Effect of pressure, membrane thickness, and placement of control volumeson the flux of methane through thin silicalite membranes: A dual controlvolume grand canonical molecular dynamics studyMarcus G. Martin, Aidan P. Thompson, and Tina M. Nenoff Citation: J. Chem. Phys. 114, 7174 (2001); doi: 10.1063/1.1360256 View online: http://dx.doi.org/10.1063/1.1360256 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v114/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 16 22 APRIL 2001

Effect of pressure, membrane thickness, and placement of control volumeson the flux of methane through thin silicalite membranes: A dualcontrol volume grand canonical molecular dynamics study

Marcus G. Martina) and Aidan P. ThompsonComputational Biology and Materials Technology, Sandia National Laboratories, Albuquerque,New Mexico 87185-1111

Tina M. NenoffCatalysis and Chemical Technologies, Sandia National Laboratories, Albuquerque,New Mexico 87185-0710

~Received 4 December 2000; accepted 9 February 2001!

The flux of methane through the straight channels of thin silicalite membranes is studied via dualcontrol volume grand canonical molecular dynamics. The adsorption layers on the surfaces of thethin membranes are found to provide a significant resistance to the flux of methane. This strongsurface effect for thin membranes requires that the control volumes~where insertions and deletionsare performed! must be placed far enough away from the membrane surface that they do not overlapwith the surface adsorption layer. The permeance~flux/pressure drop! of methane through thesurface layer is shown to be insensitive to both the average pressure and the pressure drop. Incontrast, the permeance through the interior of the membrane increases with decreasing averagepressure. These results are explained using a model which treats the transport through the surfacebarrier as driven by the pressure gradient and transport through the zeolite as driven by the chemicalpotential gradient. A new force field named DACNIS is presented which accurately describes theadsorption isotherms of methane and ethane in silicalite. ©2001 American Institute of Physics.@DOI: 10.1063/1.1360256#

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I. INTRODUCTION

In recent years there has been considerable interest insteady-state nonequilibrium transport of small to modersize molecules through microporous materials such aslites. The driving force behind much of this work is a desto select or design microporous materials which are capof achieving a good separation of similar molecules usinminimal amount of energy. Molecular simulation of this prcess was pioneered by Maginnet al.1 for methane in sili-calite~ZSM-5! using gradient relaxation molecular dynamiand color field nonequilibrium molecular dynamics. The susequent development of the dual control volume grandnonical molecular dynamics~DCV-GCMD! method2,3 hasled to a large body of work studying pressure driven traport through slit pores,4–13 amorphous materials,14–17 andcrystalline microporous solids.18–21

DCV-GCMD is a hybrid method which periodically peforms Monte Carlo molecular insertion and deletion attemin specific regions~the control volumes!, and then propa-gates the system through time using molecular dynamThere is not always a consensus among people in thisregarding simulation details that involve the control voumes. Some workers4,8 have advocated adding a streamivelocity to molecules which are inserted into the control vumes in order to eliminate the discontinuity in the flux whiwould otherwise occur at the boundaries of the control v

a!Electronic mail: marmart@sandia.gov

7170021-9606/2001/114(16)/7174/8/$18.00

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umes. An iterative approach is required as the observeddetermines the streaming velocity, yet the streaming veloaffects the flux. One of these groups4 has also claimed thaperiodic boundary conditions cannot be applied in the fldirection, and instead suggests using a vacuum at theends of the control volumes.

In this paper we explore the effect of an additionstreaming velocity, demonstrate that it is possible to impperiodic boundary conditions in the flow direction, and ivestigate how the position of the control volumes relativethe membrane surface can affect the observed flux. Oncesimulation details are resolved we then study the fluxmethane through several different thicknesses of silicamembranes, and present a method for extrapolating thresults to any membrane thickness in order to compare wexperiment. All simulations are performed using the nDozen Angstrom Cutoff of Nonelectrostatics for Inhomogneous Simulations~DACNIS! force field which we fitted tothe experimental adsorption isotherms of methane and ethusing Gibbs ensemble Monte Carlo simulations.

II. SIMULATION METHODS

A. Gibbs ensemble Monte Carlo

The DACNIS force field is designed for use in simultions where the system is not homogeneous and thereanalytical tail corrections are not appropriate. This force fiuses a united-atom Lennard-Jones 12–6 potential whictruncated at a cutoff (r cut) of 12 Å and shifted so that the

4 © 2001 American Institute of Physics

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7175J. Chem. Phys., Vol. 114, No. 16, 22 April 2001 Flux of methane through a silicalite membrane

energy atr cut is zero. Cross terms are computed usingstandard Lorentz–Berthelot23,24 combining rules. This forcefield is fitted using the same philosophy as the TraPPE-force field,22 so the parameters for methane and ethanefitted to their experimental vapor–liquid coexistence curv~VLCC!. Once these single component parameters are dmined the parameters for silicalite~with interaction sites centered on the oxygens! are fitted to the single component asorption isotherms of methane and ethane. Ethane is modas two united atoms connected by a fixed C–C bond lenof 1.54 Å.

The VLCC are computed using theNVT version of theGibbs ensemble,25,26while the adsorption isotherms are computed using theNpT version of the Gibbs ensemble, whevolume moves are only performed in the box that doescontain zeolite. The system size for computing the VLCwas 400 molecules of methane or ethane. The system sizcomputing the adsorption isotherms was 400 moleculesmethane or ethane and 23332 unit cells of silicalite. Eachunit cell of silicalite contained 192 interaction sites with dmensions of 19.93313.40320.07 where the straight channels run down thex direction, and the atoms were treateda rigid framework. All simulations were equilibrated forleast 30 000 cycles~where one cycle isN Monte Carlomoves!, the production runs were 10 000 cycles, andstandard deviations were computed by breaking the runfive blocks.

Sampling efficiency was enhanced by performing cofigurational bias27–30 with ten trial insertions of the first in-teraction site31,32 and eight trial positions for the second si~for ethane! when performing the molecule transfer move

B. DCV-GCMD

The DCV-GCMD simulations were performed using tmassively parallelLADERA program, version 5.58. Much likethe simulation method itself, this code is a hybrid of a msively parallel molecular dynamics code namedLAMMPS33,34

and a massively parallel grand canonical Monte Cacode.35 LADERA 5.58 follows the same general parallelizatiophilosophy as already documented for the previoversions36,37and therefore is only discussed briefly here, wemphasis on the modifications from the older versions. Tsimulations are run onPx3Py3Pz processors where eacprocessor controls a rectangular parallelepiped which1/(Px3Py3Pz) of the total volume and the minimum lengtof any dimension is at least twice the nonbonded cu(r cut). The processor decomposition is completely indepdent of the positions of the membrane and control volumA Monte Carlo cycle is run everyNstep molecular dynamicstime steps. During the Monte Carlo cycle each procesbreaks its volume into eight equal rectangular parallelepeds. The processors then performNiter move attempts ineach of the eight subdomains where all of the processorsworking in the same subdomain at the same time. Thislows the processors to insert and delete molecules indedent of what is happening on the neighboring processbecause they are guaranteed not to be working on molecwhich could affect the neighboring energy calculations as

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subdomains are separated by at leastr cut from each other. Ifa processor subdomain contains both control volumenoncontrol volume regions, then the move attempts arevided between the two regions. Any move attempts thatcur in a control volume region are equally likely to be insetions or deletions. The only move that is performed in tnoncontrol volume regions is a test particle insertion whis not accepted and is only used to compute the chempotential. Trial positions for insertion attempts are selecuniformly over the appropriate volume and the ghost atmethod37 is no longer used.

The basic system setup is shown~looking in from thenegativey direction! in Fig. 1. The system is periodic in athree dimensions, and the chemical potential gradient isforced in thex direction. The control volumes are each brken into a positive and a negative region which signifiesdirection in which the additional streaming velocity is addto the random Gaussian velocity. In our simulations contvolumea has a higher pressure~and higher chemical potential! than control volumeb, so the flux is always froma to b.Throughout the simulation the density and chemical potenare monitored in bins along thex direction in order to com-pute density and chemical potential profiles. The chempotential is computed following a modified version of Wdom’s formula for trial particle insertion38

m

kBT52 lnS L23K V

NL K expS 2U1

kBT D L D , ~2.1!

wherem is the chemical potential,kB is Boltzman’s constant,T is the temperature in Kelvin,̂V/N& is the molecular vol-ume averaged over the simulation,L is the thermal de Bro-glie wavelength, andU1 is the energy of inserting a fictionatrial particle into the system. Note that we use the productwo ensemble averages instead of combining them asingle average~with N replaced byN11), as this has beenshown empirically to be more reliable for smallN,39 which isthe case for the small bins used to compute the profiles.net number of molecules through specific planes in the s

FIG. 1. Basic system setup for the DCV-GCMD simulations. Control vumesa andb are each broken into two regions~1 and2! which signify thedirection in which the streaming velocity is added to the random Gausvelocity. Regionn is the space in between the control volumes andmembrane, and regionm is the membrane.

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7176 J. Chem. Phys., Vol. 114, No. 16, 22 April 2001 Martin, Thompson, and Nenoff

tem was monitored and then combined with the simulattime andy–z cross-sectional area in order to determineflux.

All of the DCV-GCMD systems studied in this work arlisted in Table I. Two main classes of systems are studThe ‘‘membrane’’ systems only contain zeolite in the mebrane ~m! region and are used to study the effect of tvapor–surface interfaces on the flux of methane. The ‘‘inrior’’ systems contain zeolite in every region, and are usedstudy the diffusion in the interior portions of a zeolite membrane. Systems is the only simulation in which an extrstreaming velocity was not added to molecules insertedthe control volumes. A 5 fs time step was used and the tperature was maintained at 200 K using a Nose´–Hooverthermostat40–42with a time constant of 0.01 fs21. The mem-brane simulations were run for at least 150 ns on 4 toprocessors and all hady–z dimensions of 40.20340.14 Å.The interior simulations were run for 25 ns on 72 or 1processors and hady–z dimensions of 80.40380.28 Å.Standard errors were computed by breaking the simulatinto 5 ns blocks for the membrane simulations and 2.5blocks for the interior simulations. The Monte Carlo paraetersNstepandNiter were adjusted so there were two ordersmagnitude more accepted insertions and deletions compto the net number of insertions or deletions in each convolume. This was found to be necessary in order to mainthe specified chemical potential in all of the control volumTypical values for the interior simulations wereNstep5100

TABLE I. System setup for all of the DCV-GCMD simulations performein this paper. Control volume regionA is equally divided betweena¿ andaÀ while control volume regionB is equally divided betweenb¿ andbÀas shown in Fig. 1. In the ‘‘membrane’’ systems regionn is the spacebetween the control volumes and the membrane, and regionm is the zeolitemembrane. In contrast, for the ‘‘interior’’ systems the entire simulation bis filled with zeolite ~including the control volumes! so regionn does notexist and regionm is the distance between the control volumes.pa and pbare the methane pressures imposed~via the chemical potentials! in controlvolume regionsA andB. The distances for the regions are all given asnumber of straight channel silicalite unit cells~19.93 Å! that would fit inthat region~even for regions which do not contain any silicalite!.

Name pA @kPa# pB @kPa# A B n m

Membrane systemssa 270 95 2 2 2 1u0 270 95 6 6 0 1u1 270 95 4 4 1 1u2 270 95 2 2 2 1m1a 200 95 2 2 2 1m1b 270 165 2 2 2 1m1c 270 95 2 2 2 1m1d 340 95 2 2 2 1m2 270 95 2 2 2 2m4 270 95 2 2 2 4m8 270 95 2 2 2 8

Interior systemsi5a 200 95 2 2 ••• 5i5b 270 165 2 2 ••• 5i5c 270 95 2 2 ••• 5i5d 340 95 2 2 ••• 5i12 270 95 2 2 ••• 12

aNo additional streaming velocity.

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and Niter51000, while onlyNstep5500 andNiter520 wasrequired for the membrane simulations. This is due torelatively lower acceptance rates for inserting and deletmolecules in a zeolite compared to the gas phase. A tota500 000 time steps on 144 nodes of the Cplant Siberia clu~500 MHz DEC alpha ev6 chips connected by Myrinet!43 forsystemi 8 took 8 h ofwall time and spent less than 10% of itime on communication.

III. RESULTS AND DISCUSSION

The final DACNIS force-field parameters for methanethane, and silicalite oxygen are shown in Table II. Therameters were manually adjusted until good agreementfound with the experimental VLCC44 for methane andethane. Then, a systematic search of parameter spaceperformed centered on the initial good parameters. Figurshows that the extrapolated critical points and coexisteliquid densities are well reproduced by DACNIS. Recall ththe only difference between the DACNIS and TraPPE-Uforce fields is the use of a cut-and-shift potential vs cut wanalytic tail corrections. Therefore, it is not surprising ththe s values are very similar for DACNIS and TraPPE-UAwhile thee values are higher for DACNIS to make up for thattractions that are lost by neglecting the tail corrections.

Since we were only interested in methane diffusionthis paper, we initially tried to fit the silicalite interactioparameters to three different pressures on the experimeadsorption isotherm45 of methane at temperatures of 277 a353 K. The mean unsigned deviation of the simulated loings from the experimental values was computed for a ra

x

TABLE II. Lennard-Jones parameters for the DACNIS united-atom fofield. Alkane parameters for the TraPPE-UA force field are listed for coparison.

United atomDACNISe/kB @K# s @Å#

TraPPE-UAe/kB @K# s @Å#

CH4 158.5 3.72 148 3.73CH3 108.0 3.76 98 3.75OSi0.5 82.0 3.20 N/A N/A

FIG. 2. Vapor–liquid coexistence curves for methane and ethane. Simtion results and extrapolated critical points are shown for methane as ciand ethane as squares. Experimental data are shown as dashed lines~meth-ane! solid lines~ethane! and crosses.

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7177J. Chem. Phys., Vol. 114, No. 16, 22 April 2001 Flux of methane through a silicalite membrane

of possible silicalite parameters and is shown in Fig. 3. Ufortunately, it appears that silicalites and e can be playedoff against each other along the linee'267s1296 to yieldsimilar values of the methane adsorption accuracy. Thislationship appears to agree with the OSi0.5 parameters that aLorentz–Berthelot combining rule would predict using tforce fields of Juneet al.46 or Vlugt et al.47 Therefore, oneshould expect the simulation results for methane in silicatbe similar when using any of these three force fields. Hoever, it is hoped that the DACNIS force-field parameters wwork better for longer linear and branched alkanes wheregeometry of the molecule means thats and e cannot beplayed off against each other as easily.

In order to fit the OSi0.5 parameters to a single pair ofsand e values, we also computed the adsorption isothermethane. Since ethane no longer has a simple spherical getry it does not follow the same trend as observed for meane, as shown in Fig. 4. The DACNIS parameters for OS0.5

were selected using Fig. 5, which is the average of the mane and ethane adsorption isotherm errors. The single c

FIG. 3. Silicalite united-atom parameters error field for methane adsorpisotherms. The size of the bar is proportional to the mean unsigned deviof the six simulations with vapor pressures of 41.35, 516.84, and 205kPa at a temperature of 277 K, and vapor pressures of 82.00, 514.772071.83 kPa at a temperature of 353 K. Thee ands values are those of theOSi0.5 united atom.

FIG. 4. Silicalite united-atom parameters error field for ethane adsorpisotherms. The size of the bar is proportional to the mean unsigned deviof the six simulations with vapor pressures of 6.89, 206.50, and 2108.69at a temperature of 277 K, and vapor pressures of 6.89, 133.69, and 20kPa at a temperature of 353 K. Thee ands values are those of the OSi0.5

united atom.

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ponent adsorption isotherms for methane~Fig. 6! and ethane~Fig. 7! using the final values of the DACNIS parameters ain good agreement with the experimental results.45 Finally,Table III shows the computed gas phase chemical potentand the corresponding loadings in silicalite, for the pressustudied in the DCV-GCMD simulations at a temperature200 K.

Once the DACNIS force field was determined, we begexploring how the simulation setup in the DCV-GCMsimulations affected the observed flux. System s waswithout imposing any streaming velocity on atoms inserinto the control volumes and the resulting methane denprofile is shown in Fig. 8. Notice that there is a suddchange in the density at the boundaries of the control vumes. The discontinuity in the flux caused by inserting mecules which have no net flux also translates into a disconuity in the density and the chemical potential. System m1c isidentical to system s, except that it uses an additional streing velocity. Figure 9 shows that the streaming velocitymoves the discontinuities at the control volumes boundarIt also increases the observed flux from 10086109 to 1378645 mol m22 s21. We believe the increase in flux is not duto the additional velocity the molecules have when theythe membrane surface, but instead to the change in the etive pressure drop across the membrane. In system s it

FIG. 6. Adsorption isotherms for methane in silicalite. DACNIS simulatiresults are shown for 277 K~circles! and 353 K~squares!. Experimental dataare shown at 277 K~dashed line! and 353 K~solid line!.

non9nd

nonPa.49

FIG. 5. Silicalite united-atom parameters error field for methane and ethadsorption isotherms. The size of the bar is proportional to the averagthe methane and ethane mean unsigned deviations shown in Figs. 3 a

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7178 J. Chem. Phys., Vol. 114, No. 16, 22 April 2001 Martin, Thompson, and Nenoff

pears that the driving force for diffusion is not the pressuin the control volumes, but instead the pressures thatobserved in the region between the control volumes andmembranes.

With the need to use an additional streaming velocconfirmed, we then turned to the issue of whether convolume placement affects the observed flux through thetem. Since the resistance to flow of the methane vapoconsiderably less than that provided by the zeolite mebrane, the observed flux should be independent of thetance between the control volumes and the membranelong as the control volumes are not inside any regions whcontribute significantly to the resistance. In systems u0, u1,and u2 the total system size is held constant, while the edof the control volumes is either zero, one, or two unit ceaway from the zeolite surface. The observed fluxes w22786396, 12026278, and 13176239 mol m22 s21, re-spectively. Clearly there is a significant change in the flwhen the control volume is near the surface of the zeoThis is because transport through the methane layer adsoon the surface of the zeolite is a significant resistance toflux and inserting molecules inside this layer allows mecules to avoid part of this barrier. All of our subsequemembrane simulations had a distance of two unit cellstween the control volumes and the membranes in ordeavoid any problems with the surface layer.

In order to understand the effect of pressure drop ontransport through the surface layer, and through the intewe looked at four different pressure drops across the onecell thick membrane~systems m1a, m1b, m1c, m1d) andthrough five unit cells of the interior~systems i5a, i5b, i5c,

FIG. 7. Adsorption isotherms for ethane in silicalite. DACNIS simulatiresults are shown for 277 K~circles! and 353 K~squares!. Experimental dataare shown at 277 K~dashed line! and 353 K~solid line!.

TABLE III. Chemical potentials and zeolite loadings for methane at a teperature of 200 K. The subscripts show the standard deviation of thedigit.

Pressure@kPa# m/kBT Loading @molec/unit cell#

95 213.81405 15.912165 213.26606 16.522200 213.07495 16.716270 212.77962 16.984340 212.55304 17.145

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i5d). Figure 10 shows that the permeance~flux/pressure drop!is constant for the four pressure drops in the one unitthick membrane. In contrast, Fig. 11 shows that the pmeance is not the same for any of the four pressure drthrough the interior. Two results are shown for each systas there are two equivalent paths to get from control voluA to control volumeB, and these two results agree to withthe standard error bars for all of the systems. To understthese results we consider the isothermal diffusion of meththrough silicalite48

Jmx 52

Dm,scm

RT

dmm

dx, ~3.1!

which can be rewritten using the Gibbs–Duhem relation49 as

Jmx 52

Dm,s

RT

dp

dx, ~3.2!

whereJmx is the flux of methane in thex direction,Dm,s is the

diffusion coefficient between methane and silicalite,R is thegas constant,p is the pressure,cm andmm are the concentration and chemical potential of methane, respectively.

We analyzed the interior simulations using Eq.~3.1!,assuming thatdmm/dx could be replaced byDmm/Dx, andcomputedDm,s. Table IV shows that the diffusion constanagree well with each other for the four pressure drothrough the five unit cells of silicalite, butDm,s

Dm is signifi-

FIG. 8. Methane density profile for system s. The methane density goescale in the two membrane regions.

FIG. 9. Methane density profile for system m1c . The methane density goeoff scale in the two membrane regions.

-al

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7179J. Chem. Phys., Vol. 114, No. 16, 22 April 2001 Flux of methane through a silicalite membrane

cantly higher for the 12 unit cell simulation. Looking closeat the chemical potential distribution in Fig. 12 shows theis a sharp drop inm/kBT at the boundaries of the controvolumes, but otherwise the chemical potential changesearly with position. We computeddm/dx from the slope ofthe chemical potential distribution for each system and uthat value to computeDm,s

dm ~see Table IV!, and now the dif-fusion constant is statistically identical for all five interiosimulations. It is not clear whether the sudden drop in checal potential at the control volume boundary is due to ourof a single streaming velocity for the entire control volum~where the inhomogeneous density profile suggests theof a position-dependent streaming velocity!, or to an insuffi-cient number of accepted insertions/deletions, despitefact that the total number of insertions and deletions in econtrol volume is already two orders of magnitude more ththe net change. Whatever the cause, it is clear that fittinstraight line to the measured chemical potential profile yiediffusion coefficients that are more consistent, so our bestimate of the silicalite straight channel diffusion constfor methane is 1.9060.0331024 cm2/s.

With the transport through the interior portion of thzeolite now understood, we turn to modeling the transpor

FIG. 10. Methane permeance through a one unit cell thick silicalite mbrane. The observed permeance and its standard error are shown for ethe two membranes that occur in each system. Simulation data are showsystems m1a ~circles!, m1b ~diamonds!, m1c ~squares!, and m1d ~crosses!.

FIG. 11. Methane permeance through five unit cells of the silicalite interThe observed permeance and its standard error are shown for eachtwo membranes that occur in each system. Simulation data are showsystems i5a ~circles!, i5b ~diamonds!, i5c ~squares!, and i5d ~crosses!.

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the membrane simulations. Figure 13 shows that therestrong surface adsorption peak that is not affected bymembrane thickness. In addition, the methane density dibution inside the zeolite matches the interior simulationsall places except for the first 4 Å from the membrane surfaceSince the permeance was constant for the four thin mbrane simulations this suggests that pressure is the driforce for transport through the surface layer. Nitta aFurukawa21 observed similar density distributions in thesimulation studies of methane transport through silicalite athey interpreted the nearly constant density distributacross the membrane as implying a large resistance adownstream surface of the membrane. Analysis ofchemical potential distribution in Fig. 14 shows a stechange in the chemical potential occurs at both the upstrand downstream surfaces of the membrane. The similar mnitude of the chemical potential change suggests thatupstream adsorbing surface and the downstream desorsurface provide approximately equal resistances to the flof methane.

Exchanging the derivatives for differences in Eq.~3.2!gives

Jmx 52

Bm

RTDp, ~3.3!

where the mass transfer coefficientBm5Dm/Dx, Dm is thediffusivity of methane in the surface layer,Dx is the lengthof the surface layer, andDp is the pressure drop across thsurface layer. It is convenient to lumpDx into Bm because itis not obvious what the correct distance is for the surfa

-h offor

r.thefor

FIG. 12. Methane chemical potential distribution for system i5d .

TABLE IV. Results for the DCV-GCMD interior simulations. The valuereported are the average of the results from the two membranes insystem and subscripts indicate the average standard error in the final dThe diffusion constants were computed using Eq.~3.1! and either replacingdm/dx with Dm/Dx ~Dm,s

Dm), or computingdm/dx directly from the slope ofthe chemical potential profile~Dm,s

dm ).

Name J@mol m22 s21# Dm,sDm @cm2 s21# Dm,s

dm @cm2 s21#

i5a 6366129 1.68331024 1.94431024

i5b 409998 1.65431024 1.87531024

i5c 8687182 1.65331024 1.88431024

i5d 10632108 1.63231024 1.90231024

i12 3982101 1.81531024 1.92531024

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7180 J. Chem. Phys., Vol. 114, No. 16, 22 April 2001 Martin, Thompson, and Nenoff

layer. We then model our membrane transport as threegions @two surfaces following Eq.~3.3! and one interior us-ing a difference version of Eq.~3.1!# which must have equaflux at steady state. Figure 13 indicates that the methdensity in the membrane simulations is identical to the dsity of the interior simulations except in the first 4 Å from asurface. Thus, we subtracted 8 Å from the zeolite thicknessto determine the interior distance, and also used the avedensity from the interior simulations. Pressure was conveinto chemical potential using the relationp5a exp(bm),whereb51/kBT, and the value ofa observed in the Gibbsensemble simulations is 9.6853107 kPa. This yields a system of three equations with four unknowns

Jmx 52

Bm

RT~phi2a exp~bmads!!

52Dm,scmb~mads2mdes!

RTDx

52Bm

RT~aexp~bmdes!2plo!, ~3.4!

FIG. 13. Methane density profiles in silicalite. Simulation data are shofor systems m1c ~dashed line!, m2 ~long dashed line!, m4 ~dot-dashed line!,m8 ~thick solid line!, and i5c ~thin solid line!. Systems m2 , m4 , and m8

overlap completely on this graph. The small tick mark at 59.79 Å showsbeginning of the zeolite in the membrane systems. The large density pcentered on the zigzag channels, go off this scale to a height o31023 molecules/Å3.

FIG. 14. Methane chemical potential distribution for system m8 . The binsize for this plot was 1.993 Å in order to reduce the noise observed inportions of the system which contain zeolite.

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e-

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ged

wheremadsandmdesare the chemical potential of methanethe adsorbing surface and the desorbing surface, restively. These equations can be rearranged to solve forBm asa function ofDm,s

Bm

RT5

Jmx @11exp~Jm

x Dx/Dm,scm!#

phi2plo exp~Jmx Dx/Dm,scm!

. ~3.5!

Table V shows the results forBmint using theDm,s of 1.90

31024 cm2/s computed from the interior simulations. Thresulting Bm

int decreases as the thickness of the membrincreases. Since Fig. 13 shows no difference in the surdensity distribution with membrane thickness, one wouldpect a constant value of the mass transfer coefficient. Adjing the size of the surface layer in the range 0 to 8 Å has noeffect on the trend, and neither does the use of a mowhich utilizes different mass transfer coefficients on the ustream and downstream surfaces. Further investigationvealed that the only way to produce a model with a constmass transfer coefficient was to use an interior diffusion cstant of 0.5131024 cm2/s, which is shown in Table V asBm

fit . Close inspection reveals a very slight change~about10%! in the height of the surface adsorption peaks wchanging membrane thicknesses, but this change ismuch smaller than those observed when varying the presdrop for the one unit cell thick membranes, and that systyields a constant mass transfer coefficient. Clearly the mois missing an important process, as there is no satisfacexplanation for the change in the diffusion constant, orthe change in the mass transfer coefficient.

IV. CONCLUSIONS

The new DACNIS united-atom force field was deveoped for use with the cut-and-shift potential truncation thamost suitable for molecular dynamics simulations of inhmogeneous systems. This force field was fitted to singcomponent vapor–liquid coexistence curves to determinealkane parameters, and to adsorption isotherms to determthe silicalite parameters. This was then used to exploresensitivity of DCV-GCMD simulations upon system detaiIt was shown that the flux is independent of the placementhe control volumes so long as they do not intersect withsurface adsorption layers. When the control volumes aredense region, such as inside a zeolite, then the numbe

n

eks,8

e

TABLE V. Results for the DCV-GCMD membrane simulations. The valureported are the average of the results from the two membranes insystem and subscripts indicate the average standard error in the final dThe mass transfer coefficients were computed using either the obseinterior diffusion constant (Bm

int) or a diffusion constant fitted to these resulto yield a constant mass transfer coefficient (Bm

fit).

Name J@mol m22 s21# Bmint @m s21# Bm

fit @m s21#

m1a 84039 26.212 27.313

m1b 82840 27.213 28.914

m1c 137945 26.79 28.19

m1d 195243 26.86 28.56

m2 117943 23.49 26.410

m4 103533 21.47 28.19

m8 78033 17.07 28.112

license or copyright; see http://jcp.aip.org/about/rights_and_permissions

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7181J. Chem. Phys., Vol. 114, No. 16, 22 April 2001 Flux of methane through a silicalite membrane

accepted insertions and deletions must be at least two orof magnitude larger than the net change in order to mainthe desired chemical potential distribution. Even in that cthere is a slight drop in the chemical potential acrosscontrol volume boundaries, so it is more reliable to compthe diffusion constants from the observed linear slope inchemical potential. An additional streaming velocity mustadded to inserted molecules in order to avoid sharp disctinuities in the density and chemical potential across the ctrol volume boundaries.

Simulations performed with gas phases on either sidea silicalite membrane revealed a strong resistance to metflux occurs on both the upstream adsorbing surface, andthe downstream desorbing surface. The flux throughstraight channels of very thin silicalite membranes~19.93 Å!is shown to be proportional to the pressure difference.contrast, the flux through the interior of the zeolite is proptional to the chemical potential gradient. A model is prposed which explains the observed trend in methanethrough membranes over a range of pressures, and for mbrane thicknesses from one to eight unit cells thick. Hoever, when this model used the diffusion constants obsefrom interior simulations, the mass transfer coefficientthe surfaces decreased with increasing membrane thicknIt was possible to produce a model with a diffusion constthat does not match the interior simulation, but yieldsthickness-independent mass transfer coefficient. We susthat the equation of Mason48 fails to capture some of theessential details of transport through the surface of a mbrane, but at this time we are unable to construct a viaalternative.

ACKNOWLEDGMENTS

We wish to acknowledge useful discussions wEdward Maginn. Sandia is a multiprogram laboratory opated by Sandia Corporation, a Lockheed Martin compafor the United States Department of Energy under contNo. DE-AC04-94AL-85000.

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