J. Phys. Chem. B 1999, 103, 4949-4959
4949
Intracrystalline Diffusion of Linear and Branched Alkanes in the Zeolites TON, EUO, and MFI Edmund B. Webb III,*,† Gary S. Grest,† and Maurizio Mondello‡ Corporate Research Science Laboratories, Exxon Research and Engineering Company, Annandale, New Jersey 08801 ReceiVed: NoVember 24, 1998; In Final Form: February 11, 1999
Diffusion constants D and activation energies for diffusion Ea were obtained for linear and branched alkanes inside the zeolites TON, EUO, and MFI via molecular dynamics simulations. Molecules with carbon numbers in the range n ) 7-30 were studied in the dilute limit. The zeolites used have channels formed by 10member silicate rings, but the diameter and connectivity of the channels differ between zeolites. Because the zeolites’ channel features are different, it was observed that the influences of channel structure on molecular transport, or lattice effects, were also different between zeolites. For linear alkanes in the relatively uniform channels of TON, lattice effects were least pronounced so the highest values for D were observed and D scaled as 1/n. Furthermore, Ea ≈ 1 kcal/mol at small n and decreased to nearly zero for large n. The features of the channels in EUO and MFI were such that lattice effects were observed for linear alkanes. Generally, D was lower than in TON and decreased faster than 1/n; in EUO, Ea increased with n, while in MFI the dependence of Ea on n was anisotropic. Monomethyl-branched alkanes diffused slower than their linear counterparts. Branched molecules are effectively bulkier making interactions with the channel walls more influential. Lattice effects were observed for branched molecules in all the zeolites; especially in the trends of D with branch position.
I. Introduction Zeolites are microporous crystalline materials used in various chemical processes including molecular separation and catalysis. These materials are well-suited to such applications since their channel structures have dimensions comparable with the size of relevant molecules. As such, zeolites can act as molecular sieves by restricting the diffusion of certain molecules through the zeolite channels. They can also impose molecular shape selectivity in catalytic reactions by restricting diffusion of reactant or product molecules to or away from a reaction site. Alternatively, reactions may be suppressed if the transition state for a given reaction is too bulky to exist in the zeolite channels.1 Molecular shape selectivity is used to create product distributions that, in the absence of selectivity, are not attainable. A thorough understanding of systems incorporating zeolites therefore requires knowledge of the behavior of relevant molecules inside these structures. The channels in zeolites are created by the periodic repetition of 8-, 10-, or 12-member aluminosilicate rings (where the number refers to the cations in the ring). The ideal zeolite for a given application depends on the molecules involved, the chemistry desired, and the zeolite’s channel structure. An example is found in the catalytic isomerization of medium-sized linear alkanes to branched molecules which is a desired component of certain hydrocarbon fluid upgrading processes.1,2 The zeolites often used in this process have 10-member ring (10 MR) channels because their dimensions are very similar to the size of linear and branched alkanes. As such, shape selectivity mechanisms are present and more useful product † Present address: Sandia National Laboratories, P.O. Box 5800, MS 1411, Albuquerque, NM 87185. ‡ Present address: Tech Hackers Inc., 5 Hanover Square, New York, NY 10004.
distributions are realized. Isomerization in the presence of a zeolite is a process involving adsorption and desorption, surface and intracrystalline diffusion, and complex reaction mechanisms. Thorough descriptions have been proposed3 wherein a consensus has been expressed that the influential selectivity mechanism is transition-state shape selectivity. There is debate about this, however, since some studies have proposed that reactant or product shape selectivity, that is, diffusion limitations on certain molecules, significantly influences product distributions.4 Aside from this debate, it is clear that isomerization requires diffusion of molecules to and away from reaction sites in or on the zeolite. Therefore, it is important to better understand diffusion of molecules such as linear and branched alkanes inside zeolites. Since many catalytic processes operate across a temperature range, it is also useful to study the activation energy for diffusion. Experimental methods for measuring the diffusion constant D and activation energy for diffusion Ea of molecules in framework structures exist5,6 and have been applied to a variety of molecules and structures.7-15 For the same guest molecule and host structure, order of magnitude differences between diffusion constants have been reported.5,6 A compilation of representative experimental data and discussion of such discrepancies has been presented previously.16-18 The lower values of D were typically obtained by experimental methods which probe more macroscopic features of mass transport and not the molecular motion itself.17 These methods, such as uptake rate measurements, are considered macroscopic diffusion measurements. Experimental techniques such as pulsed field gradient nuclear magnetic resonance (PFG NMR) and quasi-elastic neutron scattering (QENS) directly probe aspects of molecular motion and are therefore considered microscopic diffusion measurements. While agreement between macroscopic and microscopic methods has been obtained for some molecule/
10.1021/jp9845266 CCC: $18.00 © 1999 American Chemical Society Published on Web 05/29/1999
4950 J. Phys. Chem. B, Vol. 103, No. 24, 1999 zeolite systems, in controlled experiments on the same samples, large differences between uptake rate and PFG NMR diffusivities were observed.5 Furthermore, diffusion measurements via microscopic methods are often restricted to subcatalytic temperatures, so as to remove the influence of reactions. At low temperatures, PFG NMR and QENS are only able to study small molecules so that the values of D are within the limit of resolution of the experiment. Consequently, there still exists considerable uncertainty in the experimental ability to accurately describe the dynamics of medium-length alkanes in zeolites at relevant temperatures. Computer simulation is an attractive alternate method for studying molecules in zeolites. The environment and state point of a simulation can be specified precisely and chosen free of the restraints that are imposed on experiments. Prior studies have used Monte Carlo (MC) simulations to assess thermodynamic properties for linear alkanes in various zeolites. Adsorption thermodynamics, molecular conformations, and molecular occupancies have been calculated.19-24 While MC techniques provide a description of pertinent thermodynamics, system properties dependent upon molecular motion are often probed with molecular dynamics (MD) simulations which compute the trajectory of molecules in time. This provides a direct prediction of D from the observed mean square displacement (MSD) for a molecule through the Einstein equation for diffusion.25 This technique has been applied in a number of studies of molecular diffusion in zeolites.16-18,26-34 The primary limitation to calculating D is the long simulation time needed to obtain reliable statistics. The MSD curve from which D is calculated should demonstrate good linearity in time to allow a reliable measurement of the slope. As discussed in ref 18, there is an additional requirement in that the MSD should be calculated over a long enough duration to allow the molecule to completely sample the unit cell of the zeolite. For a diffusion constant of 10-10 m2/s and a unit cell length of 20 Å, the time necessary for this is of the order 20 ns. In addition, accuracy is improved when multiple time segments are used to calculate an average MSD. This requires a total simulation duration that is greater than the time needed to sample the entire unit cell. MD simulations on the order of tens of nanoseconds for chain molecules of any appreciable length are very computationally intensive. As such, many previous simulations to predict D have been performed on small molecules and used trajectories that did not allow the entire unit cell to be sampled. However, recent advances in computational resources have begun to lift this barrier, and a recent study18 was able to observe diffusion for molecules as long as eicosane in MFI. There are very few prior results for simulations of branched molecules in zeolites, and these were limited to very small n.26,35 These molecules diffuse more slowly than their linear counterparts so the computational demands for accurate D calculations are even greater. To our knowledge, no results for medium-length branched alkanes in multiple zeolites have been presented. Herein, we present results from simulations of linear alkanes (Cn), with n ) 7-30, and monomethyl-branched isomers (xmethyl Cn-1), with n ) 7-16 and x ) 2-5, in three zeolites. The zeolites chosen are listed in Table 1 by their structure type codes which are used to identify them in this work. Also given there are the full type names, the largest and smallest free diameters of channels in each applicable crystallographic direction, and the zeolite unit cell parameter in the corresponding direction; all the data presented in Table 1 are from ref 36. The zeolites chosen have 10 MR diffusion channels but differ from one another in the diameters and connectivity of their channels.
Webb et al. TABLE 1: Properties of the Zeolites Studieda code
name
[100]
[010]
[001]
a
b
c
TON ZSM-22 4.4 × 5.5 13.8 17.4 5.0 EUO EU-1 4.1 × 5.7 13.7 22.3 20.2 MFI ZSM-5 5.1 × 5.5 5.3 × 5.6 20.1 19.9 13.4 a The structure type codes, the full type names, the largest and smallest free diameters for the 10 MR channels, and the unit cell parameters are given; the latter two are quoted in A.
Figure 1. . Cross sections of the structure of zeolites (a) TON, (b) EUO, and (c and d) MFI. Sufficient material has been removed from the pictures to clearly illustrate the channels in which alkane molecules diffuse. The channels run in the crystallographic direction indicated.
For clarification, cross sections of the channels in each zeolite are depicted in Figure 1. Only the bonds between atoms of the zeolite are shown, and a sufficient amount of material has been removed to illustrate differences in the structure of the channel wall. Each picture is oriented such that the diffusion channel direction is horizontal in the page. Figure la shows that TON has a system of one-dimensional uninterrupted channels with small periodic undulations in the walls. The channels run in the [001] direction so only the Dz component of the diffusion tensor is applicable for molecules in TON. Figure lb shows that EUO has a system of one-dimensional channels as well. These run in the [100] direction with a corresponding diffusion constant Dx. The channels in EUO differ from TON because their walls are periodically interrupted by 12 MR windows on alternating sides of the channels. These windows open onto pockets in the wall that are large enough for segments of linear or branched alkanes to enter. Diffusion in the pocket direction, however, is not possible. The two channel systems in MFI run in the [100] and [010] directions (Dx and Dy). This zeolite differs from TON and EUO because the channel systems intersect. The intersections are such that a path is created allowing diffusion in the third dimension as well (Dz). Since the path in the third direction is not created by one specific channel, no channel diameters
Intracrystalline Diffusion in TON, EUO, and MFI are provided for that direction in Table 1. Figure 1c shows that the walls of channels running in the [100] direction have wellpronounced oscillations. For this reason, these channels are described as sinusoidal. Observation of Figure 1d indicates why channels running in the [010] direction are called straight. It is important to bear in mind, though, that the figure does not provide a clear picture of the intersections that periodically occur between straight and sinusoidal channels in MFI. This means that even the straight channels have periodic interruptions in their wall structure. It was anticipated that the nature of the channel systems influences transport behavior for alkanes, inducing what we term lattice effects. An example of such an effect was described in ref 18 where diffusion of linear alkanes in the straight channels of MFI was seen to decrease with increasing chain length except for certain “resonant” chain lengths. Runnebaum and Maginn demonstrated that the enhanced diffusion at these lengths resulted from commensurability between the molecular length and the spacing between intersections of the straight and sinusoidal channels in MFI. It is a matter of debate as to whether this phenomenon has been observed experimentally,18 and it is clear that the conditions of a rigid lattice, dilute limit, and low temperature magnify such lattice effects (in ref 18 the lattice effect was seen to decrease with increasing T). Nonetheless, the large selection of zeolite candidates available for a given chemical process provides motivation to better understand the differences between zeolites with regard to molecular transport through them. Even if certain lattice effects are washed out at high temperature or loading, they may influence behavior during some stages of a process. Furthermore, the referenced lattice effect was observed for linear alkanes for which the constraint imposed by 10 MR channels is not extreme. In the case of branched molecules, constraint is much more pronounced. Experimental diffusion constants for small monomethylbranched molecules in zeolites are as much as 1 order of magnitude lower than similar mass linear molecules.7,14,37 Therefore, lattice effects may be more influential for branched molecules and less easily removed by temperature or loading. Here, we study the effect of monomethyl branches on diffusion. Our attempts to study dimethyl branches were unsuccessful due to the very low diffusion of these molecules. The geometry and connectivity of channels in TON, EUO, and MFI are sufficiently different that comparisons could be made between systems where lattice effects are expected to be a minimum and systems where they are not. In section II, we describe our model and simulation methodology as well as the levels of expected error in our measurements. Results for DR and Ea are presented in section III where we also compare transport behavior in the three zeolites studied. We summarize our conclusions in section IV. To our knowledge, this is the first work examining the effect of branching and branch position on the transport of mediumlength alkanes in multiple zeolites. By studying the same molecules in multiple zeolites, a fairly systematic comparison of lattice effects is possible. II. Simulation Model and Methodology The alkane molecules were simulated using an united atom (UA) model in which CHn groups are treated as single particles. Intramolecular interaction between particles, or united atoms, is represented by bonded and nonbonded forces. Bonded forces are of three types: constraint forces which keep intramolecular nearest neighbors at a fixed bond distance, bending forces arising
J. Phys. Chem. B, Vol. 103, No. 24, 1999 4951 TABLE 2: Lennard-Jones Potential Parameters group
σ (Å)
(kcal/mol)
CH3 CH2 CH O
3.930 3.930 3.850 2.789
0.227 0.093 0.064 0.240
from the term
Vb(θ) )
kb (θ - θb)2 2
(1)
which maintain the equilibrium angle, θb, between successive bonds, and torsional forces from
Vt(φ) )
∑i ai cosi(φ)
(2)
characterizing preferred orientations and rotational barriers around all nonterminal bonds. The nonbonded forces are described by Lennard-Jones (LJ) interaction sites located at the position of each carbon atom center of mass. The LJ potential is defined by
[(σr ) - (σr ) ]
VLJ(r) ) 4
12
6
(3)
This nonbonded interaction is between intramolecular sites except for those which are separated by less than four bonds and which therefore interact through one or more of the bonded interaction terms. The LJ interaction was truncated at rc ) 10 Å, and the potential was shifted so that VLJ(rc) ) 0. The interaction parameters are due to Siepmann et al.38 with torsional potentials from Jorgensen et al.39 The parameters for the LJ term are given in Table 2; parameters for cross terms in the LJ potential are calculated from the Lorentz-Berthelot rules.40 The bonded interaction parameters used here have been presented as model A in prior publications on linear and branched alkane simulations.41 The zeolites used were obtained directly from the zeolite structure database contained in the Solids Builder module of the Insight II software package.42 As such, the lattices used represent idealized stoichiometry and crystal structure. For a zeolite with an n-dimensional channel system, the unit cell was replicated in the appropriate n crystallographic directions. A minimum amount of zeolite material was used that would still permit appropriate application of n-dimensional periodicity. Since alkane molecules were unable to diffuse in nonperiodic directions, it was only necessary to include zeolite material in those directions sufficient to ensure saturation of the alkane/ zeolite interaction. The zeolite lattice was held rigid throughout the simulations, and the LJ potential was used for interactions between the CHn groups and the zeolite. Furthermore, it was assumed that the large oxygen anions would strongly dictate the shape of the force field experienced by the alkanes so network-forming cations were not included in the LJ calculation. The LJ parameter used for oxygen interactions therefore represents an effective parameter for calculating the overall zeolite force field. These same approximations for representing the zeolite and the alkane/zeolite interaction have been used in prior simulation studies.16,18,22,24 The effect on D of the use of a rigid lattice has been examined.26,43 For highly constrained molecules such as branched alkanes, allowing for lattice vibrations can significantly affect D. However, in the current study we compare various trends in D and Ea between the three zeolites studied to examine the influence of channel structure
4952 J. Phys. Chem. B, Vol. 103, No. 24, 1999
Webb et al.
on alkane transport. Since all the zeolite lattices are held static, we believe this approximation still permits this comparison. Different from prior works, we distinguish between CHn groups for the alkane/alkane and alkane/zeolite interactions. This change was made since we included branched alkanes in the study. The LJ parameters for zeolite oxygens were calculated from the values used in ref 17 by normalizing with respect to normal decane in our model. Simulations presented here were done in the range T ) 300900 K, and all were in the dilute limit. The condition of dilute limit allowed us to focus on lattice effects and at the same time lowered computational demands thereby providing the ability to perform longer simulations. Since the zeolite was static and the molecules did not interact with one another, a Langevin equation employing a damping term and a random noise term was used to thermostat the alkanes.44 The time constant used for the thermostat was τ ) 1 ps. The equations of motion were integrated using the velocity Verlet algorithm, and bond lengths were kept constant using the RATTLE algorithm.25 For data gathering runs at T g 600 K δt ) 2.5 fs was used, while for T < 600 K δt ) 5 fs. For branched molecules, δt ) 5 fs for all data gathering regardless of T. Initially 64 all-trans versions of a molecule were inserted in the diffusion channel with the molecular axis collinear with the channel axis. For MFI, molecules were inserted in the straight channels only. The 64 molecules did not interact so that we are strictly in the dilute limit. Each of the molecules was given a random rotation about the channel axis and shifted along the channel a random amount. Initial high-energy configurations were relaxed by a 2-ps run using a 1-fs time step (δt ) 1 fs) at T ) 900 K. The simulations were continued at 900 K for 1 ns using δt ) 5 fs to randomize the molecular positions further. Another 1 ns run with δt ) 5 fs was then performed to reach the desired temperature. After equilibration, runs in the range of 6-40 ns were done to measure DR, where R ) x, y, or z direction. Since our goal was to predict diffusion constants, our primary concern was not the starting positions of the 64 molecules. Rather, as discussed below, we strove to compute trajectories over a long enough duration to allow molecules to sample the zeolite unit cell in all applicable crystallographic directions. Nonetheless, for certain test cases, the MSD was calculated using different starting configurations. As long as the conditions for good accuracy outlined below were met, starting configuration did not affect the predicted diffusion constant. The diffusion constant was obtained, as described in section I, from the MSD through the Einstein equation for self-diffusion:
Ds )
〈(r(t) - r(0))2〉 6t
(4)
The braces are used to denote an ensemble average over the 64 (isolated) molecules and all available initial times. The duration over which a MSD calculation was performed ideally allowed molecules to diffuse, on average, a distance at least equal to the length of the zeolite’s unit cell in the direction of diffusion. It therefore depended upon the unit cell of the zeolite and how fast the molecules diffused. For fast moving molecules inside TON (unit cell length c ) 5 Å), a 400-ps duration for computing the MSD was sufficient and this was done using molecular trajectories that were at least 4 ns in length. For the slowest molecules in EUO and MFI (with larger unit cells), MSD calculations 20 ns in length were performed from 40-ns simulation durations. Due to the anisotropic nature of diffusion in zeolites, it was useful to consider diagonal terms of the
Figure 2. MSD in the channel direction at T ) 600 K as a function of time for n-alkanes in EUO (n ) 10, 16, and 30). The square of the value of the unit cell parameter in the diffusion channel direction is indicated on the y-axis.
diffusion tensor. For the x direction (simplifying notation slightly),
Ds,xx ) Dx )
〈(x(t) - x(0))2〉 2t
(5)
with similar expressions for the y and z directions and
Ds )
Dx + Dy + Dz 3
(6)
The activation energy for diffusion Ea was computed assuming
( )
Ds ) D0 exp
-Ea kBT
(7)
where D0 is a preexponential factor and kB is the Boltzmann constant. Arrhenius plots of diffusion then yielded predictions of Ea. For TON and EUO, Ds in eq 7 should be replaced with Dz and Dx, respectively. For MFI, the activation energy from Ds and the diagonal components of the diffusion tensor was computed. In ref 41, the same model used here was used to calculate Ds for melts of linear and branched alkanes, and uncertainty in the data was around 5-10%. By using the same measure of statistical uncertainty here, we are able to obtain an idea of our error. However, two additional sources of systematic error should be addressed for the current calculations. As discussed above, the overall MSD should be great enough to demonstrate that molecules have sampled the entire unit cell of the zeolite yet short enough, relative to the simulation duration, to provide sufficient statistics. For the sake of discussion, MSD () curves for linear alkanes in EUO at T ) 600 K are shown in Figure 2. Data are shown for carbon numbers n ) 10, 16, and 30. All plots demonstrate the desired linearity for computing a reliable slope and, therefore, diffusion constant. The value of the squared lattice parameter in the direction of diffusion is indicated on the y-axis. From this it can be seen that both C10 and C16 molecules move, on average, approximately the length of the unit cell within the duration of the MSD calculation, which is good evidence that the entire unit cell is being sampled by molecules and properly represented in the calculated diffusion constant. For C30 this is not the case; however, doubling the MSD duration to 10 ns (providing the slope remains the same) would allow for these longer molecules to move a unit cell’s length. Our longest simulation durations were 40 ns, and from
Intracrystalline Diffusion in TON, EUO, and MFI
J. Phys. Chem. B, Vol. 103, No. 24, 1999 4953
TABLE 3: Diffusion Constant D for n-Alkanes in the Three Zeolites Studieda zeolite TON
EUO
MFI, [100]
MFI, [010]
MFI, [001]
T (K)
n-C7
n-C10
n-C15
300 400 500 600 700 400 500 600 650 700 400 500 600 700 400 500 600 700 400 500 600 700
89.4 103.7 121.9 146.8 166.4 4.0 5.5 9.3 10.8 11.0 16.3 27.9 34.7 38.5 61.4 57.1 72.2 84.0 5.1 6.6 8.8 10.6
53.2 75.0 85.2 95.8 112.6 3.1 5.7 8.8
50.9 65.2 68.8 71.5 79.3 1.4 2.3 4.0 4.2 4.5 2.3 5.8 9.4 14.0 45.6 34.1 32.6 33.3 0.5 1.3 2.2 3.3
10.8 6.6 13.6 18.4 27.3 50.7 54.5 55.9 58.9 1.9 3.5 4.7 6.9
n-C16
70.5
3.9
6.8
33.6
2.0
n-C24 45.6 43.3 44.1 41.8 49.3 0.6 1.0 1.7 2.7 0.3 1.6 3.5 5.5 31.2 18.6 18.3 18.1 0.1 0.5 0.9 1.4
n-C30
32.7
1.1
3.3
16.4
0.5
a Values are reported for the [100], [010], and [001] crystallographic directions in MFI. The values for D quoted are × 1010 m2/s.
these, we performed 20-ns MSD calculations. Using a 20-ns MSD duration and the unit cell parameter(s) of a given zeolite, a straightforward calculation yields an approximate value of DR below which error from this source may arise. For TON and EUO, this is 0.06 and 0.5 × 10-10 m2/s, respectively. For the [100] and [010] directions in MFI, this is 1 × 10-10 m2/s, while for the [001] direction it is 0.5 × 10-10 m2/s. So for Dx of C30 in EUO, computing a 10-ns MSD is within our capability and error imposed by insufficient trajectory sampling is minimized. For nearly all of the linear alkane data, this is true, and in these instances, the error in our measures are of the order 5-10%. For diffusion in TON, this is true for the branched molecule data as well. For some branched molecule data in EUO and MFI, this error is more pronounced; we estimate it is as large as 10-25%, and in a few worst cases (Dz for monomethylbranched molecules in MFI), our predictions are reliable only in order of magnitude. However, Ds in MFI is dominated by the Dy component. Furthermore, this source of error is reduced in all cases by the use of 64 noninteracting molecules distributed throughout the channel structures. The other source of systematic error can arise from the selection of time step for the simulations (δt) due to the high temperatures studied herein. Because of the ensemble and thermostating method used, energy conservation is not an applicable criterion for time step selection. In ref 41 δt ) 5 fs was used throughout, but the highest temperature studied was T ) 448 K. In the present work it was determined that above approximately T ) 600 K a time step of δt ) 5 fs can potentially give rise to larger systematic errors. So for linear alkanes in zeolites with T g 600 K δt ) 2.5 fs was used, minimizing this error. Since the diffusion constant for many of the branched molecules was near our statistical limit, dt ) 5 fs was used for all branched alkane simulations. III. Results and Discussion A. Linear Chains. From data such as those shown in Figure 2, diffusion constants were calculated for linear alkanes with carbon number n ) 7, 10, 15, 16, 24, and 30 in TON, EUO, and MFI at T ) 600 K. The results are collected in Table 3 and
Figure 3. nDR versus 1/n for linear alkanes inside the zeolites TON, EUO, and MFI at T ) 600 K. For TON (b) and EUO (9) results are presented for Dz and Dx, respectively. For MFI, results for Dx (0), Dy (4), and Dz (O) are presented.
shown in Figure 3 where nDR is plotted versus 1/n. For TON nDz and for EUO nDx are plotted, while for MFI nDx, nDy, and nDz are shown. Linear alkanes diffuse the fastest in TON, and this is attributed to the absence of any channel features which pose a significant hindrance to diffusion, that is, minimal lattice effects. In a dilute gas, alkane diffusion scales as 1/n so in the absence of lattice effects we expect similar scaling. As shown in Figure 3 this is very nearly the case for TON (a least-squares fit to the data for nDz gives a slope of 0.006). Interactions with the channel walls do dictate the drag or friction coefficient on diffusing molecules, but there are no specific barriers to diffusion due to the TON structure. Furthermore, the length of the smallest molecule studied here is already greater than the unit cell length in the channel direction. Therefore, the whole length of the unit cell is experienced by even the smallest molecule in any conformation. This acts to further remove lattice effects from the n dependence. For linear alkanes in EUO, diffusion is the slowest of the three zeolites studied. While Dz in MFI is lower than Dx in EUO for most n, it should be kept in mind that overall diffusion is dominated by the fastest component of the Ds tensor and both Dx and Dy in MFI are faster than Dx in EUO. The side pockets of EUO are readily sampled by the ends of linear alkanes. Analysis of molecular trajectories demonstrated that diffusion in EUO occurs by molecules hopping from pocket to pocket. The first observed effect of this lattice feature is an overall decrease in diffusion relative to the other zeolites. As seen in Figure 3, for n larger than 7, Dx is seen to drop faster than what would be predicted by 1/n scaling. This, as well, can be explained in terms of a lattice effect. Longer molecules are able to have both ends of the chain in a side pocket so diffusion can occur only when both ends come out of their pockets for a coordinated jump. As n increases, so does the floppiness of the alkane molecule. That is, the motion at opposite ends of the chain is increasingly decorrelated so the frequency of coordinated jumps is lowered. A final point about the EUO data should be mentioned: for n ) 7 nDx is less than for n ) 10 even though Dx for n ) 7 is higher than for n ) 10. This indicates lower diffusion for n ) 7 than is expected from the data for all larger n. Molecular trajectories demonstrated that at n ) 7 an alkane is able to just fit inside a side pocket and so is quite efficiently trapped. For slightly larger n, one end of the molecule must stick out of the pocket so it is more likely to sample adjacent side pockets, thereby reducing the trapping efficiency of a pocket. For still slightly larger n, both ends of the molecule are able to sample two side pockets. This indicates that a small-
4954 J. Phys. Chem. B, Vol. 103, No. 24, 1999 molecule regime might exist where, inside the zeolite EUO, such molecules experience long residence times due to adsorption in the pockets. For slightly larger molecules, residence times are lower because being fully adsorbed into a pocket is no longer possible. The data for Dy (straight channel diffusion) in MFI are quite comparable to those for diffusion in TON. This is true for both the magnitudes and dependence on n (a least-squares fit to the MFI nDy data gives a slope of 0.084). This is sensible since the only hindrance to diffusion along the straight channels comes from intersections with sinusoidal channels. Unlike in EUO, these intersections do not represent traps for molecules but, rather, a position where diffusion might be slowed while a molecule samples the entrance to a sinusoidal channel. If the molecule enters the other channel, its motion contributes to the Dx component of the diffusion tensor. Thus, overall transport is not greatly hindered by the intersections. Diffusion in the sinusoidal channels is subject to lattice effects for obvious reason. Longer molecules must adopt higher energy conformations to move down these undulating channels. This explains the greater than 1/n dependence for Dx in MFI. Since diffusion in the [001] direction is achieved by successive combinations of motion in the [100] and [010] directions, it is expected that the values of Dz will be lower and the n dependence will be dictated by Dx since this is the rate-limiting direction for diffusion. While Ds is dominated by Dy for MFI, there still is a significant contribution from Dx so that Ds shows greater n dependence than 1/n. For longer molecules the likelihood of part of the molecule being affected by intersections and the sinusoidal channels is greater. The lattice or “window” effect created by intersection spacing described in ref 18 was not expected at this high temperature since the effect had practically disappeared even at T ) 400 K. From the observed diffusion data, it is not surprising that activation energy for diffusion of the linear alkanes Ea as well as the dependence of Ea on n is different between the three zeolites. Diffusion constants were obtained at different temperatures for selected chain lengths and are tabulated in Table 3. Arrhenius plots were then formed, and as described in section II, Ea was computed. Examples of such plots are illustrated in Figure 4a for n ) 7, 10, 15, and 24 in TON and EUO and in Figure 4b for n ) 10, 15, and 24 in MFI. Data for both Dx and Dy are shown to illustrate the anisotropy of transport in MFI, while Dz is not shown for the sake of clarity (data for n ) 7 are also omitted from Figure 4b to improve clarity). More will be said about the data specific to MFI below, but it should be pointed out that the activation energy obtained from experiment would generally reflect the change in Ds with T. However, carefully designed experiments using single crystals of MFI have been able to explore the anisotropy in transport properties.45-47 For TON and MFI, the lowest temperatures studied were not included in the calculation of Ea. Runnebaum and Maginn discussed the difficulty in obtaining single activation barriers for linear alkanes in MFI at lower temperature where there are several competing processes, often giving rise to an apparent negative activation barrier.18 By using data at high temperature, we avoid this difficulty. So, in TON and EUO, only values of DR for T g 400 K were used, while in MFI we use values for T g 500 K. Figure 4b shows the deviation from linearity at low temperature is most pronounced for large n. Observation of Figure 4 also facilitates discussion about the error in our Ea calculations. Because most of the Arrhenius plots were constructed from three or four points only, the lines fit to this data, and therefore the Ea obtained, can be significantly changed by
Webb et al.
Figure 4. Arrhenius plots for the diffusion constants DR of n-alkanes inside the zeolites (a) TON and EUO and (b) MFI. For TON and EUO, data are shown for n ) 7 (asterisks), 10 (triangles), 15 (squares), and 24 (circles); except for n ) 7, open symbols are for TON and filled are for EUO. In MFI, data are shown for n ) 10 (triangles), 15 (squares), and 24 (circles); filled symbols represent Dx and open symbols Dy.
a 5-10% shift in values of DR. It is easy to imagine errors in DR combining, resulting in even larger errors for Ea. While we do not offer a quantitative measure of this error, a conservative estimate would be that it could be twice as large as the error in our diffusion measurements or 10-20%. For branched molecules, it could be even larger. For this reason, only general statements are made about comparisons of Ea and the trends observed with n. Figure 5 shows our results for Ea at n ) 7, 10, 15, and 24 for the three zeolites; Table 5 contains the calculated values of Ea. For MFI, the Ea computed for Dx, Dy, Dz, and Ds is plotted in the figure but only Ds is tabulated. As was mentioned above, lattice effects for linear alkanes in TON were minimal and the Ea data support this notion. The value obtained at small n is roughly 1 kcal/mol, a low barrier to diffusion. This is in alignment with the absence of any structural features that hinder diffusion along the zeolite channel. As n increases, Ea decreases to values close to zero for large n. We study temperatures high enough to allow for molecular flexibility and, thereby, generally observe increasing DR with T. However, for large n in TON, the influence of T on Dz is small because the increase in motion with higher temperature is countered by more frequent interactions with the channel walls. In EUO, Ea is significantly larger (≈2-3 kcal/mol) for all n. This is a direct result of the pockets in EUO. The increased barrier to diffusion exists because molecules must withdraw from a pocket to move down the diffusion channel. Within the error of our calculations, it is difficult to ascertain if a systematic dependence of Ea on n exists.
Intracrystalline Diffusion in TON, EUO, and MFI
J. Phys. Chem. B, Vol. 103, No. 24, 1999 4955
TABLE 4: Diffusion Constant D for Monomethyl-Branched Isomers in the Three Zeolites Studieda zeolite
T (K)
TON
EUO MFI, [100]
MFI, [010]
MFI, [001]
a
2M-C6
400 500 600 650 700 600 600 700 800 900 600 700 800 900 600 700 800 900
3M-C6
2M-C9
3M-C9
4M-C9
17.2
0.5
2.7 7.7 14.8
0.8
0.2
0.5 0.3
0.5 0.1
0.9 0.04
0.9
0.2
1.2 2.3 3.4
0.1
0.03
0.10 0.20 0.40
0.2 0.08 0.30 0.40 0.70 0.2 0.4 1.0 1.9 0.02 0.07 0.20 0.24
23.9 0.3 0.4 0.6 1.2
0.1
0.02
5M-C9 0.1 0.6 1.1 1.7 1.1 0.07 0.15 0.24 0.50 0.1 0.4 0.5 0.8 0.02 0.05 0.10
2M-C15
3M-C15
10.8
0.5
0.3 0.1
0.1 0.03
0.5
0.2
0.05
0.01
0.15
Values are reported for the [100], [010], and [001] crystallographic directions in MFI. The values for D quoted are ×1010 m2/s.
Figure 5. Activation energy Ea for diffusion versus chain length n for linear alkanes in TON (b), EUO (9), and MFI. For MFI, Ea calculated from Dx (0), Dy (4), Dz (O), and the three-dimensional diffusion constant Ds (×) are shown.
TABLE 5: Activation Barriers Ea for Diffusion of Linear and Branched Alkanes in the Zeolites Studied zeolite
n-C7
n-C10
n-C15
n-C24
2M-C9
TON EUO MFI
0.9 2.0 1.3
0.7 2.4 0.9
0.3 2.3 0.7
0.2 2.8 0.6
4.0 5.5
3M-C9
5M-C9 8.7
7.7
7.2
If it does, it is small. Since the hindrance to diffusion is withdrawal of a molecule end from a pocket, there can be at most only two such events that need occur for a molecule to diffuse, regardless of length. Therefore, at least for n large enough to bridge two pockets, one does not expect a significant n dependence. Since the spacing between pocket mouths is 6.8 Å, bridging two pockets is possible even for relatively small n. This again indicates the possibility of a small-molecule regime for which the molecule is entirely inside a pocket in EUO which could result in a different activation barrier. Data presented here are insufficient to describe such a regime if it exists. As discussed before, motion in the [001] direction inside MFI is rate-limited by motion in the [100] direction so it is not surprising to see the Ea for Dz in MFI closely follow Ea for Dx. Furthermore, it is not surprising to see the significant increase of the barrier with n. For a molecule in the sinusoidal channels of MFI, high-energy bond conformations must be introduced along the molecule’s length for it to move along the channel. For large n, more such conformations must exist, thus a higher barrier to diffusion. The Ea for Dy is small for n ) 7 (≈1.3
kcal/mol) and decreases with increasing n. This behavior is similar to what was seen for TON which is sensible since the MFI channels that run in the [010] direction are straight like those in TON. Despite studying higher temperatures than those in ref 18, we still obtain a slightly negative Ea for Dy at the largest values of n in MFI. This behavior is probably very dependent on the conditions of rigid lattice and dilute limit.18 However, the Ea calculated for Ds does not show this peculiarity. Similar to TON, Ea obtained for Ds in MFI is fairly small and decreases slowly with increasing n. B. Branched Chains. Another aspect we examined was the effect of branching on the transport behavior. This was done by studying x-methyl Cn-1 molecules with x ) 2-5 and n ) 7, 10, and 16 in the three zeolites (for n ) 7 only 2- and 3-methyl C6 were applicable). Diffusion constants were calculated and are collected in Table 4. The plots in Figure 6 allow us to examine the n dependence for DR of branched alkanes. Figure 6a shows data for molecules inside TON at T ) 600 K; Dz is plotted versus n for 2-methyl Cn-1 and 3-methyl Cn-1 for n ) 7, 10, and 16. The data for linear alkanes in TON are also included for comparison. The first observation one can make is that even adding one methyl branch lowers diffusion 1-2 orders of magnitude relative to linear alkanes. This is true in all the zeolites studied as evidenced by Figure 6b,c for EUO and MFI, respectively. These data are in qualitative agreement with experiment7,14,37 and explain why branched molecules are described as bulkier. This also illustrates why we were unable to study diffusion of molecules with longer branches or more than one methyl branch. Experimental diffusion constants for these molecules are even lower, in some cases by orders of magnitude, than monomethyl-branched molecules. Therefore, we expect that the simulation times needed to obtain a reliable diffusion constant for more complex branched molecules are currently beyond our capability. For TON, Figure 6a shows that the 1/n dependence observed for linear alkanes does not hold for monomethyl-branched molecules. This is most obvious for 3-methyl Cn-1 where Dz increases from n ) 7 to n ) 10. As mentioned previously, the long simulations and small unit cell in TON combined to minimize error for measurements in this zeolite, so we feel this increase is significant, albeit small, within the model. An explanation for this behavior was evident from graphical analysis of the molecules’ trajectories. From Figure 1a, one can see the undulations in the walls on opposite sides of the TON channel. They are 90° out of phase with a wavelength equal to the unit
4956 J. Phys. Chem. B, Vol. 103, No. 24, 1999
Figure 6. Diffusion DR versus carbon number n at T ) 600 K for linear alkanes Cn (4) and monomethyl isomers 2-methyl Cn-1 (0) and 3-methyl Cn-1 (O). Figures are shown for molecules in the zeolites (a) TON, (b) EUO, and (c) MFI. Results for MFI are shown only for the fastest direction [010] (Dy).
cell length. Observation of the trajectories for 3-methyl C6 and 3-methyl C9 shows that the CH3 united atoms preferentially orient themselves at a position of maximum amplitude in the undulations; this is particularly true for the CH3 branch. The 3-methyl C6 molecules are able to adopt conformations in which all the CH3 groups are at these energetically favorable positions, and so the molecule is fairly well bound. The fit of the CH3 groups on the 3-methyl C9 molecules to the undulations in the channel wall is not as good. As a result, hops along the channel are less frequent for 3-methyl C6 than 3-methyl C9 molecules and Dz is lower. This effect is small but helps to illustrate that for constrained molecules like branched alkanes, even subtle channel features can induce lattice effects. A final observation for TON can be made in comparing the data for 2-methyl Cn-1 and 3-methyl Cn-1. The effect of branch position in this structure
Webb et al. is quite significant as there is 1 order of magnitude decrease in Dz when the branch is moved from position 2 to position 3 at all n studied. Again, analysis of molecular trajectories provides an explanation for this. A branch at the 2 position means a molecule has a symmetric tail of two CH3 groups at one end. Because the wall undulations are 90° out of phase, it is not possible for both CH3 groups to occupy the energetically favorable positions near amplitude maximums in the walls. With the branch at the 3 position, the branch CH3 group is shifted along the channel from the end CH3 group so that they are both able to occupy energetically favorable positions. As a result the hop frequency for molecules with the branch at the 2 position is significantly higher than when the branch is at the 3 position. Figure 6b presents data for molecules in EUO. For branched molecules, Dx decreased more quickly with n than Dz in TON but this difference is small. What is most prominent from Figure 6b is the lack of dependence on branch position. Within the error of our calculations, Dx is equal for 2- and 3-methyl Cn-1 molecules at all n tested. Trajectory analysis demonstrates that both types of molecules can fit the end with two CH3 groups into a side pocket in EUO. The decrease in Dx upon adding a branch is associated with the increase in difficulty of pulling the branched end of the molecule out of a side pocket for a diffusive hop. Thus, a branch at position 2 or 3 disrupts diffusion in a roughly equal fashion in EUO. Figure 6c shows branched molecule data in MFI; only Dy is plotted, but similar behavior was observed for the other directions. Adding a branch to a linear molecule has the greatest effect in MFI in that there is a nearly 2 order of magnitude drop in Dy between the linear molecule curve and the 2-methyl Cn-1 data. There is nearly another order of magnitude drop when the branch is moved from the 2 to the 3 position. The reason for this pronounced effect is the channel intersections and the increase in void space found there. The bulky end of the branched molecules preferentially resides in the intersection region because the void space allows the CH3 groups to optimize their interactions with the channel walls. This is more pronounced for molecules with the branch at position 3 because those molecules have effectively bulkier ends. Dependence of Dy on n is small because the diffusion-limiting process is moving the branched end from an intersection region. There appears to be a slight enhancement of Dy at n ) 10, and at least for the 2-methyl Cn-1 data, we believe this is significant within the model. We believe the explanation for this is the “window” effect described by Runnebaum and Maginn18 who found enhanced diffusion for linear alkanes at n ) 8 and 16. They observed linear alkane diffusion along [010] in MFI as resulting from hops over intersection regions. Molecular lengths at n ) 8 and 16 were found to be nearly integer multiples of the intersection separation distance. These molecules exhibited hops of multiple intersection spacings resulting in enhanced diffusion at these resonant lengths. That work, however, was done in the range T ) 300-400 K, and the window effect was nearly gone at T ) 400 K. We believe that for branched molecules the relevant energy scale for such an effect would be wider reflecting the increased energetics associated with their diffusion. With a branch added to a molecule at n ) 10, the backbone length is very near the length of linear C8. Thus we propose that the window effect can operate for branched alkanes but does so to a higher temperature than for linear alkanes. In Figure 7 we examine the effect of branch position more directly. Figure 7a shows data for molecules in TON at T ) 600 K; relative diffusion (D/Dnz, where Dnz is the diffusion constant of the linear molecule) is plotted versus branch position.
Intracrystalline Diffusion in TON, EUO, and MFI
Figure 7. Relative diffusion DR/DnR versus monomethyl branch position at T ) 600 K for carbon numbers n ) 7 (4), 10 (0), and 16 (O). Figures are shown for molecules in (a) TON, (b) EUO, and (c) MFI (Dy).
A branch position of 1 refers to the linear molecule; therefore, Dz/Dnz ) 1 at this position. Curves are plotted for n ) 7, 10, and 16; for n ) 7 there are data for branch positions 2 and 3, while for n ) 10 and 16 there are data for branch positions 2-5. The successive order of magnitude drop in Dz going from a linear molecule to a 2-methyl branch and then to a 3-methyl branch is again observed in this figure. What is most interesting is the significant upturn in the n ) 10 and 16 curves going from a branch position of 4 to 5. When the branch is moved from the 3 position to the 4 position, there is a decrease in Dz though not as great as occurs from the 2 to 3 position. It appears, then, that branched molecules are increasingly bulkier in TON as the branch is moved toward the center of the chain. The upturn in the curve at branch position 5 reverses this notion. Here again, the constraint on branched molecules is such that the small undulations in the TON channel walls induce lattice
J. Phys. Chem. B, Vol. 103, No. 24, 1999 4957 effects. Graphical analysis of molecular trajectories shows that the enhancement in diffusion for branch position 5 relative to position 4 is caused by the degree of alignment of the CH3 groups with maxima in the wall undulations. For branch position 5 this is poor relative to branch position 4 so hops are more frequent in the former case. As before, the effect is not large but of interest considering the fairly uniform nature of the TON channels. Figure 7b,c show similar data for molecules in EUO and MFI, respectively. The decreased dependence of diffusion upon branch position for molecules in EUO relative to TON is again observed. For this structure peculiar behavior in Dx/Dnx is also observed as a branch is moved toward the center of the chain. At n ) 10, diffusion is significantly higher for branch position 4 than for positions 2 and 3; it is higher still at branch position 5. This lattice effect is related to the side pockets in EUO but in a different way than has been seen so far. Clearly the pockets in EUO strongly dictate transport behavior since they act as diffusion traps. When a branch is at the 2 or 3 position, the branch end of the molecule can still reside in a pocket unhindered. However, trajectory analysis shows that as the branch is moved further from the end of the molecule, insertion of that end into a pocket requires that the branch CH3 group reside very near the entrance to the pocketsan energetically less favorable conformation. As a result, the molecule end does not get trapped by the pocket as efficiently and diffusion is faster. This is most pronounced when the branch is at the 5 position. Figure 7c for molecules in MFI shows different behavior from that seen in TON and EUO in that Dy/Dny successively drops as a branch is moved toward the center of the chain and appears to be leveling out by branch position 5. In this zeolite, molecules do become effectively bulkier as a branch is moved toward the center. The data presented in Figure 7 can be related to experimental catalytic data for characterizing zeolites. Briefly, the ratio of yields of 2-methyl C9 to 5-methyl C9 from a clean linear C10 feed is used to characterize the level of constraint in a zeolite. A higher ratio indicates higher constraint, and the underlying notion for this is that the transition state for forming a 5-methyl C9 is bulkier and therefore more restricted in constrained environments than the transition state for forming 2-methyl C9. Diffusive limitations on the different molecules have been discounted as not playing a role in determining this ratio. We discuss this matter in detail elsewhere,48 but the diffusion data we have gathered challenges some of these notions. That is, we find for some zeolites such as TON and MFI that diffusion limitations may be significantly influential in determining the ratio of yields mentioned above. Finally we studied the effect that branching had on the activation energy for diffusion Ea. For this part of the study, only TON and MFI were compared. Figure 8 contains Arrhenius plots for linear and branched molecules in TON and MFI. For both zeolites, data are shown for linear C10, 2-methyl C9, and 5-methyl C9; for MFI data for 3-methyl C9 are shown as well. The diffusion data used in these figures are collected in Table 4, and the data for Ea are in Table 5. It was said before that the energetics of branched alkane transport are greater than for linear alkanes, and the Ea data support this statement. Even within the error of our calculations, it is clear that Ea is much higher for branched molecules than for linear. This is reasonable since the constraint on branched molecules is larger and lattice features are more influential. Therefore, there are more pronounced energetic barriers to diffusion. It also appears that Ea for 2-methyl C9 is lower than for 5-methyl C9 in both zeolites. In
4958 J. Phys. Chem. B, Vol. 103, No. 24, 1999
Webb et al. were present for all the zeolites, demonstrating the high constraint imposed on branched molecules in 10 MR zeolites. References and Notes
Figure 8. Arrhenius plots for the diffusion constants of n-decane (4) and n-methylnonanes where n ) 2 (0), 3 (O), and 5 (*). Figures are shown for (a) TON and (b) MFI (Ds).
MFI, Ea values for 3-methyl and 5-methyl C9 are comparable. This trend appears consistent with the diffusion data in Figures 6 and 7. The large drops in DR with branch position are a result of an increase in the energetics for diffusion. Recall, also, that the drop in Dy for MFI from branch position 2 to 3 is much more pronounced than the drop from portion 3 to 5. This too is in line with the energetics for diffusion of 3-methyl and 5-methyl C9 being similar. These results further support the notion that the window effect for diffusion in MFI discussed in ref 18 would be effective to a higher temperature regime for branched molecules. IV. Conclusion Diffusion constants and activation energy for diffusion of linear and branched alkanes in the zeolites TON, EUO, and MFI have been calculated. The channel systems in these zeolites differed in dimension, wall structure, and connectivity such that trends in much of the data collected could be compared based on lattice differences. The channels in TON have no distinctive features which significantly affect the diffusion of a linear molecule, and as such, lattice effects were least pronounced. This was demonstrated by the 1/n dependence of Dz and the low Ea values obtained. For linear alkanes in MFI and EUO, this was not the case. The intersections in MFI are regions of increased void space which linear alkanes preferentially avoid; therefore, these influence transport behavior. The anisotropy in the channel structure for MFI shows up in the data with regard to both magnitude and dependence on n. In EUO there are large pockets off the diffusion channels which strongly influence linear alkane transport. This was especially evident at the smallest values of n studied. For branched alkanes, lattice effects
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