THE JOURNAL OF CHEMICAL PHYSICS 132, 144507 共2010兲
Levitation effect in zeolites: Quasielastic neutron scattering and molecular dynamics study of pentane isomers in zeolite NaY Bhaskar J. Borah,1 H. Jobic,2 and S. Yashonath1,3,a兲 1
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India Institut de Recherches sur la Catalyse et l’Environnement de Lyon, Université de Lyon, CNRS, 2 Ave. A. Einstein, 69626 Villeurbanne, France 3 Center for Condensed Matter Theory, Indian Institute of Science, Bangalore 560012, India 2
共Received 30 December 2009; accepted 28 February 2010; published online 12 April 2010兲 We report the quasielastic neutron scattering 共QENS兲 and molecular dynamics 共MD兲 investigations into diffusion of pentane isomers in zeolite NaY. The molecular cross section perpendicular to the long molecular axis varies for the three isomers while the mass and the isomer-zeolite interaction remains essentially unchanged. Both QENS and MD results show that the branched isomers neopentane and isopentane have higher self-diffusivities as compared with n-pentane at 300 K in NaY zeolite. This result provides direct experimental evidence for the existence of nonmonotonic, anomalous dependence of self-diffusivity on molecular diameter known as the levitation effect. The energetic barrier at the bottleneck derived from MD simulations exists for n-pentane which lies in the linear regime while no such barrier is seen for neopentane which is located clearly in the anomalous regime. Activation energy is in the order Ea共n-pentane兲 ⬎ Ea共isopentane兲 ⬎ Ea共neopentane兲 consistent with the predictions of the levitation effect. In the liquid phase, it is seen that D共n-pentane兲 ⬎ D共isopentane兲 ⬎ D共neopentane兲 and Ea共n-pentane兲 ⬍ Ea共isopentane兲 ⬍ Ea共neopentane兲. Intermediate scattering function for small wavenumbers obtained from MD follows a single exponential decay for neopentane and isopentane. For n-pentane, a single exponential fit provides a poor fit especially at short times. Cage residence time is largest for n-pentane and lowest for neopentane. For neopentane, the width of the self-part of the dynamic structure factor shows a near monotonic decrease with wavenumber. For n-pentane a minimum is seen near k = 0.5 Å−1 suggesting a slowing down of motion around the 12-ring window, the bottleneck for diffusion. Finally, the result that the branched isomer has a higher diffusivity as compared with the linear analog is at variation from what is normally seen. © 2010 American Institute of Physics. 关doi:10.1063/1.3367894兴 I. INTRODUCTION
Fluids when confined to the narrow pores of molecular dimensions often show a markedly different behavior as compared with bulk. Fluid-wall interaction plays a predominant role in confined fluids and their strong interaction results in an inhomogeneous fluid and alters its structural and dynamical properties. Several changes in physical properties have been observed for confined fluids. For example, many interesting surface-driven phase changes such as layering, wetting, and commensurate-incommensurate transitions are seen. These phenomena are not seen in bulk phases. The temperature of glass transition is lowered on confinement and is inversely proportional to the pore size.1 In one dimensional zeolite channels, the mean square displacement of the adsorbates is proportional to 冑t responsible for the so-called single file diffusion.2 Such changes in properties make study of confined fluids worthwhile from a fundamental as well as industrial viewpoint. Zeolites are aluminosilicates interlinked in such a fashion that they form a framework with molecular sized pores. These pores accommodate small molecules within them. For a兲
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every aluminum which replaces the silicon atom, the framework acquires a negative charge. In order to compensate for this, positively charged cations also exist within the pores but these are not part of the aluminosilicate framework. Zeolites are known for their catalytic and separation properties. They are often used as molecular sieves where molecules of different sizes move through the column of zeolite at different speeds enabling their separation. In the past two decades, there has been a significant increase in interest in investigations relating to fluids confined in zeolites, metal organic frameworks, and other porous substances.3–5 In particular, hydrocarbons confined to zeolites have been widely investigated due to their use in petroleum industries for separation, cracking, and conversion of hydrocarbons. More often, adsorption and diffusion properties of hydrocarbons confined to zeolites have been studied. A molecular dynamics 共MD兲 simulation study of propane and propylene mixture in pure silica zeolites with eightmembered rings such as CHA 共e.g., chabazite兲 and SAS 共e.g., magnesioaluminophosphate, STA-6兲 has been reported by Combariza et al.6 These studies suggest that at low temperatures both propane and propylene have very low diffusivities. At high temperatures, propylene has considerably higher diffusivity as compared with propane in both CHA
132, 144507-1
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J. Chem. Phys. 132, 144507 共2010兲
Borah, Jobic, and Yashonath 0.25
0.25
(a)
2
0.2
150 K LR
0.15
(b) 0.2
LR
AR AR
0.15
-8
and SAS. Gribov et al.7 made a comparison of heat of sorption of hexane in several one-dimensional siliceous zeolites and concluded that the maximum heat of sorption is seen when hexane fits perfectly in the channel. Diffusion and adsorption of hexane and its isomers have been studied in zeolite MOR 共e.g., mordenite兲.8 It is seen that the branched isomers of hexane have lower self-diffusivity as compared with n-hexane up to very high loading. Jobic9 reported a study of linear and branched alkanes in ZSM-5. They find that the branched alkanes have significantly lower self-diffusivities as compared with their linear analog. Schuring et al.10 studied the diffusion of n-hexane and 2-methylpentane in zeolite silicalite. They reported that n-hexane has a higher selfdiffusivity than 2-methylpentane at all loadings. Krishna and co-workers11 investigated siting and diffusion of linear hydrocarbons such as n-hexane and compared the results with those of 2,2-dimethylbutane in MFI 共e.g., ZSM-5兲 zeolites at 300 K. It is seen that the linear molecule prefers the channels while the branched hydrocarbons prefer the channel intersections. Gump et al.12 carried out experimental studies of hexane isomers across MFI membranes. Isomers of butane have been studied by Krishna and coworkers in MFI and suggested that isobutane effectively blocks pore junctions and diffuses at least 100 times slower than n-butane. Subsequently, it was shown by pulsed field gradient 共PFG兲-NMR measurements by Kärger and co-workers13 that isobutane indeed blocks the pores by occupying the junctions of the pores in MFI. Most of the above studies attempt to study differences between linear and branched hydrocarbons in zeolites whose pore network consists of cylindrical channels where usually single-file diffusion is seen. The results often can be predicted due to the tight-fitting nature of the guests. The differences in diffusion between the linear and branched hydrocarbons in three dimensional networks consisting of pores of relatively larger diameters as compared with the diffusant where single-file diffusion is not possible have not been investigated in detail. The study by Bárcia et al.,14 however, reports adsorption equilibrium and kinetics of hexane and its isomers in zeolites with large pore diameter made up of 12membered rings, the beta zeolite. They obtained selfdiffusivities and activation energies of n-hexane, 2- methylpentane, 2,3-dimethylbutane, and 2,2-dimethylbutane. They found that self-diffusivities of branched isomers are lower than that of linear alkanes. They also found that the activation energies of branched isomers were lower than those of linear alkanes. Systems with large pores are of interest in many separation processes as well as hydrocarbon cracking reactions. Previous studies have shown that the self-diffusivity within a confined medium such as a zeolite or carbon nanotube exhibits a maximum when the diameter of the diffusant is comparable to the diameter of the pore or void of the host.15,16 More precisely, the maximum is seen when the dimensionless quantity known as the levitation parameter defined as
Ds(10 m /sec)
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0.1
0.1
0.05
0.05
0 0.6
0.8
γ
1
0
0.1
0.05 2
-2
1/σs (Å )
FIG. 1. Self-diffusivity Ds as a function of 共a兲 ␥ and 共b兲 1 / s2, obtained from Lennard-Jones fluid in NaY zeolite at 150 K. s is the size of the monatomic sorbates and ␥ is the ratio of diffusant diameter to the neck diameter. Also shown are the values of 1 / s2 and ␥ for n-pentane 共filled circle兲, isopentane 共filled square兲 and neopentane 共filled triangle兲. The vertical dashed line indicates the boundary between the linear regime and the anomalous regime. 共c兲 12-ring window along with a large diffusant of diameter comparable to the window. Also shown in a small diffusant. The reason for the observed maximum can be traced to the mutual cancellation of forces for the diffusant whose diameter 共shown as larger circle兲 is comparable to the bottleneck diameter. The smaller diffusant 共smaller circle兲 is closer to one part of the bottleneck and therefore is acted upon by a larger magnitude of force.
␥=
2opt w
共1兲
is unity. Here, opt is the distance at which the interaction between the diffusant and the medium is most favorable and w is the window diameter, the diameter of the narrowest part of the void network. For Lennard-Jones 共LJ兲 interaction potential, the most favorable interaction occurs at a distance of 21/6 with an energy of interaction equal to −⑀. The dependence of self-diffusivity Ds in zeolite Y on ␥ as well as reciprocal of square of the diffusant diameter is shown in Fig. 1. There are two distinct regimes of dependence of Ds on the diameter of the diffusant s, as can be seen from Fig. 1. First, when Ds varies as a function of 1 / s2. This is called the linear regime 共LR兲. Then as s increases further, the Ds shows an anomalous maximum. Such a maximum is seen when the size of the diffusant s is comparable to the window diameter w. This is the anomalous regime 共AR兲. The value of the levitation parameter at which the self-diffusivity is maximum is always close to unity irrespective of the diffusant-zeolite system. The existence of diffusivity maximum has been seen in all types of zeolites with widely differing geometry and topology of the pore network provided by the host. The diffusivity maximum has its origin in the dispersion interaction between the diffusant and the zeolite; in the absence of this attractive dispersion interaction, there is no diffusivity maximum.17 Kemball18 observed experi-
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QENS and MD study of pentane isomers in NaY
mentally that some specific adsorbates in certain adsorbents have high mobility or diffusivity. Further, it has been seen by Kemball that such high mobility is associated with minimum loss in entropy upon adsorption of the diffusant in the confining medium. Derouane19 analyzed the nature of dependence of the force on the size of the diffusant and showed that the force exerted by the zeolite on the diffusant is a minimum when the diameter of the diffusant and the void are comparable. They suggest that this is responsible for high diffusivity. The lowered force is a consequence of the fact that the forces along diagonally opposite directions are equal in magnitude leading to weakly bound adsorbate which therefore floats. Many computer simulation results in a wide variety of mediums such as zeolites, carbon nanotubes, even simple liquids, close-packed solids, polar solvents, and so forth show that such maximum exists in all these widely differing condensed phases of matter.17 Since the maximum in diffusivity is rather universal, existing in a wide variety of condensed phases of matter, it would be interesting to obtain an experimental proof of the existence of a maximum in diffusivity. We recently presented preliminary experimental evidence to show that such a maximum exists for pentane isomers in zeolites.20 We report here the detailed quasielastic neutron scattering 共QENS兲 measurements as well as MD investigations into diffusion of linear and branched pentanes. The system has been ingeniously chosen to be pentane isomers in zeolite NaY. The isomers of pentanes provide three different adsorbates whose masses are all identical while only their molecular diameters are different. This has been deliberately chosen so as to mimic the conditions employed in previous MD simulations where only the diameter of the adsorbate was allowed to vary.16 We report different quantities such as selfdiffusivities, energetic barrier at the 12-ring window, and activation energy. We report here the radial distribution functions 共rdf’s兲, cage residence times, velocity autocorrelation functions 共vacf’s兲, decay of intermediate scattering functions at small Q, relaxation times, as well as Q dependence of the width of the dynamic structure factor for the first time. These properties provide a coherent and complete view of the motion within zeolites for the three isomers along with the differences between the motion from one isomer to another.
TABLE I. Nonbonded LJ parameters for interaction between pentane isomers.
Type of interaction CH3 – CH3 CH3 – CH3 CH3 – CH3 CH2 – CH2 CH2 – CH2 CH–CH C–C
Molecule
共Å兲
⑀ 共kJ/mol兲
Isopentane Neopentane n-pentane Isopentane n-pentane Isopentane Neopentane
3.910 3.960 3.905 3.905 3.905 3.850 3.800
0.669 44 0.606 825 0.732 375 0.493 712 0.493 830 0.334 720 0.209 25
lated using Lorentz–Berthelot combination rules,
ij = 21 共i + j兲,
共3兲
⑀ij = 冑⑀i⑀ j .
共4兲
Neopentane is modeled as a rigid molecule with no internal degrees of freedom. The C–C bond lengths in isopentane and n-pentane are constrained to the equilibrium bond length of 1.53 Å. The bond angle bending is modeled using a harmonic cosine potential of the following form: Ubend = 21 k共cos − cos 0兲2 ,
共5兲
where k is the force constant and 0 is the angle at equilibrium. 0 and k / kB for the CCC bond angle is taken as 112° and 20 545.008 K for isopentane and n-pentane. In case of isopentane and n-pentane the torsional motions are included using OPLS 共Optimized Potentials for Liquid Simulations兲 potential,21 V共兲 = V0 + +
V1 V2 共1 + cos 兲 + 共1 − cos 2兲 2 2
V3 共1 + cos 3兲. 2
共6兲
The parameters V0, V1, V2, and V3 for isopentane are 11.354共0.0兲, 6.386共5.905兲, 2.231共⫺1.134兲, and ⫺4.691共13.162兲 in kJ/mol. The values in parenthesis are for n-pentane. B. Interaction potential for zeolite
II. EXPERIMENTAL AND COMPUTATIONAL DETAILS A. Interaction potential for hydrocarbons
Pentane isomers have been modeled using united atom approach. Each group of CH3, CH2, and CH is represented by a single interaction site or bead. Thus a pentane isomer consists of five beads. The nonbonded interactions are modeled using LJ potential, U共rij兲 = 4⑀ij
冋冉 冊 冉 冊 册 ij rij
12
−
ij rij
6
,
NaY zeolite crystallizes in the cubic space group ¯ m.22 In one unit cell of zeolite NaY, there are eight suFd3 percages connected tetrahedrally via 12-membered ring windows of free diameter 7.5 Å. The diameter of each supercage is approximately 11.8 Å. In NaY, the sodium ions are distributed over three different sites.22 The potential model proposed by Demontis et al.23 has been employed. Bond stretch is modeled using a harmonic approximation,
共2兲
where ij is the diameter, ⑀ij is the well depth, and rij is the distance between two beads. The parameters have been taken from Jorgensen et al.21 and are listed in Table I. The cross interaction parameters between two unlike beads are calcu-
U共r兲 =
kr 共r − r0兲2 , 2
共7兲
where kr is the force constant and r0 is the bond length at equilibrium. Bond angle bending is modeled with a potential function having higher terms than the harmonic term alone,
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Borah, Jobic, and Yashonath
TABLE II. Bond stretch and bond-angle bending parameters for modeling of the zeolite framework. Units for the force constants are in kJ/mol. 0 is in degrees and r0 is in Angstrom. Molecule Zeolite
U共兲 =
kr
k
k⬘
k⬙
r0
0
2090.0
371.2258
⫺390.7054
41.7207
1.605
109.2801
k⬘ k⬙ k 共 − 0兲 2 + 共 − 0兲 3 + 共 − 0兲 4 , 3 4 2
共8兲
where k’s are the force constants and 0 is the equilibrium angle. The topology of the initial framework is preserved throughout the whole simulation. The values of various parameters for the zeolite are listed in Table II. The sodium cations are held at fixed positions throughout the simulation. The LJ parameters of hydrocarbon-zeolite cross interactions were calculated using Lorentz–Berthelot combination rules. The self-parameters of the zeolite atoms have been taken from Fuchs et al.24 and these are listed in Table III. C. Details of molecular dynamics simulation
One unit cell of zeolite NaY with a = 24.8536 Å has been employed. The simulation cell consists of 2 ⫻ 2 ⫻ 2 unit cells with an edge length of 49.7072 Å. Pentane isomers were initially distributed uniformly in the cages of zeolite NaY so as to have a concentration of one molecule/cage. The integration of the equations of motion was carried out with the help of velocity Verlet algorithm. An integration time step of 1 fs was used which gave good energy conservation. A spherical cutoff of 15 Å was employed. Periodic boundary conditions were applied in all the three directions. All the simulations were performed in NVE ensemble at four different temperatures 共150, 200, 250, and 300 K兲. Equilibration is 500 ps long and production is for 5 ns. Positions and velocities of the molecules were stored every 0.2 ps for further analysis. All simulations were performed using DLPOLY 共Ref. 25兲 package. D. Quasielastic neutron scattering
QENS experiments were carried out on the time-of-flight spectrometer IN6 at the Institut Laue-Langevin, Grenoble, France. The incident neutron energy was taken as 3.12 meV, corresponding to a wavelength of 5.1 Å. After scattering by the sample, the neutrons are analyzed as a function of flight time and angle. Spectra from different detectors were grouped in order to obtain reasonable counting statistics. This also helps to avoid the Bragg peaks of the zeolite. The average wave vector transfer Q of the selected spectra varied between 0.25 and 1.2 Å−1. Time-of-flight spectra were then converted to energy spectra. The line shape of the elastic TABLE III. Nonbonded LJ parameters for zeolite atoms 关taken from Fuchs et al. 共Ref. 24兲兴.
Atom type Na–Na O–O
共Å兲
⑀ 共kJ/mol兲
2.584 3.00
0.418 52 0.777 60
energy resolution could be fitted by a Gaussian function, the half width at half maximum 共HWHM兲 of which varied from 40 eV at small Q to 50 eV at large Q. Hydrogenated pentane isomers were used to take advantage of the large neutron cross section of hydrogen. Since this cross section is mostly incoherent, QENS monitors the self-diffusion of the adsorbates.26 The spectra obtained at the different Q values were first fitted individually with a rotational scattering function convoluted with a translational scattering function and with the instrumental resolution.26 The NaY sample was activated by heating under flowing oxygen up to 720 K. The zeolite was cooled and pumped to 10−4 Pa, then heated up to the activation temperature while pumping 共final pressure inferior to 10−3 Pa兲. The zeolite was transferred inside a glovebox into a slab-shaped aluminum container, which was connected to a gas inlet system. After recording the scattering of the dehydrated zeolite, a concentration of about one pentane isomer per supercage was introduced. QENS spectra were measured at three different temperatures of 300, 330, and 360 K to derive an activation energy for diffusion. III. RESULTS AND DISCUSSIONS A. Structure and dynamics 1. Diffusion
The system has been carefully chosen to consist of three isomers of pentane diffusing within the pores of zeolite NaY. The choice of the system is dictated by the necessity to 共a兲 keep the masses and the interaction strength of the three isomers with the zeolite nearly the same.21,24 Only the crosssectional diameter of the three isomers varies significantly. This will ensure that the system mimics previous simulation conditions.16 共b兲 The choice of zeolite NaY has been made so as to ensure that the levitation parameter for the three isomers is such that some of them lie in the AR while others lie in the LR. Since for polyatomic nonspherical systems, the opt 关in Eq. 共1兲兴 is different for the three principal directions of the molecule, molecular diameter perpendicular to the long molecular axis represented by ⬜ is the relevant quantity as it determines the molecular cross section perpendicular to the direction of diffusion. This is different for the three isomers. ⬜ therefore replaces 2opt for molecular systems in Eq. 共1兲. The center of mass-center of mass 共c.m.兲 rdf’s obtained from MD simulation are shown in Fig. 2 for the three isomers at four different temperatures between 150 and 300 K. With increase in the temperature, the height of the first peak decreases. It is seen that the decrease in the height of the first peak is most significant on going from 150 to 200 K. Also note that the value of the g共r兲 at the first peak is highest for n-pentane, which is followed by isopentane and then neopen-
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neopentane 2
gcom-com(r)
0
isopentane
150 K 200 K 250 K 300 K
4
0
n-pentane
6
FIG. 4. HWHM vs Q2 corresponding to the translational motion of the pentane isomers in zeolite NaY at 300 K, neopentane 共triangles兲, isopentane 共squares兲, and n-pentane 共circles兲. The symbols correspond to the individual fits of the QENS spectra and the solid lines to simultaneous fits with all spectra at different Q values using a jump diffusion model.
3 0
0
5
10
15
20
r, Å FIG. 2. The c.m.-c.m. radial distribution functions of the pentane isomers at different temperatures.
tane. These changes and the self-diffusivities are related. As we shall see below, the diffusivities of the three isomers are in the reverse order 共Ds共neo兲 ⬎ Ds共iso兲 ⬎ Ds共n-兲兲. Thus, the intensity of first peak in the rdf is higher whenever the isomer is less mobile. While several techniques are available to measure the self-diffusivity of the pentane isomers confined to zeolite NaY, only neutron scattering provides intracrystalline selfdiffusivities without any contribution from internal transport barriers. This is because the other techniques such as NMR or uptake measurements probe motion on longer length and time scales during which time the isomers would necessarily sample the internal transport barriers.26 Further, QENS probes length and time scales similar to the MD simulations. Therefore, they can be directly compared with results from simulations. Experimental and calculated QENS spectra are shown in
Fig. 3 for different values of Q. The fits are quite good, no additional elastic contributions were required. The broadening due to the translational motion alone of the three isomers in NaY zeolite at 300 K is shown in Fig. 4 for Q2 below 0.3 Å−2 or Q below 0.55 Å−1. The self-diffusivities have been determined from small Q values, where the contribution from the rotational motion is negligible and where the broadening of the elastic peak tends to a linear function of Q2 and is thus model independent.26 The self-diffusivities at different temperatures as measured from the broadening of the spectra are listed in Table IV. Also listed are the values obtained from MD simulations. These values were obtained from the slope of the mean square displacement. We see that both QENS and MD exhibit the same trend at 300 K: Ds共neo兲 ⬎ Ds共iso兲 ⬎ Ds共n-兲. The values from QENS and MD compare well with each other when we note the approximations in the intermolecular potential. These values may be compared with the selfdiffusivity values of the three isomers in the liquid phase: neopentane at 25 ° C 共density= 0.585 g / cm3兲 Ds = 4.86 ⫻ 10−09 m2 / s, isopentane at 25 ° C 共density= 0.615 g / cm3兲 Ds = 5.3⫻ 10−09 m2 / s, and n-pentane at 25 ° C 共density = 0.621 g / cm3兲 Ds = 5.62⫻ 10−09 m2 / s.27 Note that the trend TABLE IV. Diffusivities of pentane isomers obtained from QENS measurements and MD simulation at different temperatures 共in units of 10−9 m2 / s兲. QENS Pentane isomers
T, K
Ds
T, K
Ds
Neopentane
300 330 360 ¯ 300 330 360 ¯ 300 330 360 ¯
3.17 3.71 4.24 ¯ 2.46 3.01 3.56 ¯ 2.27 2.95 3.56 ¯
150 200 250 300 150 200 250 300 150 200 250 300
0.84 2.40 4.16 6.13 0.55 1.96 3.96 5.40 0.11 0.81 2.27 3.99
Isopentane
n-pentane FIG. 3. Comparison between experimental 共crosses兲 and fitted 共solid lines兲 QENS spectra obtained for neopentane at 300 K for different values of the wave vector transfer Q.
MD
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TABLE V. Levitation parameter ␥, activation energies Ea, and entropy in the adsorbed state for different pentane isomers from QENS measurements and MD simulation. Activation energy Ea is in kJ/mol and entropy S is in J/共K mol兲.
n-pentane Isopentane Neopentane
␥
Ea 共QENS兲
Ea 共MD兲
S共ads.兲
0.71 0.86 0.96
7.1 5.9 5.8
9.4 5.9 5.0
74.5 79.6 81.1
ln Ds
Isomer
-19
-19 n-pentane isopentane neopentane -19.2
-20
-19.4
-21
-19.6
-18
3
4
(b)
QENS
MD
-22 -23
in the liquid phase is what one expects based on the molecular diameter perpendicular to the long molecular axis: Ds共n-兲 ⬎ Ds共iso兲 ⬎ Ds共neo兲. This order of diffusivity is seen in spite of the fact that the densities of the three isomers are in the order 共n-兲 ⬎ 共iso兲 ⬎ 共neo兲. Clearly, molecular cross section dictates the trend in diffusivity rather than the density. If ⬜ is higher for a given isomer then one expects frequency of collisions to be higher and mean free path to be lower, and therefore a lower self-diffusivity. One expects n-pentane to have the highest self-diffusivity since it has the least cross-sectional area perpendicular to the long molecular axis. Indeed the different investigations by several groups discussed in Sec. I confirm that this is in fact the case: linear alkane typically has higher diffusivities.8–13 The present result is unexpected since Ds is larger for the isomer with larger cross section. This surprising dependence can be understood in terms of dimensionless levitation parameter ␥ defined earlier 关see Eq. 共1兲兴. The numerator ⬜ in this equation is a function of molecular geometry as well as dz the LJ interaction parameter between the diffusant and the oxygens of the zeolite. More specifically, the relevant parameters in the present case are the CH3–O, CH–O, and C–O 共and not X–Na, X = CH3, CH2, CH, and C兲, since the hydrocarbon approaches closest to the oxygens during its passage through the 12-ring bottleneck and since the hydrocarbons are not in close proximity to the Na+ during its passage through the bottleneck. For the 12-ring window, w = 10.11 Å. The values of ␥ for the three isomers are listed in Table V and their positions are indicated in Fig. 1共a兲 in the plot of Ds-␥. Ds-␥ is a generic feature and is the same across different guest-zeolite 共gz兲 systems. It has been observed that for ordered solids such as zeolites, ␥ ⬍ 0.78 belongs to LR and ␥ ⬎ 0.78 belongs to AR. It is different from the previous estimate of 0.83 for the dividing line between LR and AR. The value of 0.78 is a more accurate estimate and we have obtained it by carrying out longer simulations at smaller intervals of diffusant diameter. In the present study both neopentane 共␥ = 0.96兲 and isopentane 共␥ = 0.86兲 lie in the AR, whereas n-pentane 共␥ = 0.71兲 lies entirely in the LR 关see Table V and Fig. 1共a兲兴. Values of ␥ can vary depending on the way it is computed, but what is clear is that isopentane and neopentane have larger ␥ and lie in the AR while n-pentane lies in the LR of the dependence of Ds on ⬜. Clearly, neopentane has a diameter comparable to the 12-ring window of the medium 共namely, the host zeolite兲, leading to a mutual cancellation of forces exerted on this isomer by the medium, the zeolite. The force cancellation also occurs for isopentane but to a lesser degree. This
(a)
-19.8
6
5
1000/T, K
-1
7
-20
2.8
3
3.2
1000/T, K
-1
FIG. 5. Arrhenius plots of the self-diffusivities of pentane isomers in NaY zeolite from 共a兲 MD simulation data and 共b兲 QENS data.
explains the counterintuitive result that Ds is higher for isomers with larger ⬜ leading to the trend Ds共neo兲 ⬎ Ds共iso兲 ⬎ Ds共n-兲. Results from MD are also listed in Table IV and the observed trend is the same. There is a good agreement with QENS results given the approximations in the intermolecular potentials for the zeolite as well as hydrocarbons. Later, we shall discuss the wavenumber dependence of selfdiffusivity where it will become clear that neopentane is somewhat smaller in diameter than the 12-ring window. Previous studies have shown that the anomalous maximum in diffusivity disappears at higher temperatures.28 Thus, at sufficiently high temperatures, the self-diffusivity will decrease with increase in ⬜. Then, the observed trend in selfdiffusivities will be Ds共n-兲 ⬎ Ds共iso兲 ⬎ Ds共neo兲. Thus, although the present result confirms the existence of levitation effect, a more reliable confirmation of levitation effect is obtained by the trend in the activation energies. As shown previously,16,17 molecules in the AR have lower activation energy 共Ea兲 than those from LR; Ea共AR兲 ⬍ Ea共LR兲. The lowest Ea is seen for the isomer close to diffusivity maximum. From this we see that neopentane should have lower Ea than n-pentane, with Ea for isopentane somewhere in between. The Arrhenius plots from MD simulation data and QENS data at different temperatures are shown in Figs. 5共a兲 and 5共b兲, respectively. Ea have been calculated from the slope and they are listed in Table V. We see that QENS and MD, respectively, report 5.8 and 5.0 kJ/mol as the activation energy for neopentane. Given the approximations in the intermolecular potential the agreement between MD and QENS is good. Activation energies for isopentane are only marginally higher. This is to be expected since isopentane also lies within the AR. Ea for n-pentane is significantly higher. This may be attributed to the fact that it lies in the LR. The trend seen in Ea is in conformity with the previous predictions of the levitation effect.29 The QENS results for pentane isomers in NaY zeolite provide a direct confirmation of the existence of the levitation effect for hydrocarbons in zeolites. We note that differences in the values of Ea for the three isomers ensure that they intersect at some temperature. The temperature of intersection 共Ti兲, also known as inversion temperature,17 is the temperature at which the diffusivities of those two isomers change order: if isomer 1 had lower dif-
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T = 200 K
n-pentane neopentane isopentane
-25
-30
-2
-1.5
-1
-0.5
0
1
0.5
1.5
2
d, Å FIG. 6. Potential energy landscape of different isomers in zeolite Y at 200 K: the gz interaction is plotted vs d, the distance of molecular c.m. from the window plane. Full, dashed, and dotted lines correspond to n-pentane, isopentane, and neopentane, respectively.
the isomer. Once each of the molecules was assigned to the cage at each time step, it was easy to find out when the cage of residence underwent a change. The residence times were then computed as the difference between the exit and entry times. The distribution of cage residence times is plotted for the three isomers in Fig. 7. The average values of cage residence times obtained from the distribution are 13.6, 8.0, and 6.3 ps for n-, iso-, and neopentane, respectively. These are consistent with the energy profile and activation energies reported earlier. The entropy of the three isomers has been calculated using the method proposed by Goddard et al.32 The entropy values at 300 K are listed in Table V. It is observed that the entropy of neopentane is highest and of n-pentane least and the isopentane entropy lies in between. Experiments by Kemball showed that the entropy loss suffered on adsorption by an adsorbate in an adsorbent is minimum whenever its mobility is high. In other words, the change in entropy on adsorption ⌬S = S共ads.兲 − S共free兲, where S共ads.兲 is the entropy in adsorbed state and S共free兲 is the entropy of the adsorbate before adsorption in its free state, is a minimum. Since S共free兲 is greater than S共ads.兲, ⌬S is negative. Further, S共free兲 is essentially a constant quantity and therefore variations in S共ads.兲 are reflected in changes in ⌬S. Thus, our result that the highly mobile neopentane is also associated 0.07
T = 200 K
0.06
n-pentane isopentane neopentane
0.05
f(τs)
fusivity than isomer 2 below Ti, then isomer 1 will have higher diffusivity than isomer 2 above Ti. The inversion temperature for n-pentane and isopentane is 385 K, for n-pentane and neopentane is 380 K, and for neopentane and isopentane is 360 K. It is important to note that studies by Bárcia et al.14 of hexane and its isomers in beta zeolite reported that the linear isomer had a higher diffusivity than branched isomer. However, they report a lower activation energy for the branched isomer as compared with the linear hexane. This result is in agreement with the present results, both being in 12-ring zeolites where the branched isomers normally lie in AR while the linear isomer lies in the LR of the dependence of Ds on ⬜. Since the activation energy of the branched isomer is lower than the linear analog, it follows that below the inversion temperature, Bárcia et al.14 will also observe that the branched isomer will have a higher diffusivity than the linear isomer. Thus, the results by Bárcia et al. are in complete agreement with the present findings. More studies are required to see that the higher diffusivity of the branched isomer in 12-membered ring zeolites is a generic result. We also like to compare the activation energies for diffusion of the three isomers in the liquid phase at 25 ° C. Ea of n-pentane is 6.3 kJ/mol, isopentane is 7.1 kJ/mol, and that of neopentane is 10.9 kJ/mol.27 These clearly show that the larger the molecular diameter perpendicular to the long molecular axis, the higher the activation energy. This is in agreement with most of the results of hydrocarbon diffusion in zeolites discussed in Sec. I. The only exception arises when diffusant diameter and the pore diameter of the bottleneck are comparable leading to the levitation effect. In general, there is a strong correlation between activation energy and underlying potential energy landscape. In fact, it is possible in principle to compute the activation energy if we know the potential energy landscape.16,30,31 Generally, the potential energy landscape associated with the AR is shallower and has only small amplitude undulations in energy as compared with the LR guest.16,17 This is particularly reflected in the energetic barrier at the bottleneck. In zeolite NaY, 12-ring windows have a free diameter of about 7.4 Å as compared with the cage diameter of about 11.8 Å. Thus, the 12-ring serves as a bottleneck. The energy profile as the molecule approaches the bottleneck can be computed by averaging hydrocarbon-zeolite potential energy of interaction 共Ugh兲 over all the cage-to-cage crossover events found during the MD run.16 In Fig. 6, the potential energy profile is plotted as a function of d, the distance of the molecular c.m. from the plane of the bottleneck for different isomers of pentane. We can see that for n-pentane, the bottleneck 共d = 0 Å兲 is associated with an energy maximum. In contrast, the potential energy profile for neopentane shows a minimum at the bottleneck. Isopentane is associated with a weak maximum. These results are consistent with the results of monatomic guests16 diffusing in zeolites A and Y confirming the generic nature of the results. These energy profiles at the window determine the cage residence times. In order to obtain cage residence times, we first found the cage where each isomer is located at each time step by finding the cage whose center is closest to the c.m. of
Ugh, kJ/mol
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0.04 0.03 0.02 0.01 0
0
5
10
τs, ps
15
20
FIG. 7. Distribution of cage residence times for the three isomers at 200 K. The average residence times obtained from the distribution are 13.6 共npentane兲, 8.0 共isopentane兲, and 6.3 ps 共neopentane兲.
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1
1 0.95 0.9 0.85
n-pentane
0.6
isopentane neopentane
Fs(Q,t)
0.8
0.4
0 2
t, ps
3
4
5
Fs(Q,t)
1
0.4 0
0.2
0
0.6
n-pentane
0 20 40 60 80 100
0.2
T = 200 K
Fs(Q,t)
/
0.8
FIG. 8. vacf’s of all the three isomers at 200 K. The vacf for neopentane decays smoothly whereas the vacf of isopentane and n-pentane has undulations.
with large entropy 关S共ads.兲兴 implies that ⌬S is small. Thus, the present result is in agreement with Kemball’s experimental findings.18 We now discuss the related dynamical properties obtained from the MD trajectory to understand the unexpected dependence of Ds upon ⬜. The vacf’s for all the three isomers are calculated at a temperature of 200 K and are shown in Fig. 8. The vacf for neopentane exhibits a single minimum around 0.7 ps. Isopentane exhibits only a shoulder without a distinct minimum around 0.5 ps followed by a pronounced minimum around 1.2 ps. n-pentane shows a fast decay leading to a distinct minimum around 0.5 ps, which is followed by another minimum at around 1.2 ps. The multiple minimum suggests multiple collisions and bumpy or rough motion of n-pentane as compared with neopentane. Isopentane has only a slightly bumpy ride. Neopentane has particularly smooth ride since the potential energy landscape for this isomer has only undulations of small amplitude. The observed behavior of the three isomers is consistent with trend in diffusivity. 2. Q dependent properties
The intermediate scattering function Fs共Q , t兲 computed from the MD generated positions is shown in Fig. 9. The decay of these functions with time is shown for the three isomers for Q = 0.252 Å−1. Also shown are the biexponential and single exponential fits to the MD curve for all the three isomers. For clarity and to enable distinction between the fit and MD curve, only discrete points have been shown for the MD. We see that single exponential provides a good fit to the MD curve only for neopentane and isopentane. Both these isomers lie in the AR where previous studies on monatomic species in zeolites as well as liquids have shown single exponential decay.17,33,34 Thus, the present results are in excellent agreement with earlier studies on model guests. For n-pentane, a single exponential does not provide a good fit, especially at short times 共see the inset兲. A biexponential fit provides a good fit over the whole range of time up to 1500 ps. From the decay of the curves, we see that the decay is slowest for n-pentane and fastest for neopentane. From the fit to the MD curve, we obtained the relaxation times 1 = 99.6 ps and 2 = 620.6 ps for n-pentane and 1 = 117.1 ps
0
0.8 0.6 0.4 0.2 0 0 1 0.8 0.6 0.4 0.2 0 0
500 isopentane
1000
1500
2000
MD double exp. fit single exp. fit
1000
500
neopentane
1500
T = 200 K -1
Q = 0.252Å
500
t, ps
1000
1500
FIG. 9. Self-part of the intermediate scattering function Fs共Q , t兲 obtained from MD for the three isomers as a function of time for Q = 0.252 Å−1 at 200 K. Also shown are single and biexponential fits to the Fs共Q , t兲. Note the excellent fit of the single exponential function for isopentane and neopentane 共both lie in the AR兲 but not for n-pentane from the LR. The inset shows the poor fit obtained from single exponential fit at small times for n-pentane showing the need for a biexponential fit.
for isopentane and 85.17 ps for neopentane. These are listed in Table VI. The two relaxation times obtained here for n-pentane are in agreement with the two distinct minima seen in the vacf for n-pentane. The single decay times for neopentane and isopentane are in agreement with single distinct minimum in the vacf for these two isomers. These are also consistent with the potential energy profile across the bottleneck; the presence of a positive energy barrier across the bottleneck for n-pentane leads to motion within the supercage being facile and motion past the supercage being difficult. This leads to two distinct relaxation times obtained in Fs共Q , t兲 for small Q. At Q = 0.252 Å−1, the length scale corresponds to around 24 Å, which means we are looking at diffusion well beyond the supercage and therefore two relaxation times are seen. Thus, all properties computed show trends consistent with each other and give us an overall picture of the motion within the zeolite. By Fourier transformation of the intermediate scattering function, we obtained the dynamic structure factor at different wavenumbers Q. The Q dependence of the width ⌬ of the self-part of the dynamic structure factor can yield inforTABLE VI. The biexponential relaxation times 1 and 2 for n-pentane and single exponential relaxation time 1 for iso- and neopentane obtained from the fitting of Fs共Q , t兲 with an exponential function of the form A1 exp共−t / 1兲 + A2 exp共−t / 2兲 or A1 exp共−t / 1兲. Q = 0.252 Å−1
Pentane isomer Neopentane Isopentane n-pentane
1 共ps兲
2 共ps兲
85.2 117.1 99.6
¯ ¯ 620.6
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1
(a)
n-pentane isopentane neopentane
0.6
2
(b)
AR LR
0.8
0.4
∆ω/2DsQ
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0.6
0.2 0.4 0
0
1.3
0.5
1
2
1.5
2.5
3
(c)
1.2 ρ* = 0.84
1.1
T* = 0.72 1 0
1
2
3
Q, Å
4 -1
5
6
0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3
1
2
3
4
ρ* = 0.65, T* = 1.87
(d)
0
1
2
3
4
Q, Å
5
6
7
-1
FIG. 10. ⌬ / 2DsQ2 共a兲 for the three pentane isomers as a function of Q at 200 K, 共b兲 for a binary solute-solvent mixture 共Ref. 34兲, 共c兲 for a single component argon liquid 共Ref. 35兲 at ⴱ = 0.84 and Tⴱ = 0.72, and 共d兲 for a single component argon liquid 共Ref. 36兲 at ⴱ = 0.65 and Tⴱ = 1.87.
mation about the Q dependence of self-diffusivity. In the hydrodynamic limit 共Q → 0 , → 0兲, the width is 2DsQ2. In Fig. 10共a兲, we have plotted ⌬ / 2DsQ2 as a function of Q. Since we have employed two unit cells of zeolite Y as the simulation cell, we have points every 0.126 Å−1. It is observed that the curve for neopentane decays with only a weak minimum near Q = 0.6 Å−1. Isopentane also exhibits a decay similar to neopentane. In contrast, the Q dependence of ⌬ / 2DsQ2 for n-pentane exhibits a more pronounced minimum and maximum. The minimum is seen at Q = 0.5 Å−1. Previously, such a minimum has been seen by Nijboer and Rahman35 in liquid argon at high density and low temperature 共ⴱ = 0.84 and Tⴱ = 0.72兲. Levesque and Verlet36 carried out simulations on liquid argon at relatively lower density and higher temperature 共ⴱ = 0.65 and Tⴱ = 1.87兲. They computed ⌬ / 2DsQ2 and found that it exhibits a monotonic decrease with Q. Boon and Yip37 provided a possible microscopic reason for these two distinct behaviors. They suggest that in the high density liquid argon at low temperatures, the first neighbor shell is well defined and argon has difficulty to getting past this shell. As a result there is a slowing down of motion while the argon atom is getting past this shell 关see Figs. 10共c兲 and 10共d兲兴. This leads to a minimum at around the Q value corresponding to the first neighbor shell. Our own previous simulations of binary solute-solvent mixtures have shown that the solute in the LR does exhibit a pronounced minimum as a function of Q, for Q corresponding to first solvent shell.34 The behavior of ⌬共Q兲 / 2DsQ2 is also shown in Fig. 10共b兲 for both LR as well as AR solutes. The AR solute which is larger in diameter exhibits a near monotonic decay of ⌬ / 2DsQ2 with Q. The results obtained on isomers of pentane can be interpreted in the light of these previous findings. Here, in the case of guests in zeolites, there is no solvent shell but the supercages provide the envelope that significantly slows down the guest motion. We see that the minimum occurs around 0.5 or 0.6 Å−1, which corresponds to a distance of l ⬇ 2 / Q = 11.5 Å. This is the diameter of the supercage. Thus, here the passage past the supercage is the rate determining step which is difficult normally for the isomer from LR. For neopentane, which is not
FIG. 11. Trajectory of n-pentane 共blue, dark in black and white兲 and neopentane 共red, gray兲 within zeolite NaY framework obtained from MD simulation at 200 K over 200 ps. Only the c.m. of the isomers is shown. Neopentane traverses significantly a longer distance over the same period of time than n-pentane.
exactly at the maximum in self-diffusivity of the AR but close to it, the diffusion past 12-ring window is relatively facile. This is also consistent with the energy profile presented earlier. The reason for the near monotonic decay 共and not a monotonic decay兲 of ⌬ / 2DsQ2 with Q seen for neopentane arises from the fact that neopentane has a diameter that is slightly less than the 12-ring window. The present study provides additional insight into reasons for the minimum seen in ⌬ / 2DsQ2 with Q for n-pentane. It suggests that apart from higher density at the supercage periphery, the energetic barrier slows down the diffusing n- pentane. Neopentane, which does not have a barrier, does not encounter any difficulty in getting past the supercage. It is worthwhile to compare ⌬ / 2DsQ2 in zeolites with those of liquid argon. We see that the minimum is more pronounced in the case of high density liquid argon but the minimum for n-pentane is less pronounced or less deep. This is because in liquid argon the first solvent shell provides a greater difficulty since the g共r兲 at first peak is several times bulk density. This makes it more difficult to get past the first shell. In the case of zeolites, the difficulty provided by the supercage is relatively less being not so dense and characterized by only weak energy barriers. The difference in diffusivity between n-pentane and neopentane can also be seen from the trajectory shown in Fig. 11. The trajectories are plotted for a period of 200 ps. It is clearly seen that neopentane explores a larger amount of void space within the zeolite, whereas n-pentane is confined mostly in a smaller region during the same period of time. IV. CONCLUSIONS
The results show that self-diffusivities are in the order Ds共neo兲 ⬎ Ds共iso兲 ⬎ Ds共n-兲. More importantly, activation energies exhibit the trend Ea共n-兲 ⬎ Ea共iso兲 ⬎ Ea共neo兲. The levitation parameter ␥ for the three isomers is approximately 0.71 共n-兲, 0.86 共iso兲, and 0.96 共neo兲. The value of ␥ separating LR from AR is 0.78. Thus, n-pentane lies in the LR while
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Borah, Jobic, and Yashonath
the other two lie in the AR. The energy profile for neopentane exhibits a minimum in energy at the bottleneck while for n-pentane a distinct positive energy barrier is seen. Our analysis in terms of Arrhenius behavior suggests that the self-diffusivities are determined principally by the activation energies. The reasons for changes in activation energies with ␥ or molecular cross section can be understood in terms of the levitation effect by analyzing the changes in force and energy with diffusant diameter. When the diameter of the isomer is small relative to the void or bottleneck diameter, the isomer prefers to stay close to the internal surface of the supercage of the zeolite. When this happens, there is a significant attraction exerted on the isomer by the zeolite atoms closest to it while atoms far away only exert a weak attraction. A two-dimensional representation of this is depicted in Fig. 1共c兲. We see that the net force on the isomer in this case is large. The interaction energy between the isomer and zeolite at the bottleneck, however, is small in magnitude although favorable 共negative兲. When the isomer has a diameter comparable to the bottleneck diameter, then the attraction exerted on the isomer by the zeolite from any given direction is large. But the force exerted from any given direction is equal and opposite to the force exerted on it in the diagonally opposite direction. This leads to a negligible net force on this isomer. At the bottleneck this larger-sized isomer optimizes the interaction better with the zeolite leading to a lower energetic barrier at the bottleneck. For these reasons, neopentane has a lower barrier at the 12-ring window. Effectively, neopentane experiences a shallower energy landscape whereas n-pentane has to traverse over an energy landscape with deeper valleys and high barriers. Activation energy is related to the potential energy landscape and in fact, as has been pointed out by Truhlar and Zwanzig,30,31 the activation energy can be obtained by taking a weighted average over the different trajectories if we know the potential energy landscape. Thus, a shallower potential energy landscape is associated with neopentane and the average over this will lead to lower activation energy. n-pentane is associated with large amplitude variation in energy as it traverses, and therefore has a higher activation energy. Fs共Q , t兲 at small Q共0.252 Å−1兲 exhibits single exponential decay for neopentane and isopentane, while a biexponential decay is seen for n-pentane. ⌬ / 2DsQ2 exhibits a minimum as a function of Q at around 0.5 Å−1, whereas no distinct minimum is seen for the other two isomers. This behavior is attributed to slow down of motion of n-pentane near the bottleneck, the 12-ring window. We now discuss the exact conditions under which branched isomers exhibit a higher diffusivity as compared with linear counterparts as seen in this study. In general, a guest hydrocarbon will have three different molecular diameters along three mutually perpendicular directions. The direction along which the diameter is largest is usually referred to as the molecular axis. For example, in the case of n-pentane, this is parallel to the line connecting the two terminal methyl groups. Usually, during diffusion the molecule diffuses with its long molecular axis parallel to the direction of displacement. Hence, the molecular diameter perpendicu-
lar to the molecular axis is of importance since this determines the collision diameter of this molecule. Further, this diameter is also of importance for the purpose of deciding whether or not it will lie in the AR of the levitation effect. This diameter is to be compared with the diameter of the bottleneck of pore network within which the diffusant diffuses. If the levitation parameter computed from these two diameters is greater than 0.78 then it lies in the AR. If smaller than 0.78, that diffusant lies in the LR. Many branched isomers have diameters comparable to the diameter of the 12-ring bottleneck 共10.11 Å兲 and hence lie in the AR. Linear alkanes have much smaller diameters and hence have ␥ values less than 0.78. As a result, many of the branched isomers in zeolites with 12-membered ring bottlenecks will exhibit a higher diffusivity than their linear counterparts. Thus, branched alkanes in zeolites with 12-ring bottlenecks will exhibit normally a higher diffusivity than their linear analog. In zeolites with eight- or ten-ring bottlenecks this will not be the case. In these zeolites, the branched isomers will have slightly larger diameter than the bottlenecks which will lead to larger barriers or activation energies than the linear alkane. In 18-member ring zeolites, the levitation parameter for the branched alkane is likely to be less than 0.78 and therefore may lie in the LR. From this discussion, it is evident that zeolites other than those with 12membered ring bottlenecks will normally not exhibit a higher diffusivity of the branched isomer especially for alkanes. In conclusion, the present study demonstrates a direct experimental verification of the levitation effect. Second, it has been shown that the intracrystalline diffusivity of the branched hydrocarbon such as neopentane or isopentane is higher than the linear n-pentane. Preliminary studies on n-decane and 3-methyl pentane in zeolite NaY by Pazzona et al. suggest that the branched isomer exhibits a higher diffusivity.38 It appears that the higher diffusivity of branched isomers in zeolites with 12-ring bottlenecks may be generic. ACKNOWLEDGMENTS
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