1.4.4 Covariance Matrix Adaptation Evolutionary Strategies . ...... para-bromobenzonitrile (pbbn), 2,4-hexadiyne (hex), ortho-dicyanobenzene ...... which are dependent up on its configuration, include the mean-square dipole moments, the.
Chemical Physics and the Condensed Phase: NMR Studies in a Liquid-Crystal Testing Ground by Adrian C. J. Weber
B.Sc., University of Manitoba, 2005
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Chemistry)
THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2010 c Adrian C. J. Weber 2010
Abstract Liquid crystals are an excellent media for the study of the condensed phase by NMR spectroscopy since the highly accurate proton dipolar couplings do not average to zero as they do in the isotropic condensed phase. Of course we can also take the opposite view and seek to understand the behavior of individual molecules and the effect of the condensed phase on them and so the impetus for studies of solutes in liquid crystals is two fold. By coupling theory to experiment via dipolar couplings one can gain insight into aspects of chemical physics and the condensed phase provided the spectra can be solved. As the number of spins of a molecule and its lack of symmetry increase so do the complexity of NMR spectra of solutes in orientationally ordered phases. Covariance Matrix Adaptation Evolutionary Strategies (CMA-ES) have proven to be remarkably useful towards the end of obtaining dipolar couplings from congested spectra. In essence this algorithm uses the principles of natural selection coupled with an aspect of cross-generational memory to find the set of spectral parameters at the global minima of an error surface which reproduce the experimental spectrum. It is not an overstatement to say this tool has significantly altered the allocation of efforts in the area of research presented here. In the research herein two approaches are employed which are complimentary. In the first chapters we use a diversity of solutes to test postulated interaction Hamiltonians intended to describe the intermolecular environment of nematic and smectic A phases. The putative Hamiltonians are fitted to solute order parameters obtained from dipolar couplings. Once an explicit form is obtained, reasonable speculation is made concerning what the Hamiltonian can tell us about the intermolecular environment of the condensed phases studied. In the latter chapters the complimentary view is taken. Specifically we attempt to understand how internal rotations of molecules are affected by the condensed phase enviii
Abstract ronment. To this end is considered the simplest example in n-butane. Again by obtaining dipolar couplings we can use a variety of theoretical tools in an attempt to exploit the full accuracy of these anisotropic spectral parameters and gain insight into the effect of a condensed phase on configurational statistics. These phenomena are also studied as a function of temperature.
iii
Table of Contents Abstract
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ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
List of Tables
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vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Dedication
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xi
Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Nematic Liquid Crystals . . . . . . . . . . . . . . . . . 1.1.3 Other Liquid-Crystal Phases . . . . . . . . . . . . . . . 1.2 Anisotropic Intermolecular Interactions . . . . . . . . . . . . . 1.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Probing the Potential . . . . . . . . . . . . . . . . . . . 1.3 Model Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Maier Saupe . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Size and Shape . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Kobayashi McMillan . . . . . . . . . . . . . . . . . . . 1.4 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Dipolar Couplings and Orientational Order Parameters 1.4.4 Covariance Matrix Adaptation Evolutionary Strategies 1.5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . .
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1 1 1 1 2 3 3 5 6 6 7 8 10 10 10 11 13 14 16 18
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Table of Contents 2 Solute Order Parameters: Application of MSMS-KM Theory . . . . . . 2.1 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24 31
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 The 3.1 3.2 3.3 3.4 3.5 3.6 3.7
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38 39 41 42 45 47 51 52
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 The Butane Condensed Matter Conformational Problem . . . . . . . . 4.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64 74 74
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
5 Condensed Phase Configurational Statistics and Temperature Effects 5.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85 89 89
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
6 Conclusion
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95
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
Smectic Effect Rationalized by Introduction . . . . . . . . . . . . Experimental . . . . . . . . . . . . The Nematic Potential: MSMS . . The Smectic Potential: MSMS-KM Results and Discussion . . . . . . Conclusion . . . . . . . . . . . . . Tables and Figures . . . . . . . . .
MSMS-KM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Appendices A Experimental detail concerning chapters 2 to 5 . . . . . . . . . . . . . . .
99
B Experimental order parameters . . . . . . . . . . . . . . . . . . . . . . . . .
100
C Structural details of n-butane calculated with Gaussian 03
106
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v
List of Tables 3.1
Maier-Saupe solute parameters . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Spectral parameters (Hz). The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for simultaneous fits to all four sets of dipolar couplings. The CCH angle was varied in the top set of calculations and held constant in the bottom set. The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculated and experimental dipolar couplings . . . . . . . . . . . . . . . . RIS Chord. The numbers in brackets are the errors in the last digit. . . . . Calculated conformer order parameters using the RIS approximation for the Cd and CI(2k) models. The calculated order parameters shown are from fits to n-butane dipolar couplings when dissolved in the MM and while varying the CCH bond angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
4.3 4.4 4.5
5.1
5.2 5.3
Experimental dipolar couplings of n-butane in 1132 as a function of temperature. The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CCd fitting parameters as a function of temperature. The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . CI(2k) fitting parameters as a function of temperature. The numbers in the brackets are the errors in the last one or two digits. . . . . . . . . . . . . .
B.1 B.2 B.3 B.4 B.5 B.6
Experimental Experimental Experimental Experimental Experimental Experimental
order order order order order order
parameters parameters parameters parameters parameters parameters
of of of of of of
sample sample sample sample sample sample
1 2 3 1 2 3
in in in in in in
C.1 C.2 C.3 C.4 C.5
Atom labels of n-butane nuclei and bonding Bond lengths and angles of trans n-butane . Dihedral angles of trans n-butane . . . . . . Bond lengths and angles of gauche n-butane Dihedral angles of gauche n-butane . . . . . .
8OCB 8OCB 8OCB 8CB . 8CB . 8CB . . . . . .
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53 75
77 78 79
80
90 91 92
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100 101 102 103 104 105
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106 107 108 109 110
vi
List of Figures 1.1 1.2 1.3 2.1
2.2
2.3
2.4 3.1
3.2
3.3
Example liquid crystal structure of EBBA . . . . . . . . . . . Liquid-crystal phases running left to right and top to bottom; matic, smectic A, smectic C, chiral nematic, crystalline . . . . The first four generations of an ES taken from [32]. . . . . . .
. . . . . . . . isotropic, ne. . . . . . . . . . . . . . . .
The upper plot is of the experimental 400MHz NMR spectrum while the lower is found using the CMA-ES. The peaks of the solutes (from left to right: tcb, clpro, fur and thio shown above in the molecule fixed coordinate system) are intersperced with one another. . . . . . . . . . . . . . . . . . . . The asymmetry in the order parameters (R) for fur, thio and clpro are plotted against their respective Sxx . Although vibrational corrections have been neglected the error in R is small. An arrow marks the phase transition and the lines are the best fit to the points in the nematic phase. . . . . . . . . . G8CB,ZZ (1) is plotted against G8CB,ZZ (2) where the black points signify the use of the smectic Hamiltonian and the open points are obtained with the nematic potential only. The ten points closest to the origin are from measurements in the nematic phase while the rest are in the smectic phase. Inset: RMS of fits to the potentials in both phases are plotted against G8CB,ZZ (2). The smectic order parameters τLs are plotted against temperature for each of the solutes tcb, fur, thio and clpro. . . . . . . . . . . . . . . . . . . . . . The structures of solutes in the molecule fixed (with the exception of hex and clpro for which the z-axis lies along the C-C and Cl-Cl directions) frame where the z axis protrudes from the plane of the page. . . . . . . . . . . . . The asymmetry in the order parameters R (Eq. 3.1) of each solute is plotted against its Sxx . Although vibrational corrections have been neglected the error in R is small. The open points correspond to measurements in the liquid crystal 8CB while the filled points are from 8OCB. Nematic points are to the left and smectic to the right. . . . . . . . . . . . . . . . . . . . . . . . GL,ZZ (1) is plotted against GL,ZZ (2) where the filled points signify the use of the smectic Hamiltonian and the open points are obtained with the nematic potential Eq. 3.7 only. The seven (8CB) or eight (8OCB) points closest to the origin are from measurements in the nematic phase while the rest are in the smectic phase. Inset: The RMS of fits to either potential in both phases are plotted against GL,ZZ (2). . . . . . . . . . . . . . . . . . . . . . . . . . .
19 20 21
32
33
34 35
54
55
56
vii
List of Figures 3.4
3.5 3.6
3.7
3.8
4.1
4.2
4.3 5.1
bs,MSMS is plotted against temperature for pdcb and dcnb in 8OCB and 8CB. Points in the smectic phase are to the left and those in the nematic to the right. The open circles were obtained using GL,ZZ (i) from the nematic potential Eq. 3.7 and the filled circles used GL,ZZ (i) from the smectic potential Eq. 3.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The κ′L (i) are plotted with error bars versus temperature with error bars for each liquid-crystal solvent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . mod (i) The modulation of each nematic mechanism from smectic layering HN tot in 8CB at 298.0K and 8OCB at 327.0K (see Eq. 3.11) and their sum HN for tcb (θs =π/2) and hex (θs =0) where the plane of the ring and symmetry axes are aligned along the director, respectively. The centre of the layer is at the origin. The dashed line is the total smectic Hamiltonian HA,Ls (Z) (Eq. 3.11) for each solute’s given orientation. The Hamiltonian prefactors used are G8CB,ZZ (1)=0.952, κ′8CB (1)=0.456, G8CB,ZZ (2)=0.369, κ′8CB (2)=-3.145, G8OCB,ZZ (1)=0.911, κ′8OCB (1)=0.160, G8OCB,ZZ (2)=0.326, κ′8OCB (2)=-1.303, τ8CB tcb =-0.182, τ8OCB tcb =-0.417, τ8CB hex =-0.147, and τ8OCB hex =-0.182. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mod (i) The modulation of each nematic mechanism from smectic layering HN tot in 8CB at 298.0K and 8OCB at 327.0K for pdcb where and their sum HN each molecule-fixed axis is, in turn, oriented along the director. The centre of the layer is at the origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . The smectic order parameters τLs are plotted against temperature for all solutes with error bars in each liquid crystal solvent. The odcb in 8OCB and phac in 8CB τLs values are fixed. . . . . . . . . . . . . . . . . . . . . . . . .
57 58
59
60
61
Calculated a) and experimental b) NMR spectrum of n-butane in 1132 at 298.5K. c) and d) show an expanded region of the calculated and experimental spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energies, calculated from Gaussian 03 for n-butane as a function of dihedral angle φ. The points are for n-butane in the gas phase. The dashed line for the gauche conformer is shifted due to the isotropic part of the intermolecular potential in the condensed phase when the orientational order is described by the CI(2k) model. The solid line for the gauche conformer is shifted downward and obtained when describing the orientational order with the gas CCd model. The Etg calculated by Gaussian 03 is 651 cal mol−1 . . . . . . The probability P (φ) of finding n-butane at dihedral angle φ for the CCd model (solid line) and the CI(2k) model (dashed line) in the MM at 301.4K.
82
The Etg of n-butane as a function of temperature when the orientational potential is described by the CI (open points) and Cd (filled points) models. gas The dashed line represents the constant Etg calculated from Gaussian 03. .
93
76
81
viii
Acronyms CMA-ES = Covariance matrix adaptation evolutionary strategies Cd = Modified chord model C = Circumference model CI = Circumference integration model DFT = Density functional theory ES = Evolutionary strategy hEF Gi = Average electric field gradient GA = Genetic algorithm I = Integration model KM = Kobayashi McMillan L = Liquid crystal ME = Maximum entropy MS = Maier Saupe NMR = Nuclear magnetic resonance RIS = Rotational isomeric state RMS = Root mean square s = Solute
ix
Acknowledgements I would like to thank Elliott, Ron, Kees and Leo for all the valuable advice over coffee.
I would also like to thank my Parents, Abbey and C´eline, for instilling in me the value of dedicating oneself to producing works of quality.
x
To Andrea, my wife
And what is good, Phaedrus, And what is not good– Need we ask anyone to tell us these things?
Robert Pirsig
Qu’on ne dise pas que je n’ai rien dit de nouveau: la disposition des mati`eres est nouvelle.
Blaise Pascal
xi
Statement of Co-Authorship Chapter 2 is a co-authored publication of A. C. J. Weber, X. Yang, R. Y. Dong, W. L. Meerts and E. E. Burnell. The research programs for this chapter were designed by A. C. J. Weber, R. Y. Dong and E. E. Burnell with technical assistance from W. L. Meerts. The research and data analysis were performed by A. C. J. Weber and X. Yang with the paper being written by A. C. J. Weber. Chapter 3 is a co-authored publication of A. C. J. Weber, X. Yang, R. Y. Dong and E. E. Burnell who were involved in similar capacities as in Chapter 2. Chapter 4 is a co-authored publication of A. C. J. Weber, C. A. de Lange, W. L. Meerts and E. E. Burnell. The research programs for this chapter were designed by A. C. J. Weber and E. E. Burnell with input from C. A. de Lange and technical assistance from W. L. Meerts. The research and data analysis was performed by A. C. J. Weber with guidance and suggestions from E. E. Burnell and the manuscript written by A. C. J. Weber. Chapter 5 is a work in progress by A. C. J. Weber and E. E. Burnell.
xii
Chapter 1
Introduction 1.1 1.1.1
Liquid Crystals General
In simpler times the condensed phase was understood as a dichotomy whereby there could be liquids which flow easily and crystals that do not. The liquid, with no regularity in any direction was the anti-thesis of the crystal which has regularity in all spatial directions. In 1888 Reinitzer [1] found that a turbid liquid was formed when solid cholesteryl benzoate was melted and became a clear isotropic liquid upon further heating. When the turbid liquid was characterized by Lehmann [2] it was found to be birefringent and therefore anisotropic. These phases which are anisotropic and still exhibit some degree of fluidity are described as ‘liquid-crystalline’. So it seems as though the condensed phase was actually a trinity of isotropic liquids, anisotropic liquid crystals and solids but this, as it turns out, is not quite so. Specifically, certain organic materials do not show a single anisotropic phase between liquid and solid, but instead a cascade of transitions involving new phases. These various phases are a rich source of fascinating physical behavior and have been well documented over the years [3, 4].
1.1.2
Nematic Liquid Crystals
Liquid crystal molecules forming nematic phases, or nematogens, are typically elongated with semi-rigid cores and flexible alkyl chains towards the end(s), an example of which is seen in Fig. 1.1. Within this phase are found ‘domains’, of the order a million molecules each, where there is long-range orientational ordering that can extend over distances of up to 1µm [3]. This orientational ordering describes the tendency of a nematogen’s long axis
1
1.1. Liquid Crystals to lie along the domains shared axis known as the director. The director is defined as the unit vector, ~n, parralel to the direction in which the aggregate mesogens long axis in a given domain is pointed toward. In the absence of external fields the orientation of the directors will vary throughout the sample. Thus, the turbid appearance comes from the scattering of light as it propagates through the phase, the dimensions of the domains being on the order of the wavelength of visible light. If a magnetic or electric field is applied all the nematic directors will align parallel or perpendicular to the field. The direction of the director alignment depends on the magnetic or dielectric susceptibility anisotropy of the domain. Since the interaction energy for the susceptibility anisotropy coupled with the field is very small compared to the thermal energy, the magnetic or electric field has a negligible effect on the relative orientational ordering of the individual molecules [5]. Over the entire collection of molecules within a domain however, there is sufficient energy to cause alignment of directors. Since a strong magnetic field will not significantly influence the relative orientational ordering of molecules within a phase we can attempt to use Nuclear Magnetic Resonance (NMR) parameters to learn about the intermolecular forces that create this ordering by exploiting the interplay between theory and experiment. The directors of the liquid crystals studied here all align with the magnetic field. Since there is no positional ordering of the component molecules, the nematic phase is cylindrically symmetric about the nematic director. The probabilities of a molecule aligning parallel or anti-parallel with the laboratory fixed Z direction are equal and so the nematic phase is apolar.
1.1.3
Other Liquid-Crystal Phases
Some liquid crystals, often after sufficient cooling of the nematic phase, form layers whose planes are normal to the director. In this smectic A phase then there is, in addition to orientational order, positional order in one of the spatial dimensions. In some cases the director of each layer will be tilted from the normal of the layers plane and is called the smectic C phase. Chiral phases (often used in displays) have the layers staked such that each director is rotated in the plane with respect to those above and below so as to trace 2
1.2. Anisotropic Intermolecular Interactions out a helix in the direction normal to the layers plane. In Fig. 1.2 can be found schematics for these phases. There are many more interesting phases one can consider [6] but here the focus will be on the relatively simple nematic and smectic A phases.
1.2 1.2.1
Anisotropic Intermolecular Interactions General
Anisotropic intermolecular interactions are responsible for the orientational ordering of liquid-crystal phases. The interaction can be characterized as anisotropic short-range repulsive or as anisotropic long-range. Short-range repulsive interactions depend on the details of the molecular structure such as size, shape and flexibility. Long-range interactions involve dipoles, quadrupoles, polarizabilities and other properties that describe the distribution of charges over a molecule and can be either attractive or repulsive. The average of any single-molecule property X(Ω) over the orientations of all molecules is defined by hXi =
Z
dΩX(Ω)f (Ω)
(1.1)
L where f (Ω) can be expanded in terms of Wigner rotation matrices [7], Dm ′ m (Ω), of rank L
f (Ω) =
∞ X
L X
L=0 m,m′ =−L
2L + 1 L aLm′ m Dm ′ m (Ω). 8π 2
(1.2)
L∗ (Ω) and integrating over the angles, it follows that the Multiplying both sides by Dm ′m
expansion coefficients aLm′ m are L∗ aLm′ m = hDm ′ m (Ω)i.
(1.3)
The averages are the microscopic orientational order parameters. The orientational ordering of an inflexible molecule is completely described by the orientational distribution function f (Ω) where Ω denotes the Eulerian angles that describe the orientations of the molecular fixed axes relative to the nematic director which is parrallel to the magnetic field direction in 3
1.2. Anisotropic Intermolecular Interactions the experiments reported here. Since these molecules are in a fluid phase f (Ω) is an average over all molecular reorientations and f (Ω)dΩ is the probability of finding the molecule in a small solid angle dΩ at the direction defined by Ω. The f (Ω) and ultimately the molecular orientational order parameters are supposed to originate from an orientational pseudo-potential U (Ω) which is characterized by the short-range repulsive and long-range interactions
exp −UkT(Ω) f (Ω) = R . exp −UkT(Ω) dΩ
(1.4)
U (Ω) is the potential of mean torque experienced by a single molecule and is defined by eq. 1.4 [6]. The potential of mean torque is responsible for making molecules preferentially align parallel to each other and to the director. Because of the apolar nature of the nematic phase any measured properties are therefore invariant to rotations about the nematic director and all odd components of f (Ω) are necessarily zero. In principle, all components of the function can be assessed by X-ray diffraction techniques [8]. Up to the fourth rank component of the distribution function can be determined from neutron diffraction techniques [3] but this components effect is small compared to the second order Legendre polynomial. In practice though these techniques prove difficult to do on account of poor resolution and instrumental limitations [3]. The average second rank component, identified as the second rank orientational order parameter, of f (Ω)dΩ is accessible via the analysis of NMR spectra. 2 i) is then the leading The second rank orientational order parameter (analagous to hD0m
term in the expansion of the anisotropic components of f (Ω) and can be written as
Sαβ =
R
3 2
cos(θαZ ) cos(θβZ ) − 12 δαβ exp( −UkT(Ω) )dΩ R exp( −UkT(Ω) )dΩ
(1.5)
where α and β represent the molecule fixed axes and θαZ is the angle between these axes and the laboratory Z direction which in the studies presented here is coincident with the magnetic field and director direction.
4
1.2. Anisotropic Intermolecular Interactions
1.2.2
Probing the Potential
An important way to learn about U (Ω) is to compare real experimental Sαβ ’s with ones calculated from theory, models or simulations. Orientational order parameters of the constituent molecules of a liquid crystalline phase are difficult to study because these molecules are normally devoid of symmetry and exist in a number of symmetrically unrelated conformers. A proper description would require a multitude of orientational parameters as well as conformer probabilities and so to proceed one must assume some model for the pair potential in order to relate experimental measurements to single-molecule properties. Whatever molecular properties are involved in the potential it has become clear that size and shape anisotropy plays a dominant role for molecules of sufficient size [9]. However the ordering of very small and highly symmetric molecules such as D2 and CH4 show there are other effects to be considered. When NMR spectra where obtained of D2 , HD and DT it was suprising to find a positive order parameter in some nematic phases and a negative one in others. What was also intriguing was that the experimental spectra involving deuterons could not be reproduced with an intramolecular electric field gradient alone. Specifically there was an extra contribution to the quadrupolar splittings in these NMR spectra which suggested the notion of an external average electric field gradient (hEF Gi) [10, 11]. With these observations in mind the quadrupolar coupling constant of the deuteron was written as
Bobs = −
3eQD ¯ (FZZ − eqS) 4h
(1.6)
where eQD is the deuteron nuclear quadrupole moment, F¯ZZ is the ZZ component of the hEF Gi parallel to the director of the cylindrically symmetric nematic phase, S is the order parameter of the cylindrically symmetric solute and eq is the average electric field gradient due to the intramolecular charge distribution around the deuteron nucleus. With the S determined from the dipolar coupling the F¯ZZ could be adjusted for a given liquid crystal sample in fits to Bobs . The fitted F¯ZZ [12] could then be used to calculate orientational order parameters assuming a solute quadrupole to liquid-crystal hEF Gi interaction Hamiltonian 5
1.3. Model Potentials which then reproduced the observed order of isotopomer orientational order [9]. Further confirming the idea of liquid crystal phases possessing an hEF Gi was the discovery of ‘magic mixtures’ (MM) of the liquid crystals ZLI-1132 (see [13] for chemical composition) and p-ethoxybenzylidene-p ′ -n-butylaniline (EBBA) possessing F¯ZZ ’s of opposite sign that resulted in essentially zero hEF Gi and orientational order of hydrogen and its isotopomers [14]. It is also found that descriptions of orientational ordering using molecular size and shape anisotropy work best in the MM [9]. Methane is not expected to orient on account of its small size and tetrahedral symmetry. Yet this molecule does show significant orientational order according to NMR spectra. This seemingly strange result can be rationalized by realizing that there is a coupling between vibrational and orientational molecular motions [15]. One can easily see that there can be a variety of significant contributions to molecular orientational order and there is still much debate as to the form that would apply equally well to all orientationally ordered molecules. For many intents and purposes one can postulate a potential that is not inconsistent with the various putative underlying mechanisms but is ambiguous as to which one.
1.3 1.3.1
Model Potentials Maier Saupe
One persistently popular way to render complicated problems, like liquid crystals, tractable is the ‘mean field’ approximation. In the theory of Maier and Saupe (MS) a single-molecule potential is sought assuming each molecule is moving in a field generated by its interactions with all the surrounding molecules and that this field is independent of the degrees of freedom of every molecule except the one being considered. The theory also assumes the liquid-crystal molecules can be well approximated as axially symmetric rods. The physical basis of the anisotropic mean field does not have to be specified and it is assumed that the pair potential between two rod molecules is of the form
UA,B = UA,B (~rAB , θA , φA , θB , φB )
(1.7) 6
1.3. Model Potentials where θ and φ are the polar and azimuthal angles that the rods make with the vector ~rAB joining their centres. To obtain the single-molecule potential of a molecule interacting with its environment the intermolecular forces potential is averaged over all degrees of freedom of the other molecules as well as over the translational degrees of freedom of the given molecule. Short-range orientational effects are neglected. The most important feature of the MS mean field is its dependence on molecular orientational order leading to the longrange order characteristic of the nematic phase. Maier and Saupe calculated the mean field seen by one particle among many and showed [16, 17] that for a potential that varies with angle as P2 (cos θ) it can be written as
HN (θ) = −νS
3 1 cos2 (θ) − 2 2
(1.8)
where ν is a scale parameter that indicates the strength of the rod-rod interaction and θ is the angle between the ‘rod’ long axis and the director. This form (which is identified with the leading term in the potential of mean torque) of the interaction can represent short-range size and shape effects or longer-range interactions such as that involving the liquid-crystal hEF Gi interacting with the solute quadrupole or the solute polarizability with the mean square electric field although this list is not exhaustive. Although the MaierSaupe theory does not specify the exact physical nature of the intermolecular potential, its relative success demonstrates that second-rank interactions dominate.
1.3.2
Chord
An extension of the potential of mean torque is to assign a ‘potential of mean torque’ to each bond or group of atoms in a molecule. This assumes a molecule is built up of rigid subunits each independently interacting with the liquid-crystal mean field to produce an orientational torque [18, 19]. A draw-back of this model is that molecules with substantially different shapes, and therefore different orientational ordering, can have the same potential of mean torque [20]. To better account for short-range interaction, Photinos, Samulski, and co-workers have
7
1.3. Model Potentials extended the idea of a potential of mean torque to a ‘chord’ model which attempts to account for the size and shape of the molecules [20–22]. This mean field model for molecular orientation in a uniaxial phase is specially constructed for molecules comprised of repeating identical units like alkanes [20, 21] and is derived from the leading terms in a rigorous expansion of the mean-field interaction
Un (Ω) = −
X w ˜0 P2 (si , si ) + w ˜1 P2 (si , si+1 )
(1.9)
i=1
The si is a unit vector describing the orientation of the ith C-C bond of the hydrocarbon chain and the sum is over all bonds in the chain. The factors P2 (si , si+m ) are given by P2 (si , si+m ) =
The fitting parameters w ˜i =
3 2 Swi ,
3 1 cos(θZi ) cos(θZi+m ) − si · si+m 2 2
(1.10)
where S is the liquid-crystal order parameter. The
first term in the chord model corresponds to the independent alignment of separate C-C bonds while the second term incorporates correlations between adjacent bond orientations and therefore distinguishes between conformations that may have equal numbers of trans and gauche bonds but significantly different shapes. In other words it accounts for shapedependent excluded-volume interactions.
1.3.3
Size and Shape
Simulations have shown that nematic phases can be realized based on size and shape anisotropy interactions alone [9, 23]. This suggests a large role for size and shape contributions to the potential of a liquid-crystal environment. Often times a molecule is approximated by a collection of van der Waals spheres that are centered on the nuclei. It is the anisotropy in the shape of the molecule interacting with the uniaxial nematic field that gives rise to the orientational dependence of the potential energy. In one size and shape model the liquid crystal is taken to be an elastic tube that must stretch to accommodate a solute. The Z or long axis of the tube is coincident with the nematic director. One can
8
1.3. Model Potentials imagine a restoring force resulting from this stretching which can be described with Hooke’s law F = −kdC(Ω)
(1.11)
where C(Ω) is the circumference about the tube perimeter for the solute oriented at the angle Ω. The energy associated with this distortion is 1 U (Ω) = kC 2 (Ω) 2
(1.12)
and is referred to as the Circumference (C) model. Another size and shape model was inspired by the idea that short-range interactions are those between the solute surface and the liquid-crystal mean field. In this model the circumference CZ (Ω) is calculated at points along the length of the molecule in the Z direction and integrated along Z such that 1 U (Ω) = − ks 2
Z
Zmax
CZ (Z, ω)dZ
(1.13)
Zmin
This is known as the Integration (I) model. The I and C models give roughly equally good fits to the experimental order parameters calculated from NMR experiments [9]. However, for longer solutes the C model tends to overestimate, and the I model underestimate, the experimental order parameters. Given this observation it is reasonable to suspect that the combination of these two models would better describe experimental order parameters and so we write the CI model as 1 1 U (ω) = k(C(ω))2 − ks 2 2
Z
Zmax
C(Z, ω)dZ
(1.14)
Zmin
This potential can be thought of as describing both an elastic distortion of the liquid crystal and an anisotropic interaction between a liquid-crystal mean field and the solute surface [24]. Modelling short-range forces this way can also successfully fit order parameters from Monte Carlo simulations [25].
9
1.4. NMR
1.3.4
Kobayashi McMillan
In the smectic-A phase the liquid crystal field is periodically modulated along the Zdirection. The Kobayashi-McMillan (KM) [6] model is mean field and includes both this periodic modulation as well as a term that accounts for the change in nematic order arising from the coupling of the nematic and smectic-A order parameters. The Hamiltonian is then
HA = −τ ′ cos(
2πZ ) + HN d
1 + κ′ cos(
2πZ ) d
(1.15)
where HN is the nematic Hamiltonian from eq. 1.8, κ′ is the magnitude of smectic-nematic coupling and τ ′ scales the pure periodic smectic effect. Z maps the direction normal to the layer planes and d is the repeat distance of the smectic-A translational periodicity. All of the above models are employed in the studies presented here.
1.4 1.4.1
NMR Introduction
The NMR spectra of solutes in isotropic liquids familiar to most chemists are usually interpreted in terms of scalar quantities such as the chemical shieldings (σ) and the indirect spin-spin couplings (J). Although in ‘normal’ NMR these properties appear as scalars, they are in fact tensorial properties. The Brownian movement of the molecule and the resulting isotropic tumbling leads to a situation where only the isotropic part of the tensor is expressed and so the Hamiltonian of NMR in isotropic liquids is
H=−
X hBZ X γi (1 − σiiso )Ii,Z + hJijiso I~i · I~j 2π i
i 12 . The classical interaction energy between a magnetic ~ and a magnetic moment µ field B ~ leads to the interaction energy ~ W = −~ µ·B
(1.19)
causing the compass needle to point north when we turn in the Earths magnetic field. The magnetic moment of a nucleus is related to the spin operators via
µ ~ =γ
h ~ I. 2π
(1.20)
11
1.4. NMR The splitting of otherwise degenerate spin states by a magnetic field along the Z direction can then we written in operator form as
HZ = −γ
h IZ BZ . 2π
(1.21)
Of course the field at the site of the nucleus is not exactly that of the external field BZ due to the electron cloud surrounding a given nucleus and so we must add the effects of local chemical environments HZ = −
hBZ X γi (1Z − σi,Z )Ii . 2π
(1.22)
i
The indirect spin-spin coupling Hamiltonian in Cartesian tensorial form is written as
HJ = h
X
I~i · Jij · I~j = h
i
