. (2.68) and (2.69) we have ignored the vibrational motion of the lattice, other than in its contribu‡ tion to W1,2 , treated the guest vibrations classically, and have assumed a unit transmission

coefficient. By using Eqs. (2.68) and (2.69) for the saddle points and paths shown in Fig. 2–5 in Eq. (2.67), we obtain PTST 0 = 5.439 × 1013 s/(mkg), n∞ eβWBulk which is clearly different from the value obtained with our method. While part of the difference could be blamed on our use of high-temperature, harmonic, partition functions for the vibrational motion transverse to the reaction coordinate in the transition state approach, it is clear that the basic assumptions of the transition state theory aren’t satisfied very well, as was discussed above. In fact, the large contributions to the permeability from regions other than the steepest descent lines manifests itself in other ways; for example, if we define an apparent activa‡ = −4.38, compared tion energy as ∆E ‡ ≡ ∂ ln P /∂(−β), we find that, at 300K, β∆ETST

with β∆E ‡ = −2.48 using our method. (In both cases, the number is reported with respect to the bulk energy, and in our method, we have ignored the temperature dependence of D(z)eβW (z) ). 2.6 Concluding Remarks One important result of this work is that D(z)eβW (z) is not only constant in the vicinity of the barrier tops, it is roughly constant throughout the channel for our system. This means that the diffusion of xenon in Theta-1 is well described as a Smoluchowski [49]

78 process, which says that D(z) ∝ e−βW (z) , and, as we saw above, not by transition state theory. Our expression for the potential of mean force, Eq. (2.57), includes the relaxation of the lattice and a temperature correction term. It is a common practice to neglect both of these effects. For example, experimental evaluation of diffusion in silicates (see, e.g., Ref. [1]), often assumes that the activation energy is temperature independent. Also, a rigid lattice is often used in simulations and in the calculation of the available volume for a guest inside a zeolite (see, e.g., Ref. [17]). We have verified that neglecting the temperature dependence of the potential of mean force does not lead to large errors. The use of a rigid lattice leads to larger, but still acceptable errors on the shape of the potential for the system investigated. There is a more important problem associated with the dynamical studies of a guest in a channeled structure using a rigid lattice: the lattice cannot dissipate the guest’s energy. This problem becomes more important when the activation energy is large. We computed velocity correlation functions at some test points using a rigid lattice, and it turns out that these are similar to the ones that are obtained with flexible lattice. The decay of the velocity correlation functions occur on the same timescale in both cases, and in the rigid lattice arises solely because of the dephasing associated with the average over initial velocities (kinetic energy correlations are quite different). There are two main mechanisms leading to the decay of the velocity correlation functions. The first is the randomization of the direction of vG (t) . The second arises from fact that total energy of the guest is not conserved in a flexible lattice and this manifests itself in the magnitude of the guest velocity. For our system, the energy exchange between the guest and the lattice occurs on a somewhat longer timescale compared to that associated randomization of the direction, and thus, this latter effect is captured by the rigid lattice calculations at short and intermediate times. On the other hand, the relaxation of the lattice will have bigger effects on the shape of the potential in small crystal structures (i.e., β-quartz) where the guest is often in the

79 strongly repulsive part of the pair potential, where the energies are larger, and where lattice distortions are larger. In addition, anharmonicities and energy exchange will be more important. In this work, we used generalized Langevin equations to simulate the target equations of motion. It is clear from Secs. 2.2.1 and 2.2.2 that the evaluation of the macroscopic permeability can be obtained from standard MD. We opted for the GLE approach because the consistency of our method relies in part in the accurate description of the infinite crystal vibrations. In MD this is achieved when the number of simulated atoms is large. Using GLEs allows us to drastically reduce the number of simulated degrees of freedom . Note that the macroscopic permeability of the Theta-1 interface may be hard to get experimentally. In fact, Theta-1 crystals are usually needle-like crystallites with length ranging from 0.6 to 1.0 µm and width from 0.06 to 0.10 µm [4]. It may therefore be difficult to construct a macroscopic interface where all the channels are aligned. Also, as noted by K¨arger et al.[92] in their work on single-file diffusion, 1d channels can easily be blocked, and hence, in an experiment, not all channel will participate in the transport, thereby giving a lower apparent single-channel permeability. In conclusion, we briefly summarize the main features of our approach. First, we believe that our theory is well-suited for diffusion studies in systems containing large potential barriers where hopping events are rare, and moreover, does not make a priori assumptions about steepest descent paths (and which turn out to be unwarranted for the example considered here). Second, the Langevin equation is exact to the extent that the guest does not interact directly with the bath, that all the forces within the crystal are harmonic, and the vibrational density of states of the full crystal is accurately reproduced by our approximation for the force-force time correlation function. This leads to a practical simulation that incorporates the full vibrational motion of the crystal. In addition, the required time correlation functions are obtained on a ps timescale.

80 Third, we introduced an accurate and simple way of obtaining the guest potential of mean force for the system and we tested it against the simulation results. Fourth, the expression for D(z), cf. Eq. (2.11) which requires the evaluation of plane averages is general and can be applied to any crystal system with connected channels. With such a system the guest is allowed to travel in different channels and the target zone must be large enough such that the guest does not escape during the simulation time. This would make the problem more difficult numerically. In addition, some assumptions about the range of the correlations that appear in the memory functions in Eq. (2.4) made in obtaining the expression for the permeability in Ref. [6] might break down if the structure is too porous and contains solvent. Finally, we have seen that transition state theory gives a very different prediction at room temperatures in Theta-1, in part due to the very anharmonic nature of the potentials, and in particular, due to the contribution of other regions of the channel to the permeability. In a subsequent paper, we will investigate the diffusion of noble gases in β-quartz where the energy barriers are large. For such a system, the flexibility of the lattice plays a crucial role. On the other hand, transition state theory is expected to be more accurate. Another interesting aspect would be to investigate the role of quantum mechanics in our analysis. 2.7 Appendix: Equations of motion In this section, we demonstrate that −˙y(t) replaces the noise and memory term in Eq. (2.13). In what follows, the T subscript will be omitted for matrices and vectors in the target space. We rewrite Eq. (2.46) as, ˙ ¯ Y(t) = −MY(t) + N(t) + P0 (t)

(2.70)

81 where 



 y(t)  Y(t) ≡  , y˙ (t)   0   N(t) ≡  , C−1 η(t)   0   P0 (t) ≡  , −1 −1 C M P(t)

(2.71)

(2.72)

(2.73)

and  ¯ ≡  M 

 0

−1  , −1 −1 C A C B

(2.74)

¯ is a square matrix. The formal solution of Eq. (2.70) is where each block of M Z ¯ −Mt

Y(t) = e

t

Y(0) +

¯

dt1 e−M(t−t1 ) N(t1 )

0

Z

t

+

¯

dt1 e−M(t−t1 ) P0 (t1 ).

(2.75)

0

The last term of this equation is very similar to the memory term in our Langevin equation, Eq. (2.13), and, in fact, it contains that term. Since the last term is a convolution, when Laplace transformed, it becomes, ¯ + s)−1 P0 (s). (M If we keep in mind that the upper half part of P0 is zero (the first 3NTarget elements) and if we denote the inverse as, 



q11 q12  ¯ + s)−1 ≡ Q(s) ≡  (M ,  q21 q22

(2.76)

82 then only the 12 and the 22 part of the inverse will contribute. Evaluating the inverse, we find that 1 A A + Bs + Cs2

(2.77)

s C, A + Bs + Cs2

(2.78)

q12 = − and q22 =

which when multiplied by C−1 M−1 P(s), shows that q22 C−1 M−1 P gives the expected term, i.e., q22 C−1 M−1 P =

s M−1 P(s), A + Bs + Cs2

and hence, to the accuracy of our approximate representation of the memory function, Eq. (3.71), Z Y2 (t) ≈ [e

¯ −Mt

t

Y(0) +

¯

dt1 e−M(t−t1 ) N(t1 )]2

0

Z

t

+β 0

dt1 hF†T (t − t1 )F†T T iM−1 P(t1 ),

(2.79)

where the subscript 2 refers to the lower half of the column vectors. Therefore, in order to include friction in the equations of motion, we will need to subtract the second part of ˙ Y(t) to the equations for P(t). This is indeed done in Eq. (2.45). The Y(t) must also describe the random force [cf. Eq. (2.13)] through the white noise term. From Eq. (2.75) and from Eq. (2.13), it is clear that the random force F† (t) should be represented by, Z †

¯ −Mt

F (t) = [e

t

Y(0) +

¯

dt1 e−M(t−t1 ) N(t1 )]2 .

(2.80)

0

A few manipulations show that this is indeed the case if   0   0 hNNT i =  , −1 −1 0 kB T C BC

(2.81)

83 and if the random initial conditions for Y satisfy hY1 Y1T i = kB T A−1 ,

(2.82)

hY2 Y2T i = kB T C−1 ,

(2.83)

and

where the averages of N(t), Y1 and Y2 are zero. 2.8 Appendix: Energy cost of a local displacement in an elastic medium This appendix discusses the energy cost for a local displacement in an infinite isotropic continuum. As shown in Ref. [111], the vector field u describing the displacement of the continuum lattice at position r obeys the equation, (1 − 2σ)∇2 u(r) + ∇(∇ · u(r)) = 0,

(2.84)

where σ is Poisson’s ratio. The free energy cost per unit volume is given by, 1 1 E = µ(uik − δik ull )2 + κu2ll , 3 2

(2.85)

where summation over repeated indices is implied. The quantities κ and µ are the bulk modulus and the modulus of compression, respectively, and the elements of the symmetric strain tensor uik are given by, uik =

∂ui (r) ∂uk (r) + . ∂rk ∂ri

(2.86)

To get the full energy of the system, Eq. (2.85) is integrated over the whole system. For the case, of interest, the following boundary conditions will apply in d-dimensions: u(r, Ω) = 0 as r → ∞

(2.87)

u(a, Ω) = δu

(2.88)

84 where Ω represents the angular coordinates. The boundary condition at infinity comes from tethering the edges of the lattice, while that at r = a represents, for our problem, a uniform displacement of the target zone along an arbitrary axis. We solved this problem in 1, 2 and 3 spatial dimensions using the following ansatz: u(r) = ∇φ(r) + ∇ × A(r).

(2.89)

Since we are using linear elasticity, these potentials must have forms: φ(r) = δu · rf (r)

(2.90)

A(r) = δu × rg(r).

(2.91)

and

The boundary conditions are then expressed in terms of the f and g functions, the differential equation is solved and the energy cost for such a local displacement is computed. We found that there was zero energy cost in 1 or 2 spatial dimensions, while in 3-dimensions it becomes E = 6πµ

3κ + 4µ 2 δu a. 6κ + 11µ

(2.92)

For typical values of the moduli and a, this energy is large compared with kB T , thereby confirming our hypothesis that the channel cannot uniformly translate to reduce its energy.

CHAPTER 3 Diffusion in channeled structures II: Systems with large energy c barriers [112](°American Chemical Society, 2005) The formalism developed in the previous chapter will now be used for a different system. We expect most of the conclusions of the last chapter to be system dependent; in particular, the crystal system (Theta-1) contained large enough channels and concomitantly small energy barriers such that neither the flexibility of the lattice, nor its dynamics played a large role in the diffusion of the guest atom (Xenon). On the other hand, we saw that the very soft potential barriers allowed the guest to sample large regions inside the channels. This contradicts one of the assumptions of transition state theory where one or many reaction coordinates are supposed to dominate the motion of the guest. We will now use our formalism on a crystal system that contains very narrow channels and large energy barriers (Argon in α-quartz). Within this system, we expect the guest-crystal atoms collisions to be stronger and more frequent, and, therefore, increase the rate of energyexchange between the guest and the lattice. The role of the lattice vibrations will be more important here than for the previous system. Also, the fact that this system contains narrow channels (which means that the motion of the guest is more confined) suggest that a reaction coordinate description of the guest motion should be more appropriate for this system. These questions are examined and answered in this chapter. 3.1 Introduction In a previous paper [53], henceforth referred to as I, we formulated a theory for calculating the permeability of guest molecules in channeled crystals and applied it to the diffusion of xenon through the zeolite theta-1. The energy landscape inside the theta1 crystal channels had relatively small energy barriers. In this paper, we will apply the methodology described in I to a system where the energy barriers are large. 85

86 In the general approach that we use, based on the work of Ronis and Vertenstein [6], equations of motion for Gibbs surface excess quantities describing the dynamics in the energy barrier regions were derived using a multipole expansion, first formulated by Ronis and co-workers for a system with one interface [50]. Specifically, if one starts from a generalized diffusion equation that is non-local in space, e.g., as given by linear response theory, the multipole expansion results in time-correlation function expressions for the parameters appearing in macroscopic boundary conditions, and in particular, gives one for the permeability. This approach allows us to predict the macroscopic permeability that governs the diffusion (in the linear regime) through a crystalline interface from a microscopic evaluation of various correlation functions, and naturally gives rise to a transition state sampling scheme. As was mentioned in I, previous simulation studies have used either frozen or dynamic lattices (see, e.g., [93, 46, 45, 113]). The importance of the flexibility and especially of the lattice vibrations was investigated by us in I, by Kopelevich and Chang [54] and by Suffritti and coworkers [39, 114, 115]. We include the lattice vibrations, as if the crystal was infinite, through a set of exact generalized Langevin equations that describe the motion of selected crystal atoms. Energy transfer between the lattice and the diffusing particle is then included in the simulations. We found in I, for a small energy barrier system, that the energy exchange took place on a relatively long time-scale, and the vibrations only had small effects on the diffusive dynamics. In this work, we show how this changes when the coupling to the lattice is stronger, as happens in higher barrier systems. A more common approach that is used to study intra-crystalline diffusion is transition state theory (TST). When the diffusion is described by a hopping model, TST predicts the values of the rate constants describing the hops. In I we found that, although the permeability that TST predicted was in the right range, its underlying assumptions were not justified. Here, we will also calculate the permeability using TST and we will see how accurately it describes the diffusion process for a large energy barrier system.

87 The paper is divided as follows. In Sec. 3.2, we summarize the theory that leads to an expression for the macroscopic permeability and we briefly describe how the necessary microscopic correlation functions were simulated. In Sec. 3.3, we describe the large energy barrier system that we will study and we report the calculated permeabilities at different temperatures. In Sec. 3.4, the permeabilities are calculated using TST and compared with our predictions. In Sec. 3.5, we relate and compare the permeability with the diffusion coefficient that is measured experimentally, and finally, in Sec. 3.6, we discuss the role of lattice vibrations for the diffusion in this system and we make some concluding remarks. 3.2 Theory The techniques that we will use in this paper have been described extensively in I or in previous work [6]. Here, we will therefore briefly state the governing equations that describe our system and simulation procedure. First, recall that we are looking at a system where an interface separates two regions of a guest component with different chemical potentials. This interface has a macroscopic thickness, 2d, and is made of a porous crystalline material, idealized here as an aligned single crystal. The steady-state flux, j, through this material is then characterized by the following equation, j = −P 0 [µ(d) − µ(−d)],

(3.1)

where µ(±d) are the chemical potentials at the interface boundaries and where P 0 is the permeability intrinsic to the material and is given by, 1 1 = 0 P β

Z

d

dz −d

1 , D(z)

(3.2)

where β ≡ 1/kB T , T is the temperature, D(z) is a space-dependent Onsager diffusion coefficient given by, 1 D(z) = A

Z

Z

∞

dt 0

Z dr0k

dr hJ†z (r, t)J†z (r0 )i,

(3.3)

88 where J†z is the z component of the irreversible or dissipative part of the current, and A is the area of the crystal. Also note that the integration variable, z, is defined to be normal to the interface (i.e. parallel to the net flux) and is perpendicular to rk . The time correlation function that appears in Eq. (3.3) is not directly amenable to simulation since the dynamics in J†z (r, t) includes a projection operator that projects out the slow parts of the motion (in practice, the parts not included explicitly in the macroscopic equations) [7]. By manipulating the well known integral equations relating correlation and memory functions, Vertenstein and Ronis [6, 7] obtained an approximate expression for D(z) by assuming infinite dilution inside the material and that D(z)eβW (z) is approximately constant near the W (z) barrier tops; the result is that R∞ R n∞ 0 dt1 drk hvz (t1 )vz ir e−βW (r) R∞ R D(z) = , A + 0 dt1 drk hβF (z(t1 ))vz ir e−β[W (r)−W (z)]

(3.4)

where n∞ is the number density of the guest in the bulk, vz is the z-component of the guest velocity, h...ir denotes a conditional equilibrium average under the constraint that the initial position of the guest is r, and W (z) and F (z) are the plane averaged potential of mean force and mean force of the guest in the plane z, respectively (we distinguish the plane average quantities and their 3d counterparts by the argument to the function, z in the former, and r in the latter); in particular, Z e

−βW (z)

−1

≡A

drk e−βW (r) .

Note that the denominator in Eq. (3.4) approximately reproduces the effect of the projection operator that is implicit in J† , the irreversible part of the current, cf. Eq. (3.3). It follows from Eqs. (3.3) or (3.4) that D(z) will be small when z is a barrier-top plane, and it is just these planes that give the main contributions to the permeability, cf. Eq. (3.2). Hence, provided that a practical way is found for calculating the potential of mean force near the transition states (e.g., analytically or using umbrella sampling), our

89 theory becomes a transition state sampling method, only requiring a knowledge of dynamical correlations for initial states near the transition state. Clearly, it is necessary to simulate the guest dynamics inside the crystal in order to calculate the time correlation functions appearing in Eq. (3.4). As described in I, we assume that the guest interacts with a limited number of crystal atoms (referred to as the “target atoms”, or T for short). The effect of the rest of the crystal (the bath) will be described using a projection operator approach [55], where the projection operator is an average over the bath degrees of freedom, and gives rise to the following generalized Langevin equation, dU [r(t), RT (t)] dp(t) =− dt dr

(3.5)

and dPT (t) dU [r(t), RT (t)] = −Kef f RT (t) − + F†T (t) dt dRT Z t −β dτ hF†T (t − τ )(F†T )T iR˙ T (τ ),

(3.6)

0

where U is the potential interaction between the guest and the target atoms, RT and PT are, respectively, the target position and momentum column vectors, Kef f is an effective force constant matrix, F†T is colored Gaussian noise with zero mean and variance hF†T (t)(F†T )T i and the T superscript indicates a transpose. The projection operator drops out of the time dependence of the memory function, which is obtained by considering the harmonic motion of the bath in the presence of frozen target and guest. Note that these equations are exact provided that the interaction between the crystal atoms is harmonic and provided that the guest does not directly interact with the bath (this last point can be relaxed somewhat to allow for direct harmonic interactions between the guest and bath). In I, we described in detail the procedures that allow us to calculate Kef f and βhF†T (t)(F†T )T i. When the Laplace transform of the memory function

90 is approximately written as, βhF†T (s)(F†T )T i =

s , A + Bs + Cs2

(3.7)

the equations of motions simplify to a set of coupled stochastic differential equations with additive white noise, that, by properly choosing A, B and C, reproduces the basic features of the infinite crystal vibrational density of states (cf. I and Fig. 3–1). Finally, a low temperature expansion for the potential of mean force was developed, and to linear order in T gives 1 (0)T (0) (0) W (r) = U (r, RT ) + RT Kef f RT 2 (0) 2 kB T −1 ∂ U (r, RT ) + ln(det[1 + Kef f ]), 2 ∂R2T

(3.8)

where (0) Kef f RT

(0)

∂U (r, RT ) =− , ∂RT

(3.9)

(0)

defines new equilibrium lattice positions, RT , where the net force on the target atoms is zero in the presence of a guest fixed at r. 3.3 Results 3.3.1 Specification of the system The system that we will be working with is Argon diffusing in α-quartz. This crystal contains multidimensional interconnected channels that are roughly the size of an Argon ˚ Note that quartz shows a phase transition from the α to the β structures atom (3–4A). at 846K [116], hence, we will work below this temperature. The coordinates of the nine crystal atoms in the unit cell were obtained from Ref. [117]. The inter-atomic crystal potential, which we describe solely by stretches and bends, is described in I. The vibrational density of states obtained from our model is shown in Fig. 3–1 and it agrees well with other calculations [118]. Our target zone is centered in the middle of the primitive unit ˚ 14.7402A, ˚ 16.2156A). ˚ It contains 279 cell and is rectangular with dimensions (17.7402A,

91

Figure 3–1: The exact density of states (full line) for α-quartz obtained from a Brillouin zone calculation is compared with the approximate density of states (dashed line) that is reproduced from Eq. (3.7)).

92

Figure 3–2: Constant potential of mean force surfaces for argon in α-quartz at 300 K. The surface energy is indicated in the corner of each part. Two symmetry equivalent low energy reaction coordinates are shown as path 1 (red) and 2 (blue). The kink is the position of a local minimum along the path. Also shown is a high energy reaction coordinate (green) connecting different channels in different unit cells (the saddle point is indicated by an “X”). atoms and is large enough such that the interaction energy between the guest and bath is small. In I, the target-guest interactions were described in terms of a Lennard-Jones potential and a lattice partial charge guest polarization interactions. Here, we use the Lennard-Jones

93 term only, with parameters that differ slightly from I (cf. Sec.3.5). We have tested our approximate expression for the potential of mean force, cf. Eq. (3.8), by comparing the resulting mean force against the numerically simulated average force on a frozen guest after the lattice has equilibrated. The agreement was excellent for the three temperatures we will consider, namely 100K, 300K and, surprisingly even at 800K [remember that Eq. (3.8) is a low temperature expansion]. Some constant potential of mean force surfaces are shown in Fig. 3–2 at 300 K. The three binding sites in the unit cell are equally separated along the z axis in a spiral staircase conformation. Each pair of nearest neighbor binding sites are connected by two symmetry related steepest descent paths. They go through a local minimum and have two saddle points along the way (see the 25.7kB T contour of Fig. 3–2 and Fig. 3–3, which shows the potential of mean force along the path). The guest, at least at 300K, will preferably travel along the reaction paths 1 and 2 shown in Fig. 3–2 and undergo a net displacement along z. From Fig. 3–2 at 28.5kB T , we see that the channels lie along z and are disconnected at moderate energies. To travel from one channel to another, the guest must overcome a larger barrier. This is seen from the dashed line in Fig. 3–3 and by the pocket labeled “A” in Fig. 3–2 that makes a connection with the channel only at 41.0kB T . Hence, the channels can be assumed to be disconnected, at least at 100K and 300K. Also shown in Fig. 3–2 is the high energy reaction coordinate that connects neighboring channels whose saddle point is indicated by an “X”. The plane average potential of mean force, W (z), at the temperatures of interest is shown in Fig. 3–4. Note the significant increase with temperature in the absolute potential of mean force, a feature we did not observe in I. On the other hand, the activation energy, obtained from W (z), for the individual hops between the binding sites, decreases, albeit very slightly, with an increase in temperature.

94

Figure 3–3: The potential of mean force along the reaction coordinate (Path 1 and 2 in Fig. 3–2) is shown as the full line. The dashed line shows the potential of mean force for a reaction coordinate that connects different channels, cf. Fig. 3–2. The saddle point at “X” is indicated in Fig. 3–2. The rate constants used in transition state theory, cf. Eq. (3.10), are also indicated at the two saddle points 1 and 2.

95

Figure 3–4: Plane average potential of mean force. The temperatures are, starting with the bottom line, T = 100K, 300K and 800K.

96 Table 3–1: D0 and P 0 as a function of temperature. PT0 ST is the number obtained from transition state theory. TK 100 300 800

D0 × 109 0.81 3.46 8.69

m2 s

s P0 n∞ eβWBulk mkg −24

1.17×10 9.76×10−2 1.20×106

D0,T ST × 109 0.05 0.21 0.71

m2 s

PT0 ST s n∞ eβWBulk mkg −25

2.46 ×10 2.95 ×10−2 5.49 ×105

3.3.2 Permeabilities The first step in calculating the permeabilities, P 0 , is to obtain the time-correlations functions appearing in Eq. (3.4). This was done by simulating Eqs. (3.5) and (3.6) with a second order stochastic Runge-Kutta integrator [110]. The required correlation functions at any initial guest position were obtained from an ensemble average of 2000 trajectories. Each of the trajectories was “aged”, keeping the guest frozen while the lattice equilibrated. The aging and running times for the individual runs were 4.096 × 10−12 s. The simulations and ensemble averaging were performed on a Beowulf Cluster containing 64 processors. The plane averages appearing in Eq. (3.4) where computed using finite grids that contained from 35 to 72 points. Understanding that D(z) must scale with the probability of being in the z plane, which is proportional to e−βW (z) , it is clear from Eq. (3.2) that the main contribution to the permeability will come from the planes where W (z) is large. We have checked numerically that D(z) = n∞ D0 (T )e−βW (z) is approximately true near the barrier tops and the proportionality constant, D0 (T ), is reported in Table 3–1. In the same table, we report the values of P 0 /n∞ eβWBulk at the three temperatures. Note that the potential of mean force in the bulk was kept unspecified here. The log of the permeability is shown in Fig. 3–5 as a function of 1/T . This figure shows that the permeability has an Arrhenius form. 3.4 Transition state theory As discussed in I, an easy and much more common way of obtaining an estimate for the permeability is transition state theory (TST). Of the many assumptions involved in TST, a crucial one, is that the trajectory of the diffusing particle on average follows

97

Figure 3–5: The intrinsic permeability, P 0 , shows Arrhenius behavior in the temperature range,100-800K. The top line shows ln (P 0 /n∞ eβWBulk ) while the bottom line shows ln (PT0 ST /n∞ eβWBulk ).

98 the reaction coordinate or a steepest descent path that links different binding sites. In I, we showed that this is invalid for small energy barrier systems. We now present the permeabilities for the present system using TST. In a hopping model that incorporates the 2 paths shown in Fig. 3–2, and that takes in account the local minima that these paths cross, the steady state flux through the material is given by, j=−

2κ+ Keq n∞ (µ+ − µ− ), kB T N (1 + κ− /k− )

(3.10)

where κ± and k± are the rate constants whose respective rate processes are indicated in Fig. 3–3. The equilibrium constant Keq describes the equilibrium between the gas in the bulk and the first layers of binding sites (assuming that their energies are the same as those for sites well inside the crystal). Finally, N , is the number of binding sites in a channel of a given length, here 3 per unit cell. In obtaining Eq. (3.10), we have assumed that the gas in the bulk is ideal, hence δβµ± ∼ δn± /n∞ . TST gives analytical expressions for all rate constants in Eq. (3.10) while Keq can be obtained from a Langmuir adsorption model, that assumes equilibrium between the gas and binding sites in the outermost layer of the crystal; this gives · κ+ Keq =

2π(kB T )3

and κ− = k−

¸1/2

mη1‡ ζ1‡ Ã

η2‡ ζ2‡ η1‡ ζ1‡

‡

ρc e−β(W1 −WBulk ) ,

(3.11)

!1/2 ‡

‡

e−β(W1 −W2 ) ,

(3.12)

‡ where Wi‡ is the potential of mean force while ηi‡ and ζ1,2 are the two positive potential of

mean force curvature eigenvalues at the ith saddle point, ρc is the channel density per unit area, and m is the mass of the guest. Note that partition functions for the outer binding sites drop out of the final expression, cf. Eq. (3.11). The permeability predicted by TST is

99 obtained from Eqs. (3.10) and (3.11) and gives, ρc κ+ Keq PT0 ST ρc κ+ Keq =2 ≈2 , n∞ kB T N (1 + κ− /k− ) kB T N

(3.13)

where the last approximation is valid when W1‡ À W2‡ (this is the case for 100K and 300K, but not for 800K, and the full expression will be used for the three cases). In this limit, the effective hopping rate constant between two binding sites is unaffected by the local minimum. The permeabilities can now be calculated. They are reported in Table 3–1 and plotted in Fig. 3–5 for comparison with the permeabilities using the theory in I. It is seen that the temperature dependence predicted by TST is close to our value while the actual permeabilities are consistently lower. 3.5 Comparison with experiment We now compare our results with experimental argon diffusivities obtained by Watson and Cherniak [32]. This group studied the diffusion of argon into α and β-quartz. In this study, the silica surface was exposed to pressurized argon (around 100 MPa) for several hours. Then, the near surface region was analyzed using Rutherford back-scattering spectrometry, a technique capable of measuring the concentration profile of various species inside the material[31]. These profiles where fitted to the solution of the 1D diffusion equation, ∂n(z, t) ∂ 2 n(z, t) =D , ∂t ∂z 2

(3.14)

which describes the density n(z, t) of argon inside the material as a function of time, t, and depth, z, to obtain the diffusion coefficient, D. The diffusion equation applied to our interface system under steady-state condition, gives for the flux, j=−

D [n(d) − n(−d)], 2d

(3.15)

where, as usual, 2d is the thickness of the crystalline interface and n(±d) the density at the boundaries. By rewriting the last equation in terms of the difference in chemical potentials,

100 we obtain, D j=− 2d

µ

∂n∞ ∂µ

¶ [µ(d) − µ(−d)],

(3.16)

β

in the linear regime. It is easily shown that the permeability and the diffusion coefficient are related by,

·

¸ ∂ P0 βWBulk D = 2dkB T (n∞ e ) , ∂n∞ n∞ eβWBulk

(3.17)

where WBulk = ∆µ is the non-ideal part of the guest chemical potential in the bulk. For ideal gases, the factor in square brackets in Eq. (3.17) is equal to one, but at the experimental pressures (around 100MPa), the non-ideal corrections are non-negligible. We computed the quantity appearing in the square brackets by numerically solving the hypernetted chain equation (HNC) for a bulk Lennard-Jones fluid and noting that µ

∂β∆µ ∂n

and

¶ = −˜ c(β, n)

(3.18)

β

Z

n

dn n˜ c(β, n),

βp = n −

(3.19)

0

where c˜(β, n) is the zero wave-vector Fourier transform of the direct correlation function[27] and p is the pressure. Note that Eq. (3.19) is the well known compressibility expression for the equation of state. Finally, we can use the HNC results for c˜(β, n) to numerically integrate Eqs. (3.18) and (3.19) and the result is shown in Fig. 3–6. We are now in position to compare our predicted D with the experimental ones. The experiment reports the full diffusion coefficient as follows, ‡

D = D0 e−β∆E .

(3.20)

In terms of TST, the diffusion constant for multiple barrier channel hops takes the form: kB T −β∆A‡ 4dρc D= e h N ρ0

µ

T0 T

¶3/2 ,

(3.21)

101

Figure 3–6: The non-ideal bulk correction factor appearing in Eq. (3.17) for a LennardJones fluid with Argon-like parameters [27] as a function of pressure, p, using the HNC approximation and the compressibility relation for the pressure (σ is the Lennard-Jones size-parameter). Note that the HNC solution becomes unphysical for pressures beyond those shown in the figure at 100K, possibly signaling a phase transition.

102 Table 3–2: Comparison with experimental diffusion coefficients for the two potential models, (a) and (b)

∆E ‡ /kB (K) ∆S ‡ /kB 0 ( m2 ) D800 s − log10 D800

Our method (a) (b) 7863 20304 -11.80 -13.28 7.80 × 10−10 1.75 × 10−10 13.38 20.78

TST (a) (b) 7912 20601 -12.52 -13.72 3.80 × 10−10 1.13 × 10−10 13.72 21.13

Ref. [32](α-quartz) 6150 ± 750 -32.47 +8.8 8.2−4.2 × 10−19 21.43

where h is Planck’s constant and where ρ0 and T0 are the bulk gas density and temperature, respectively, in the standard state (here 300 K and 1 atm). The activation free energy is easily obtained by using either our or TST’s result for D, cf. Eq. (3.17), in the last equation. The activation Helmholtz free energy takes the usual form, ∆A‡ = ∆E ‡ − T ∆S ‡ , where, as usual, ∆E ‡ is obtained from the slope of a plot of ∆A‡ /T versus 1/T , cf. Fig. 3–5. In Table 3–2, we report ∆E ‡ , ∆S ‡ and D0 and compare them with the experimental values. We report both calculated and experimental D0 only at 800K because the experiment never considered room temperature or lower. The Lennard-Jones parameters that we used are the same as in I, but with modified ˚ and rSi = 0.6380 A) ˚ such that our ∆E ‡ agrees well target atoms radii (rO = 0.9720 A with the experiment. This set of potential parameters, used everywhere unless specified otherwise, is denoted as “a” in Table 3–2. For this potential, the Arrhenius pre-exponential 0 factor, i.e., D800 differs by 8 orders of magnitude from that reported by Watson and Cher-

niak and the overall D at 800K also differs by 8 orders of magnitude. Such a big difference very likely arises from a Boltzmann factor, and strongly suggests that the current potential parameters (“a”) underestimate the strength of the actual guest-lattice interactions, and that the already large energy barriers of the system are in fact even larger. The limited free volume for argon to diffuse in α-quartz implies that argon is always in regions of strong repulsive potential of the closest crystal atoms. In this regime, small changes in the potential parameters can drastically change the shape of potential surfaces.

103 In fact, the depth of our Lennard-Jones potential was obtained in I from the London formula [14], and it is well known that this approach tends to give smaller depths than others (like the Kirkwood-Muller [15, 16] formula). In Ref. [12], it is shown that the London formula can give depths that are too small by a factor of 3. Moreover, these formulas model the attractive part of the potential. Here, it is the repulsive part that would need to be modeled accurately. As a test, we have redone the calculation with an increased well depth (3 times larger), increased oxygens radii (15% larger) and increased force constants (20% larger). While the basic features of the energy landscape are preserved with these parameters, albeit at higher energies, the local minimum separating the binding sites disappears. The results with this potential are denoted as “b” in Table 3–2. The overall diffusion constant at 800K agrees well with the experiment, but the ∆E ‡ is now three times too large. 0 For both set of potential parameters, the factor D800 is not well reproduced. This indicates

that both potentials fail to describe accurately the entropy loss as the lattice rearranges in the presence of the guest. In I, we pointed out that our potential model was only qualitative, but, since the typical energies were small, the accuracy was reasonable. Here, mainly because the energies are large, small changes in the potential parameters lead to large changes in D. An accurate potential model would be necessary for good agreement with the experiments. Of the limitations of our potential model are the facts that the repulsive part is poorly described, that we have neglected all long-range interactions and that the lattice may start to show anharmonic effects considering the large displacements that are induced by the guest. Note that there are certainly some sources of errors in the experiments coming from surface effects, defects, impurities and nano-channels, but Watson and Cherniak carefully checked that these were small, and in any event cannot be used to explain an O(107 ) difference. 3.6 Discussion The effect of the crystal vibrations on the guest motion was also addressed in I, where we found that neglecting the lattice vibrations introduced only small errors for a system

104

Rt Figure 3–7: The correlation function hv(t)vi and the integral 0 ds hv(s)vi are shown for an argon atom starting at a saddle point in an adiabatic, flexible and frozen lattice simulation.

105 with small energy barriers. Here, we consider again the role of lattice vibrations. The effect of the crystal motion appears in several ways. First, at the level of the potential of mean force, allowing the lattice to relax to a new equilibrium position in the presence of the guest (cf. Eq. (3.9)) significantly reduces the energy barriers in W (z), e.g., by 60kB T at 300 K or 18kB T at 800 K. As was mentioned before, this happens because the argon atom is always in the steep repulsive part of the closest lattice atoms before they are allowed to relax. Also note that the flexibility of the lattice is responsible for the temperature dependent part of W (r). Second, aside from mean potentials, the question of the importance of the lattice vibrations on the guest dynamics can be addressed by comparing the velocity auto-correlation functions for an argon atom released from a saddle point in our dynamic lattice and in an adiabatically equilibrated lattice. In the later case, we mean that the lattice follows the guest adiabatically such that the net force on the lattice remains zero at all times. This is (0)

(0)

achieved using an initial RT that solves Eq. (3.9) and propagating RT in time according to,

" #−1 (0) (0) (0) dRT ∂ 2 U (r, RT ) ∂ 2 U (r, RT ) dr = − Kef f + . (0) (0) (0) dt ∂RT ∂RT ∂RT ∂r dt

(3.22)

(0)

In both simulations, the force felt by the guest is simply −∂U (r, RT )/∂r (the temperature dependence of W (r) is ignored in the adiabatic lattice case and is implicit in the vibrating lattice case). The result is shown in Fig. 3–7. The data shows that for the adiabatic lattice, the correlation time is much longer and the integral of hv(t)vi is larger compared with the case where the vibrations are included. As mentioned in I, there are two mechanisms that contribute to the decay of hv(t)vi. The first is randomization of the velocity directions, while the second, is guest energy transfer to and from the lattice vibrations. The results of Fig. 3–7 suggest that, at least for our system, the energy exchange with the lattice is strong enough to shorten the velocity relaxation time, leading to a reduction in the overall diffusion rate. In I, we found the

106 opposite, i.e., that this energy exchange time scale was too long and did not contribute strongly to the decay of the velocity correlations. The behavior of hv(t)vi at short times is very different in the two cases. For the adiabatic case, the guest is initially accelerated down the barrier, producing a peak at short time, while for the real calculation, the collisions with the randomly moving lattice initially slow down the guest. In Fig. 3–8, we show the distributions of positions along z inside the crystal as a function of time. From these figures, it is clear that the trapping of the guest in binding sites is much more efficient for the unfrozen lattice where energy exchange is possible. In the vibrating case, if the guest starts at a saddle point, it travels mainly to the nearest binding sites, and only a small fraction go to second nearest binding sites. It travels farther and with bigger proportion in the adiabatic case. Also shown in Fig. 3–7 is the velocity auto-correlation obtained from the simulation of the guest released at a saddle point in a completely rigid lattice where the crystal atoms are frozen at their guest-free, equilibrium lattice positions. The short-time behavior shows a much stronger acceleration down the barrier; this is not surprising given the much larger barrier. In addition, we see higher frequency motions, indicative of the rattling motion of the guest in the narrow rigid channel. In testing the validity of transition state theory, we observed that the temperature dependence predicted by TST is close to the one obtained from our method. The activation entropy, ∆S ‡ , is also close for the two methods. This is not surprising, since the energy landscape that we have in this system combined with the theory, cf. Eq. (3.2), gives much more importance to the saddle point region; TST only includes information at the saddle point (which of course is determined from the same free energy surface, W (r)). In Fig. 3–9, we compare the steepest descent path, that is assumed to describe the dynamics in TST, against simulated trajectories that start at the high energy saddle point and that are averaged under the condition that they end up in one of the 2 binding sites (the black path in the Fig. 3–9). From this figure, we conclude that the steepest descent

107 path accurately describes the motion for this system (the oscillations in the black curve are explained by the fact that guest particles with kinetic energy cannot take the sharp turn in the steepest descent path). Note that, starting from the saddle point, 98% of the trajectories end up in one of the two binding sites (almost in a 1:1 ratio). The remainder either get trapped in the local minimum or go to farther binding sites. These observations, combined with the fact that the permeabilities obtained from transition state theory are systematically lower, tells us that something is still missing in TST (note that the TST transmission coefficient in this case is close to unity, and in any event, including it would only make the agreement worse). This is surprising since TST usually gives an upper bound to Smoluchowski processes (cf. Ref. [119]). The difference cannot be explained from the cooperative motion of the guest and the lattice atoms since, as shown in Fig. 3–7, the vibrations tend to reduce the diffusion. One possible explanation of the discrepancy between TST and our calculation is the following. Recalling that D0 is obtained from R∞ the plane average of 0 dt hvz (t1 )vz ir e−β(W (r)−W (z)) . This factor is shown in Fig. 3–10 for the W (z) maximum energy plane (which is very close to the plane that contains the a saddle point). In the context of TST, this factor is assumed to be harmonic with the maximum at the saddle point. It is clear from the figure that the harmonic approximation is not accurate. In conclusion, while TST seems to describe more effectively the averaged diffusion process in our large energy barriers system (18kB T at 300 K) than it did in I for smaller energy barriers (2kB T at 300K), although the assumption about what regions of phase space contribute to the diffusion process is still wrong. Most experiments (cf., Ref. [32] and [33]) claim that the pre-exponential factor, D0 , as defined in Eq. (3.20), does not appreciably change with temperature. The prefactor is directly proportional to D0 and the non-ideal corrections appearing in Eq. (3.17). By assuming the gas is ideal in the bulk and that the saddle-point curvatures are independent of temperature, TST predicts that D0 ∝ T 3/2 , cf. Eq. (3.11). This is not supported by the data, cf. Table 3–1 since there is significant temperature dependence in the curvatures.

108 Indeed the TST results are well described by a T 1.28 power law while our method predicts T 1.1 , albeit with larger deviations. In both models, the temperature dependence is clear, but this effect is very hard to see in an Arrhenius plot (cf. Fig. 3–5), and would be even harder to see in a real experiment given the large error bars reported for D0 . In conclusion, we have shown that the driving forces for the diffusion of a guest in a large energy barrier system are different than in I where we considered a small energy barrier system. The permeability is very sensitive to the accuracy of the activation energy, and underlying potential model, and on the flexibility of the lattice. For a more accurate evaluation of the permeabilities, a potential model that describes the repulsive interactions better, resulting in a more negative activation entropy, would be required. The goal of this work was to establish what phenomena contribute to the prefactor and we shown that the prefactor depends on the cooperative motion between the guest and the lattice. This effect is absent in TST and all frozen lattice or adiabatically equilibrated lattice simulations and can lead to non-negligible errors in systems with large energy barriers, and concomitantly large forces.

109

Figure 3–8: The distribution of positions along z of the 2000 trajectories as a function of time. The left figure is for the adiabatic case while the right figure is for the unfrozen lattice.

110

Figure 3–9: The steepest descent path (blue) is compared against the average trajectories (black) as obtained from our simulations started at the high energy saddle point.

111

R∞ Figure 3–10: The factor 0 dt hvz (t1 )vz ir e−β(W (r)−W (z)) in the maximum W (z) plane ˚ × 3.46A ˚ is shown where an area of 2.757A

CHAPTER 4 Diffusion in Channeled structures III: Quantum corrections induced by the lattice c vibrations. [120](°American Physical Society, 2007) In the previous two chapters, we have developed a theory that allows the evaluation of the macroscopic permeability of a guest in a channeled material through the evaluation of microscopic time correlation functions. We successfully applied this formalism to two systems: one where the crystal contains wide channels, resulting in small energy barriers (Xenon in Theta-1), and the other where the crystal contains narrow channels, resulting in large energy barriers (Argon in α-quartz). For the latter system, we found that the lattice vibrations played an important role in the diffusion process, and that, at least for that system, the guest mobility was reduced compared to a rigid framework. Another question naturally arises. Should we treat the problem quantum-mechanically? In fact, it is well known that, at room temperature, many of the crystal vibrations are not excited (which results in a heat capacity that is well-below the classical prediction). In this chapter, we reformulate our theory to include some leading order quantum effects and we recalculate the permeability semi-classically. This allows us to obtain the first quantum corrections to the diffusion process and permeability. Of course we will work in regimes where the classical guest approximation is expected to hold, which means that the temperature should not be too low. 4.1 Introduction In two recent papers [53, 112], hereafter referred to as I and II, we studied the intracrystalline diffusion of a guest inside selected zeolite structures. There, we used the formalism initially developed by Ronis and Vertenstein [6] to calculate the permeability through a crystalline interface of finite thickness that separates two bulk regions containing

112

113 the guest component at different chemical potentials. The advantage of using this formalism is that it is fully microscopic in the sense that the permeability is obtained from the integration of a space-dependent Onsager diffusion coefficient which, in turn, is calculated from simulations of specific guest time-correlation functions. In I and II, we compared our method with transition state theory (TST), which is often used to describe the diffusion in these types of systems [44, 45, 46] and where it is assumed that the guest diffusion follows one or more reaction coordinates. Within our formalism, we were able to test the underlying assumptions of transition state theory and we found it to be inappropriate for more open channels. A major part of our work in I and II was to understand the role that lattice vibrations play in the diffusion process. Many earlier studies of diffusion in channeled structures used standard molecular dynamics (MD) with a frozen lattice [93, 38, 43] to simulate the motion of the guest. Following these studies, the motion of the crystal atoms was included by simulating the vibrational motion of a relatively small number of lattice atoms [44, 45, 46, 96]. The role of the lattice vibrations on the guest diffusion had been studied, before us, by Kopelevich and Chang [54] and by Suffritti and coworkers [39, 114, 115]. In Ref. [54], only the motion of the guest was simulated from a generalized Langevin equation (GLE) where the energy exchange with the lattice was described by dissipative (memory) and random terms which were approximated phenomenologically using a simple one parameter model for the memory function. In I and II we used a model similar to that of Ref. [54]. In our formalism, the motion of the guest and selected target atoms (the only ones that interact directly with the guest) was explicitly simulated using a set of GLE’s. The energy exchange with the infinite bath (the part of the crystal that does not interact directly with the guest) was included by the dissipative and random force terms that typically arise in GLE and that are related by generalized Einstein-Nyquist relations or the 2nd fluctuation-dissipation theorem (see, e.g., Ref. [55]). The periodicity of the lattice and the approximation that the crystal is

114 harmonic allowed us to compute the memory function exactly. This memory function was then fitted to a general simple function that allowed us to include the memory and noise terms in the equations of motion for the target atoms. As shown in I and II, the success of our procedure is that the approximate memory function we use is exact for long and short times and hence, it reproduces the full crystal vibrational density of states very well. Within this framework the role of the lattice vibration on the diffusion process was studied. In I, the guest (Xenon) was diffusing in a crystalline zeolite that has wide channels (Theta1). There, we found that the lattice vibrations did not affect the motion of the guest insofar as the permeability is concerned. On the other hand, in II, we explored the diffusion of Argon in α-quartz, a crystal with very narrow channels. For this other system, we found that the flexibility and vibrational motion of the lattice played a significant role in the guest motion, most importantly, on the free energy landscape. For systems like the one studied in II, where the lattice vibrations play a significant role in the diffusion, another question naturally arises, namely, should we treat the problem, or at least parts of it, quantum mechanically? After all, it is well known that most vibrational modes in typical crystals are not thermally excited at room temperature. Specifically, ω ∗ ≡ h ¯ ω/kB T is the dimensionless parameter that characterizes how quantum mechanical a vibrational mode with frequency ω at temperature T is. The vibration is classical when ω ∗ ¿ 1 and quantum mechanical when ω ∗ À 1. At room temperature, ω ∗ = 1 when ω = 0.393 × 1014 sec−1 . A typical vibrational density of states is shown in Fig. 4–1, and shows that roughly 75% of the vibrational modes are in the quantum regime at 300K. In this paper, we will use an entirely different formalism based on path integrals to approximately include the quantum mechanical nature of the lattice. The goal is to calculate the quantum corrections to the potential of mean force of the guest in the presence of

115 the quantum lattice, to obtain the quantum corrections to the space-dependent Onsager diffusion coefficient, and, ultimately, to see what kind of correction yields at the macroscopic level, i.e. on the permeability. The paper is divided as follows, in Sec. 4.2 we show how the permeability is expressed in terms of Kubo averages and we recall what types of correlations have to be calculated in order to obtain the space-dependent Onsager diffusion coefficient. In Sec. 4.3 we describe the path integral approach that allows us to obtain the required correlations semi-classically. In particular, a closed form expression for the density matrix is obtained and we show how Forward-Backward path integrals approximately define a set of second order differential equations that can be turned in a stochastic initial value problem. We explicitly show how, in the classical limit, these equations reduce to the classical generalized Langevin equations of motions obtained in I and II. We conclude this section by showing how the space-dependent Onsager diffusion coefficient expression simplifies when the guest is assumed to be classical. In Sec. 4.4, we show how to implement the formalism developed in Sec. 4.3. More precisely, we describe how the potential of mean force is obtained from our expression for the density matrix and explain how the new (non-classical) terms are calculated numerically using the theory of defects, Brillouin zone sums and contour integrations. We then describe how these techniques can be further used to simulate the non-classical terms in the generalized Langevin equations. In Sec.4.5, we apply our theory to the study of Neon in α-quartz. We will first show that our formalism is exact in the absence of the guest, and then report the main results of this paper. Specifically, we show how the quantum corrections slightly increase the potential of mean force of the guest inside the crystal and we show that this difference is increased as we lower the temperature. The appropriate guest time correlation function, the space-dependent diffusion coefficient and the permeability are reported and compared with their classical counterparts at room temperature. We show that the lattice effectively vibrates more (the velocity correlations have larger amplitudes) when quantum mechanics

116 is included and, for this system, this results in a slower diffusion for the guest and a permeability which is about 25% smaller compared to the classical one. Sec. 4.6 contains a discussion and concluding remarks.

Figure 4–1: The exact density of states (full line) for α-quartz is compared with the one obtained from our approximate form for the memory function (dashed line), cf. Eq. (4.89). 4.2 The Time-Correlation Function Form of the Permeability In I and II, we studied diffusion through a macroscopic crystalline material by understanding the microscopic motion of the guest inside the crystal insofar as what contributes

117 to the permeability of the macroscopic system. As was discussed in I and II (and references therein), the permeability intrinsic to the material is given by 1 1 = 0 P β

Z

d

dz −d

1 , D(z)

(4.1)

where β ≡ 1/kB T , 2d is the width of the interface separating the two bulk regions containing the guest component, and where 1 D(z) ≡ A

Z

Z

∞

dt

Z drk

0

dr0 hJ†z (r, t)J†z (r0 )i,

(4.2)

is a space-dependent Onsager diffusion coefficient. In this last expression, r and r0 are spatial coordinates, t is the time, and A is the area of the crystalline interface. In addition, J†z (r) ≡ exp[i(1 − P)Lt](1 − P)Jz (r) is the z-component of the dissipative part of the guest diffusion current, where P is a projection operator (see below) and L is the Liouville operator. The integral over rk , i.e., (x, y), is a consequence of the fact that the average current through the interface was chosen to lie along the z axis. In I and II, where we assumed everything to be classical, hJ†z (r, t)J†z (r0 )i in Eq. (4.2) is a simple correlation function that, modulo effects associated with the dagger, can be easily evaluated from classical molecular dynamics simulations. When quantum effects are taken into account one convenient approach is to replace the standard correlation functions by Kubo averages [121], a procedure fully consistent with linear response theory. For a recent review where Kubo averages are used to study quantum liquids, see, e.g., Ref.[122]. Our expression for D(z) now becomes 1 D(z) = A

Z

Z

∞

dt

Z dr

0

ˆ † (r0 )iK , ˆ † (r, t)J dr0k hJ z z

(4.3)

where the Kubo average is defined by Z

1

ˆ Bi ˆ K≡ hA(t) 0

i h ˆ ˆ ˆ λβ H −λβ H ˆ Be ρˆ , dλTr A(t)e

(4.4)

118 ˆ is the Hamiltonian operator, Tr is a quantum mechanical trace, and ρˆ ≡ e−β Hˆ /Q where H i h ˆ −β H is the density matrix, with the partition function, Q ≡ Tr e . The quantum or classical mechanical expressions for D(z) are not particularly useful in the above form without a prescription that takes care of the effects that the projection operators (i.e., the daggers) have on the current operators and on their time dependence. For the classical case, Vertenstein and Ronis [6, 7] obtained a relation, expressed in terms of standard correlation functions, that approximately accounts for the projection operators. They used a projection operator formalism [94, 95] with the projection operator PA ≡ hAN i ∗ hN N i−1 ∗ N,

(4.5)

where the N is the fluctuation in number density of the guest component and where the ∗ indicates any spatial integrations. Note that hN N i−1 is not an algebraic inverse of the hN N i correlation function, rather it is the solution to Z dr1 hN (r)N (r1 )ihN (r1 )N (r0 )i−1 = δ(r − r0 ). With this in hand, they showed, assuming infinite dilution for the guest in the crystal, that D(z) could be accurately approximated by R∞ n∞ 0 dt hvG,z (t)vG,z iz(t=0)=z e−βW (z) R∞ D(z) = , 1 + 0 dt hβF (z(t))vG,z iz(t=0)=z

(4.6)

where n∞ is the number density in the bulk, vG,z is the z-component of the guest velocity, W (z) and F (z) are, respectively, the plane average potential of mean force and the mean force. Specifically, −βW (z)

e

1 ≡ A

Z drk e−βW (r)

(4.7)

where r is the guest position, W (r) is the usual potential of mean force and F (z) ≡ −∂W (z)/∂z. All correlations in Eq. (4.6) are evaluated conditional to the z component of the guest position being initially equal to z (and is indicated by the subscripts

119 on the averages). Finally, note that this derivation uses the fact that hN (r)N (r0 )i = n∞ δ(r−r0 )e−βW (r) and further assumes that D(z)/e−βW (z) is approximately constant near the barrier top (this approximation is exact for Smoluchowski diffusion processes [49]). Quantum mechanically, an appropriate and equivalent choice of projection operator is ˆN ˆ iK ∗ h N ˆN ˆ i−1 ∗ N ˆ ˆ ≡ hA PA K

(4.8)

ˆ is the fluctuation in the number density operator and where all correlations are where N Kubo averages. It is shown in Ref. [123] that the above projection operator is reasonable in that it satisfies all the required properties of projection operators. By repeating the standard steps that led to Eq. (4.6), one soon obtains the following expression for the current-current time correlation function: ˆ t)J(r ˆ 0 )iK = hJ ˆ † (r, t)J ˆ † (r0 )iK hJ(r, Z t Z Z ´ ∂ ³ ˆ −1 † † ˆ ˆ ˆ hN (r1 )N (r2 )iK − dτ dr1 dr2 hJ (r, τ )J (r1 )iK ∂r1 0 ˆ 0 )iK . ˆ (r2 , t − τ )J(r × hN

(4.9)

ˆ (r1 )N ˆ (r2 )iK 6= n(r1 )δ(r1 − Unlike the classical case, here, at infinite guest dilution, hN ˆ (r1 )N ˆ (r2 )i = n(r1 )δ(r1 − r2 )]. Hence, a solution for the r2 ) [although we still have hN ˆ † (r, t)J ˆ † (r0 )iK , in terms of the other standard correirreversible current correlations, hJ ˆ t)J(r ˆ 0 )iK and hN ˆ 0 )iK is more involved, but could in principle be ˆ (r2 , t)J(r lations, hJ(r, ˆ (r1 )N ˆ (r2 )iK , obtained by solving an integral equation. Nonetheless, it is clear that hN ˆ t)J(r ˆ 0 )iK and hN ˆ 0 )iK are the objects that we need to calculate. ˆ (r2 , t)J(r hJ(r, The next section will show how any of these Kubo averages can be obtained within a semi-classical picture that relies on the fact that the guest dynamics is slow compared to the vibrations, thereby simplifying the problem considerably. For the types of averages that appear in Eq. (4.9), where both operators depend explicitly solely on guest degrees of freedom, we will show that D(z) is obtained from an expression identical to Eq. (4.6),

120 but with a corrected potential of mean force and a new set of dynamical equations, used to calculate the desired correlations; these take into account the effect of the quantum degrees of freedom (the crystal). In the subsequent sections, we describe how the Kubo averages are computed. We will work with the anti-commutator correlation defined as ˆ ˆ ≡ Tr h{A(t), B}i

h³

´ i ˆ ˆ ˆ ˆ A(t)B − B A(t) ρˆ .

(4.10)

This anti-commutator correlation can then be used to obtain the Kubo transform using the relation

Z ˆ Bi ˆ K= hA(t)

∞

1 ˆ 0 ˆ dt0 γ(t − t0 ) h{A(t ), B}i, 2 −∞

(4.11)

where γ(t) is most easily represented spectrally as 1 γ(t) ≡ 2π

Z

∞ −∞

dω eiωt

tanh(β¯ hω/2) . β¯hω/2

(4.12)

This connection between the Kubo average and the anti-commutator was first established by Kubo [121]. From this, it is easy to show that the Kubo average and the anti-commutator correlation become identical in the classical limit (i.e., β¯hω → 0). Note that the projection operator, Eq. (4.8), could directly be defined in terms of anti-commutator correlations replacing the Kubo averages. We choose the later because it is consistent with response theory [90]. 4.3 Evaluation of anti-commutator correlation functions using Path integrals We now turn to the evaluation of the desired anti-commutator correlations. As in I and II, we assume that the guest component is anharmonically coupled to a limited number of atoms in the crystal, which we call target atoms . These target atoms are allowed to further interact with themselves and with any other atoms of the macroscopic crystal that do not interact with the guest directly, the so-called bath atoms (note that, this last assumption could be relaxed to include harmonic interactions between guest and bath). The interaction

121 between any two crystal atoms is assumed to be harmonic. This model is described by the following Hamiltonian ˆ ≡ H

pˆ2g 1 ˆ T −1 ˆ 1 ˆ T −1 ˆ + P t Mtt Pt + Pb Mbb Pb 2mg 2 2 1 ˆ t) + R ˆ T Ktt R ˆt + R ˆ T Ktb R ˆb + 1 R ˆ T Kbb R ˆ b, +V (ˆrg , R t t 2 2 b (4.13)

ˆ t,b and R ˆ t,b are the ˆ g and ˆrg are the guest momentum and position operator, P where p target/bath momentum operator vectors that contain all the atoms in the target/bath space, K with subscript tt, tb and bb is the appropriate block of the force constant matrix of the harmonic crystal, and the superscript T indicates a matrix transpose. The anti-commutator correlation function can be formally written in the coordinate representation as 1 ˆ 1 ˆ h{A(t), B}i = 2 2

Z

Z dR1

Z

n ˆ 2 )K+ (R2 , R3 ; t)B(R ˆ 3) dR3 K− (R1 , R2 ; t)A(R ¾ ρ(R3 , R1 ) ˆ ˆ + B(R1 )K− (R1 , R2 ; t)A(R2 )K+ (R2 , R3 ; t) Q dR2

(4.14) ˆ operators in coordinate space, where Ri ≡ (rT , RT , RT )T , for i = for diagonal Aˆ and B g t b i 1, 2, 3, and where, using bra-ket notation, ˆ

K± (R1 , R2 ; t) ≡ hR1 |e∓iHt/¯h |R2 i

(4.15)

and ˆ

ρ(R1 , R2 ) ≡ hR1 |e−β H |R2 i,

(4.16)

each of which can be expressed in terms of Feynman path integrals, cf. Ref. [58]. Also ˆ 1 ) in the second term of the right-hand side of Eq. (4.14) only note that the operator B(R acts on K− (R1 , R2 ; t) and not on ρ(R3 , R1 ). The partition function, Q, that appears in

122 Eq. (4.14) is, as usual, expressed as a trace, Z Q≡

dR ρ(R, R).

(4.17)

4.3.1 The density matrix We first focus on the density matrix, ρ(R1 , R2 ). It can be written in terms of path integrals as,

Z ρ(R1 , R2 ) =

1

D[R(τ )] e− h¯ A ,

(4.18)

where the Euclidean action is given by Z β¯h " mg r˙g2 R˙ Tt Mtt R˙ t R˙ Tb Mbb R˙ b A ≡ + + dτ 2 2 2 0 1 + V (rg , Rt ) + RTt Ktt Rt + RTt Ktb Rb 2 ¸ 1 + RTb Kbb Rb , 2

(4.19)

where the time argument was omitted for all position variables and where r˙ g (τ ) ≡ drg /dτ , etc.. All paths are constrained to start at R1 and end at R2 . We will now use, for the first but not the last time, the fact that the guest is slow at room temperature and higher and we will argue, as in Ref. [58], that the position of the guest barely changes in the complex time interval, β¯ h. Effectively, for this path integral, we treat the guest as a free particle (0)

evolving in a constant potential. Therefore, we set rg = rg (to be specified later) in the (0)

last equation and, as in I and II, we Taylor expand V (rg , Rt ) with respect to the crystal (0)

(0)

coordinates around Rt = Rt (rg ), a point at which there is no net force (in the classical sense) on the crystal atoms; i.e., where ³

´T

(0) Rt (r(0) g )

(0)

Kef f

(0)

∂V (rg , Rt ) =− ∂Rt

(4.20a)

and (0)

(0)

Rb = −K−1 bb Kbt Rt ,

(4.20b)

123 (0)

where, as in I and II, Kef f ≡ Ktt − Ktb K−1 bb Kbt . It is easy to show that the best choice of rg

is the mid-point (r1,g + r2,g )/2, in that this choice makes the first corrections in h ¯ for the (0)

guest vanish. As in I and II, the terms in the Taylor expansion for V (rg , Rt ) above second order are neglected because the oscillators, even in the quantum mechanical case, do not strongly deviate from their equilibrium positions because of the strength of the harmonic part of the potential. Within these approximations, the only path integrals that have to be evaluated are a simple free-propagator for the guest and standard harmonic potential path integral for the crystal (see, e.g., Refs. [58, 57]). The manipulations are standard, and the final expression for the density matrix is µ ¶#1/2 ¶3 µ mg yg2 mg Ξ(D) (0) (0) exp −βV (r , R ) − ρ(R1 , R2 ) = det t g 2π¯h 2πβ¯ h2 2β¯ h2 ³ ´T (0) (0) (0) (0) Rt Kef f Rt (R1,c − Rc )T Λ(D)(R1,c − Rc ) − −β 2 2¯h ! (0) (0) T (0) (0) (R2,c − Rc ) Λ(D)(R2,c − Rc ) (R1,c − Rc )T Ξ(D)(R2,c − Rc ) − + , 2¯h h ¯ "µ

(4.21) where ˜ + D) ˜ 1/2 coth (β¯h[K ˜ + D] ˜ 1/2 )M1/2 , Λ(D) ≡ M1/2 (K

(4.22)

˜ + D) ˜ 1/2 csch(β¯h[K ˜ + D] ˜ 1/2 )M1/2 , Ξ(D) ≡ M1/2 (K

(4.23)

and

where Rc ≡ (RTt , RTb )T and K is a matrix defined in the crystal subspace and contains all the harmonic force constants. The Λ(D) and Ξ(D) are matrices defined in the full crystal subspace and are functions of the curvature matrix, (0)

D≡

(0)

∂ 2 V (rg , Rt ) , ∂RTt ∂Rt

(4.24)

124 which by assumption is nonzero only in the target subspace and which depends on the guest position. We have introduced a difference coordinate yg ≡ r2,g − r1,g in Eq. (4.21). Finally, henceforth, we use a tilde over various matrices to indicate that they have been ˜ ≡ Mc−1/2 KMc−1/2 , where M ≡ diag(mg , Mt , Mb ). mass scaled, e.g., K A guest potential of mean-force can be defined, as in I and II, in terms of the diagonal part of the reduced density matrix as −βW (rg )

e

ρR (rg ) ≡ Qharmonic

where

µ

mg 2πβ¯h2

¶−3/2 ,

(4.25)

Z ρR (rg ) ≡

dRc ρ(R, R),

(4.26)

and where Qharmonic is the partition function of the crystal decoupled from the guest. This definition makes W (rg ) zero when the guest is far from the crystal. Using this definition, we find that (0)

W (rg ) = V (rg , Rt ) +

1 ³ (0) ´T (0) Kef f Rt Rt 2

+∆W (rg ),

(4.27)

where all temperature dependence is carried by kB T ∆W (rg ) ≡ log 2

µ

¶ det [(Λ(D) − Ξ(D))Ξ(D)−1 ] . det [(Λ(0) − Ξ(0))Ξ(0)−1 ]

(4.28)

The first two terms on the right-hand side of Eq. (4.27) appear in the classical potential of mean force obtained in I and II. The temperature dependence of the potential of mean force is different here than what we had in I and II and accounts for the quantum nature of the crystal (we give a detailed comparison in Sec. 4.5.1). Also note that in I and II the potential of mean force was expressed as an expansion in powers of the temperature, where the nonlinear temperature corrections resulted from the higher order terms in the (0)

Taylor expansion of V (rg , Rt ). Here, we work within the same approximation as in I

125 (0)

and II; namely, we only keep terms up to second order in the V (rg , Rt ) expansion. However, now, the resulting potential of mean force is a nonlinear function of the temperature because of quantum mechanical effects. Obviously, in the high temperature limit all the vibrational modes are excited, and Eq. (4.27) reduces to the expected classical expression, 1 ³ (0) ´T (0) Wclass (rg ) = V + Rt Kef f Rt 2 £ ¤ kB T + ln (det 1 + K−1 D ). ef f 2 (0) (rg , Rt )

(4.29)

Recall that, deriving Eq. (4.21), the guest was assumed to behave like a free-particle in the time interval β¯h. Other techniques that approximately include quantum mechanical effects in the calculations of partition functions and density matrices have been developed, and, in principle, could have been used above. Of these, the variational approach invented by Feynman and Kleinert [71, 74] and its extensions (for examples, see [74, 124, 125]) is very accurate, but is useful for non-singular potentials only. Also, Ermakov, Butayev and Spiridonov [67] suggested an approximate expression, exact for harmonic potentials, which gives the right high-temperature (classical) limit and which can be applied to any potentials. This method has later been improved by Mak and Andersen [68], Cao and Berne [69] and by Chao and Andersen [70]. We will show below that these less restrictive approximations result in small differences with our fully classical guest approximation, at least for temperatures for which the permeability experiments could be performed. (The experiments that measure the diffusion of gases through crystals are rarely performed at/or below room temperature). As a final comment, note that many authors, in their calculations of the density matrix, assume that, at time t = 0, the system (here, guest and target) and the bath are decoupled [61, 126]. It is clear that here, this approximation is inappropriate. Remember that in our formalism, the guest has two effects on the crystal degrees of freedom. First, it modifies the force constant matrix in the tt block, but also shifts the oscillator centers, even in the bath. This last effect would be absent if the bath was decoupled.

126 4.3.2 Forward-Backward path integrals and connection with MSR Now that we have a semi-classical approximation for the density matrix, we show how to get the same type of semi-classical approximation for the real-time propagators, K± (R1 , R2 ; t). Each of these propagator is written in terms of path integrals as Z K± (R1 , R2 ; t) =

i

D[R± (τ )] e± h¯ S± ,

(4.30)

where K± is the forward (backward) propagator defined in terms of the action, Z

µ

t

S± ≡

dτ 0

1 1 2 mg r˙g,± + R˙ Tc,± Mcc R˙ c,± 2 2 ¶ 1 T −V (rg,± , Rt,± ) − Rc,± KRc,± , 2 (4.31)

where, for the +(−) propagator, all paths start(end) at R2 and end(start) at R1 . We then have to consider these path integrals in terms of a semi-classical picture consistent with our treatment of the density matrix in the previous section. The formalism that we will use to evaluate these propagators is inspired from the work of Schmid et al. [59, 60] and of Kleinert and Shabanov [61] who obtained quantum Langevin equations for simpler systems. Here, rather than considering the two propagators individually, we will work with the ˆ 2 )K+ (R2 , R3 ; t) appearing in Eq. (4.14). We first realize combination K− (R1 , R2 ; t)A(R ˆ 2 ) only acts on the end point of the positive time propagator. With that the operator A(R this in mind, we will replace R2 in the argument of Aˆ and K+ by Z and, after the operator ˆ A(Z) has acted on the combination K− (R1 , R2 ; t)K+ (Z, R3 ; t), we will set Z back to R2 . This combination of Forward-Backward path integrals is rewritten in terms of sums and

127 differences of the forward and backward path variables, Z K− (R1 , R2 ; t)K+ (Z, R3 ; t) =

µ Z t h i D[X(τ )]D[Y(τ )] exp dτ mg x˙ g · y˙ g + X˙ Tc Mcc Y˙ c h ¯ 0 yg Yc − V (xg + , Xc + ) 2 2 ¸¶ yg Yc T +V (xg − , Xc − ) − Xc Kcc Yc , (4.32) 2 2

where X ≡ (R+ + R− )/2 and Y ≡ R+ − R− . Note that the boundary condition for Y(t) = Z − R2 becomes equal to zero when Z is set back to R2 . We use the same approximations as above and Taylor expand the potential in the action in Y. All the even terms in Y cancel and we keep only the linear ones, neglecting terms of order O(Y3 ). Again, for the guest, this is justified because, for the temperatures considered, the paths are localized around the classical trajectory which, as will be shown below, is given by the classical equations of motion for xg (τ ) and Xc (τ ). For the crystal variables, the expansion is again approximately valid since, for any forward or backward path, the crystal position will not deviate strongly from their mean because of the strength of the harmonic part of the potential. After this expansion is carried out, the argument in the exponential becomes linear in Y. This results in a path integral which closely resembles that which is obtained in the Martin-Siggia-Rose (MSR) formalism [127, 128] that represents stochastic processes as path integrals. This suggest that we should be able to revert the MSR formalism and simulate the path integrals from a set of stochastic differential equations. We now show that this transformation can indeed be carried for our system. This derivation follows that of Jensen’s [128] derivation of MSR in reverse. We start by writing the path integrals in their discretized form and make use of the approximations described

128 in the preceding paragraph. This gives, ( ¶N +1 Z Y µ N ³ m ´3(N +1) Mc g dXj dYj K− (R1 , R2 ; t)K+ (Z, R3 ; t) ≈ lim det N →∞ 2π¯ h² 2π¯h² j=1 " N µ i² X mg (xg,m+1 − xg,m ) · (yg,m+1 − yg,m ) × exp h ¯ m=0 ²2 +

(Xc,m+1 − Xc,m )T Mcc (Yc,m+1 − Yc,m ) ²2

!#)

+ Fg (xg,m , Xt,m ) · yg,m + Ft (xg,m , Xt,m )T Yt,m − XTc,m Kcc Yc,m

, (4.33)

where ² ≡ t/N , the j = 0 and j = N + 1 subscripts represent the boundary conditions on the paths at τ = 0 and τ = t, respectively, and where the forces are defined by ∂V (xg,m , Xt,m ) , ∂xg,m

(4.34)

∂V (xg,m , Xt,m ) , ∂Xt,m

(4.35)

Fg (xg,m , Xt,m ) ≡ − and Ft (xg,m , Xt,m ) ≡ − as usual.

Clearly, the fact that we have dropped all cubic terms and higher in Y(τ ) allows us to perform all the yg,j and Yc,j integrals for j = 1...N , which gives a product of delta functions that can be trivially manipulated to give, ( µ ¶Z Y N ³ m ´3 Mc g K− (R1 , R2 ; t)K+ (Z, R3 ; t) ≈ lim dXj det N →∞ 2π¯ h² 2π¯h² j=1 · µ ¶ N Y ²2 × δ xg,m+1 − 2xg,m + xg,m−1 − Fg (xg,m , Xt,m ) mg m=1 µ ¶# ²2 × δ Xc,m+1 − 2Xc,m + Xc,m−1 − (Ft (xg,m , Xt,m ) − Kcc Xc,m ) Mcc mg

mg

T Mcc Y T Mcc Y c,0 +i(Xc,N +1 −Xc,N ) c,N +1 h ¯² h ¯²

×e−i h¯ ² (xg,1 −xg,0 )·yg,0 +i h¯ ² (xg,N +1 −xg,N )·yg,N +1 −i(Xc,1 −Xc,0 )

o .

(4.36)

129 ˆ In order to proceed, note that after the operator A(Z) acts on K− (R1 , R2 ; t)K+ (Z, R3 ; t) and after Z is set back to R2 , we obtain a function, A(xg (t), vg (t)) that multiplies K− (R1 , R2 ; t)K+ (R2 , R3 ; t) where xg (t) and vg (t) are, respectively, the guest final position and velocity (the final velocity is defined by vg (t) ≡ (xg,N +1 − xg,N )/²)1 . We will show examples of this function for specific choices of the operator Aˆ in the following section. We now use the fact that the final position phase vector, R2 T = (xTg,N +1 , XTt,N +1 , XTb,N +1 ) is integrated over all possible R2 in the anti-commutator expression (cf., Eq. (4.14)) to realize that, in the N → ∞ limit, the product of delta functions combined with the integrations over X2 , X3 ,...,XN forces XN +1 = X(t) = R2 to be the solution of the initial value problem, d2 xg = Fg (xg , Xt ), dt2

(4.37a)

d2 Xt = Ft (xg , Xt ) − Ktt Xt − Ktb Xb , dt2

(4.37b)

d2 Xb = −Kbb Xb − Kbt Xt , dt2

(4.37c)

mg Mtt and Mbb

˙ with initial position X(0) = X0 = (R1 + R3 )/2 and velocities X(0) = V(0) = (X1 − X0 )/² (recall that the integrals over X1 were not performed yet). The remaining integrations in the anti-commutator expression, Eq. (4.14), are over R1 , R3 and X1 . A simple change of

1

This result is simple to demonstrate for typical velocity and position operators as shown in Appendix 4.7. One has to be more careful when the operator Aˆ contains higher derivatives of Z. Nonetheless, a careful analysis shows that the result still holds.

130 variables allows us to reexpress the anti-commutator as µ ¶Z ³ m ´3 1 M 1 ˆ cc g ˆ h{A(t), B}i = det dX(0)dV(0)dY 2 2 2π¯h 2π¯h ³ mg T Mcc ˆ g (0) + yg ) × e−i h¯ vg (0)·yg −iVc (0) h¯ Yc Apath (xg (t), vg (t))B(x 2 ´ mg y T Mcc Y g −i v (0)·y −iV (0) g g c c ˆ h ¯ h ¯ Apath (xg (t), vg (t)) + B(xg (0) − )e 2 ρ(X(0) + Y2 , X(0) − Y2 ) × , (4.38) Q where Y ≡ R3 − R1 . In this last equation, Apath (xg (t), vg (t)) is the function defined by the operator Aˆ which is evaluated in terms of the guest and target final position and velocities which are obtained by solving Eqs. (4.37a)–(4.37c) with the proper initial conditions (these initial conditions are then averaged over with a weight that is prescribed by the density matrix). Note that Eqs. (4.37a)–(4.37c) are completely deterministic and are identical, modulo the weight on the initial conditions, with the classical equations of motion found in I and ˆ are set to one, the Y integrations in Eq. (4.38) simply give the II. Also, when Aˆ and B Wigner distribution [129] form of the density matrix. As was shown earlier by Deutch and Silbey [55], these equations can be transformed to a reduced set of generalized Langevin equations (GLE) after the bath degrees of freedom are projected out. Here, this is done by noting that the equation of motion for the bath, Eq. (4.37c), is easily solved in terms of the bath initial conditions and in terms of a Green’s function that couples to the target degrees of freedom. Then, using the fact that the correlations we want to compute depend explicitly only on the guest degrees of freedom and using Eq. (4.21) for the density matrix, we can perform all remaining bath integrals as well as the target Yt integrals. Following these two operations Eq. (4.38) is rewritten in terms of a reduced density matrix and the differential equations that are used to calculate Apath (xg (t), vg (t)) reduce to a smaller set of stochastic coupled equations where the target

131 dynamics are governed by a GLE. The anti-commutator correlation then becomes Z 1 ³ mg ´3 1 ˆ ˆ h{A(t), B}i = dxg (0)dvg (0)dyg dXt (0)dVt (0) 2 2 2π¯ h ³ mg ˆ g (0) + yg ) e−i h¯ vg (0)·yg Apath (xg (t), vg (t))B(x 2 ´ m ˆ g (0) − yg )e−i h¯g vg (0)·yg Apath (xg (t), vg (t)) + B(x 2 ρ0g (xg (0), yg )ρt|g (Xt (0), Vt (0); xg (0)), (4.39) where ¶ µ ³ ´ mg yg2 (0) (0) T (0) β Kef f Rt det exp −βV (xg (0), Rt ) − 2β¯h2 − 2 Rt ¶, µ ³ ´ ³ ´ R (0) (0) T (0) Ξ(D) β Kef f Rt dxg (0) det (Λ(D)−Ξ(D)) exp −βV (xg (0), Rt ) − 2 Rt ³

ρ0g (xg (0), yg ) ≡

Ξ(D) (Λ(D)−Ξ(D)

´

(4.40) (0)

(0)

with Rt obtained from Eq. (4.20) for xg (0) = rg , and ρt|g (Xt (0), Vt (0); xg (0)) ≡ · µ ¶¸1/2 1 1 1 det (Gtt (D) − Gtb (D) Gbt (D))(Htt (D) − Htb (D) Hbt (D)) π2 Gbb (D) Hbb (D) µ 1 (0) (0) × exp −(Xt (0) − Rt )T (Gtt (D) − Gtb (D) Gbt (D))(Xt (0) − Rt ) Gbb (D) ¶ 1 T − Vt (0) (Htt (D) − Htb (D) Hbt (D))Vt (0) , Hbb (D) (4.41) where the subscripts on these matrices indicate the appropriate t/b, t/b blocks, and where we have introduced two new matrices G(D) ≡ h ¯ −1 (Λ(D) − Ξ(D)) and

µ −1

H(D) ≡ h ¯ M

1 Λ(D) + Ξ(D)

(4.42a)

¶ M.

(4.42b)

132 In Eq. (4.39), the xg (0) dependence of ρt|g (Xt (0), Vt (0); xg (0)) comes from the matrices (0)

G(D) and H(D) as well as from Rt ). We have separated the density matrix into two parts in order to highlight the fact that ρt|g (Xt (0), Vt (0); xg (0)) can be used as a (conditional) distribution function for the initial target positions and velocities (cf., Eq. (4.41)) that is easily shown to reduce to the classical equilibrium distribution function for high temperaˆ are both unit operators, the anti-commutator, tures. It is easy to show that, when Aˆ and B Eq. (4.39), is, as expected, also unity. The reduced set of differential equations that has to be solved in order to obtain Apath (xg (t), vg (t)) is expressed as a set of generalized Langevin equations, i.e., mg

d2 xg = Fg (xg , Xt ) dt2

(4.43)

and 2

Mtt

d Xt = Ft (xg , Xt ) − Kef f Xt − dt2

Z

τ 0

1/2 ˜ ds Mtt K tb

³ ´ ˜ 1/2 (τ − s) cos K bb ˜ bb K

˙ ˜ bt M1/2 K tt Xt (s)

†

+Υ (τ )

(4.44)

where ¸ ³ ´ ´· 1 1 ˜ 1/2 (0) 1/2 ˜ ˜ Υ (τ ) ≡ cos Kbb τ Kbt − Gbt (D) Mtt Xt (0) − Xt ˜ bb ˜ bb (D) K G ³ ´ ˜ 1/2 τ sin K bb 1 ˜ 1/2 ˜ 1/2 +Mtt Ktb (4.45) Hbt (D)Mtt Vt (0) + F † (τ ), 1/2 ˜ ˜ Hbb (D) K bb 1/2 ˜ −Mtt K tb

†

³

and where F † (τ ) is a Gaussian random force with zero mean and co-variance · ³ ´ 1 ´ ³ 1 1/2 ˜ ˜ 1/2 τ ˜ 1/2 τ 0 hF (τ )F (τ ) i = Mtt Ktb cos K cos K bb bb ˜ bb (D) 2 G ´ ³ ´ ³ ˜ 1/2 τ ˜ 1/2 τ 0 sin K sin K bb bb 1 ˜ bt M1/2 K + tt . 1/2 1/2 ˜ ˜ ˜ Hbb (D) Kbb Kbb †

†

0 T

(4.46)

133 These expressions are very similar to the ones in I and II; the differences are in the terms denoted by Υ† (τ ) on the right-hand side of Eq. (4.44). The first two of these terms, the (0)

ones containing the factors (Xt (0) − Rt ) and Vt (0), are completely absent in the classical case. Indeed, a careful analysis of the G(D) and H(D) matrices shows that these terms vanish in the high temperature limit. Although these new terms look strange at first glance because the dynamics of the target seems to “remember” the initial positions and velocities through them, we show in Appendix. 4.7 that, when the formalism is used to calculate target-target correlations in the absence of the guest, the resulting correlations are nonetheless“exact”. Also, the bath dynamics contains an infinite number of frequencies such that these terms will eventually go to zero at long times due to the superposition of terms with different phases. Furthermore, these two new terms can be considered, like F † (τ ), as colored noise since the target initial positions and velocities are randomly sampled from ρt|g (Xt (0), Vt (0); xg (0)). The random force term in Eq. (4.44), F † (τ ), also appears in the classical GLE in I and II, albeit with different correlations, cf. Eq. (4.46). It is also easy to show that these correlations identically reproduce the classical generalized Einstein-Nyquist relation in the high-temperature limit; specifically, in the classical limit, the random force correlations are proportional to the memory function, lim hF † (τ )T F † (τ 0 )i

T →∞

1/2 ˜ = kB T Mtt K tb

³ ´ ˜ 1/2 (τ − τ 0 ) cos K bb ˜ bb K

˜ bt M1/2 K tt .

(4.47)

Hence, while classically the generalized Einstein-Nyquist relation predicts that the noise vanishes at absolute zero, Eq. (4.46) shows that the noise remains finite at very low temperature; this is expected because of zero-point motion.

134 To summarize, the differential equations, Eq. (4.44), contain quantum generalizations to the classical equations of motion obtained in I and II and reduce to them in the hightemperature limit. It is probably too strong to interpret them as equations of motion (e.g., they cannot be used to predict the evolution of a wave-packet, even in the harmonic case), they are simply differential equations that approximately leads to the correct correlation functions, but that are nonetheless exact for purely harmonic systems. We conclude this subsection with some remarks on the approximations we made in the above derivation and compare with earlier work. For example, instead of our approximations on the combination of the forward and the backward propagator, one could have obtained each propagator within the semi-classical WKB approximation [57] in the coherent state-representation [130]. This approach was taken by Makri et al. [62, 63, 64, 65] in the semi-classical evaluation of correlations functions for non-harmonic quantum systems. In short, both this approach and ours are exact for harmonic systems, but in the later, a classical simulation of the forward trajectories is followed by another simulation for the backward propagators. In both cases, for non-harmonic systems, terms of order O(Y3 ) are neglected in the evaluation of the propagator. The reason we choose our approach is that it requires half the simulation effort (we simulate the forward and backward path integrals at the same time). Also, our stochastic boundary value problem directly reduces to the classical Langevin equation for high temperatures. 4.3.3 The permeability revisited We conclude this section by explicitly showing how the path integral formalism developed above can be used to obtain the anti-commutator correlations needed to calculate the space-dependent Onsager diffusion coefficient, D(z), and through it, the permeability. ˆ 0 )}/2i ˆ g , t), J(r ˆ 0 )}/2i, h{N ˆ (rg , t), J(r As seen from Eq. (4.9), these correlations are h{J(r g g ˆ (rg , t), N ˆ (r0 )}/2i where, as usual, the anti-commutator is related to the Kubo and h{N g average through Eq. (4.11).

135 ˆ (rg , t), N ˆ (r0g )}/2i. As in I and II, We start with the density-density correlations, h{N ˆ (rg , t) = δ(rg − ˆrg (t)). Because this we work in the infinite dilution regime and take N operator simply multiplies the other factors in Eq. (4.39), we trivially obtain Z 1 ˆ 1 ³ mg ´3 0 ˆ h{N (rg , t), N (rg )}i = dxg (0)dvg (0)dyg dXt (0)dVt (0) δ(rg − xg (t)) 2 2 2π¯ h ³ mg yg yg ´ δ(r0g − xg (0) − ) + δ(r0g − xg (0) + ) e−i h¯ vg (0)·yg 2 2 ρ0g (xg (0), yg )ρt|g (Xt (0), Vt (0); xg (0)).

(4.48)

The sum of the two delta functions is then Taylor expanded in yg to give + δ(r0g − xg (0) + y2g ) 2 ∂ 2 δ(r0g − xg (0)) yg2 0 ∼ δ(rg − xg (0)) + + O(yg4 ). : ∂r0g ∂r0g 8

δ(r0g − xg (0) −

yg ) 2

(4.49) It is easy to show that, when the yg integrations are performed, the Taylor expansion in yg becomes an expansion in h ¯ . To be consistent with our density matrix approximation, cf. Eq. (4.21), we only keep the leading term in the expansion, and Eq. (4.48) becomes, Z ˆ (rg , t), N ˆ (r0g )}/2i h{N

=

dxg (0)dvg (0)dXt (0)dVt (0)

×[δ(rg − xg (t))δ(r0g − xg (0)) × ρg (xg (0), vg (0))ρt|g (Xt (0), Vt (0); xg (0))], (4.50) where ρg (xg (0), vg (0)) ≡ · µ ¶¸ ³ ´ ³ ´ ³ ´ (0) (0) T (0) βmg 3/2 Ξ(D) 1 1 2 det (Λ(D)−Ξ(D)) exp −β 2 mg vg (0) + V (xg (0), Rt ) + 2 Rt Kef f Rt 2π · µ ¶¸ , ´ ³ ´ ³ R (0) (0) T (0) Ξ(D) 1 Kef f Rt dxg (0) det (Λ(D)−Ξ(D)) exp −β V (xg (0), Rt ) + 2 Rt (4.51)

136 which is very similar to the classical distribution function modulo the Λ(D) and Ξ(D) matrices which contain the quantum mechanical effects of the lattice vibrations. Note that Eq. (4.51) is essentially Eq. (4.40), but with the yg transformed to the initial velocity distribution. In Eq. (4.50), xg (t) is obtained by integrating Eq. (4.44) with the noise and initial target parameters sampled from Eqs. (4.46) and (4.41), respectively. The initial position of the guest is fixed at r0g and its initial velocity sampled from Eq. (4.51). ˆ (rg , t), N ˆ (r0 )}/2i, must be evalAs seen from Eq. (4.9), the required correlation, h{N g uated for t = 0. In this case, xg (t) = r0g and 1 ˆ ˆ (r0g )}i = δ(rg − r0g ) h{N (rg ), N 2

Z dvg (0) ρg (rg , vg (0))

= δ(rg − r0g )nbulk e−βW (rg ) ,

(4.52)

where W (rg ) is the potential of mean force as defined in Eq. (4.27). When t = 0, this result is identical with what is obtained when all terms in the Taylor expansion, cf. Eq. (4.49), are kept. There, the only approximation is at the level of the effective potential, which comes from our approximation to the density matrix, Eq. (4.21). Both the classical and quantum expressions have the same functional forms, and differ only in W (rg ). Note ˆ (rg , t), N ˆ (r0g )}/2i must be known in order to that the full time dependence of the h{N compute the Kubo average according to Eq. (4.11). On the other hand, we will show below that, at room temperature, the Kubo average and the anti-commutator of guest operators become indistinguishable. Therefore, the equal time Kubo average will also follow ˆ (r1 )N ˆ (r2 )iK = δ(rg − r0 )nbulk e−βW (rg ) . hN g The remaining two anti-commutator correlations that appear in Eq. (4.9) can be obtained by applying the same approximations that led to Eq. (4.52). We present some of the

137 details in Appendix 4.8; here, we simply state the final results, namely 1 ˆ ˆ 0 )}i = h{J(rg , t), J(r g 2

Z dxg (0)dvg (0)dXt (0)dVt (0)

×δ(rg − xg (t))δ(r0g − xg (0))vg (t)vg (0) ×ρg (xg (0), vg (0))ρt|g (Xt (0), Vt (0); xg (0)) (4.53) and 1 ˆ ˆ 0 )}i = h{N (rg , t), J(r g 2

Z dxg (0)dvg (0)dXt (0)dVt (0)

×δ(rg − xg (t))δ(r0g − xg (0))vg (0) ×ρg (xg (0), vg (0))ρt|g (Xt (0), Vt (0); xg (0)). (4.54) The functional forms of these expressions are again identical to their classical counterparts, although here too, the dynamics, the noise distribution and the initial condition distribution are different from what is obtained classically. Finally, since the expressions that we obtained for the three necessary anti-commutator correlations have the same properties as the classical correlations, the steps that led to the approximate expression for D(z), cf. Eq. (4.6), are equally valid. More precisely, ˆ g , t), J(r ˆ 0 )}i, 1 h{N ˆ 0 )}i and ˆ (rg , t), J(r the functional forms of our semi-classical 12 h{J(r g g 2 1 ˆ (rg , t), N ˆ (r0 )}i h{N g 2

greatly simplifies Eq. (4.9) and, if we again assume D(z)eβW (z) to

be approximately constant close to the barrier tops (implying that the motion is a Smoluchowki diffusion process in that region), the classical derivation for D(z) goes through unchanged (for more details, see Ref. [6]). Again, the correlations that appear in Eq. (4.9) are Kubo averages and here, we have worked with the anti-commutator. In order to be consistent, we should apply the transformation, Eq. (4.11), to the anti-commutators before solving Eq. (4.9). On the other hand, because the guest dynamics is slow, we expect the

138 Kubo average and the anti-commutator to be approximately equal, as is shown numerically in Sec. 4.5. 4.4 Computational Details In this section we show how the formalism developed above can be implemented numerically; specifically, we consider two issues: First, the potential of mean force is defined in terms of large matrices that are functions of the harmonic force matrix in the presence of the guest, i.e., Λ(D) and Ξ(D) in Eq. (4.27). Here, the effect of the guest is not as simple as in I and II because quantum mechanically, the guest also modifies the bath-bath blocks of these matrices. We will show how we can overcome this problem by using the theory of vibrational defects [52] and some well-chosen contour integrations. Second, we need an efficient procedure to simulate the new random terms that appear in the differential equations, i.e., Υ† (τ ) in Eq. (4.44). We will show that these new terms are easily simulated by implementing a Brillouin zone summation during the simulation process and by using our approximation, which was tested in I and II, for the memory function. 4.4.1 Potential of mean force Here, we show how the new terms in the potential of mean force, i.e., ∆W (rg ), cf. Eq. (4.28), can be calculated. These terms are most easily written in terms of the vibrational normal modes of the crystal, 3N X ∆W (rg ) = kB T (f (ωj0 ) − f (ωj )),

(4.55)

j=1

with f (ω) ≡

β¯ hω + ln(1 − e−β¯hω ), 2

(4.56)

and where ωj0 is the j’th normal mode frequency of the lattice in the presence of the frozen guest [i.e., where D is added to K, cf. Eqs. (4.22) and (4.23)], ωj is the j’th normal frequency of the pure lattice (D = 0), N is the number of atoms in the crystal.

139 We can reexpress Eq. (4.55) as a contour integral as µ ¶ I 3N X 1 1 1 β∆W (rg ) = dω ωf (ω) − , iπ C1 ω 2 − ωj02 ω 2 − ωj2 j=1

(4.57)

where C1 is a counter clockwise contour that includes all positive poles of the integrand. From here, it is easy to show that ∆W (rg ) can be written in terms of the vibrational Green’s function of the crystal, 1 β∆W (rg ) = iπ

I dω ωf (ω)Tr[G(ω, D) − G(ω, 0)],

(4.58)

C1

where Tr is a trace and where the Green’s function matrices are defined by G(ω, D) ≡

1 ˜ −D ˜ ω2 − K

(4.59)

[G(ω, 0) is the original Green’s function of the pure crystal]. The advantage of writing ∆W (rg ) in terms of the Green’s function is that we can use the theory of defects, as we did in I, to compute the trace using matrices that are defined in the target subspace only. Recall that, as shown in I, in complex crystals, like zeolites, the vibrational density of states of the crystal starts to be accurate when roughly 104 crystal atoms are included. As was shown in I, G(ω, D) can be calculated in terms of the perfect lattice Green’s function, G(ω, 0), 0 G(ω, D) = @

˜ −1 Gtt (ω, 0) (1 − Gtt (ω, 0)D)

˜ −1 Gtb (ω, 0) (1 − Gtt (ω, 0)D)

˜ − Gtt (ω, 0)D) ˜ −1 Gtt (ω, 0) + Gbt (ω, 0) Gbt (ω, 0)D(1

1

A. ˜ − Gtt (ω, 0)D) ˜ −1 Gtb (ω, 0) + Gbb (ω, 0) Gbt (ω, 0)D(1 (4.60)

What we really need in order to compute ∆W (rg ) is the trace of the difference between G(ω, D) and G(ω, 0), and using the last equation and the invariance of the trace under cyclic permutations, we finally obtain 1 β∆W (rg ) = iπ

I dω ωf (ω) C1

˜ − Gtt (ω, 0)D) ˜ −1 ], ×Tr[(G 2 (ω, 0))tt D(1

(4.61)

140

Im (ω) C2

(Λ,ε) C1

Re (ω)

Figure 4–2: The two contours, C1 and C2 described in the text, are illustrated in the complex ω plane. Here, the absolute value of the real part of ω is bounded by Λ and the absolute value of the imaginary part is bounded by ε. where (G 2 (ω, 0))tt = (Gtt (ω, 0)Gtt (ω, 0) + Gtb (ω, 0)Gbt (ω, 0))

(4.62)

is the tt block of G(ω, 0)2 . We show the contour, C1, that is used to calculate ∆W (rg ) in Fig. 4–2. Note that the maximum of Re(ω) along the contour (Λ in Fig. 4–2) must be larger than the largest frequency in the crystal vibrational density of states. The knowledge of the Green’s function allows us to compute more than the potential of mean force. In fact, in order to perform the simulations described above, we also need the G(D) and H(D) matrices that characterize the distribution of target initial positions and velocities, cf. Eq. (4.41). These matrices satisfy 1 Gbt (D) = (Ltt (D))−1 Gbb (D)

(4.63a)

1 Hbt (D) = (Ttt (D))−1 , Hbb (D)

(4.63b)

Gtt (D) − Gtb (D) and Htt (D) − Htb (D)

141 and where the two new matrices are defined in the full space as follows L(D) ≡ G(D)−1 , and T(D) ≡ H(D)−1 .

(4.64)

We can also reexpress these matrices in terms of the Green’s function and contour integrals; the derivation is very similar to what was done for ∆W (rg ) and we simply state the final results for the mass-scaled matrices, namely, µ ¶−1 1 ˜ tt (D) − G ˜ tb (D) ˜ (D) G G = ˜ bb (D) bt G µ ¶ I h ¯ β¯hω Gtt (ω, D) dω coth 2πi C2 2 2 + ˜ ef f βK

(4.65)

and µ ¶−1 1 ˜ ˜ ˜ Htt (D) − Htb (D) H (D) = ˜ bb (D) bt H µ ¶ I β¯hω h ¯ 2 dω w coth Gtt (ω, D), 2πi C2 2 (4.66) where the counter clockwise contour, C2, shown in Fig. 4–2 now encloses both positive ˜ ˜ and negative poles of G(ω, D). This can be done because both G(D) and H(D) are even functions of ω. Note that in Eq. (4.65), the residue at ω = 0 (coming from the hyperbolic cotangent), does not represent any vibrational modes in the crystal and is explicitly canceled by the last term on the right hand side of Eq. (4.65). Finally, note that the value of the imaginary part of ω along the horizontal parts of the contour, ε in Fig. 4–2, must satisfy ε < π/(β¯h). This guarantees that the poles of coth (β¯hω/2) that lie on the imaginary axis are excluded from the contour. In practical terms, ε should not be too small because small ε means that the contour passes near the poles on the real axis thereby causing numerical integration problems.

142 The formalism we just described relies on knowing the Green’s function of the perfect crystal in the target space. This was obtained from standard Brillouin zone sums that we briefly review in the next section, but that are extensively described elsewhere (e.g., see Refs. [131, 51]). 4.4.2 Noise terms We now consider the noise terms, Υ† (t), in Eq. (4.44). Recall that Υ† (t), not only includes F † (t), but also the two other non-classical terms that depend on the initial target positions and velocities. A closer look at Eqs. (4.44) and (4.46) shows that these terms only depend on the guest degrees of freedom through the matrices H(D) and G(D), and (0)

through Rt . Even if this guest dependence is responsible for the temperature dependence of the potential of mean force, its effect on H(D) and G(D) is quite small (this was checked by evaluating the guest dependent matrices using the Green’s function combined with a contour integration as described in the previous section). Hence, here we will neglect the guest dependence of H(D) and G(D) and evaluate the noise terms as if the lattice was perfect. This will allow us to again use Brillouin zone sum techniques. We start with the hF † (τ )T F † (τ 0 )i correlations, cf. Eq. (4.46), and perform a double Laplace transform to obtain M−1/2 hF † (s)T F † (s0 )iM−1/2 ≈ µ 0 ¶ 1˜ ss 1 1 1 ˜ , Ktb + K ˜ bb G ˜ bb H ˜ bb s02 + K ˜ bb bt 2 s2 + K

(4.67)

where, for the remainder of this section, we omit the D argument on matrices that are evaluated at D = 0. This last relation becomes an equality when Gbb and Hbb are replaced by Gbb (D) and Hbb (D). This result is not particularly useful because it expresses the noise correlations in terms of matrices in the bath space. Simple matrix manipulations allow us

143 to reexpress this result in terms of matrices in the target space only, i.e., M−1/2 hF † (s)T F † (s0 )iM−1/2 ¤ 1 1 £ ¯ ¯ 0 T ≈ hξ(s)ξ(s ) i + h¯ η (s)¯ η (s0 )T i , G˜tt (s) G˜tt (s) (4.68) where we have defined two new sets of uncorrelated Gaussian noise vectors of length ¯ and η¯(s) which have 3Ntarget (Ntarget is the number of atoms in the target space), ξ(s) zero mean and variances ¯ ξ(s ¯ 0 )T i ≡ hξ(s) µ ¶ ss0 1 ˜ 0 ˜ 0 ˜ ˜ ˜ ˜ ˜ (G(s)LG(s ))tt − (G(s)L)tt (G(s )L)tt ˜tt 2 L (4.69) and h¯ η (s)¯ η (s0 )T i ≡ µ ¶ 1 1 ˜ 0 ˜ 0 ˜ ˜ ˜ ˜ ˜ (G(s)TG(s ))tt − (G(s)T)tt (G(s )T)tt , ˜ tt 2 T (4.70) where the Green’s function, ˜ ≡ G(s)

s2

1 , ˜ +K

(4.71)

is the frequency analytic continuation of Eq. (4.59) for the pure lattice, and where L and T matrices where defined above by Eq. (4.64) with D = 0. ¯ and η¯(s) in terms of pure lattice matrices is that their The advantage of defining ξ(s) respective correlations can be expressed exactly in terms of Brillouin zone sums. For

144 example, ¯ ξ(s ¯ 0 )T i = hξ(s)

1 X ss0 L(ωj (k)) ut (k, j)ut (k, j)† 2 2 02 2 2N k,j (s + ωj (k) )(s + ωj (k) ) · 1 XX ss0 L(ωj (k))L(ωj 0 (k0 )) + 2N 2 k,j k0 ,j 0 (s2 + ωj (k)2 )(s02 + ωj 0 (k0 )2 ) 1 ut (k, j)ut (k, j) ut (k0 , j 0 )ut (k0 , j 0 )† ˜ Ltt †

¸ (4.72)

where X ˜tt = 1 L(ωj (k))ut (k, j)ut (k, j)† , L N k,j

(4.73)

N is the number of wave-numbers used in the discrete sum, L(ωj (k)) = h ¯ coth(β¯hωj /2)/ωj [as can be obtained combining Eqs. (4.64), (4.42a), (4.42b), (4.22) and (4.23)] and where the sum over j goes from 1 to 3Nu.c , where Nu.c is the number of atoms in the primitive unit cell. In order to have the correct number of vibrational modes, in an exact calculation, N should be equal to Nc /Nu.c where Nc is the total number of atoms in the crystal. In practice, N is much smaller. The ωj (k)’s introduced in the last equation are the positive square roots of the eigenvalues of the discrete Fourier transform of the dynamical matrix, ˜ n,m (k) ≡ K

X

˜ n,m (R), e−ik·R K

(4.74)

R

˜ n,m (R) is an element of the mass-scaled force constant matrix that couples the where K n’th atom in one unit cell to the m’th in another cell, the cells being separated by a lattice ˜ vector R. The eigenvectors of K(k), ej (k), define ut (k, j) according to (ut (k, j))m = (ej (k))n eik·R ,

(4.75)

where n refers to the atom within the primitive unit cell and R is the translation vector to the actual m’th target atom. Since these Brillouin zone sums are real, we can replace ut (k, j)ut (k, j)† by Re(ut (k, j)ut (k, j)† ) in all of the above expressions.

145 We now postulate the following form for the noise, X s ¯ = 1 ξ(s) Re(ut (k, j)ut (k, j)† )ξ¯k,j , 2 2 N k,j s + ωj (k)

(4.76)

where ξ¯k,j is a vector of random variables. In other words, here, the amplitude of each phonon contribution is randomly sampled from a distribution that is appropriately chosen such that Eq. (4.72) is reproduced. It is simple to show that this is accomplished by choosing hξ¯k,j ξ¯kT0 ,j 0 i =

1 N L((ωj (k)))Ak,j δj,j 0 δk,k0 2 1 ˜−1 , − L(ωj (k))L(ωj 0 (k0 ))L tt 2

(4.77)

where Ak,j is a matrix of rank 3Ntarget defined as Bk,j Ak,j Bk,j ≡ Bk,j ,

(4.78)

¯ ξ(s ¯ 0 )T i exactly, cf. Eq. (4.72). Note where Bk,j ≡ Re(ut (k, j)ut (k, j)† ), reproduces hξ(s) that Eq. (4.78) is not trivially solved because Bk,j is not invertible. On the other hand, it is easy to show that it has at least one non-zero eigenvalue. Therefore Eq. (4.78) is first transformed to a basis where Bk,j is diagonal, Ak,j is then obtained by solving Eq. (4.78) in the space of nonzero eigenvalues and finally the solution is transformed back to the original space. The above procedure, when implemented numerically, is memory intensive because it requires the simulation of all the random numbers, ξ¯k,j for all k and j. Recall that each of these ξ¯k,j ’s is a vector of length 3Ntarget . We can reduce the number of noise variables by considering the scalar ξk,j = ut (k, j)† ξ¯k,j as the random variable. In general, ξk,j are complex numbers fully determined by the hξk,j ξk∗0 ,j 0 i and hξk,j ξk0 ,j 0 i correlations; which are themselves easily obtained from Eq. (4.77). Using these complex random numbers,

146 ¯ as a simple Brillouin zone sum, we can now write ξ(s) X s ¯ = 1 ξ(s) Re(ut (k, j)ξk,j ), 2 N k,j s + ωj (k)2

(4.79)

or equivalently in the time-domain X ¯ = 1 ξ(t) cos (ωj (k)t)Re(ut (k, j)ξk,j ). N k,j

(4.80)

In practical terms, we simulate all the ξk,j complex random numbers in advance and we ¯ to the equations of motion at any time of the simulation by can add the noise variable ξ(t) performing the last Brillouin zone sum, cf. Eq. (4.80). The same procedure can be applied for η¯(t) and gives η¯(t) =

1 X sin (ωj (k)t) Re(ut (k, j)ηk,j ), N k,j ωj (k)

(4.81)

where the ηk,j ≡ ut (k, j)† η¯k,j correlations are obtained, as above, from h¯ ηk,j η¯kT0 ,j 0 i, h¯ ηk,j η¯kT0 ,j 0 i =

1 N T (ωj (k)))Ak,j δj,j 0 δk,k0 2 1 ˜ −1 , − T (ωj (k))T (ωj 0 (k0 ))T tt 2

(4.82)

and where X ˜ tt ≡ 1 T T (ωj (k))Re(ut (k, j)ut (k, j)† ), N k,j

(4.83)

with T (ωj (k)) = h ¯ ωj coth(β¯ hωj /2). We now have a formalism that fully reproduces the correlations that appear in the square brackets in Eq. (4.68). We still have to include the effects of the Green’s function Gtt (s)−1 that multiplies the random variables, cf. Eq. (4.68), but before doing so, we first return to the other random terms that appear in Υ† (t), namely, the ones that depend on Xt (0) and Vt (0) in Eq. (4.45). Again, these terms can be rewritten in terms of matrices in the target space. The matrix manipulations that are performed are quite similar to what we described above and, after a Laplace transform, these terms can be written as

147 1/2 Mtt

³

´−1 ˜ Gtt (s) Z(s), where µ

¶ ³ ´ 1 1 1/2 (0) −1 ˜ ˜ ˜ ˜ Z(s) ≡ s (G(s)L)tt − (G(s)K )tt Mtt Xt (0) − Rt ˜tt ˜ −1 )tt L (K ¶ µ ˜ tt 1 − G˜tt (s) M1/2 ˜ T) (4.84) + (G(s) tt Vt (0), ˜ tt T where ˜ −1 )tt = (K

1 X 1 Re(ut (k, j)ut (k, j)† ). 2 N k,j ωj (k)

(4.85)

Clearly Z(s) can be written in terms of Brillouin zone sums similar to those introduced ¯ ξ(s ¯ 0 )T i and h¯ for hξ(s) η (s)¯ η (s0 )T i. Here, we report the final expression in the time domain, i.e., Z(t) =

à " # ³ ´ 1 1 1 X 1/2 (0) † cos (ωj (k)t)Re(ut (k, j)ut (k, j) ) L(ωj (k)) − Mtt Xt (0) − Rt ˜tt ˜ −1 )tt N L ωj (k)2 (K k,j ¸ ¶ · sin (ωj (k)t) 1 1/2 − 1 Mtt Vt (0) . (4.86) + Re(ut (k, j)ut (k, j)† ) T (ωj (k)) ˜ tt ωj (k) T

At this point, all the noise terms that appear in Υ† (t), cf. Eq. (4.44), are formally written as ³ ´−1 1/2 ¯ + η¯(s)) (Z(s) + ξ(s) Υ† (s) = Mtt G˜tt (s) µ ¶ s2 1/2 2 ˜ ˜ ˜ ≡ Mtt s + Kef f + Ktb K Q(s), ˜ bb (s2 + K ˜ bb ) bt K (4.87) ¯ in Laplace frequency space, where Q(s) ≡ Z(s)+ ξ(s)+ η¯(s) and where we have obtained the target-target part of Gtt (s) in terms of the force constant matrix according to Eq. (4.71). Since Eq. (4.44) will eventually be numerically simulated in time, it is convenient to write

148 the noise terms in time; they can be rewritten as µ

†

d2 Q(t) ˜ + Kef f Q(t) dt2 ³ ´  Z τ ˜ 1/2 (t − τ ) cos K bb ˜ tb ˜ bt Q(τ ˙ ) , + dτ K K ˜ K 0 bb

Υ (t) =

1/2 Mtt

˜ tb ˙ where Q(0) and Q(0) are both zero. In this equation, the memory term (K

(4.88) ” “ ˜ 1/2 (t−τ ) cos K bb ˜ bb K

˜ bt ) K

is, up to factors of mass, identical to the memory function appearing in Eq. (4.44) and to that appearing in the classical Langevin equations used in I and II. As was done in I and II, we approximate the Laplace transform of the memory function as ˜ tb K

s s ˜ bt ≈ K , ˜ bb (s2 + K ˜ bb ) ˜ + Bs ˜ + Cs ˜ 2 K A

(4.89)

˜ and C ˜ are obtained for the small and large s limit of the memory function and where A ˜ is determined from a linear least squares fit (see I). In this form, Eq. (4.88) can where B be simplified by introducing a new field, f(t), µ †

Υ (t) =

1/2 Mtt

d2 Q(t) ˜ df(t) + Kef f Q(t) + 2 dt dt

¶ ,

(4.90)

where f(t) is the solution of the differential equation 2 dQ(t) ˜ d f(t) + B ˜ df(t) + Af(t) ˜ C = , 2 dt dt dt

(4.91)

with the initial conditions, f = df/dt = 0 at t = 0. It is easy to show, by Laplace transforming, that the two representations of the noise, i.e., Eqs. (4.90) and (4.88), are equivalent (to the extent that our approximation to the memory function, Eq. (4.89), is accurate). Note that the approximation to the memory function, Eq. (4.89), is also used for the friction term in Eq. (4.44). We close this section by noting that the only approximations used in the noise term calculation were the use of the approximate memory function, as in I and II it reproduces

149 the vibrational density of states semi-quantitatively, cf. Fig. 4–1, and the neglect of the guest position dependence of various matrices. This formalism requires the Brillouin zone sums to be performed while Eq. (4.44) is simulated, and of course, while including more discrete wave-numbers improves the accuracy of the noise terms, it also makes the numerical calculation more time consuming. 4.5 Results We now apply our formalism to Neon in α-quartz. Recall that in I and II, we studied, respectively, Xenon in the sodalite Theta-1 and Argon in α-quartz. In I, our conclusions were that the crystal vibrations had very little effects on the guest motion. In II, we found the opposite, namely, the crystal vibrations and the flexibility of the lattice played a major role in the diffusion. Part of this was explained by the fact that the channels in Theta-1 are much wider than those in α-quartz, and thus, in α-quartz, the guest is always very close to one or many crystal atoms, with concomitant enhanced coupling between the guest and crystal dynamics. We therefore chose α-quartz again because we expect that any vibrational quantum effects on the guest motion will be larger in systems where the guest-lattice interaction is stronger. The spatial group of α-quartz is P 31 21 and the coordinates of the unit cell were obtained from Ref. [117]. The crystal parameters (atoms positions and harmonic force constants with stretching and bending motion only) are described in II. We used the usual Lennard-Jones interaction potentials, "µ

Ntarget

V (rg , Rt ) =

X j=1

4²j,g

σj,g dj,g

¶12

µ −

σj,g dj,g

¶6 # ,

(4.92)

where di,g is the distance between the guest and the j’th target atom. The potential pa˚ and rameters that we used are ²O,g /kB T = 0.1841, ²Si,g /kB T = 0.0385, σO,j = 1.9584A ˚ at 300K. These potential parameters were obtained in I and II and we beσO,j = 2.256A lieve that they estimate the true interaction potential within factors of two. The target zone

150

Figure 4–3: The target zone used in all simulations is shown in the box. Red and blue atoms are oxygen and silicon, respectively. The silicon atom whose correlations are reported in Fig. 4–8 appears in green. The positions labeled A and B are associated with the potential of mean force results of Fig. 4–6 and the guest correlations shown in Fig. 4–9, respectively. The locations of some of the binding sites are shown as small white spheres. In this figure, the z-axis is normal to the page.

151 ˚ ˚ 16.2156A). ˚ It contains 243 atoms, exthat we use has dimensions (14.7402A,12.7653 A, actly 27 primitive unit cells, and is large enough such that the interaction energy between the guest and bath is practically zero. In this geometry, the crystal is oriented such that the net flow is parallel to the z-axis. The target zone that we used in all simulations is shown in Fig. 4–3. 4.5.1 Potential of mean force for Neon in α-quartz We first report the results of the potential of mean force calculation. Recall that the potential of mean force is given by Eqs. (4.27) and (4.28). We computed ∆W (rg ) using the contour C1 defined in Sec. 4.4.1 with Λ = 3.56993 × 1014 s−1 and ε = 4.66048 × 1012 s−1 (here, Λ is about 30% larger than the largest vibrational frequency of the crystal). The contour integral was performed numerically using Simpson’s rule with ∆ω = 4.66048 × 1011 s−1 . The Green’s function that was used in the contour integration was pre-calculated with N = 153 dispersion points in reciprocal space. In Fig. 4–4, a potential of mean force surface is shown for a region well inside the target zone at 300K. Here, we explicitly calculated a third of the unit cell along z and generated the other two thirds from the symmetry operations of the α-quartz lattice. Because of this symmetry, there are three absolute minima in the potential of mean force each with Wmin = −0.97483kB T at 300K. We do not show the classical counterpart to Fig. 4–4 because the differences are too small to be seen given the resolution of the figure. Because the net macroscopic flux is taken to be along the z axis, the theory requires the evaluation of plane-average potential of mean force, W (z), according to Eq. (4.7). This is shown in Fig. 4–5 where it is compared against the fully classical W (z). The differences between the classical and semi-classical W (z) is very small at 300K. As we will see below, these differences will grow as the temperature is lowered, although our approximations, which treat the guest as a free-particle, becomes more problematic. The fact that, at least at 300K, the quantum-corrections to the potential of mean

152

Figure 4–4: Potential of mean force energy surface is shown for Neon inside α-quartz. The surface is drawn for W (rg ) = 3.5kB T at 300K. force are small already suggest that the corrections to the permeability will not be dramatic. Remember that, the permeability is obtained from D(z) whose dominant contribution comes from the potential of mean force that appears in the exponential. The remaining parts of D(z), which are determined from the evaluation of microscopic correlation functions, will only introduce pre-exponential corrections. We now consider the full temperature dependence for the potential of mean force, W (rg ), at a point chosen to be close to that of the minimum energy in the barrier top ˚ 0.0A, ˚ 0.0A) ˚ T (the point labeled “A” in Fig. 4–3). plane, specifically at rg = (−0.2305A, The temperature dependence of the potential of mean force for the classical and quantum case are compared in Fig. 4–6. The quantum calculation includes zero-point motion, and

153

Figure 4–5: Plane average potential of mean force is shown for Neon inside α-quartz at 300K. Our semi-classical approximation is compared against the classical expression.

154

Figure 4–6: The potential of mean force of Neon inside α-quartz at the point ˚ 0.0A, ˚ 0.0A) ˚ is shown as a function of temperature. The two sets of points (−0.2305A, are obtained from approximations similar to those in Refs. [68, 69, 70]. The points are obtained using reference frequencies that include curvature corrections at either the closest local minimum (circles) or global minimum (squares).

155 this contribution persists even at very low temperatures. Again, we stress that the guest approximations become ad hoc at sufficiently low temperatures. In Fig. 4–6, we also show a few points for which the semi-classical potential of mean force was obtained from an approximate method similar to the one developed by Ermakov, Butayev and Spiridonov [67] and later improved by Mak and Andersen [68], Cao and Berne [69] and by Chao and Andersen [70]. The details of the calculation for W (rg ) within this approximation are shown in Appendix 4.9. Briefly, an approximate form for the density matrix of the full system is used. This approximate density matrix is determined by a set of reference vibrational frequencies that are chosen ad hoc, it has the right high temperature limit and it becomes exact for completely harmonic systems (provided the reference frequencies are appropriately chosen). In Fig. 4–6, two sets of data for the potential of mean force are shown within this approximation for the same point, ˚ 0.0A, ˚ 0.0A) ˚ T . The points shown as circles are obtained from reference rg = (−0.2305A, frequencies determined from the curvature of the potential at a local minimum close to rg while the squares are obtained from reference frequencies determined from the global minimum of the potential. Clearly, these approximations are justified because, at low temperature, the paths are expected to sample regions around the various minima. Here, the final answer should be an average of the two data sets. Also note that the curvature at rg was not used to determine the reference frequencies because, for some cases, this could result in imaginary reference frequencies. In Fig. 4–7, W (rg ) is reported for a point close to a global minimum. As in Fig. 4–6, the discrete data points are obtained from the approximation described in Appendix 4.9. Here, the only choice of approximate harmonic reference potential is taken at the global minimum. The problem with this approximation is that it contains many ad hoc steps and there is no apparent reason why this method should do better than ours at very low temperature. In this paper, we will report the permeability and calculate the correlations

156

Figure 4–7: The potential of mean force of Neon inside α-quartz close to a global minimum is shown as a function of temperature. The set of points are obtained from approximations similar to those in Refs. [68, 69, 70].

157 at room temperature, 300K. As seen in Figs. 4–6 and 4–7, at that temperature, the difference between both methods is small, henceforth, we use the approximation presented in Sec. 4.3. 4.5.2 Guest-Free correlations In Sec. 4.3.2, we claimed that the formalism we developed is exact when there is no guest, and as shown in Appendix. 4.7, the exact anti-commutator correlations for some target quantities are obtained. Of course, in practical terms, we use the numerical methods described in Sec. 4.4 to simulate these correlations, and here, we show how typical anticommutator correlations are reproduced using our simulation procedure compared with the “exact” result obtained by standard methods. Note that the only sources of error here come from our approximation for the memory function, cf. Eq. (4.89), from the numerical methods associated with the integrator, from the limited number of terms used in the Brillouin zone sum, and from the statistical error associated with the finite number of ensemble members used. We performed our guest-free simulation using a time step of 2.5 × 10−16 s with a second order stochastic Runge-Kutta integrator [110] and the noise terms were obtained from Brillouin zone sums with N = 53 . The correlation function was obtained from an ensemble of 2000 trajectories. All simulations reported in this work were performed on ˆ 1 (t), X ˆ 1 }i and a Beowulf cluster containing 64 processors. In Fig. 4–8, we report 21 h{X 1 ˆ 1 (t), V ˆ 1 }i h{V 2

where the 1 subscript denotes the green silicon atom shown in Fig. 4–

3. These two correlations are shown at 300K and at 30K. In each panel, the classical correlation is shown for comparison. Fig. 4–8 clearly shows that our simulation procedure and MSR mapping combined with our approximate methods for simulating Eq. (4.44) is very accurate throughout the displayed time window. Except the small error that builds up for the 300K position correlation at later time, the agreement is still very good for larger times.

158

ˆ

ˆ

ˆ

ˆ

X1 } V1 } Figure 4–8: The anti-commutator correlations, h {X1 (t), i and h {V1 (t), i, are shown for 2 2 T = 300K (the left) and T = 30K (right) for the silicon atom in green in Fig. 4–3. In each panel, we show the “exact” anti-commutator correlation, our simulated result and the classical correlations.

159 As expected, Fig. 4–8 clearly shows that the quantum corrections are larger at 30K than at 300K for both types of correlations. Perhaps more interestingly, Fig. 4–8 also shows that the quantum corrections to the velocity-velocity correlations are noticeably larger than the quantum corrections to the position-position correlations. Even at 300K, the envelope of the velocity-velocity correlation is about twice as large in the quantum mechanical case, while, at the same temperature, the classical and quantum position-position correlation function only differ slightly. 4.5.3 Diffusion and the permeability We know from Fig. 4–8 that the quantum corrections to the lattice dynamics are significant even at room temperature. In this section, we examine their coupling to the guest motion. In Fig. 4–9, we show hvG,z (t)vG,z ixg (t=0)=rg calculated by simulating the GLE, Eq. (4.44), and averaging over 2000 ensemble members at 300K, and where we chose ˚ −0.93A, ˚ 0.0A) ˚ (a point in the barrier top plane, position “B” in Fig. 4– rg = (−0.2305A, 3). As above, this was done using a time-step of 2.5 × 10−16 s and the noise was obtained from Brillouin zone sums with N = 53 . Note that aging is not necessary in this formalism since the target atoms initial positions are sampled directly from Gaussian distribution given in Eq. (4.41), which includes the relaxation of the lattice to a new equilibrium position. The total length of the simulations was 2.048 × 10−12 s. As seen from Fig. 4–9, the quantum mechanical effects on the guest correlations are small, but noticeable. The velocity-velocity correlation function decorrelates faster in the quantum mechanical case, as seen from the faster decrease of the quantum correlations at short time as well as from the lower value of the time integral of the quantum velocityvelocity correlation functions (the inset in Fig. 4–9). This effect can be explained from the fact that, in the quantum case, the lattice effectively vibrates “more” (The amplitude of the position and velocity correlation functions showed in Fig. 4–8 are larger in the quantum case). Note that we have shown in II that the lattice vibrations tends to slow down the diffusion (i.e., make the velocity auto-correlation function integrals smaller). We believe

160

Figure 4–9: The simulated velocity-velocity time correlation function for a trajectory ˚ −0.93A, ˚ 0.0A) ˚ is reported for our semiwhere the guest is initially at xg = (−0.2305A, classical treatment and for the completely classical case at 300K. In the inset, we compare the time integral of the same correlation function.

161 that the reason these effects are small comes from the guest-free position correlations that are shown in Fig. 4–8. Remember that, even if the velocity correlations of the quantum lattice is very different compared to the classical lattice at 300K, the position correlations are not. This means that the actual extent of the motion of the crystal atoms increases in the quantum case, but not by much, at least at 300K.

Figure 4–10: The simulated velocity-velocity time correlation function for a trajectory ˚ −0.93A, ˚ 0.0A) ˚ is reported for our semiwhere the guest is initially at xg = (−0.2305A, classical treatment and for the completely classical case at 30K. In the inset, we compare the time integral of the same correlation function. When the temperature is dropped to 30K, the differences become larger. This is shown in Fig. 4–10 where we again compare hvG,z (t)vG,z ixg (t=0)=rg in the quantum and

162 classical case. Again, the time integral of the velocity correlation function is bigger for the classical calculation. The plateau value of the inset in Fig. 4–10 is equal to (1.8199 ± 0.46) × 10−10 m2 /s in the quantum case and (4.2822 ± 0.45) × 10−10 m2 /s in the classical calculation (also note that both numbers are about 1 or 2 orders of magnitude below their values at 300K). Again, for such low temperatures, the classical guest approximation is suspect.

Figure 4–11: The anti-commutator velocity-velocity correlation shown in Fig. 4–9, obtained within our semi-classical formalism, is compared with the velocity-velocity Kubo average. The region where the two curves differ the most is shown in the inset.

163 Before going further, recall that the correlation that is shown in Fig. 4–9 (in the semiclassical approximation) is in fact an anti-commutator correlation. On the other hand, the space-dependent diffusion coefficient is given in terms of Kubo averages. Here, we take the anti-commutator correlation of Fig. 4–9 and, from it, we obtain the Kubo average using Eq. (4.11) (using numerical Fourier transforms). The results are shown in Fig. 4–11. At 300K, for this system and within the approximation that only the lattice is quantum mechanical, it is clear that the Kubo average and the anti-commutator are almost identical. They only differ slightly at short times. This is shown in the inset of Fig. 4–11. We have also verified that the anti-commutator correlation that reduces to hβF (z(t1 ))vG,z ixg (t=0)=rg is also almost identical to its Kubo average. Hence, all anti-commutator correlations, at this temperature, can be considered to be Kubo averages. We now examine the space-dependent diffusion coefficient, D(z). Recall that D(z) can be obtained from the long-time limit of Rt R n∞ 0 dt1 Unit Cell drk hvG,z (t1 )vG,z ixg (t=0)=rg e−βW (r) D(z, t) ≡ , (4.93) Rt R Acell + 0 dt1 Unit Cell drk hβF (z(t1 ))vG,z ixg (t=0)=rg e−β[W (r)−W (z)] where Acell is the area of the primitive unit cell. In Fig. 4–12, we show how the dynamics (the velocity correlations) changes the region in the plane that contributes to D(z) at 300K. Rt To do so, we compare the Boltzmann factor, e−βW (rg ) against the factor 0 dt1 hvG,z (t1 )vG,z ixg (t=0)=rg e−βW (rg ) for the quantum case in the maximum energy plane. For each point in the plane, we obtain the required correlation functions through simulations that we described above. The rectangular window that we used to characterize this plane is defined by its lower left corner ˚ −1.86A, ˚ 0.0A), ˚ and its upper right corner position , (0.154A, ˚ 1.86A, ˚ 0.0A). ˚ position, (−0.8072A, We used a 6 × 5 grid within this window where, at each grid point, the correlations are explicitly calculated. We then extrapolated between these grid points with a two dimensional bi-cubic spline. Outside this window, the contribution to D(z) is negligible. Note that, for this plane, W (z) = 8.4997kB T quantum mechanically and W (z) = 8.4077kB T classically at 300K. On the scale of the figure, it is hard to see the differences between

164 the quantum and classical cases, hence, the classical data was not shown. Nonetheless, a careful analysis shows that the contributions to the quantum D(z) are smaller than the classical ones. Also, the region in the plane that contributes to D(z) is a little more narrow in the semi-classical case.

Figure e−βW (rg ) in the maximum energy plane is compared R t4–12: The Boltzmann factor,−βW (rg ) with 0 dt1 hvG,z (t1 )vG,z ixg (t=0)=rg e at 300K. The second panel has units of m2 /s. The factor D(z, t)/n∞ e−βW (z) for the maximum energy plane is shown in Fig. 4–13 where we also show the uncorrected part of D(z, t), i.e., what is obtained by neglecting the denominator in Eq. (4.93). As expected, Fig. 4–13 also shows that the quantum corrections to the space dependent diffusion coefficient are not very large, but still make it smaller compared to the classical case. From the plateau value of D(z, t)/n∞ e−βW (z) , we can obtain the space dependent diffusion coefficient for that plane. In the quantum case, we find that D(z, t)/n∞ e−βW (z) = (5.3198 ± 0.10) × 10−9 m2 /s at z = 0, compared with (6.1968 ± 0.14) × 10−9 m2 /s classically. At this point, we assume that D(z)eβW (z) to be constant in the barrier region (i.e., the diffusion is a Smoluchowski process) and we calculate the permeability, cf. Eq. (4.1). We do not test this approximation here as it already was verified classically in I and II,

165

Figure 4–13: The quantum space-dependent diffusion coefficient, D(z), obtained from the plateau value of the solid curves for the maximum energy plane, z = 0 is compared with its classical counterpart. The dashed curve represents D(z, t) but where the denominator has been set to one in Eq. (4.93). and because the semi-classical formalism is much more time-consuming numerically. Remember that the goal of this work is to establish the extent of the quantum corrections, if any, on the permeability and for this purpose, the analysis of a single plane is sufficient. With this approximation, we use Eq. (4.1) to calculate the intrinsic permeability and find that P 0 (300K) = 9.91 × 108 s/(mkg) quantum mechanically, compared with 1.26 × 109 s/(mkg) classically. This predicts that, for this system, the quantum nature of the lattice vibrations decreases the intrinsic permeability by about 25%, with a little less than half the effect coming from the potential of mean force and the rest from the velocity correlations. Of course, this result depends strongly on the system under study and even more on the interaction potential that is used. 4.6 Discussion The purpose of this paper was to establish the extent of the first quantum corrections to the guest diffusion in channeled structures. In order to do this, we chose a system for which we have shown classically [112] that the lattice vibrations play a role in the diffusion process. This work was motivated from the well known fact that many crystal phonons are not excited at room temperature, but still have significant zero-point motion contributions.

166 We have developed a semi-classical formalism where, in effect, the guest is treated classically and the lattice quantum mechanically, based on the MSR relation between stochastic process and path integrals that allowed us to calculate the desired Kubo averages or anti-commutator correlations. This formalism, which depends on the fact that the guest is slow, has the advantage that the time correlation function theory of permeability developed earlier by Vertenstein and Ronis [6] remains unchanged formally. More precisely, the connection between the microscopic information, the correlation functions, and the macroscopic permeability is exactly the same as what was used in I and II, but now the time correlations functions where obtained from modified generalized Langevin equations and potential of mean force that include the quantum nature of the lattice. One of the crucial parts of this work is described in Sec. 4.4. After all, quantum Langevin equations are not new and have been studied for different systems. Here, we gave a very technical procedure that combines Brillouin zone sums techniques and contour integrals such that the new (quantum) terms in the Langevin equations and potential of mean force can be accurately calculated. We also showed how, by using appropriately chosen Gaussian random amplitudes in the Brillouin zone sums, that we could effectively simulate the noise terms in the Langevin equation. This procedure was able to reproduce the guest-free correlation with high accuracy as shown in Fig. 4–8. These pure lattice correlation functions show that even at room temperature, the crystal is far from behaving classically. In fact, the velocity-velocity correlations in Fig 4–8, in the quantum case, is almost identical at 300K and 30K, suggesting that at room temperature the velocity of the lattice is still dominated by zero-point motion. Moreover, at 300K, the amplitudes of the velocity-velocity correlations are twice as large in the quantum case. On the other hand, the amplitude of the motion, as characterized by the position-position correlations, does not change drastically, at room temperature, when quantum mechanics is included.

167 The effects of the quantum lattice were first examined for the potential of mean force. Figs. 4–5 and 4–6 show that the quantum correction to the potential of mean force are small at room temperature. Still, when the permeability is computed, according to Eq. 4.1, Rd R one has to compute −d dz eβW (z) . This integral is a multiple of u.c dz eβW (z) where the u.c subscript means that the integration region is limited to the primitive unit cell along z. For the quantum case, this equals 7.077 × 10−7 m while in the classical case, this gives 6.4323 × 10−7 m at 300K. The intrinsic permeability is inversely proportional to this factor. Note that small relative quantum corrections to W (rg ) can be significant when exponentiated if βW (rg ) is large, as seems to be the case here. We also emphasize the importance of using very accurate potential models when the diffusion occurs in large energy barriers systems. This was extensively discussed in II. Hence, at the level of the potential of mean force only, the quantum corrections already decreases the permeability by about 10%. When the dynamics are included, we get a further decrease in the permeability by another factor of about 15%. This decrease in the diffusion, compared to the classical case, is shown by the plateau value of Fig. 4–13. This effect is explained by the fact that the lattice effectively vibrates more rapidly in the quantum case. This is in agreement with what we found in II where the lattice vibrations slowed down the guest inside the crystal. We also believe that the quantum corrections to the diffusion coefficient are small because the amplitude of the vibrations (the average displacement of each crystal atoms) increases, but only slightly, compared to the classical case. We have also computed the potential of mean force of argon in α-quartz. In this case, the potential of mean force quantum corrections to the permeability decreases the later by about 25%. We did perform a limited number of simulations for this case and obtained the velocity time correlation function, hvG,z (t)vG,z ixg (t=0)=rg for some points, although not enough to be able to compute D(z). It seems that the dynamical quantum corrections to the permeability is very small for the heavier argon atom.

168 In conclusions, for the neon/α-quartz system, both the potential of mean force and the dynamics work in the same direction and ultimately decrease the crystal permeability to neon at room temperature. The total decrease is about 25%. play a role. The quantum corrections to the activated free energies or the pre-exponential dynamical factors are rationalized as follows. In this semi-classical formalism, the lattice effectively vibrates more which constrains even more the motion of the guest in an already narrow channel. This extra confinement of the guest manifest itself in larger absolute energies and slower dynamics. Two effects that reduce the permeability. 4.7 Appendix: Anti-commutator for Target-Target correlations in the absence of the guest In this appendix, we show that the formalism developed in Sec. 4.3 is exact when it is used to calculate target-target anti-commutator correlations in the absence of the guest (for completely harmonic systems). We start by writing down the anti-commutators in the full crystal space that can be obtained, e.g., from creation-annihilation operators techniques,

and

³ ´ 1 1/2 ˆ 1 1/2 ˆ ˜ 1/2 t G ˜ −1 M h{X(t), X}iM = cos K 2 2

(4.94)

³ ´ 1 1 1/2 ˆ 1/2 ˆ ˜ 1/2 t H ˜ −1 , M h{V(t), V}iM = cos K 2 2

(4.95)

where all matrices have been defined in Sec. 4.3, cf. Eqs. (4.42a) and (4.42b), and where, in the absence of the guest, D = 0. We first consider the position-position correlation. A Laplace transform of the correlation is taken and simple matrix operations allow us to write the result as i−1 sh 2 ˜ 1 1/2 ˆ 1/2 2 −1 ˜ ˆ ˜ ˜ M h{Xt (s), Xt }iMt = (s + Ktt ) − Ktb (s + Kbb ) Kbt 2 th 2 ih i ˜ bt . ˜ −1 G ˜ bt G ˜ tt − G ˜ tb G ˜ tb (s2 + K ˜ bb )−1 G ˜ −1 G 1+K (4.96) bb

bb

169 ˆ The formalism developed in Sec. 4.3 leads to the same expression. Here, both Aˆ and B represent target positions in Eq. (4.38), and thus, the anti-commutator makes the Yt terms in the pre-exponential factor cancel exactly. The Yt integrals are performed exactly and we end up with a prescription where one has to compute hXt (t)Xt (0)T i where Xt (t) is obtained integrating the target part of Eq. (4.44) in the absence of the guest and where the target (0)

initial position and velocities are distributed according to Eq. (4.41) with Rt

= 0. This

correlation function is easily obtained by Laplace transforming the target part of Eq. (4.44) followed by multiplying from the right by Xt (0)T , and finally, averaging over the initial conditions. This gives 1/2

1/2

Mt hXt (t)Xt (0)T iMt

h i−1 ˜ tt ) − K ˜ tb (s2 + K ˜ bb )−1 K ˜ bt = s (s2 + K h ˜ bt ˜ tb (s2 + K ˜ bb )−1 K ˜ −1 K × 1+K bb

i ˜ tb (s2 + K ˜ bb )−1 (K ˜ −1 K ˜ bt − G ˜ −1 G ˜ bt ) −K bb bb

1/2

1/2

×Mt hXt (0)Xt (0)T iMt .

(4.97)

When the variance matrix of initial target positions is calculated using Eq. (4.41) and used in this last expression, Eq. (4.94) is obtained. Note that the third term inside the second square brackets comes from the term in Eq. (4.44) that depends on the initial target position, and hence, this term is needed to obtain the exact result. In order to show similar agreement for the anti-commutator of the velocity operators, one has to be a little more careful as the operator Aˆ now contains a derivative. We take Aˆ = −i¯hM−1 t ∂/∂Zt , allow it to act on K− (R1 , R2 ; t)K+ (Z, R3 ; t) and then set Z = R2 in Eq. (4.36). When this is done Apath becomes Vt (t) where Vt (t) is obtained at the end of the trajectory resulting from Eq. (4.44). There is another term arising from the derivative acting on the product of delta functions in Eq. (4.36); however, it is easy to show that this ˆ can be written as vanishes when Z is set back to R2 and integrated. The other operator, B, 1 T T ˆ = −i¯hM−1 B t ( 2 ∂/∂Xt (0) ± ∂/∂Yt ) in the first (second) term on the right hand side of

170 Eq. (4.14); by integrating the first term by parts it follows that the ∂/∂Xt (0)T terms cancel leaving i2¯hM−1 t

∂ K− (R1 , R2 ; t)K+ (Z, R3 ; t) ∂YtT

= 2Vt (0)K− (R1 , R2 ; t)K+ (Z, R3 ; t).

(4.98)

As before, the Yt integrals can be evaluated and the steps leading to Eq. (4.97) repeated. This results in an expression containing hVt (0)Vt (0)T i; the exact result, cf. Eq. (4.95), is obtained when Eq. (4.41) is used to compute the equal time correlation function. ˆ g , t), J(r ˆ 0 )}/2i in the semi4.8 Appendix: The Anti-commutator correlation h{J(r g classical limit ˆ g , t), J(r ˆ 0 )}/2i. In the Here we consider the current-current anti-commutator, h{J(r g infinite dilution limit ˆ g , t) = − i¯h Aˆ ≡ J(r 2mg

·µ

¶ ¸ ∂ ∂ δ(rg − zg ) + 2δ(rg − zg ) , ∂zg ∂zg

(4.99)

ˆ acts on Eq. (4.36) where zg (t) is the guest part of the field Z and where the operator, A, before Z is set back to R2 (the derivative in the first term inside the square brackets acts on the delta function only). The first term on the right-hand side of this expression, vanishes when integrated over rg , cf. Eq. (4.3), and the remaining terms are similar to the ones discussed in Appendix. 4.7; they simply bring down a velocity factor and as a result, here Apath (xg (t), vg (t)) = δ(rg − xg (t))vg (t).

(4.100)

ˆ operators can similarly be written as The two B ¶ ∂ ∂ yg ± δ(r0g − xg (0) ∓ ) 2∂xg (0) ∂yg 2 µ ¶¸ yg ∂ ∂ ± , +δ(r0g − xg (0) ∓ ) 2 2∂xg (0) ∂yg

h ˆ g (0) ± yg ) = − i¯ B(x 2 2mg

·µ

(4.101)

171 where the operator with the + sign acts on the density matrix while the one with the − sign ˆ are made to act on the products of propagators acts on the product of propagators. Both B by integrating the one with the + sign by parts. When this is done, all delta functions are Taylor expanded, as in Sec. 4.3.3, only keeping the zeroth order terms in yg . This gives 1 ˆ ˆ 0 )}i = h{J(rg , t), J(r g 2

Z dxg (0)dvg (0)dXt (0)dVt (0) δ(rg − xg (t))δ(r0g − xg (0)) × vg (t)vg (0)ρg (xg (0), vg (0))ρt|g (Xt (0), Vt (0); xg (0)), (4.102)

which is again, formally identical with the classical expressions used in I and II, albeit with different dynamics and different weight factors containing the quantum corrections. ˆ 0 )}/2i, is easily ˆ (rg , t), J(r The last anti-commutator that appears in Eq. (4.9), h{N g obtained following the steps described above. The final expressions for all three anticommutators are given in Sec. 4.3.3. 4.9 Appendix: An alternate approximation for W (xg ) In this approximation, the potential is written as follows 1 (1) T (1) (1) U = V (r(1) Kef f Rt g , Rt ) + Rt 2 ¢ ¡ ¢ 1¡ T + R − R(1) K0 R − R(1) + δU, 2

(4.103)

where the one superscript now refers to the position of a minimum (global or local) of the total potential, K0 ≡ K + D(1)

(4.104)

and D(1) is the full matrix of curvatures of V at the minimum (it only has nonzero gg, gt, tg, and tt blocks). Clearly, if the potential is completely harmonic, δU is zero. Here, we choose δU such that, in the high temperature limit, we obtain our usual classical approximation for the

172 potential of mean force. Hence, we write (0)

(1)

δU ≡ V (rg , Rt ) − V (r(1) g , Rt ) h ¢T (1) ¡ ¢ 1 ¡ − rg − r(1) Dgg rg − r(1) g g 2 ³ ´³ ´ (0) T (0) (0) (1) + 2Rt Kef f + Dtt Rt − Rt i (1) T (0) (1) (0) T (0) (0) +Rt Dtt Rt − Rt Dtt Rt , ´T ¡ ´ ¢³ 1³ (1) (1) (1) D(0) − D R − R + Rt − Rt t t gg gg 2 ³ ´T ³ ´³ ´ (1) (0) (0) (1) − Rt − Rt Kef f + Dtt Rt − Rt ³ ´T ¢ (1) (1) ¡ − Rt − Rt Dtg rg − r(1) , g (4.105) (0)

(0)

and where Rt is obtained by solving Eq. (4.20) and where Dtt =

(0)

∂ 2 V (rg ,Rt ) ∂RT t ∂Rt

exactly as

in Sec. 4.3. All quantities with a one superscript are obtained at the minimum while all quantities with a zero superscript refer to the target and bath minimum when the guest is fixed at rg . With these defined, we postulate the approximate diagonal part of the complex time propagator,  ρ(R, R) = det  ·

1/2

1/2 ˜ 01/2

M

K

1/2

M ´ 01/2 ˜ 2π¯h sinh β¯ hK ³

β (1) T (1) Rt Kef f Rt 2 ¸ ¢ ¡ ¢ 1¡ (1) T (1) − R−R F(D) R − R − CδU , 2 (1)

× exp −βV (r(1) g , Rt ) −

(4.106) where

à −1

F(D(0) ) ≡ h ¯ M1/2 tanh

! ˜ 01/2 β¯hK ˜ 01/2 M1/2 . K 2

(4.107)

The additional parameter C in Eq. (4.106), must equal β for high temperature, but is otherwise arbitrary. Because the non-harmonic nature of the potential is mainly governed

173 by the guest, we choose

3

2X 1 C= Fgj,gj (D(1) ) (1) , 3 j=1 Dgj,gj

(4.108)

which is similar to what is done in Refs. [68, 69, 70]. Note that the choice of reference potential is justified by the fact that, at low temperatures, the path integral is dominated by the paths that spend most of the time near the minimum, no matter what rg is. Using this approximate form for the density matrix, the potential of mean-force is obtained using Eq. (4.25) where the reduced density matrix is obtained by integrating Eq. (4.106) over the crystal degrees of freedom. When this is done, we obtain, Ã ! (1) ¢ ¡ ¢ Dgg ¡ (1) (0) (1) T βW (rg ) = (β − C)V (r(1) Fgg − C rg − r(1) g , Rt ) + CV (rg , Rt ) + rg − rg g 2 Ã ! (0) (0) T T T T T (0) (0) (0) Dtt (0) (1) Dtt (1) (0) (1) (0) (0) (1) +C Rt Kef f Rt + Rt Rt + Rt Rt − Rt Kef f Rt − Rt Dtt Rt 2 2 ³ ´ ³ ´ T C2 (1) T Kef f (1) (0) (1) (0) (0) (0) (1) +βRt R − Rt − Rt (Kef f + Dtt )Ltt (Kef f + Dtt ) Rt − Rt 2 t 4 ³ ´T ³ (0) (1) (0) (0) + Rt − Rt C(Kef f + Dtt )Ltt Ftg + C(Kef f + Dtt )Ltb Fbg ¶ ¢ C2 (0) (1) ¡ rg − r(1) − (Kef f + Dtt )Ltt Dtg g 2 ¶ µ ¡ ¢ ¢ C 2 (1) (1) ¡ (1) (1) (1) T + rg − rg Dgt Ltt Dtg rg − r(1) CDgt Ltt Ftg + CDgt Ltb Fbg − Fgc Lcc Fcg − g 4 µ ¶ ¢ 1 ¢ 3 1 mg 1 ¡ 1 ¡ 0 0 0 − ln (det (Φ)) − ln det (Φ ) + ln (det (F cc )) − ln det (F ) + ln , 2 2 2 2 2 2πβ¯ h2 (4.109) where Φ≡

˜ 01/2 M1/2 M1/2 K ³ ´. ˜ 01/2 2π¯ h sinh β¯ hK

(4.110)

The matrices with zero superscript, Φ0 and F0 , are defined in the absence of the guest(in terms of the original force constant matrix) and they live in the crystal space only (hence, their rank is smaller). We have also introduced yet another matrix, Lcc , that lives in the

174 crystal space and which is defined as follows. 0 L−1 cc ≡ Fcc ,

where

(4.111)

  F0cc = Fcc + 

 (1) C (Dtt 2

−

(0) Dtt )

0

0  . 0

(4.112)

(1)

Note that, for high temperatures, Ltt = β2 (Kef f + Dtt )−1 . Eq. (4.109) can then be evaluated numerically using contour integral techniques combined with the theory of defects in a way which is very similar to what is described in Sec. 4.4.1.

CHAPTER 5 Effective classical partition functions with an improved time-dependent reference c potential. [125](°American Physical Society, 2006) In the last chapter, we showed how to calculate the first quantum correction to the permeability. This was done by evaluating the required time correlation functions within our semi-classical approximation. Clearly, the zero time value of the time correlation function was solely obtained by the density matrix, which we obtained by assuming that the guest is fully classical. We could have used another type of approximation to get the density matrix, which is exact when everything is harmonic, that we described at the end of the last chapter. In the end our method was not very different than the less restrictive, ad hoc one and we decided to stick with ours. On the other hand, while we where trying to improve our density matrix approximation, we came across a method developed by Feynman and and Kleinert [71], which we realized we could improve. Unfortunately, this method is ill-defined for diverging potentials, like the one we used in the previous chapter to model the guest-crystal interactions. Hence, we did not apply it in the last chapter. Still with our modified approach, we were able to improve the original Feynman-Kleinert method in the case of a bifurcating potential and obtain a more accurate approximation for the density matrix. 5.1 Introduction and Theory A remarkably accurate variational treatment of euclidean path integrals was first proposed by Feynman and Kleinert some years ago [71], and the accuracy of the method was investigated in detail in Ref. [72]. Kleinert [124] extended the basic approach to obtain a uniformly convergent variational perturbation theory, and the method has been applied, in its original or modified forms, to quantum crystal lattices [132], to calculate excited

175

176 states energies [73], density matrices [74], and much more. In this brief report, we suggest a modification to the first order, original, Feynman-Kleinert method and show that it improves the calculation of the partition function and the density matrix for a particle in a one-dimensional, double-well potential. The quantum mechanical partition function, Z, for a general potential, V (x), expressed as a path integral [58], is Z Z=

Z −A

D[x(τ )] e

=

D[x(τ )] e−

Rβ 0

dτ

˙ )2 +V (x(τ ))] [ 12 x(τ ,

(5.1)

where β = 1/T , A is the euclidean action and where the mass, Planck’s (¯ h) and Boltzmann’s constants have been set to unity. To make a connection with classical statistical mechanics, the partition function can be written as, Z Z=

dx √ o e−βWef f (xo ) , 2πβ

(5.2)

where Wef f (xo ) is called an “effective classical potential”. Clearly, Wef f (xo ) can be obtained from, e−βWef f (xo ) √ = Zx o , 2πβ where,

(5.3)

Z D[x(τ )] δ(¯ x − xo ) e−A ,

Zx o ≡

(5.4)

and where x¯ is some functional of the path variable, x(τ ). In the original Feynman-Kleinert paper [71], a variational approach was used to approximate Wef f (xo ), based on a harmonic reference problem with an analytic solution. Their reference euclidean action was, Z

·

β

Axo ≡

dτ 0

¸ x(τ ˙ )2 ω 2 (xo ) 2 + (x(τ ) − xo ) , 2 2

where the oscillator frequency depends on xo . They also chose xo =

(5.5) 1 β

Rβ 0

dτ x(τ ), such

that the harmonic oscillator is centered around the time average of each individual path.

177 The partition function was then rewritten as Z Z =

Z dxo

D[x(τ )] δ(¯ x − xo ) e−Axo e−(A−Axo )

Z =

< e−(A−Axo ) >xo , dxo Zx(0) o

where < F (x(τ )) >xo ≡

1

(5.6)

Z

(0) Zx o

D[x(τ )] δ(¯ x − xo ) e−Axo F (x(τ ))

(5.7)

(0)

is a restricted average relative to the reference problem and Zxo is defined by Eq. (5.4) with A replaced by Axo . Using the Gibbs-Bogoliubov-Jensen-Peierls inequality, < e−(A−Axo ) >xo ≥ e−xo ,

(5.8)

the final approximation for the effective potential is obtained, from the bound, Z

β

βWef f (xo ) ≤

dτ < V (x(τ )) − 0

− ln Zx(0) − o

ω 2 (xo ) (x(τ ) − xo )2 >xo 2

1 ln (2πβ). 2

(5.9)

This bound is then minimized with respect to ω 2 (xo ) to give the better estimate of Wef f (xo ). Note that Eqs. (5.6,5.8) are essentially the starting point of Zwanzig’s classical statistical mechanical perturbation theory [133] and that variational perturbation theories have been developed by Mansoori and Canfield [134] for simple liquids and by Ronis and coworkers [135, 136] for colloidal suspensions. As was shown in Refs. [71, 72, 74], this procedure gives remarkably good results for potentials where the restricted average of the potential makes sense. In particular, this averaging is problematic for singular potentials. Kleinert [74] later introduced a series of systematic improvements of the method that include higher order corrections. In this short paper, we will suggest a simple way of improving this first order procedure by modifying the reference potential. The choice of x¯ described above, without being completely arbitrary (it makes sense physically and conveniently makes < V (x(τ )) > independent of time), can certainly be

178 relaxed. In fact, when the Feynman-Kleinert formalism is used to obtain density matrices, one cannot use this choice and x¯ simply becomes another parameter that is varied to minimize W (xa , xb ), a two-point function that is defined by Z Z =

dxa dxb δ(xa − xb ) Z × D[x(τ )] δ(xa − x(0))δ(xb − x(β)) e−A Z

=

dxa dxb δ(xa − xb )Zρ(xa , xb ) Z

=

dxa dxb √ δ(xa − xb )e−βW (xa ,xb ) , 2πβ

(5.10)

where ρ(xa , xb ) is the density matrix1 . As above, the two-point function is bounded as Z

β

βW (xa , xb ) ≤ 0

dτ < ∆V (x(τ )) >xa ,xb

− ln (Z (0) ρo (xa , xb )) 1 − ln (2πβ), 2

(5.11)

where ρo (xa , xb ) and Z (0) are, respectively, the density matrix and the unrestricted partition function of the reference problem and < ∆V (x(τ )) >xa ,xb is a conditional average in a reference system where all trajectories start at xa and end at xb . The key modification introduced here is the use of the following reference euclidean action,

Z

·

β

Ao (xa , xb ) =

dτ 0

¸ x(τ ˙ )2 ω 2 (xa , xb ) 2 + (x(τ ) − x¯(τ )) . 2 2

(5.12)

where x¯ now depends on time. In other words, for each (xa , xb ) pair, we will minimize βW (xa , xb ) with respect to ω 2 (xa , xb ) and x¯(τ ) for all time in the interval [0, β] which

√ Only one 2πβ factor was included in this definition. This was done such that βW (xa , xb ) reduces to βWef f (xa ) when xb = xa 1

179 means that the center of reference potential will “move” and, hopefully, better describe the true paths. This is the unique difference between this work and what is done by Kleinert in Ref. [74] (there, x¯ is a parameter that is varied, but it is independent of time) and, as we show below, gives rise to an interesting equation for the classical path. Of course, our method is guaranteed to improve the Feynman-Kleinert original first order method. However, since the earlier work already gave very good estimates for βWef f (x) and for the free energy, we will not be able to do much better there. On the other hand, we expect ρ(xa , xb ) to be estimated more accurately using our reference potential, especially for cases where the off-diagonal correlations are important. For our reference potential, the path that minimizes the action (the classical path) satisfies, x¨cl (τ ) = ω 2 (xa , xb )(xcl (τ ) − x¯(τ )) and the reference density matrix is (for example, see Ref. [57]), s (0) ω(xa , xb ) Z (0) ρo (xa , xb ) = e−Acl . 2π sinh (βω(xa , xb )) where,

Z (0) Acl

β

dτ

= 0

1 [x˙ cl (τ )2 + ω 2 (xa , xb )(xcl (τ ) − x¯(τ ))2 ]. 2

(5.13)

(5.14)

(5.15)

In order to make comparison with Feynman and Kleinert, we will work with the following double well potential, V (x) = −

x2 gx4 1 + + , 2 4 4g

(5.16)

where g > 0. By Taylor expanding V (x(τ )) around the classical trajectory, x(τ ) = xcl (τ ) + δx(τ ), and noting that only even powers of < (δx(τ ))n >xa ,xb are non-zero, Eq. (5.11) can be rewritten as Z

β

·

x˙ cl (τ )2 x2cl gx4cl 3gx2cl (τ ) − 1 − ω 2 (xa , xb ) βW (xa , xb ) ≤ dτ − + + a(τ ) 2 2 4 2 0 µ ¸ ¶ 3g 1 βω(xa , xb ) β 2 + a(τ ) + − ln , (5.17) 4 4g 2 sinh βω(xa , xb )

180 where it can easily be shown that, a(τ ) =< (δx(τ ))2 >xa ,xb =

sinh (ω(xa , xb )τ ) sinh (ω(xa , xb )(β − τ )) , ω(xa , xb ) sinh (ω(xa , xb )β)

(5.18)

accounts for fluctuations around the classical path. Since x¯(τ ) appears only implicitly through xcl (τ ) in Eq. (5.17), minimizing the bound to βW (xa , xb ) with respect to x¯(τ ) is accomplished by taking a functional derivative of Eq. (5.17) and setting the result to zero, i.e., δ(βW (xa , xb )) = δ(¯ x(s))

Z

β 0

dτ [−¨ xcl (τ ) − xcl + gx3cl + 3ga(τ )xcl (τ )]

δ(xcl (τ )) = 0. (5.19) δ(¯ x(s))

The factor δ(xcl (τ ))/δ(¯ x(s)) of the last equation is proportional to the Green’s function that solves the classical equation of motion for xcl (τ ), Eq. (5.13). Hence, acting on Eq. (5.19) with d2 /ds2 − ω 2 (xa , xb ) shows that x¨cl (τ ) = −(1 − 3ga(τ ))xcl (τ ) + gx3cl (τ ),

(5.20)

with xcl (0) = xa and xcl (β) = xb , minimizes βW (xa , xb ). Eqs. (5.20) and (5.13) trivially determine x¯(τ ); namely, x¯(τ ) =

(ω 2 (xa , xb ) + 1 − 3ga(τ ))xcl (τ ) − gx3cl (τ ) . ω 2 (xa , xb )

(5.21)

Equation (5.20) describes the motion of a particle in an unstable potential where the harmonic force constants are time-dependent. In fact, the time-dependent term in Eq. (5.20), 3ga(τ ), is nothing more than the one-loop correction to the equation of motion, albeit with a variationally determined ω 2 (xa , xb ). In order to implement the theory numerically, we proceed as follows: For every pair (xa , xb ), the differential equation, Eq. (5.20), is solved numerically as a boundary value problem for a predetermined grid of ω’s, and the resulting xcl (τ ) are used in Eq. (5.17) to find the frequency that gives the smallest βW (xa , xb ). This frequency is further refined

181 with a one-dimensional minimization routine, thereby giving the best approximation to βW (xa , xb ), which is then used to calculate ρ(xa , xb ). If a(τ ) is set to zero in Eq. (5.20), it is well-known that the boundary value problem p can have multiple solutions if xa and xb both lie between the two minima at x = ± 1/g. This can also happen for nonzero a(τ ) when g is small and β is large. In such cases, we use the solution that gives the smallest βW (xa , xb ). 5.2 Results and Discussion We now compare our results against those of Kleinert [74] where x¯ is not a function of time, hereafter referred to as the “original” calculation. In Fig. 5–1, we compare the free energy A = − log (Z)/β when g = 0.4 for both approximate cases against the “exact” answer that we obtained by solving Schr¨odinger’s equation using 170 harmonic-oscillator basis functions for an oscillator having unit frequency and centered at x = 0. At the level of the free energy, the improvement that our method gives is small. Next we compare the effective potential obtained in both approaches for g = 0.4 and β = 10. Note that β = 10, is well into the quantum regime. In addition, in the classical limit, i.e., when β is small, both methods approach the exact answer. In Fig. 5–2, we compare β(Wef f (x)−A), where A is the free energy obtained with corresponding approximation for each approach, and where βWef f (xa ) = βW (xa , xa ) and where βWef f (xa ) is then used to calculate the quantum mechanical distribution function ρ(xa , xa ). Also shown is the “exact” effective potential and the one obtained within the “WKB” semi-classical approximation. Recall that, in the WKB approximation, Z βWef f (xa ) = −

dτ 0

+

·

β

y˙ cl (τ )2 ycl2 gy 4 − + cl 2 2 4

β − ln (β/f (β)), 4g

¸

(5.22)

182

Figure 5–1: The free energy A = − log (Z)/β in the two approximate cases is compared with the “exact” answer.

183

Figure 5–2: β(Wef f (x) − A) for g = 0.4 and β = 10. Only x < 0 is shown because βWef f (x) = βWef f (−x). In the inset, βWef f (x) is shown for the region where our method differ most compared to the original case.

184 where ycl (τ ) is the solution of the Euler-Lagrange equation with the full potential (the boundary condition is that the trajectory starts and ends at xa ) and f (τ ) is the solution of the Jacobi equation, f¨(τ ) = (−1 + 3gycl (τ )2 )f (τ ),

(5.23)

with f (0) = 0 and f˙(0) = 1. The distribution functions obtained with the various methods are shown in Fig. 5–3 where we also compare with the “exact” results. As Figs. 5–2 and 5–3 show, our choice of reference potential improves the already good results originally obtained, especially for xa lying between the two minima of V (x). Also note that the error in βWef f (xa ), is largely due to the error in the free-energy (cf. Fig. 5–1). Moreover, at this temperature, the WKB approximation is very inaccurate, and, in particular, fails to describe the suppression of the central potential barrier. The more interesting aspect of this work is to see what type of improvement we get for the full density matrix. After all, it is when the initial and final points of the path are far from each other (when the path is stretched) that we expect our more general potential to pay off, although the distance between the two points should not be too large, since, for large separation, the density matrix vanishes. In Fig. 5–4, we compare our method against the “exact” answer (again, g = 0.4 and β = 10). On that scale, the difference between the two calculations is small (the original approximation is also very similar). On the other hand, there really are some differences as can be seen from Fig. 5–3, which focuses on the diagonal of ρ(xa , xb ). In Fig. 5–5, we plot the difference |ρ − ρexact |, where ρ is calculated using our formalism or the original one. This figure focuses on the region around the off-diagonal peak and clearly shows that our potential more accurately describes the off-diagonal peaks of the density matrix, since the error is systematically lower. In particular, the small error region near (0, 0) is much larger in our case. Also, at the point (−0.9, 0.9) (a point close to the top of the

185

Figure 5–3: The quantum mechanical distribution function, ρ(x) for g = 0.4 and β = 10.

186

Figure 5–4: The density matrix, ρ(xa , xb ) for g = 0.4 and β = 10. The figure on the left is our calculation while that on the right is the “exact” answer. off-diagonal peaks), the exact density matrix is 0.243, our calculation gives 0.244 and the original method gives 0.233. In Fig. 5–6 we show x¯(τ ) and xcl (τ ) obtained from Eqs. (5.20) and (5.21) at the offdiagonal point (xa = 0.9, xb = −0.9). As expected, since the end points are symmetrically placed around zero, x¯ = 0 in the original approach, whereas here, the initial and final x¯(τ ) deviate strongly from zero. As seen in the figure, this difference in x¯(τ ) allows us to better describe the average paths (obtained from Monte Carlo simulations using the full potential and a discretized path containing 1000 points). In the same figure, we also show the xcl (τ ) that minimizes the euclidean action with the full potential used in the WKB approximation. As seen in Fig. 5–6, there are two energetically equivalent solutions, ycl (τ ) (they should be included in the theory with equal weight), both of them being quite far from the actual “exact” average path. The average of these two equivalent trajectories is also shown in Fig. 5–6 and turns out to be farther from the “exact” result than the other two approximate methods. Fig. 5–6 also shows that the reference potentials used in this work or in the “original” calculation replace the two equivalent solutions by one that follows the “exact” average more closely than the mean of the two WKB paths.

187

Figure 5–5: The difference |ρ − ρexact | for g = 0.4 and β = 10. The left figure is our calculation while the right figure is the time-independent reference potential of Kleinert. In the original and our calculation, the fluctuations around the classical path are included to first order and appear in a(τ ) defined by Eq. (5.18). These fluctuations will differ in our formalism and in the Feynman-Kleinert one only through the numerical value of ω, which is independently chosen in the two cases to minimize βW (xa , xb ). For example, we have ω = 0.495039 while, when x¯ is independent of time, ω = 0.460409 (again, with g = 0.4, xa = 0.9, xb = −0.9 and β = 10). This means that the fluctuations around the classical path will be larger in the original method. These fluctuations are compared in Fig. 5–7 against the “exact” result where it is seen that both approximate methods underestimate the fluctuations. Remember that the Gibbs-Bogoliubov-Jensen-Peierls inequality, Eq. (5.8) only guarantees that the two-point function, βW (xa , xb ) is bounded below by the “exact” results and that, consequently, our βW (xa , xb ) is guaranteed to be smaller than the one in the original calculation; nothing else need be improved. For example, as shown in Fig. 5–2, even though the two quantities βWef f (x) and βA are separately bounded by the inequality, their combination, β(Wef f (x) − A) is not. In conclusion, the modification to the Feynman-Kleinert method presented in this work improves the results by a small, but non-negligible amount. The improvement that

188

Figure 5–6: The classical path, xcl (τ ), defined by our reference potential and the original reference potential are compared with the “exact” average path linking the points (0.9, −0.9). Here, WKB(mean) is the average of the two energetically equivalent WKB paths (see text). In the inset, x¯(τ ) is compared for both reference potentials.

189

Figure 5–7: The fluctuations around the classical paths linking xa = 0.9 and xb = −0.9 for g = 0.4 and β = 10.

190 we get becomes more significant when trying to capture information about the microscopic details of the problem. In fact, we get very little improvement in the free energy, but a better description of the density matrix. In general, our method will be particularly useful for potentials containing energy barriers for temperatures where the off-diagonal correlations in the density matrix are large. We also think that the equation of motion defining x¯, Eq. (5.20), is, by itself, quite interesting.

CHAPTER 6 Other problems: The Jarzynski equality: Connections to thermodynamics and the c Second Law. [137](°American Physical Society, 2007) The last chapters were all related to the diffusion in channeled structures problem, which is the main part of this thesis. This last chapter is completely separate with the main theme of the thesis. This work is about the interpretations of the now well-known Jarzynski equality [75]. This equality relates the work done with the free-energy change of a system for any type of work process. Since Jarzynski proposed its equality in 1997, there has been many publications, both experimental, simulations and theoretical, that seems to confirm the equality. In this chapter, we take the equality as a mathematical identity but we show that it must be interpreted with caution. 6.1 Introduction Only a few relations are exactly satisfied for processes in systems far from thermodynamic equilibrium. Of the most recent of these relations is the Jarzynski equality, which relates the nonequilibrium average work done by a driving force on a system initially at equilibrium to the free energy difference between two equilibrium states of the system. This equality was first derived classically by Jarzynski [75, 76, 77] and later extended to stochastic system by Crooks [82]. Quantum mechanical Jarzynski equalities have also been investigated by Mukamel [81] and Esposito [138]. The Jarzynski equality has also been extensively tested numerically and analytically for various models; e.g., Marathe and Dhar [83] studied spin systems, while Oostenbrink and van Gunsteren [84] studied redistribution of charges, creation and annihilation of neutral particles and conformational changes in molecules. In each case the Jarzynski equality was confirmed. The Jarzynski equality has also been extensively tested for ideal gas expansions by Press´e and Silbey [85], Lua and Grosberg [86] and by Bena, Van den Broeck and Kawai [87]. Finally, 191

192 the Jarzynski equality has been verified experimentally by Liphardt et al. [78] by stretching single RNA molecules. On the other hand, the validity of the Jarzynski equality is still under debate. For example, see the objections of Cohen and Mauzerall in Refs. [79, 80]; in particular, one of their concerns involves the temperature appearing in the Jarzynski equality, ­ βW ® Zb (β) ≡ e−β∆AJ e = Za (β)

(6.1)

where W is the work done by the system for a process that brings the systems from an equilibrium initial state with work parameter a to a final state with work parameter b, h· · ·i is a non-equilibrium average, β ≡ 1/(kB T ), and ∆AJ is obtained, as above, by the ratio of two canonical partition functions defined at the same temperature, but for the two different work parameters, a and b. Their claim was that the temperature in the last equation is defined without foundation since the temperature is undetermined during irreversible processes, and often differs from the initial temperature. Although this statement is correct, it is equally clear that the Jarzynski equality is exact provided that the system is canonically distributed initially and that β is defined in terms of the initial equilibrium temperature of the system. In fact, the derivation by Jarzynski in Ref. [77] is fairly general and, at least in the original presentation, is based solely on Liouville’s theorem, when the free-energy change describes the system plus any bath, although in the event that the bath is explicitly included in the dynamics several complications arise; specifically, in extracting the system free energy change from the total and with the identification of work and heat exchange between system and bath 1 .

1

The motivation of a subdivision of the mechanics into system and bath is primarily to introduce a heat bath, whereby isothermal processes may be realized [77, 138]. Nonetheless, this should suffice to coarse grain the system reduced distribution. On the other hand, some of the analysis seems to have been over-interpreted in these approaches. By associating the mechanical work, which is assumed to couple directly to the system, with the

193 In this work, we accept the Jarzynski relation as a mathematical identity and note that it has been recognized as a tool for calculating free-energies [84, 88]. Here, we study the connections between the Jarzynski equality and nonequilibrium thermodynamics. We will focus on systems that are governed by Hamiltonian dynamics. For such systems, the process driven by external forces is always adiabatic since the Hamiltonian presumably contains everything, and provided Liouville’s equation is valid, the derivation of the iden­ ® tity in Ref. [77] shows unambiguously that eβW is equal to the ratio of two partition functions defined at the same temperature. As above, we call this ratio e−β∆AJ . One of the questions we will answer in this paper is the following: In general, how is ∆AJ related to the true free-energy changes at the end of the process, where, as pointed out in Ref. [79], the temperature, if it can be defined, has usually changed. Moreover, it is sometimes believed that the Jarzynski equality can be used to prove thermodynamics bounds [87] (i.e. the Second Law) from mechanics. We will show here that this is not true. These questions will be addressed both generally using thermodynamics or response theory, and within the context of some of the simple numerical models considered in the literature. We will also show how the Jarzynski equality works in the context of response theory. In particular, we will show that the non-equilibrium average of the work equals −∆AJ plus so-called dissipative terms. These dissipative terms are then successively canceled by the higher order cumulants of the work. Response theory also provides a framework which will then be used to derive the specific conditions under which −∆AJ becomes the true upper bound to the average work. More specifically, we show that when the process is quasi-static and leaves the basic quantities that define the ensemble (e.g., temperature,

thermodynamic work, and analyzing the change in the system’s energy terms identified with heat are found. This ignores the possibility of work-like interactions between the system and bath (e.g., mass transport, volume expansion, polarization effects, etc.).

194 density, chemical potential, etc.) unchanged, the Jarzynski work bound becomes the lowest upper bound to the work; in that case, the dissipative terms vanish. For other kinds of processes, in particular, for adiabatic ones like the 1d expanding ideal gas, the Jarzynski work bound is an upper bound to the thermodynamic work upper bound, no matter how slowly the process is carried out! This fact has already been noted numerically by Oberhofer, Dellago and Geissler [88] when they compared numerical schemes for calculating equilibrium free energies based on the Jarzynski equality to those computed using the Widom insertion method in a soft sphere liquid [139]. This was also observed by Jarzynski [76] for the isolated harmonic oscillator model where the natural frequency is increased as a function of time. Here, we show this results from general thermodynamic considerations or within the context of response theory. The paper is divided as follows. In Sec. 6.2, we define the one-dimensional gas model and we calculate the full non-equilibrium distribution function when the gas is expanding. Within this exactly solvable model, we characterize the actual state of the system during and after the expansion in terms of local thermodynamic quantities and we compare the final state free-energy with ∆AJ . In Sec. 6.3, we compare the work bound obtained from the Jarzynski relation, invoking the Gibbs-Bogoliubov-Jensen-Peierls inequality, against the work bound imposed by thermodynamics and we show that Jarzynski’s bound is less restrictive. Within the ideal expanding gas model, we calculate the average work as a function of the rate of the expansion and we compare the final free-energy difference with ∆AJ . In Sec. 6.4, we use response theory to show how the Jarzynski equality works and, in particular, how the non-equilibrium terms cancel. We also explain how the dissipative terms can contribute to the work even if the process is carried out quasi-statically. We also argue that this contribution of the dissipative terms could be used to explain why the average work does not equal −∆AJ in Fig. 3A of Liphardt et al. [78] when the work is performed very slowly. Sec. 6.5 contains a discussion and some concluding remarks.

195 6.2 One-dimensional expanding ideal gas In this section, we consider the one-dimensional expanding ideal gas. As has been shown by many authors [86, 87, 85], this model is fully consistent with the Jarzynski equality with, ∆AJ = −kB Ti ln (Lf /Li ),

(6.2)

where Ti is the initial temperature of the gas, kB is Boltzmann’s constant, and Li and Lf are the initial and final lengths of the box confining the gas, respectively. Here, ∆AJ represents the free-energy difference between two states at the same temperature, but having different lengths. To follow the expansion of the gas, we work with the same model as in Refs. [86, 87], which were inspired by the earlier work of Jepsen [140] and of Lebowitz and Percus [141]. In this model, the gas is initially at equilibrium in a box of length L = Li . This box is closed from the left by a hard wall and from the right by an infinitely massive piston. At time t = 0, the piston starts to move to the right with velocity V . Here, we will investigate the non-equilibrium process and see how it is related to ∆AJ . As mentioned above, this model as already been studied in detail in Refs. [86, 87], here, we use it to study several issues that were missed in these references; in particular, how the quasi-static limit arises, how the work relates to the maximum work predicted by thermodynamics, and what, if anything, the work distribution is really telling us about the actual state of the system. The ideal gas model here is used to raise questions, that will be investigated in more general terms in Secs. 6.3 and 6.4. The complete knowledge of the system for t > 0 can be obtained from the nonequilibrium distribution function, f (x, u; t), where, as usual, x is the position and u the velocity of the gas. For the expansion of the ideal gas, this distribution function can be obtained by solving the Liouville or Boltzmann equation [142, 143] with no external forces and, of course, no collisions, i.e., ∂f (x, u; t) ∂f (x, u; t) +u = 0, ∂t ∂x

(6.3)

196 with the boundary conditions, f (Li + V t, u; t) = f (Li + V t, 2V − u; t)

(6.4)

f (0, u; t) = f (0, −u; t),

(6.5)

for u > V and

for all u. These boundary conditions account for the change in velocity of a gas particle hitting the piston at x = Li + V t (cf. Eq. (6.4)) or the stationary wall at x = 0 (cf. Eq. (6.5)). As in the general presentation of the Jarzynski relation, we assume that the initial equilibrium distribution function is canonical, specifically, 1 f0 (x, u) = Li

µ

β 2π

¶1/2 e−βu

2 /2

Θ(Li − x)Θ(x),

(6.6)

where β = 1/Ti , Θ(x) is the Heaviside step function, and where, henceforth, we set Boltzmann’s constant (kB ) and the mass of the gas particles to one. The product of step functions guarantees that the gas is initially confined between x = 0 and x = Li . Rather than solving the Liouville equation, it is easier to get a more direct solution. The derivation follows the ideas of Refs. [144, 145, 86] and the solution is expressed as an infinite sum over n, where n is the number of collisions a gas particle makes with the piston. The details of the derivation are given in Appendix 6.6, and the final expression

197 for f (x, u; t) is f (x, u; t) = Θ(Li + V t − x)Θ(x) ×{f0 (x − ut, u) + f0 (−x + ut, −u) ∞ X + [f0 (−x + ut + 2nLi , 2nV − u) n=1

+f0 (x − ut + 2nLi , 2nV + u) +f0 (x − ut − 2nLi , u − 2nV ) +f0 (−x + ut − 2nLi , −u − 2nV )]}, (6.7) where the first and second terms on the right-hand side represents, respectively, free streaming and a single collision with the back wall. The four terms in the sum account for, respectively, an initially positive velocity resulting in n collisions with the piston followed by free-streaming, an initially positive velocity resulting in n collisions with the piston followed by a collision with the back wall, an initially negative velocity resulting in n collisions with the piston followed by free-streaming and an initially negative velocity resulting in n collisions with the piston followed by a collision with the back wall. The two step functions in Eq. (6.7) account for the fact that the gas is confined to the [0, Li + V t] interval during the expansion. This expression for f (x, u; t), can easily be shown to obey Liouville’s equation, satisfies both boundary conditions, cf. Eqs. (6.4) and (6.5), and the initial condition, Eq. (6.6). Note that, when V → 0, i.e., the process is carried out reversibly, it is easy to show (by converting the sums into integrals) that Eq. (6.7) reduces to the appropriate equilibrium distribution function; it is uniform in x and Gaussian in u with a reduced temperature Ti [Li /(Li + V t)]2 , and is in complete agreement with thermodynamics.

198

Figure 6–1: The initial equilibrium distribution function at t = 0 is shown on the left and the non-equilibrium distribution function at t = 4 is shown on the right. Here, Li = 10, V = 0.5 and Ti = 1 and the moving piston is on the right. The contours are drawn when f (x, u) = 0.005, 0.01, 0.015, 0.02, 0.025, 0.03 and 0.035. The initial equilibrium distribution function and the non-equilibrium distribution function (t = 4) for a piston of initial length Li = 10, piston velocity V = 0.5 and initial temperature Ti = 1 are shown in Fig. 6–1. As expected, the distribution function is affected by the expansion only in the vicinity of the moving piston. We next show some examples of what we can extract from this distribution function. While the non-equilibrium process is occurring, the state of the system can be described, at least in part, in terms of local thermodynamic quantities [146], perhaps generalized in various ways, if at all. For example, the local density, velocity, energy per particle, temperature and entropy per particle are well known (see e.g., Ref. [143]) and are defined by

Z

∞

ρ(x, t) ≡

du f (x, u; t), Z

∞

U (x, t) ≡ E(x, t) ≡ kB Ti

Z

(6.8)

−∞

du

f (x, u; t) u, ρ(x, t)

(6.9)

du

f (x, u; t)u2 , 2ρ(x, t)

(6.10)

−∞ ∞ −∞

199 Z

∞

du

kB T (x, t) ≡ −∞

and S(x, t) ≡− kB

Z

f (x, u; t)(u − U (x, t))2 , ρ(x, t)

(6.11)

f (x, u; t) ln f (x, u; t) , ρ(x, t)

(6.12)

∞

du −∞

respectively. Note that 1 E(x, t) = [T (x, t) + U 2 (x, t)]. kB 2

(6.13)

Clearly, all but the entropy per particle will be given in terms of sums of error functions upon the substitution of Eq. (6.7) into Eqs. (6.8)–(6.11). We also can evaluate the boundary conditions obeyed by the thermodynamic variables at the piston (x = Li + V t). The simplest boundary condition is the one for the velocity field at the boundary, U (Li + V t, t) = V , as expected from mass conservation. We show examples of the local thermodynamic quantities profiles as a function of time and position in Fig. 6–2. Before going on, we clarify a few points about the unit conventions that we use. We have set the mass of the particle and Boltzmann’s constant to unity; therefore, all quantities that have the units of energy, the work (W ) and the free-energy (A), and temperature all have the same units. It also follows that the particle’s velocity, u, and the piston velocity, √ V , have units of energy. We use this convention for what follows. At this stage, we consider the case where the piston stops moving, e.g., when t = 4. At this particular time, it is quite clear that the system is far from equilibrium (as shown from the inhomogeneous character of the quantities displayed in Fig. 6–2). We define a local Helmholtz free-energy per particle as A(x) = E(x) − T (x)S(x).

(6.14)

We expect that the system will eventually come to equilibrium, provided we wait long enough, and when this happens, the free-energy, and all other thermodynamic quantities, will be uniform in position. In particular, we expect that there the final free-energy per

200

Figure 6–2: The local thermodynamics quantities defined by Eqs. (6.8)–(6.11) are shown as a function of time from t = 0 to t = 4 for an expansion where Li = 10 and V = 0.5. At t = 0, all quantities are uniform for 0 < x < Li and zero elsewhere. The curves are equally spaced at time intervals of 4/9. Larger values of t have larger non-uniform regions.

201 particle is Af = −Tf ln (Lf (2πTf )1/2 ),

(6.15)

where Tf and Lf are the final temperature and length, respectively, and where Planck’s constant h has been set to one. The final temperature is easily obtained from the total final RL energy, cf. Eq. (6.13) or equivalently from Ef = 0 f E(x, 4)ρ(x, 4), which is conserved when the piston is at rest and which is related to the temperature at equilibrium in the usual manner (i.e., Ef = Tf /2). Perhaps surprisingly at first glance, the free energy of the system, as computed from RL thermodynamics, i.e., Eq. (6.15), is not the one obtained from A = 0 f dxρ(x, ∞)A(x, ∞), even at infinite time after the piston has stopped moving. This happens because the entropy, as defined above in Eq. (6.12), is the fine-grained entropy per particle. As is well known [147, 56], the total fine-grained entropy is a constant of the motion, but the final entropy of the system should equal, Sf =

1 + ln (Lf (2πTf )1/2 ) 2

(6.16)

which is obviously different from the initial entropy, unless the expansion is done reversibly and adiabatically. This well-known paradoxical result is resolved by introducing a coarse-grained entropy[147, 56]. There are many ways to do this, and in some of the works on the Jarzynski equality, this is done by averaging out so-called bath degrees of freedom2 . The model here is too simple to allow for this sort of coarse graining, and instead we consider an older approach due to Kirkwood [148] and define a time-averaged distribution function as 1 f¯(x, u; t) ≡ τ

2

see footnote on p.192

Z

t+τ

ds f (x, u; s), t

(6.17a)

202

Figure 6–3: The fine-grained distribution function (on the left) is compared with the timeaveraged distribution function that is later used to calculate the coarse-grained entropy according to Eq. (6.17b). Here, t = 1000 and τ = 100. The contours are drawn when f (x, u) = 0.005, 0.01, 0.015, 0.02, 0.025, 0.03 and 0.035. and then use it to define a coarse grained entropy as Z Lf Z ∞ ¯ S(t) dx du f¯(x, u; t) ln f¯(x, u; t). ≡− kB −∞ 0

(6.17b)

Note that this time averaging procedure only makes sense at long times after the end of the expansion, t → ∞, where the fine-grained distribution is approximately uniform (or more generally, slowly varying in time). Also, this time averaging of the distribution function preserves the important property that the energy is conserved after the expansion. Finally, it can be shown that the time averaged distribution is uniform in position as τ → ∞. In Fig. 6–3, we show the time-averaged distribution function for t = 1000 and τ = 100 with 2000 discretization points in the average and we compare it with the finegrained distribution at the same time. Note that the structure appearing in the fine-grain distribution is averaged out after coarse-graining. More quantitatively, the coarse-grained entropy equals S¯ = 3.811. This should be compared with Sf = 3.812 and Si = 3.722 as obtained from Eqs. (6.16) or (6.12), respectively.

203 We conclude these comments on fine and coarse-grained entropy by pointing out that, within this model, the system never quite comes back to equilibrium, even a long time after the expansion (this is well-known and has been pointed out in Ref. [89]). If it did, the final fine grained distribution would be completely uniform in position and defect free. This does not happen here because this one dimensional model does not contain any mechanism that will randomize the velocities efficiently and does not have strong separation of time scales. Returning to the Jarzynski equality, the final free energy appearing in Eq. (6.1), AJ , can be obtained from Eq. (6.15) with Ti = Tf and the final equilibrium free-energy can be obtained from Eq. (6.14), provided we use a coarse grained entropy or from Eq. (6.15) with the appropriate final temperature (for this model Tf = 2Ef ). For the expansion parameters defined above, ∆A = 0.466 as obtained from Eqs. (6.15) or (6.14) and ∆AJ = −0.1823 (see Fig. 6–5 of the next section for a more detailed comparison between ∆A and ∆AJ when the piston speed is varied). These differences are explained by the fact that the expansion of the gas is not isothermal. In fact, for the parameters described above, the temperature drops from 1.0 to 0.832. Thus, while the Jarzynski equality is rigorously correct, the above discussion shows that the equality does not provide much information about the non-equilibrium state of the system, neither locally or thermodynamically. Most significantly, ∆AJ is not the freeenergy change predicted by thermodynamics. Even if local thermodynamic equilibrium can only be invoked approximately during the expansion process (cf. Eq. (6.14)), the nonequilibrium distribution function is a complete description of the non-equilibrium state and it carries a lot of information that is hidden in the Jarzynski equality. Still, the Jarzynski equality holds. Recall here that it relates the non-equilibrium average of eβW to the equilibrium free-energy difference between two states at the same temperature, but with different lengths.

204 6.3 Work bounds As pointed out by Jarzynski in his original papers [75, 76], the Gibbs-BogoliubovJensen-Peierls inequality, ­ βW ® e ≥ eβhW i ,

(6.18)

combined with the Jarzynski equality automatically implies that the average work is bounded by, hW i ≤ −∆AJ ,

(6.19)

strongly reminiscent of the usual bound for work in isothermal process found in thermodynamics. Again, recall that we are using the convention that hW i is the work done by the system on the surroundings. We now compare this bound against the bound obtained from the laws of thermodynamics. This is an important question, since the degree to which the two bounds differ disproves the contention that Eq. (6.19) is essentially a proof of the Second Law of Thermodynamics from mechanics (this is claimed in Refs. [89, 87] and in the review section of Ref. [149]). On the other hand, the laws of thermodynamics also provide bounds for the average work, specifically, the First and Second Laws imply that Z hW i ≤ −∆A −

(dT S − dN µop ),

(6.20)

where µop is the opposing chemical potential and N is the number of particles. This becomes the usual work bound in terms of the Helmholtz free energy change for isothermal and constant N processes. In what follows, we will consider processes which conserves the number of particles, as this is appropriate for most of the numerical studies of the R Jarzynski equality, and drop the dN term in Eq. (6.20). In general the (dT S − dN µop ) term is not a state function, and greatly reduces the utility of Eq. (6.20). Nonetheless, one

205 can obtain a useful bound for the work by noting that in an adiabatic expansion, hW i = −∆E = −∆A − Sf ∆T − Ti ∆S

(6.21a)

= −∆A − Si ∆T − Tf ∆S.

(6.21b)

Since ∆S ≥ 0 for a spontaneous adiabatic process, Eqs. (6.21a) and (6.21b) imply that hW i ≤ − max(∆A + Sα ∆T ), α=i,f

(6.22)

where the inequality becomes an equality for reversible processes. Although in principle ∆T can have any sign in an adiabatic expansion, it must be negative for ideal gases, and thus, Eq. (6.22) implies that hW i ≤ −∆A − Si ∆T ≡ Wrev .

(6.23)

It is straightforward to relate the Jarzynski bound to that given in Eq. (6.22); specifically, it is easy to show that Z

Tf

∆A + Sf ∆T = ∆AJ +

dT Ti

CV (T, Vf ) (T − Ti ), T

(6.24)

where CV (T, Vf ) is the constant volume heat capacity at the final volume, Vf . The last integral is strictly positive for ∆T 6= 0, and this in turn means that the Jarzynski bound is greater than the one implied by thermodynamics, cf., Eq. (6.22). This general result reproduces the expected work bounds for the isolated harmonic oscillator model with an increasing natural frequency (ωi that increased to ωf ) discussed by Jarzynski (cf. Fig. 2 of Ref. [76]) and for the above ideal gas expansion. In both cases, when the work is done reversibly and adiabatically Sf = Si . This guarantees −(∆A + Sf ∆T ) to be the true upper bound to the work from Eq. (6.22). For the harmonic oscillator model, ∆AJ = Ti ln (ωf /ωi ) while, for the ideal gas model, ∆AJ = −Ti ln (Lf /Li ). In

206 the two cases Cv is independent of temperature, and the correction to ∆AJ in Eq. (6.24) becomes

Z

Tf Ti

· µ ¶¸ CV (T, Vf ) Tf Tf dT (T − Ti ) = CV Ti − 1 − ln . T Ti Ti

(6.25)

In both models, when the work is done reversibly and adiabatically it is easy to show that    ωf harmonic oscillator Tf ω = (6.26) ³ i ´2  Ti  Li ideal gas . Lf When this is used in Eq. (6.25), it is easy to see that these terms are, as expected, positive. Moreover, when they are used in Eq. (6.24), the ∆AJ term exactly cancels, and one is left with the usual thermodynamic work bound,    1 − ωf harmonic oscillator ωi Wrev = Ti 2   1 − L2i ideal gas; L

(6.27)

f

hence, in both cases, Wrev ≤ −∆AJ ,

(6.28)

where the equality only holds in the trivial case where no work is done on the system (i.e. ωf = ωi or Lf = Li ). These two special cases were used to illustrate the more general result given in Eq. (6.24) and to highlight the fact that Eq. (6.24) is in quantitative agreement with the harmonic model described by Jarzynski in Ref. [76]. In summary, the Jarzynski equality alone does not guarantee Eq. (6.20) to be satisfied, and hence, is not a proof of the Second Law. The 1d-gas model can further be used to calculate the average work done for any expansion rate or piston velocity V . Again, the average work, like the non-equilibrium distribution function (Eq. (6.7)), can be expressed as a sum over the number of collisions

207 with the piston, hW i = 1 Li

µ

2β π

¶1/2 Z

Li

dx0 0

∞ X

ÃZ

(2n+1)(Li /t+V )−x/t

2

du0 e−βu0 /2 (nu0 V − n2 V 2 )

(2n−1)(Li /t+V )−x/t

n=1

Z

(2n+1)(Li /t+V )+x/t

+

! 2

du0 e−βu0 /2 (nu0 V − n2 V 2 ) ,

(2n−1)(Li /t+V )+x/t

(6.29) where x0 and u0 are the initial position and velocities of the gas particle. This is equivalent to the expression for < W > obtained from the work distribution, P (W ), as defined R∞ by Eq. (13) in Lua and Grosberg [86] (hW i = 0 dW W P (W )). Simple but lengthy manipulations transforms this expression into, ∞ o Vt Xn α α erfc( [(2n + 1)(Li + V t) − Li )] − erfc( [(2n + 1)(Li + V t) + Li )] βLi t t n=1 n o Vt α + erfc(αV ) − erfc( (2Li + V t)) βLi t µ ¶ ∞ 1/2 nα X V 2t 2 α + (2n + 1) [(2n + 1)(Li + V t) − Li ]erfc( [(2n + 1)(Li + V t) − Li ]) Li β t t n=1 α α − [(2n + 1)(Li + V t) + Li ]erfc( [(2n + 1)(Li + V t) + Li ]) t t ¾

hW i =

α2

2

α2

2

− π −1/2 e− t2 [(2n+1)(Li +V t)−Li ] + π −1/2 e− t2 [(2n+1)(Li +V t)+Li ] V 2t + Li

µ ¶1/2 n 2 α α αV erfc(αV ) − (2Li + V t)erfc( (2Li + V t)) β t t ¾ 2 α −1/2 −α2 V 2 −1/2 − t2 (2Li +V t)2 −π e +π e , (6.30)

where α ≡ (β/2)1/2 and where erfc(x) is the complementary error function. When the expansion is done reversibly, we have αV ¿ 1, t À 1 and αLi /t ¿ 1. In this limit, the sums in Eq. (6.30) can be replaced by integrals with the result that hW i ∼

V t(V t + 2Li ) + O(V ), 2β(Li + V t)2

(6.31)

208 which, after some simple manipulations, agrees with Eq. (6.27). For the rest of this section, we will consider the case where Li = 10 and V t = 10 (the length of the box doubles). For finite piston velocities, the average work is calculated using Eq. (6.30), keeping enough terms such that the error falls within a small tolerance. In Fig. 6–4, the average work is shown as a function of the velocity of the piston for Ti = 1, Li = 10 and V t = 10. As V → 0, it is seen that hW i tends to Wrev , which, for these parameters, equals 3/8. This figure clearly shows that Wrev (the straight dotted line in Fig. 6–4) is the smallest upper bound to the process while the Jarzynski bound −∆AJ sits above (straight dashed line in Fig. 6–4). Note that, for large V (the tail region in Fig. 6–4), the Jarzynski equality still holds even if hW i is very small. This seemingly paradoxical results was investigated in Ref. [86]. Also note that there is no contradiction between the results shown in Fig. 6–4 and the results of Lua and Grosberg [86]. In fact, Fig. 6 of Ref. [86] shows that the average work becomes equal to −∆AJ for small piston velocities. The problem there is that they kept t and Li constant and reduced V . There, as V tends to zero, Lf − Li vanishes and there is no expansion; this is not what is meant by the quasi-static limit. As stated after Eq. (6.28), the two work bounds trivially agree in that limit. ∆A is compared with ∆AJ as a function of the piston velocity in Fig. 6–5. Recall that the final temperature of the system is determined from the work. The data shows that ∆A does not have a definite sign. Further, for very small V (i.e., a nearly reversible process), ∆A exhibits the largest differences from ∆AJ . On the other hand, for large V , the two free-energy difference agree with each other. This result is easily explained in terms of temperatures changes. When the process is slow, maximum work is done, and hence, the temperature changes the most. On the other hand, when the piston is pulled very quickly, only a small fraction of the particles can collide with it, very little work is done, and the temperature does not change. In this case, the free-energy that appears in the Jarzynski equality describes the final state appropriately.

209

Figure 6–4: The average work W is obtained from Eq. (6.30) with T = 1, Li = 10 and V t = 10. This is compared with the Jarzynski bound, which is independent of the piston velocity.

210

Figure 6–5: The real free energy difference ∆A between the two equilibrium states is compared with ∆AJ . The conditions of the expansion are the same as in Fig. 6–4

211 In a recent article by Baule, Evans and Olmsted [89], it was shown that the ideal gas expansion, within an isothermal model where the gas is effectively coupled to a thermostat, does satisfy the Jarzynski equality and that −∆AJ = Wrev in this case (note that this model does not fall into any category of model for which the Jarzynski equality has been derived rigorously). We conclude this section with a short remark on the experimental consequences of our observations. In many experiments, in particular, in the famous experiment by Liphardt et al. [78], it is often assumed that −∆AJ is equal to Wrev . Then, many realization of ­ ® the work are performed irreversibly and eβW is computed and compared to e−β∆AJ . We have shown that, in general, this is not exactly true. In fact, when the temperature change is significant, Wrev and −∆AJ can be very different. In the case of the experiment of Liphardt et al., the work was done on a single RNA molecule in solution, in which case, naively, the temperature change should be small. This will be further investigated below. 6.4 The Jarzynski relation and response theory In the previous sections we showed that, even though the Jarzynski equality is exactly satisfied for the 1d expanding gas, ∆AJ , in general, does not characterize the actual state of the system at any time during or after the expansion process. Further, the bound that we get from the Jarzynski equality tells us less than what we already know from thermodynamics. Therefore, one could wonder why the Jarzynski equality works. In this section, we show how, in the context of response theory, the terms that give rise to non-equilibrium effects ­ ® cancel when eβW is computed and examine the average work done. Consider a classical system that evolves under the Hamiltonian, H(t) = H0 + H1 (t),

(6.32)

212 where H0 and H1 (t) are, respectively, explicitly time-independent and time-dependent. In response theory, H1 (t) is treated as a perturbation that has the general form H1 (t) = −

XZ

dr Aj (r, X N )Fj (r, t)

j

≡ −A(t) ∗ F(t),

(6.33)

where X N is the phase point, the Aj (r, X N )’s are local observables at position r that depend implicitly on time through the motion of the particles and the Fj (r, t)’s are the external fields. We assume that the Fj (r, t)’s vanish for t ≤ 0. By using this form for the Hamiltonian, the right-hand side of the Jarzynski equality becomes, ­ ® e−β∆AJ = eβA∗F(t) 0 , where,

R h(...)i0 =

dX N e−βH0 (...) R , dX N e−βH0

(6.34)

(6.35)

is a canonical average with respect to H0 . This can be expanded in powers of the external fields as follows, −β∆AJ =

∞ X βn n=1

n!

hhAn ii0 (∗)n F(t)n ,

(6.36)

where hh(...)ii0 are cumulant averages [99]. We now show how, to second order in the external fields, the Jarzynski equality is satisfied. The first step is to define the work done by the system in terms of the external fields, Z

t

W (t) =

ds A(s) ∗ 0

∂F(s) . ∂s

® ­ Next, we rewrite eβW (t) in terms of cumulants, i.e., Ã∞ ! X 1 ­ βW (t) ® n n = exp e β hhW (t) ii , n! n=1

(6.37)

(6.38)

where the averages are non-equilibrium averages. Response theory provides a formalism that can be used to calculate these non-equilibrium averages to any desired power in the

213 perturbing Hamiltonian [99, 90]. In particular, in the linear regime, hhB(t)ii = hhBii0 Z t DD EE ˙ − s)A −β ds B(t ∗ F(s) + O(F2 ), 0

0

(6.39) where B is some observable and B˙ is the time derivative of B using the reference Hamiltonian, H0 . We will compare Eqs. (6.36) and (6.38) to second order in the external fields and since W (t) is already linear in F, the nonlinear response terms can be neglected. Hence, using Eq. (6.39), hhW (t)ii and hhW (t)2 ii are evaluated up to second order in the fields, giving β hhW (t)ii Z t ∂F(s) = β ds hhA(s)ii ∗ ∂s 0 Z t Z s DD EE ∂F(s) ˙ − s0 )A = β hhAii0 ∗ F(t) − β 2 ds ds0 A(s + O(F3 ), (∗)2 F(s0 ) ∂s 0 0 0 (6.40) and ®® β 2 β 2 ­­ 2 W (t) = 2 2

Z

Z

t

t

ds 0

0

0

ds hhA(s)A(s 0

0 2 ∂F(s) ∂F(s ) )ii0 (∗) ∂s ∂s0

+ O(F3 ),

(6.41)

where we have used that fact that F(t) = 0 for t ≤ 0 and that the equilibrium canonical distribution function is stationary. Equation (6.40) is, of course, an example of the usual fluctuation-dissipation theorem result in a classical system (see, e.g., Refs. [146] or [26]). In Eq. (6.40), the second term of the second line can be integrated by parts using the fact that,

DD EE ∂ ˙ − s0 )A A(s = − 0 hhA(s − s0 )Aii0 , ∂s 0

(6.42)

214 and we find that Z

t

∂F(s) hhW (t)ii = hhAii0 ∗ F(t) + β ds hhAAii0 (∗)2 F(s) ∂s 0 Z t Z s ∂F(s) ∂F(s0 ) −β ds ds0 hhA(s)A(s0 )ii0 (∗)2 ∂s ∂s0 0 0 β = hhAii0 ∗ F(t) + hhAAii0 (∗)2 F(t)2 Z Z t 2 ∂F(s) ∂F(s0 ) β t ds ds0 hhA(s)A(s0 )ii0 (∗)2 . − 2 0 ∂s ∂s0 0 (6.43) The last term in this expression is strictly positive and is responsible for any dissipation, and will be denoted as Wd below. The first two terms are just the ones expected using perturbation theory on a quasi-static Hamiltonian and are equal to ∆AJ to O(F2 ), cf. Eq. (6.36). With these observations, we see that the second cumulant of the work, cf. Eq. (6.41) is, up to factors of β, just the dissipative part of the work, and thus, finally, β hhW (t)ii +

®® β 2 ­­ β2 W (t)2 = β hhAii0 ∗ F(t) + hhAAii0 (∗)2 F(t)2 + O(F3 ), 2 2

(6.44)

which is in agreement with the Jarzynski relation, again, to second order in the external fields. Non-linear response theory could, in principle, be used to prove the Jarzynski equality to all orders in the external fields. Aside from the fact that this becomes messy very quickly, there is no need for such a proof. After all, the equality has already been proved by Jarzynski quite generally in Ref. [77]. The above approach is interesting because it clarifies how the equality works by showing how apparently dissipative terms cancel. In fact, even though hhW (t)ii itself is a non-equilibrium average that depends very much on how the work is performed, the dissipative terms (the ones containing ∂F(s)/∂s) in Eq. (6.43) are exactly canceled by the next term in the work cumulant expansion, Eq. (6.41). Also note that this cancellation only works if the β in Eq. (6.38) equals the one that appears in the

215 initial canonical distribution function. We have checked that, at least to third order in the fields, all the (F3 ) Jarzynski terms are contained in hW (t)i; the rest of the terms in hW (t)i are dissipative in the sense that they explicitly depend on the rates of change of the F’s (e.g., as in the last term on the right hand side of Eq. (6.43)). The response theory result seems to contradict what we found thermodynamically or for the ideal gas expansion considered in the preceding section, namely, there, even in the quasi-static or reversible limit, the Jarzynski bound was an upper bound to that predicted by thermodynamics. Here the response theory suggests that the Jarzynski bound is satisfied exactly in the quasi-static limit, i.e., where ∂F(s)/∂s → 0. There is no contradiction for several reasons. The first is trivial. The perturbing potential should be considered as a moving finite step potential with height V0 that is set to infinity at the end of the calculation (for a discussion of the orders of limits, see Ref. [85]). Response theory assumes that the perturbing potential is small, something that is not the case in Sec. 6.3. The second reason why there is no contradiction is more subtle and requires us to be more careful defining what is meant by a quasi-static process. A quasi-static process is one that takes place at a rate which is much slower than all other dynamical processes taking place in the system. Many-body systems have collective modes corresponding to various mechanically conserved quantities and these will evolve on arbitrarily long time-scales (governed by the wavelength of the mode). As we now show, there are no real quasistatic processes in the sense of the Jarzynski equality unless some addition assumptions are made about the nature of the perturbation applied to the system. We start by introducing a projection operator [95, 94], P, defined as follows, PA ≡ hhACii ∗ hhCCii−1 ∗ C,

(6.45)

216 where C is a column vector containing the densities of slowly evolving variables. At the simplest level C must contain the densities of conserved quantities (e.g., number, energy and momentum densities in a one-component system) but could also contain brokensymmetry variables [26], as well as multi-linear products of these fields should modecoupling effects be important. hh(...)ii is a cumulant average taken in the reference system, and for the rest of this section, we omit the zero subscript. Well known projection operator identities can be used to express hhA(t)Aii in terms of correlations of slow and fast quantities (here, we simply state the result since these techniques are standard). After Laplace transforming in time, we obtain DD EE DD EE ˜ ˜ ‡ (s)A‡ A(s)A = A ³ DD EE´ ˜ ‡ (s)C˙ ‡ + hhACii − A DD EE ˜ ∗ hhCCii−1 ∗ C(s)C ∗ hhCCii−1 ³ DD EE´ ˜ ‡ ‡ ˙ ∗ hhCAii + C (s)A ,

(6.46)

where the tilde denotes a Laplace transform in time, s is the Laplace transform frequency, ˜ ‡ (s) is the Laplace transform of and where A A‡ (t) ≡ ei(1−P)Lt (1 − P)A.

(6.47)

This object is referred to as the dissipative part of A and is orthogonal to the space of slow variables for all times; hence, their correlations should decay on microscopic time (and length) scales. In terms of the Laplace transforms, this means that, for long-time DD EE ˜ phenomena, i.e., small s, only the frequency dependence of C(s)C need be considered and will give all the long-time dependence. All the other correlations can be evaluated at s = 0; indeed, for the standard hydrodynamic variables, the hhC(s)Cii have been studied extensively [26, 150].

217 We now can analyze the so-called dissipative term in Eq. (6.43). Remember that in this term, there are two implicit spatial integrations and for translationally invariant equilibrium systems it is convenient to switch to a Fourier representation in space, thereby obtaining Z

Z

t

Z

t1

dk hhAk (t1 − t2 )A−k ii (2πL)d 0 0 : F˙ −k (t1 )F˙ k (t2 ) Z t Z t1 Z EE dk hDD ‡ ‡ ∼ −β dt1 dt2 A (t − t )A 1 2 k −k (2πL)d 0 ³ 0 DD EE´ ˜ ‡ (0)C˙ ‡ + hhAk C−k ii − A k −k

Wd ≡ −β

dt1

dt2

(6.48a)

× hhCk C−k ii−1 hhCk (t1 − t2 )C−k ii hhCk C−k ii−1 ³ DD EE´i ˜ ‡ ‡ ˙ × hhCk A−k ii + Ck (0)A−k : F˙ −k (t1 )F˙ k (t2 ),

(6.48b)

where d is the dimension of space and Ld is the volume of the system. The second relation follows by using Eq. (6.46) and ignoring the frequency dependence of the dissipative correlations as discussed above. Within the context of the projection operator approach, this is valid provided that the F’s don’t evolve on fast (i.e., microscopic) time scales. We now examine Eq. (6.48b) for several cases. The simplest is when the perturbing fields are spatially uniform, i.e., Fk (t) = Ld ∆(k)F(t), where ∆(k) is a Kronecker-δ (strictly speaking, we have to turn the Fourier integrals in Eq. (6.48b) into Fourier sums R P by letting dk → (2π/L)d k to handle this case). Since C˙ k (t) = ik · Jk , where the Jk ’s are fluxes, it is easy to show that Wd

DD EE Z t ‡ ‡ ˙ 1 )F(t ˙ 1) ˜ ∼ −β AT (0)AT : dt1 F(t 0

β − hhAT CT ii hhCT CT ii−1 hhCT AT ii : F(t)F(t), 2 (6.49)

218 where the subscript “T” denotes the total or space integral (i.e., k → 0 limit) of subscripted EE DD ‡ ‡ quantity and where we have assumed that the decay of the AT (t)AT correlation func˙ tion is on a much faster time scale than any characterizing the F’s. The first term on the right hand side of Eq. (6.49) is well behaved provided that F˙ is ˙ square integrable and in particular should vanish in the quasi-static limit (F(t) → 0, t → ˙ ∞, keeping F(t)t constant); the remaining terms clearly don’t vanish and are comparable to the O(F2 ) terms in the Jarzynski free energy difference, cf. Eq. (6.44). Moreover, these terms are negative semi-definite, and thus, the response theory also shows that −∆AJ is an upper bound to the actual work done, even when the process is quasi-static. Indeed, when Eq. (6.49) is used in (6.43), the latter becomes β DD ‡ ‡ EE hhW (t)ii = hhAT ii0 · F(t) + AT AT : F(t)2 2 0 DD EE Z t ˜ ‡ (0)A‡ ˙ 1 )F(t ˙ 1 ). −β A : dt1 F(t T T 0

(6.50) which shows that it is only the parts of A that are orthogonal to the conserved quantities that contribute to the quasi-static work at O(F2 ). Within the response approach, in order that −∆AJ equals the average work done quasi-statically, the last terms in Eq. (6.49) must vanish. Since hhCT CT ii is positive definite the only way to eliminate these terms for nonzero F (t) is to have ¯ ∂ hCT iF ¯¯ ∂ hAT i hhCT AT ii = = = 0, ¯ ∂βF F=0 ∂βΦ

(6.51)

where h...iF is a canonical average using the Hamiltonian H0 − AT · F, βΦ is a column vector containing the usual conjugate variables to C found in the theory of fluctuations (e.g., −β for energy, βµ for number, etc.). In other words, the quasi-static perturbation does not couple to the conserved variables, and hence, in the linear response regime, does not change the latter’s averages. For us, this implies that the process must be strictly

219 isochoric and isothermal. It is interesting to note the strong similarity of this criterion to one made by one of us some time ago [151] concerning the independence of the choice ensemble in response treatments of relaxation experiments. While the preceding analysis at k = 0 is appropriate when discussing the thermodynamics of large systems, it is easy to imagine other types of experiments. One such setup is where the perturbing fields are periodic in space. For a monochromatic (in space) perturbation the relevant time scales to compare with are those contained in hhCk (t)C−k ii, and while these can be very long, they are finite for nonzero wave-vectors. In this case, Wd would vanish in the quasi-static limit, to leading order in the response theory. We have qualified this last statement because, higher order terms in the response theory for Wd will allow for mode-couplings to zero wave-vector, and may result in problems like those just shown. This will be investigated in a later work. Finally, we consider one last example, one perhaps more appropriate to some of the recent optical tweezers experiments, namely one where the probe-field is localized, i.e., F(r, t) = δ(r)F(t). We consider the small wavenumber contributions to Wd , using hydrodynamic approximations for the hhCk (t)C−k ii correlations. The most important terms that result from this analysis are those that decay slowest in t1 − t2 and these are those corresponding to diffusive modes (e.g., thermal or mass diffusion) with time dependence e−Γα k

2 |t

1 −t2 |

. All other correlations can be evaluated at zero wave-vector. With this, the

hydrodynamic contribution to Wd can be written as: Wd ∼ −β

XZ α

Z

t

t1

dt1 0

dt2 0

Ψ∗α (t1 )Ψα (t2 )Sd γ(d/2, Γα kc2 |t1 − t2 |) , 2(2π)d (Γα |t1 − t2 |)d/2

(6.52)

where Sd ≡ 2π d/2 /Γ(d/2) is the area of a d-dimensional unit hypersphere, Γ(x) is the gamma function, γ(x, y) is the incomplete gamma function, the sum is over all diffusive modes, each with diffusivity Γα , kc is a large wavenumber spherical cutoff necessitated by

220 ignoring the k dependence of the correlation functions, and ˙ Ψα (t) ≡ u†α hhCT CT ii−1 hhCT Aii F(t),

(6.53)

where a hydrodynamic approximation for hhCk (t)C−k ii was used; i.e., hhCk (t)C−k ii ∼ Ld

X

2

uα e(ikωα −Γα k )t u†α ,

(6.54)

α

where uα is the amplitude of the α’th mode. Only purely diffusive modes (those with ωα = 0) are kept. Clearly, this part of the so-called dissipative contribution to the work contains slowly decaying (in-time) contributions, and to get a feel for how important they are, we now assume constant rates, i.e., we assume that the Ψα ’s are constant in time. With this, the integrals in Eq. (6.52) can be done and give Wd ∼ −

X

β|Ψα |2 kcd t2 f (d, kc2 Γα t),

(6.55)

α

where Sd f (d, x) ≡ (2π)d

µ

2x−d/2 γ(d/2, x) e−x − 1 + 2 (d − 2)(d − 4) x (d − 4) ¶ −x 2e + d − 4 + . x(d − 2)(d − 4)

At long times, specifically for kc2 Γα t À 1 it follows from this last result that    t2−d/2 for d < 2    X Wd ∝ |Ψα |2 t log(t) for d = 2 ,   α    t for d > 2

(6.56)

(6.57)

˙ constant, this last result In the quasi-static limit, namely, F˙ → 0, t → ∞ keeping Ft implies that Wd vanishes and the Jarzynski bound for the work becomes an equality for localized probes in any spatial dimension. However, note, Wd decays more slowly, all other things being equal, for d ≤ 2.

221 In summary, response theory shows that the free energy change defined by the Jarzynski equality in general is not the state function that arises in work bounds in thermodynamics, even for quasi-static processes. Instead it is an upper bound to the well known thermodynamic ones. Under special conditions, namely, where there is no coupling between the perturbation and macroscopic state variables, the two bounds become equivalent, but as we have shown, this is probably not the case in macroscopic, non-isothermal (and constant chemical potential, total density, etc., processes, cf. Eq. (6.51) and the discussion that follows). One important exception to this result is for broad-spectrum (in wave-vector) probes, i.e., ones that are spatially localized. Basically there, the couplings to the very long-wavelength slow modes are weak and a quasi-static limit is possible. In some sense this is similar to the observation in Sec. 6.3 for the limit Lf /Li → 1, although both here and there, the amount of work done becomes small. 6.5 Discussion In this article, we have studied the Jarzynski equality both generally and within the context of a well understood system, the one dimensional expanding ideal gas. We chose this latter system because it has been shown that the Jarzynski equality is satisfied unambiguously for all choices of system length and piston velocity. The simplicity of this system allowed us to obtain a general form for the full distribution function at any time during and after the expansion. Therefore, the non-equilibrium state of this system is completely known at all times. We showed that, although the Jarzynski equality is correct and clearly shows that ­ βW ® is an invariant, equal to a ratio of equilibrium canonical partition functions, its e applicability to non-equilibrium phenomena, and in particular to nonequilibrium thermodynamics, is more subtle. First, at the end of the expansion process, the distribution function is not canonical, and moreover, quantities like the free-energy and entropy cannot be defined in terms of their standard equilibrium definitions. Second, if equilibrium is ever attained (remember that our model system never quite reaches equilibrium), the free-energy

222 of the final state will, in general be different than what can be suggested from the Jarzynski equality. This is a consequence of the final temperature that is usually different than the initial one. In fact, in reply to the criticism of Cohen and Mauzerall [79], Jarzynski [77] pointed out that the final free-energy that appears in the Jarzynski equality will be the one of the final state, provided we wait long enough and that the overall system contains a large enough bath such that the temperature change can be assumed to be zero. This argument is correct, but should be taken with caution. In fact, there is no limit to the external work done and it is always possible to perform enough work, on a macroscopic region of the system, such that, even with a large bath, the temperature of the full system changes. In the experiment of Liphardt et al. [78], where the work is done on a single molecule in solution, it would seem safe to assume that the final and initial temperatures are the same. This also happens, not surprisingly, for our model system: when the system is very large compared to the expansion L À V t, the relative work and the temperature change tend towards zero (recall Eqs. (6.31) and (6.26)). Fundamentally, however, the 1d expanding gas model represents an adiabatic process and there must be some temperature change, unless the gas is coupled to a thermostat, in which case, ∆AJ has been shown to be equal to the actual energy change of the system [89]. In Sec. 6.3, we compared the bound implied by the Jarzynski equality against a thermodynamic bound which is obtained from the First and Second Law of Thermodynamics. As shown by Eqs. (6.24) or (6.28), the Jarzynski bound is an upper bound to the thermodynamic upper bound to the work, and hence, is does not prove the Second Law of Thermodynamics. The two bounds become close to each other in the limit where the extent of the expansion is small. This is another consequence of the fact that the work done can produce changes in key quantities that define the ensembles in the initial and the final equilibrium states (e.g., temperature, cf. Eq. (6.51) and the subsequent discussion). We also showed that the statement that −∆AJ = Wrev can be incorrect even in the quasi-static limit, and in particular, is wrong for our system (see, e.g., Fig. 6–4). This

223 is an important observation since this assumption is often made in experiments. In the experiment by Liphardt et al., this assumption may be justified because the work is done on a very small part of the system. On the other hand, as we have shown in Eq. (6.50), any coupling to long wavelength fluctuations of conserved quantities leads to a negative correction to Wd . Liphardt et al. measure quantities like −∆AJ − < W >quasi-static , hence, if their apparatus is not the ideal δ-function coupling considered in the preceding section, but contains some small, even O(1/N ), coupling to the conserved quantities, Eq. (6.50) predicts that they should obtain < WJE − WA,rev >= β − hhAT CT ii hhCT CT ii−1 hhCT AT ii : F(t)F(t), 2

(6.58)

in the linear response regime, (here we are using Liphardt et al.’s notation and sign convention for the work). Thus a quadratic, negative correction should be seen; interestingly, this is exactly what is shown in Fig. 3A of Ref. [78], although there the authors dismiss this as experimental error. If this is the explanation, the coupling must clearly arise from the macroscopic parts of the experimental device, here, probably associated with the piezoelectric actuator used to pull the molecule. A similar observation was made by Oberhofer et al. [88] in their numerical study of equilibrium free energies using the schemes based on Jarzynski equality and on the Widom insertion method in a soft sphere liquid. There, < WJE − WA,rev > was also shown to be non-zero and this result was explained in terms of adiabatic invariants. Numerically, they obtained a definite sign for < WJE − WA,rev > which agrees with our prediction. The argument is basically an example of more general problems encountered in ergodic theory and in the construction of ensembles when phase space is metrically decomposable [152, 151]. The analysis presented in Sec. 6.4 for systems where the work couples to the densities of conserved quantities offers a quantitative estimate for the difference.

224 We highlight the fact that −∆AJ 6= Wrev by briefly investigating the Jensen-PeierlsGibbs-Bogoliubov inequality which can be proved as follows. By defining a function ® ­ h(λ) ≡ ln eλW it follows that ­ ® W eλW dh(λ) = ≡ hW iλ (6.59) dλ heλW i and ® ­ ®2 W 2 eλW W eλW − heλW i heλW i2 ­­ 2 ®® ≡ W λ,

d2 h(λ) = dλ2

­

(6.60)

where the hh(...)iiλ is a cumulant average. Using the Jarzynski equality and the two averages defined above, it is straightforward to show that, Z −β∆AJ = β hW iλ=0 +

β

dλ 0

­­ 2 ®® W λ (β − λ).

(6.61)

Because the second term of the right hand side (we call it the work fluctuation term) is strictly positive, we immediately have β hW iλ=0 ≤ −β∆AJ (hW iλ=0 is just the usual average work). Hence, hW i = −∆AJ only when the work fluctuations are zero for all λ. In particular, as stated in Ref. [149], the fluctuations have to be zero in the original canonical ensemble, when λ = 0. These fluctuations can be non-zero even if the work is done quasi-statically, as we showed in Sec. 6.4. In fact, we showed that the fluctuation terms, to lowest order in the external fields, will differ from zero in the quasi-static limit only if the system variables, A(r, t), to which the fields couple, are orthogonal to the C(r, t)’s, the conserved quantities or if the fields have negligible Fk=0 (t) components. Recall that, in a case of microscopically localized field, the Fk=0 (t) contribution is small and the quasi-static limit is attainable. We also showed that the true free-energy change of the system, for a fixed change in volume, tends to free-energy change that appears in the Jarzynski equality, ∆AJ when the expansion is very fast (the piston velocity, V , is large). This result is expected, since,

225 within this model, the average work tends to zero for large piston velocities and produces a negligible temperature change. Finally, the criticism we wanted to raise in this paper is not against the Jarzynski equality, which we believe to be correct, but rather to how it is often interpreted. We worked on a system that falls in a class described by Hamiltonian dynamics, which is fundamentally adiabatic since the Hamiltonian could, in principle, include everything (in the language of Jarzynski, everything means the system and the bath). We showed that, for that class of systems, the Jarzynski equality can lead to erroneous conclusions. In particular, the free energy change that appears in the equality can have little to do with the actual non-equilibrium thermodynamics free-energy computed by standard methods (even when the system equilibrates) and the Jarzynski equality does not contain any information about the non-equilibrium state of the system. Further, we want to re-emphasize that the Jarzynski equality cannot be used as a proof of the Second Law of Thermodynamics because the bound that it provides is above the Thermodynamical upper bound to the work. 6.6 Appendix: non-equilibrium distribution function As shown in Ref. [86], the positive initial velocity u0 interval that result in n collisions with the piston at a later time t is, (2n − 1)(Li /t + V ) − x0 /t < u0 < (2n + 1)(Li /t + V ) − x0 /t,

(6.62)

where x0 is the initial position of the gas particle. For such a case, the final velocity is, u = 2nV − u0 ,

(6.63)

x = −x0 − u0 t + 2n(Li + V t).

(6.64)

and the final position,

When the final position is smaller than zero, it means that there has been a further collision with the hard wall after the last collision with the piston. In such a case, the sign of the final velocity and position in Eqs. (6.63) and (6.64) is reversed. For negative initial velocities,

226 the interval that leads to n collision is, −(2n + 1)(Li /t + V ) − x0 /t < u0 < −(2n − 1)(Li /t + V ) − x0 /t.

(6.65)

In this case, the final velocity and position are, u = 2nV + u0 ,

(6.66)

x = x0 + u0 t + 2n(Li + V t).

(6.67)

and

Again, if this results in a negative position, the sign of the last two equations is reversed. The above results can be used to obtain the final distribution function from an integration over the initial velocities and position weighted by the initial distribution function, f (x, u; t) = Z Z dx0 du0 f0 (x0 , u0 ){Θ(Li /t + V − x0 /t − u0 )Θ(x0 /t + u0 + Li /t + V ) ×[Θ(x0 + u0 t)δ(u − u0 )δ(x − x0 − u0 t) + Θ(−x0 − u0 t)δ(u + u0 )δ(x + x0 + u0 t)] ∞ X + [Θ((2n + 1)(Li /t + V ) − x0 /t − u0 )Θ(x0 /t + u0 − (2n − 1)(Li /t + V )) n=1

×(Θ(−x0 − u0 t + 2n(Li + V t)) ×δ(u − (2nV − u0 ))δ(x + x0 + u0 t − 2n(Li + V t)) +Θ(x0 + u0 t − 2n(Li + V t)) ×δ(u + (2nV − u0 ))δ(x − x0 − u0 t + 2n(Li + V t))) +Θ(−(2n − 1)(Li /t + V ) − x0 /t − u0 )Θ(x0 /t + u0 + (2n + 1)(Li /t + V )) ×(Θ(x0 + u0 t + 2n(Li + V t)) ×δ(u − (2nV + u0 ))δ(x − x0 − u0 t − 2n(Li + V t)) +Θ(−x0 − u0 t − 2n(Li + V t)) ×δ(u + (2nV + u0 ))δ(x + x0 + u0 t + 2n(Li + V t)))]}. (6.68)

227 After the integrals are performed, Eq. (6.7) is obtained.

CHAPTER 7 Conclusions and Ideas for Continued Research 7.1 Conclusions The diffusion in channeled structures studies that are presented in this work have the advantage of being more general than the one presented in earlier studies. More precisely, in this work, the vibrations of the crystal were not only included, but a general formalism, that can be applied to any crystal structures, that completely includes vibrational spectrum was developed and implemented. In previous studies, the lattice vibrations were included through the motion of specific target atoms only or from a Langevin equation formalism where the bath spectrum was assumed to follow a simple model (i.e. the Debye model) to reduce the numerical effort. Here, generalized Langevin equations were used with a simple approximation for the memory function such that, keeping a reasonable numerical effort, the complete vibrational density of states of the crystal was reproduced. This general procedure was used to study the guest motion inside porous crystal channels. From these simulations, microscopic information, based on time-correlation functions of the guest, was used to calculate the system’s macroscopic permeability. The permeability was obtained from the full microscopic dynamics. In other words, it was never assumed that the diffusion of the guest is dominated by specific pathways (reaction coordinates). This approximation is implied by transition state theory, a method which, combined with a hoping model for the diffusion, is often used to study the guest motion in channeled structures. The formalism that is developed here could then be used to test the validity of this particular assumption of transition state theory. This was done for two specific systems. One containing large channels and small energy barriers, Xenon in Theta-1, and the other containing narrow channels and large energy barriers.

228

229 The study of these two systems clearly showed that the various approximations that were made in previous studies are often not justified. In fact, for Xenon diffusing in Theta1, the guest motion is not well described by a hopping process with hopping rate constants obtained from TST. For that system, the results showed that the crystal vibrations played a negligible role in the diffusion process. On the other hand, for Argon diffusing in αquartz, the lattice vibrations reduced the overall diffusion of the guest while the guest motion appeared to be better described by a reaction coordinate. The formalism that was developed here is free from these approximations. For systems where the lattice vibrations coupled strongly enough with the guest such that it affected its diffusion, it was only natural to investigate the role of quantum mechanics in the diffusion process. If every degree of freedom was treated quantum mechanically, the problem would be intractable. In fact, the problem would effectively reduce to solving, numerically, the time-dependent Schrodinger equation for a very large number of atoms. Here, this problem was avoided by going to the path integral representation and taking a semi-classical approximation. Within this approximation, the problem was reduced to the simulation of a set of modified generalized Langevin equations that could be used to calculate the necessary correlations functions. This semi-classical formalism was used to obtain the quantum corrections for the permeability of α-quartz to neon. These quantum corrections reduced the permeability by a factor of about 25%. Prior to this work, quantum corrections were always neglected when studying diffusion in and through channeled structures. In the semi-classical calculation, an approximation for the density matrix had to be used. There, the density matrix was calculated treating the guest as a free-particle evolving in constant potential. There exists other types of approximate methods that relax this approximation and that are exact for harmonically interacting guest. One of the best of these methods is the one suggested by Feynman and Kleinert [71]. Even if this method is not applicable to the problem here, because it assumes that the interaction potential is

230 bounded everywhere, an improvement of this approximate method was suggested. This was done by using a reference harmonic potential whose center is allowed to move in time. With this modification, the original Feynman-Kleinert method was especially improved in the calculation of the off-diagonal parts of the density matrix. The last part of this thesis was completely separated with the theme of diffusion in channeled structures and contained an analysis of the Jarzynski equality [75] and its connection with thermodynamics. The Jarzynski equality has recently attracted a lot of attention in the statistical mechanics community since it is one of the very few exact relations that hold far from equilibrium. Here, a simple ideal expanding gas model was studied together with general thermodynamics or response theory arguments. This resulted in the conclusion that, even if the Jarzynski equality provides a work bound and predicts a free-energy change for processes that carry a system arbitrarily far from equilibrium, the obtained work bound and free-energy changes are different than what is predicted from the Laws of Thermodynamics. The fact that the work bound predicted from the Jarzynski equality differs from the Thermodynamics work bound is particularly important because it disproves the contention that the Jarzynski equality is a proof of the Second Law. A projection operator formalism combined with response theory was also used to derive conditions under which the Jarzynski and the Thermodynamic work bounds differ. 7.2 Suggestions for Continued Research Clearly, the formalism that was developed to study the diffusion in channeled structures could be obtained with a myriad of systems. The systems that were studied here were chosen because of their simplicity. It would be relatively easy to slightly modify the formalism in order to study the diffusion of small molecules. As an example of this, the formalism would be well-suited for a system like H2 molecules in carbon nanotubes. This system is relevant because it is directly related to energy storage processes, which is drawing a lot of attention from the scientific community.

231 Another aspect that was not consider here is the crystal orientation. For both crystal systems that were studied, Theta-1 and α-quartz, the net diffusion was forced along the z axis of the respective crystal systems. In Theta-1, in practical terms, no net flux is possible along the other directions since the channels are disconnected. In α-quartz, flow is possible along other directions, albeit with larger potential barriers to overcome as shown in Fig. 4– 4. The permeability could, in principle, be obtained for any other crystal orientation using the formalism as is. In practice, however, this may not be easily feasible. The reason is that the permeability theory relies on the evaluation of various plane averages that are obtained from specific correlation functions, the dynamical information, multiplied by a Boltzmann factor. As expected from transition state theory, we expect that, in the plane that contains the transition state, the plane average are more or less dominated by a region around the transition point. For the crystal orientations that were considered here, this was the case. For other crystal orientation, it is possible that, within the transition point plane, there is a region with a much lower potential of mean force. In this case, we expect the dynamical factors to be small in the low potential regions and larger close to the transition state such that, overall, the plane is still dominated by the transition state region. This may never happen if the energy difference between the two regions is too large. In that case, the Boltzmann factor will outweigh the dynamical factors unless very accurate correlation functions are obtained, something that is numerically very demanding. For such a situation, one option would be to reformulate the theory of the permeability in terms of curvilinear coordinates such that situations like the one described above can be avoided. The prescription that would define the new set of coordinates should be chosen such that within this coordinate system, a minimum is never found in a transition state plane. Such a reformulation was done for single barrier system [50] but still has to be done for the more general case. The last idea for continued research that is presented here is related to the work on the Jarzynski equality. In fact, at the end of the last chapter, linear response theory is used to

232 establish under which conditions the Jarzynski work bound becomes a lowest upper bound. There, we showed that when the external fields couple to the k = 0 mode of densities of conserved quantities, the Jarzynski work bound becomes larger than the lowest upper bound. It would be interesting to carry the analysis to higher order in response theory where mode-coupling effects appear. In fact, it would be interesting to see how, for some fields (like the one that is a delta function in space) where the Jarzynski work bound seems to be the lowest upper bound, higher order terms couple to the k = 0 component of the densities of conserved quantities. If this coupling is strong enough, it is possible that this introduces some correction between the lowest work bound and the Jarzynski work bound, therefore making the later larger than the lowest upper bound to the work. Here, this is left as an open question.

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