J. Phys. Chem. 1993, 97, 7660-7664
7660
Sorption of Xenon in Zeolite Rho: A Thermodynamic/Simulation Study A. V. Vernovt and W. A. Steele' Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802
Lloyd Abramst Dupont Company, Central Research and Development, Experimental Station, P.O. Box 80228, Wilmington, Delaware 19880-0356 Received: November 25, 1992; In Final Form: April 19, 1993
Molecular dynamics simulations of the sorption of xenon at 300 K in zeolite rho are reported. The xenon-solid potential is taken to be a pairwise sum of xenon-xide interactions, with parameters adjusted to give agreement with the experimental Henry's law constants. Energies of adsorption and the distributions of sorbed atoms in the cage/window system were evaluated as a function of the xenon loading in the zeolite. It is shown that increasing loading produces an increase in the (negative) xenon-xenon average energy which is compensated by a decrease in the (negative) xenon-solid energy. The atomic distributions show that the xenon atoms are initially tightly sorbed in the cage windows. Once these are filled, additional xenons show chaotic, fluidlike trajectories in the cages. However, at the highest xenon loadings, even the atoms in the cages become localized into an ordered structure.
1. Introduction
The physical insights obtained from measurements of the thermodynamic properties of simple gases sorbed by have benefited greatly by the computer simulations of these systems.gJ0 By appropriate modelingof thegas-zeolite potentials, one can generate realistic models which provide interesting microscopic information when used in simulation studies. The models are generally based on the assumption that the interaction between a gas atom and the solid is given by a pairwise summation over the atoms in the zeolite. Most often, the oxides in the silicate crystal lattice make such a large contribution to the pairwise total that one can omit the Si or A1 atoms from the summation^.^ For zeolites containing a significant fraction of cations, the gascation interaction should also be included in the summations.gb The direct summation of the pairwise gas-oxygen energies is readily performed in simulations run on the current generation of computers, allowing one not only to evaluate thermodynamic properties of the sorbed gas but also to study the effects of changing the cationic content or the zeolite structure. This paper reports a combined experimental/simulationsstudy of xenon sorbed in a zeolite rho of relatively low ionic content other than protons. The unit cell formula for an ideal rho zeolite is H1~All~Si36096 assuming that H+ is the sole charge compensating cation. Samples used for sorption studiesgb had a portion of their charge compensation using Cs+ ions (e.g., H11.~5C~~.75A112Si36096or Hlo.oaCsz.ooAl1zSis609~). The crystal structure of this material is known to be1.7cubiccentric Im3m;for the purposes of a sorption study, it consists of two interpenetrating but unconnected body-centered-cubic lattices of roughly spherical cages of diameter 12 A. The cages are connected by windows formed by two 8-rings of T-0 groups, with T = A1 or Si. Thus, the windows can be approximated as short cylindrical tubes of length -4.0 %I and diameter -3.6 A. This crystal readily sorbs atoms as large as xenon. (Our parametrization of the xenonoxide potential suggests that atoms larger than 4.5 A will not readily pass through theentrance windows to thecages.) Sorption
-
Contribution No.5972. t Peoples' Friendship University, Department of Natural Science, Chair of Physical & Colloidal Chemistry, Lab. of Mathematical Simulation of Physical & Chemical Processes, M.-Maklaya, 6 Moscow,Russia 117198. t
isotherms and heats are reported here and elsewheregb for temperatures ranging from 195 to 650 K. From these, sorption Henry's law constants KH are calculated. These constants are related to the xenon-solid interaction energy us@) by11
where V, is the sorption volume of the crystal and thus is equal to nunitus,where nunit is the number of unit cells per gram of zeolite and v, is the sorption volume per unit cell. Units for KH are mol atm-l g-l, and experimental values are obtained from lim ( N , / p ) = KH p-0
where N, = moles sorbed per gram of zeolite. The integral of eq 1 over the cage plus window volume is straightforward and, consequently, can be done for various parametrizations of u,(r). One therefore adjusts this potential to optimize agreement with experimental data taken over a wide range of temperature and then uses the resulting model to simulate sorption at higher coverages. One can use the observed agreement of the simulations and the experiment as a test of the accuracy of the interactionlaw; however, the better the agreement, the more likely that the simulation can predict microscopic information. Particular attention will begiven here to the probable locations of the sorbed xenon atoms in the pore structure of this solid, to the sorption energy of these atoms, and especially, to the way in which the gas-solid and gas-gas components of the total energy change with increasing amounts of sorbed xenon. We will show that the simulations yield several interesting insights concerning the nature of xenon sorption in this material. 2. Interactions
Crystallographic data are available that produce atomic positions for zeolite rho containing various amounts of Cs+ ion.197 We have considered the set of atomic positions corresponding to the case of 0.2 Cs+ per unit cell. For simplicity, we have utilized this structure in simulations of xenon in a model of the [Cs+] = 0 system wherein only the oxygens were included in the Xe-solid summation of pairwise interactions, and secondly, in simulations of xenon in the same oxide lattice but with one Cs+ added per
0022-365419312097-7660%04.00/0 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7661
Sorption of Xenon in Zeolite Rho
6-
4-
2I
Y
4
O-2
0
-4 -
-6
-
-*- 2
3
4 5 IOOOIT
6
7
F'igwe 1. ExperimentaldatafortheHenry'slawconstantKHarecompared
with values calculated using xenon-zeolite interaction potentials with the parameters listed in Table I.
Position
0.5
0
='s [exp(-u*x4(r*)/P) P r
KH*(P)
8-ring
Figure 3. A schematic picture of the cage/window system for zeolite rho is reproduced here from the paper of Parise et a1.' Also shown are the paths followed in the calculationof Xe-zeolite interaction energy (Figure 2).
TABLE I: Parameters of the Pairwise Interactions Used Xe-Xe
- 11 dr* (4)
The fit of the calculations to the experimental Henry's law values is shown in Figure 1. A value for ax& = 3.3 A was taken from the 1iterature.l' This is only slightly smaller than would be estimated using the arithmetic mean of the accepted uxbxe (Table I) and the often-used UW = 2.7 A (3.04/21/6).13 The well depth ex&/k that matched the slopes of the experimental and the theoretical lines was found to be 151 K, which is considerably different than some previous estimates of this parameter. For example,237,4221,5and 127 K9havebeenusedbyvariousauthors in connection with calculations of xenon sorption in X and Y4and in silicalite.S+' Figure 2 shows the potential energy of an isolated xenon atom in a zeolite cage plotted as a function of the Xe position. In these calculations,the Xe was moved along straight lines drawn through the cage, as illustrated in Figure 3. The solid curve in Figure 2 shows the energies along a line passing through the centers of two opposing &rings, and the dashed curve, for a line passing through the centers of opposing6-rings. Position equal to zero corresponds to an atom in the cell center, where the 8-ring and the 6-ring lines in Figure 3 intersect. For the potential used in this work, it is evident that the 6-rings are too small to allow insertion of a xenon atom but that the 8-rings are the right size to be strong adsorption
0.5
Figure 2. Plots of the variations in xenon-zeolite interaction energy as the atom moves through the unit cell. The xenon position relative to the cell center is shown, on a scale where 0.5 correspondsto the atom located on the unit cell face, as indicated in Figure 3. Changes in energy are shown for an atom moved toward and into the 8-ring window (solid line) andalso the6-ringwindow (dashedline). (a) iscalculated for thepotential used in this work, and (b) is for an alternative suggested by June et d 9 in which the Xe-O well depth and size parameters are set equal to 127 K and 3.47 A, respectively. It is evidentthat an unphysicallylarge energy barrier to motion through the window is obtained for the June potential.
cage; in agreement with the experimental information: this ion was located in the center of one of the windows formed by the pairs of oxygen 8-rings. At this point, one needs only to parametrize the pairwise functionsto produce a gas-solid potential for use in the simulations. In general,
with u* = u/exbx and ri* = ri/axbx; here, X = 0,Xe, or Cs+. The X e X e potential was assumed to be a Lennard-Jones 12-6 function with "standard" parameters12 that are listed in Table I. Evaluation of the Xe-O e/ k was essentially empirical; that is, calculations of KH*(P)were fitted to the experimental data for Here,
Position
X
4
xe-cs+
22 1 151 140
4.1 3.3 3.74
sites, producing sorption energy of -6.6 kcal/mol when a xenon is situated at the midpoint between two neighboring 8-rings.There is a small barrier entrance into the windows formed by these 8-rings that amounts to -0.5 kcal/mol. The sorption energy on the inner wall of the cage varies somewhat with position but is roughly 4 kcal/mol. This is to be compared with a value of 1.5 kcal/mol calculated for the Xe-solid energy at the cage center. On the basis of these numbers, one may expect a strong preference for xenon filling in the windows, followed by cage filling. This is in general accord with previous conclusions drawn from X-ray data for this system.6 Parameters for Cs+-Xe pair potential were estimated using standard techniques: the van der Waals radii (rdw)of Xe and Cs+were used to calculate u x d S +(= [rvdw(cs+) + rdw(Xe)]/2'j6, and the Kirkwood-Muller formulas.11 was used to estimate ex&+. Parameters used in this calculation were tvdw(cs+) = 2.02 A,13 rdw(Xe) = 2.16 A,14 polarizabilities axe = 4.01 A3 and a , y = 2.79 A3 15, and an effective number of electrons = 7.63 for Xe and for Cs+.16 The results of this calculation are listed in Table I. Of course, the Cs+ ion polarizes Xe atoms in its vicinity. However, the fact that this ion is "buried" in the window and that there is a small number of such ions per cage means that the
1662 The Journal of Physical Chemistry, Vol. 97, No. 29, 19‘93
Vernov et al.
A 6
I-
rho rho+Cs+
0
---- Experiment01
b
--
3
I
I
4
8
I
I
I
I
12
16
20
24
1
i
no molec/cell-
Figure 5. Experimental and simulated values for the integral energies of adsorption for xenon in zeolite rho are plotted as a function of the number of sorbed atoms per unit cell. Data are shown for Cs+ concentration of 0.2 ion per cage. The simulations are for zero and one ion per cage. The data terminate at the highest practical loadings for 300 K.
F i p e 4. Computer-generatedtrajectories are shown for eight xenons
in the zeolite unit cell. Two cage + windows are shown, plus a replica
of the center cage which is given to allow one to see intercage jumps, if any. Because of periodic boundary conditions, each cage is surrounded by replicas of itself. For clarity, the trajectory is shown for only one of each of the real/image atoms; however, all of the 12 windows for the two
cages actually contain xenon atoms. energy associated with the ion-polarized xenon interaction is quite small and thus it has been assumed to be negligible. Calculations of the Henry’s law constant for Xe in the Cs+rho pore structure are also shown in Figure 1, where it can be seen that the insertion of a single Cs+ per cage produces a small, roughly temperature-independent decrease in In KH relative to the “pure” rho. This change is consistent with the idea that the added Cs+ can be viewed primarily as producing a reduction in the adsorption volume for Xe by occupying the strong sites in the windows. 3. Simulations
The simulation technique used was molecular dynamics for varying numbers of xenon atoms placed in the cages plus windows in a single unit cell. (Periodic boundary conditions were utilized throughout.) Since the unit cell contains one cage window from each of the two interpenetrating lattices, the actual run consisted of a computation of trajectories for two essentially independent groups of xenons in the two cage + window systems. The use of periodic boundaries allows for intercage diffusion as well as intracage motion, and plots of the trajectories show that this diffusion can occur at high temperature during the duration of the simulation. The trajectory plots shown in Figure 4, etc., show atoms adsorbed in two cages, one from each independent subsystem. (A periodic image of one of the cages is also shown.) The centers of the cages for the second lattice are located on the corners of the cube surrounding the cages of the first lattice. (This cube is shown in Figure 3.) The two lattices have surfaces in common which are made up of 6-rings (also shown in Figure 3). Several possible molecular dynamics algorithms can be used depending upon the constraints imposed on the system. Algorithms for constant energy, constant kinetic energy, or trajectories generated for the system in contact with an abstract heat bath are available. This choice was of particular importance since severedifficultieswere observed in obtaining thermal equilibrium in the simulations at low values of the pore filling. A glance at the trajectories shown in Figure 4 for eight xenon atoms per unit
+
cell of the [Cs+] = 0 zeolite at 300 K shows the source of the problem: the first six atoms are tightly held in the windows and only vibrate on their adsorption sites. The remaining two move around in their separate cages, and there is little opportunity for the interactions or “collisions” that are needed to come to a statistical equilibrium state. In effect, the zeolite sequesters the first six sorbed atoms so effectively that they behave as isolated bodies moving in an external potential. At higher values of the pore filling, the xenon motions are coupled, causing the problem to disappear as one might expect. None of the algorithms used were adequate toovercomethis difficultyin obtainingequilibrium. Not even the isokinetic,17which constrains the total kinetic energy of the xenon to be a constant, or the Nos6,1* which employs a coupling to an abstract heat bath to achieve a canonical ensemble, was successful in giving believable results at low pore fillings. However, the fact that molecular dynamics can give erroneous values of the energy of adsorption at low coverage is not as much of a problem as it seems. The insertion of the first five or six xenon atoms per unit cell into equivalent, isolated locations in the windows means that the average xenon-solid energy per molecule will be constant at thesecoverages and equal to the zero-coverage Henry’s law limit. Also, the average X e X e interaction energy will remain small until filling exceeds six or seven xenons per unit cell. With this thought in mind, one can concentrate on simulations at pore fillings of eight or more xenons per unit cell where our tests show that the Nos6 algorithm produces configurations equivalent to those for a canonical ensemble.
4. Comparison of Simulation and Experiment The most accessiblethermodynamic quantity ob_tainabJe from the simul_ationsis t_hemolar heat of adsorption, AH = AE RT = RT - U,where U is the average potential energy per mole of sorbed xenon. (This quantity can be numerically differentiated to give-q,,, the isosteric heat of adsorption, since_q, = AH + n,(a( AHpftd.) Figure 5 shows values obtained for Uas a function of the number of xenon per unit cell in the two zeolites studied. It is evident that these energies do not vary much either with Cs+ loading or with Xe. In fact, the curve for the zeolite rho plus one Cs+ per cage can be overlaid on the one for the Cs+-free zeolite merely by translating the coverage scale by two atoms to the right. This illustrates the point that theCs+in thismodeloccupies a cage window, so that two fewer xenons can occupy the cage window system in the Cs+-rho than in the Cs+-free rho. It is important to realize that the constancy of the simulated energies is actually due to the cancellation of two terms that change significantly with xenon number. This point is illustrated
+
+
Sorption of Xenon in Zeolite Rho
4
8
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7663
12 16 20 no moleclcell -+
24
Figure 6. Computed values of the average potential energies per xenon atom are shown. The xenon-zeolite and the xenon-xenon values are plotted separately (with different vertical scales). When added to give the total energy of adsorption, the resulting curve is nearly independent of xenon loading and, thus, is in good agreement with the data shown in Figure 5 .
in Figure 6 where the average Xezeolite interaction 0, and the average Xe-Xe interaction ,E are plotted. As indicated above, the-initial window filling corresponds to nearly constant values of U, r -5.4 kcalLmo1 and i, 0. Figure 2 indicates that the window value of U , z -6.6 + ’/2RT -5.7 kcal/mol. (It is probable that part of the discrepancy here arises from the assumption of harmonic oscillator motion for a xenon in the window, whereas the curves in Figure 2 indicate a steeper variation in energy than harmonic for a transverseoscillationin the window.) However, when sorption in the cage commences at nxe > 6/cell the (negative) Xe-Xe interaction increases rapidly while the (negative) Xe-zeolite decreases from the window value toward the cage-wall value of -4kcal/mol. At the largest nxe, the X e X e interaction begins to show signs that the cavities are full in the sense that more xenons have much less attractive energies. Also, at this point, the Xesolid curve indicates a rapi! decrease that corresponds to capture in the cage center where U, -1.5 kcal/mol, as indicated in Figure 2. These conclusions are reinforced by viewing the computergenerated trajectories of the xenons in the Cs+-freezeolite. Figure 4 has already illustrated these for nxc = 8; Figures 7 and 8 show them for nxc = 16 and nXe = 24. At nxe = 16, the cages are reasonably full, but the motion is chaotic, as expected for a dense gas or a liquid. There is some indication that the trajectories are following the atomicallyuneven surfacesof the interior cage walls. A striking change occurs at the highest filling, where the Xe structure in the cages is seen to be reminiscent of the solid. Considering that 300 K is 10 K higher than the bulk critical temperature of Xe, this is a remarkable observation which is, however, quite consistent with theidea that thecages have reached their maximum occupancy at this point. Figure 9 shows the microscopic density of the xenon within the cages plotted as a function of distance from the cage center. Curves are shown for the two highest xenon loadings (16 and 24 atoms per unit cell). The cage walls are located at -5.5 A from the center, so a xenon on the wall will be located roughly at 5.5 - 2.0 r 3.5 A from the center. It can be seen that the xenons are essentially all on the cage walls for the loading corresponding to the liquidlike case of Figure 7. For the highest loading of nine atoms per cage, Figure 9b,c indicates that there is one atom at the cage center and eight on the walls. (To evaluate the number of atoms, multiply density by 47rB (d is the distance from the center) and integrate over the range of d covered by the thermal motion.) Thus, both Figures 8 and 9b,c indicate a body-centeredcubic cage filling at the limit of the highest practical xenon loading.
=
‘‘6”
Figure 7. Same as Figure 4, but for a xenon loading of 16 atoms per unit cell. Note the presence of a intercagediffusional event due to the jumping of an initial window atom into the neighboring cagethis is followed by its replacement in the window by an initial cage atom.
=
=
Figure 8. Same as Figure 4, but for a xenon loading of 24 atoms per unit cell. Even though the xenon atoms are rather tightly held in the short tubes that form the windows between cages, the simulations indicate an occasional intercage diffusional event. This is clearly shown in Figure 7, where the atom initially in the upper window of the central cage jumps into a neighboring cage. It is rather quickly replaced by an atom initially in the central volume of the center cage. Figure 8 shows two such diffusional events. In both cases, the trajectories are shown for a time of 34 ps, which is the simulation time. Clearly, the statistics are very poor for such a small number of events, so no attempt was made to evaluate self-diffusion coefficients. No doubt, this diffusion will occur more frequently at higher temperatures, but this has not yet been investigated. Finally, it should be noted that theXe-0 pair parameters used here are “effective”in the sensethat the ions (protonsand cesiums),
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The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 (a)
(bl
(C)
Y)
0 Distance from CSII Center t i )
Figure 9. Local densities for xenon atoms within the cages are plotted as a function of xenon position within the cage. For the highest loading ( m = 24), panel b shows an atom in the cage center-this is shown separately from the density for atoms on the wall (panel c) because of the large change in vertical scale between the two plots.
the silicons, and the aluminums have not been explicitly included in the pairwise energy sums. For this reason, one should not expect that the Xe-0 parameters would be transferable to other xeolites of different levels of ionicity or to other oxides. Computationally, inclusion of these ions and atoms in the pairwise sumsisquite feasible, but lackof knowledgeof the proton locations in these solids has prevented inclusion of such a refinement.
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