Friction based modeling of multicomponent transport at the nanoscaleSuresh K. Bhatia and David Nicholson Citation: The Journal of Chemical Physics 129, 164709 (2008); doi: 10.1063/1.2996517 View online: http://dx.doi.org/10.1063/1.2996517 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/129/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Computational study of pressure-driven methane transport in hierarchical nanostructured porous carbons J. Chem. Phys. 144, 044708 (2016); 10.1063/1.4940427 Sub-additive ionic transport across arrays of solid-state nanopores Phys. Fluids 26, 012005 (2014); 10.1063/1.4863206 Uncertainty quantification in MD simulations of concentration driven ionic flow through a silica nanopore. II.Uncertain potential parameters J. Chem. Phys. 138, 194105 (2013); 10.1063/1.4804669 Structure and dynamics of water confined in silica nanopores J. Chem. Phys. 135, 174709 (2011); 10.1063/1.3657408 Molecular transport in nanopores J. Chem. Phys. 119, 1719 (2003); 10.1063/1.1580797

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Friction based modeling of multicomponent transport at the nanoscaleSuresh K. Bhatiaa� and David Nicholsonb�

Division of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia

�Received 5 July 2008; accepted 11 September 2008; published online 29 October 2008�

We present here a novel theory of mixture transport in nanopores, which considers the fluid-wallmomentum exchange in the repulsive region of the fluid-solid potential in terms of a species-specificfriction coefficient related to the low density transport coefficient of that species. The theory alsoconsiders nonuniformity of the density profiles of the different species, while departing from amixture center of mass frame of reference to one based on the individual species center of mass. Thetheory is validated against molecular dynamics simulations for single component as well as binarymixture flow of hydrogen and methane in cylindrical nanopores in silica, and it is shown that purecomponent corrected diffusivities, as well as binary Onsager coefficients are accurately predicted forpore sizes sufficiently large to accommodate more than a monolayer of any of the components. It isalso found that the assumption of a uniform density profile can lead to serious errors, particularly atsmall pore diameter, as also the use of a mixture center of mass frame of reference. The theorydemonstrates the existence of an optimum temperature for any fluid, at which the fractionalmomentum dissipation due to wall friction is a minimum. © 2008 American Institute of Physics.�DOI: 10.1063/1.2996517�

I. INTRODUCTION

A fundamental understanding of the processes affectingfluid behavior in nanoscale confinements is crucial to numer-ous emerging applications in nanofluidics, materials science,membrane science, and biology.1 There are also a host ofexisting applications in catalysis, gas-solid reactions, and ad-sorptive separations that have for over a century stimulatedresearch on molecular motion and flow in confined spaces.Nevertheless, despite the long history the modeling of suchflows is still routinely based on the early concepts ofKnudsen2 and Smoluchowski,3 which neglect intermolecularcollisions as well as van der Waals fluid-solid interactionswhile analyzing particle trajectories for molecular flow incylindrical pores. The subsequent analysis of Pollard andPresent4 extends the description to include intermolecularcollisions, but still excludes dispersive and other long rangeinteractions. The consideration of more realistic interactionshas long been a goal of theoreticians, but has proved elusivebecause rigorous mechanical models rapidly become intrac-table when such interactions are considered.5–7 For a systemof Lennard–Jones �LJ� particles at low density, where onlygas-solid interactions are important, attempts to circumventthis theoretical difficulty have been made by numerical inte-gration of trajectories,8 but the need for a more generallyapplicable tractable model has remained. As a result approxi-mations, such as the dusty gas model,9,10 arbitrarily superim-posing diffusive and viscous flows, have been in widespreaduse, but these generally utilize porous medium-specific ad-justable parameters and therefore lack predictive abilities.

In the past 2 decades a vast array of new nanomaterialshaving nanopores of ideal and well defined geometry, such astemplated periodic MCM-41 silicas,11 carbon nanotubes,12 aswell as various aluminophosphates and aluminosilicates,have been developed,13 all considered to hold promise for avariety of novel applications. The infiltration of fluid mix-tures into the nanopores in these materials is a common fea-ture of most applications being investigated, and this hascatalyzed several new theoretical developments14–20 in theunderstanding of fluid equilibrium and transport at the nano-scale, while considering realistic interactions. Among these,a significant achievement, arising from this laboratory,15,17 isthe development of the oscillator model for transport in na-nopores with diffuse wall reflection, which is exact in thelow density limit for an arbitrary gas-solid interaction poten-tial. At higher densities, where intermolecular interactionsare significant, the oscillator model result is augmented by aviscous contribution to the total flux in a manner similar tothe dusty gas model,9,10 while utilizing the local average den-sity model21 �LADM� to account for nonuniformity of theadsorbate density profile. Although the theory has showngood agreement with molecular dynamics simulation results,such addition of viscous flux is somewhat arbitrary. Further,all of these recent investigations have been devoted to purecomponent systems, with little attention to mixture transport.

For long, the modeling of mixture transport has relied onhighly respected statistical mechanical theories utilizing amixture center of mass based frame of reference, in whichthe local hydrodynamic stress tensor for a given componentis related to the local rate of strain for the mixture motion asa whole.5,22 Such theories, founded on the Liouville equationframework, involve expansion of the species velocitiesaround the mixture center of mass velocity in solving theensuing Boltzmann equation. Nevertheless, despite their fun-

a�Author to whom correspondence should be addressed. Electronic mail:s.bhatia@uq.edu.au.

b�Also at Computational and Structural Group, Department of Chemistry,Imperial College, London SW72AY.

THE JOURNAL OF CHEMICAL PHYSICS 129, 164709 �2008�

0021-9606/2008/129�16�/164709/12/$23.00 © 2008 American Institute of Physics129, 164709-1

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damental rigor they have failed to provide satisfactory solu-tions to problems involving mixture transport, that are appli-cable over a wide range of densities. Indeed, there exists nodefinitive treatment even for a simple classical experimentknown as the Stefan tube. The existing theoretical treatmentshave been critically reviewed by Kerkhof and Geboers23,24

who instead suggest expansions around the individual com-ponent center of mass velocities. When the components dif-fer widely in their mobility, their individual velocities can bevery different from the mixture center of mass velocity, par-ticularly at low density. Convergence of the expansionsaround the mixture center of mass velocity may then not bepossible, leading to unreliable results. It should be noted thatexpansion around the individual mean component velocitieshas also been considered earlier in a rigorous treatment bySnell et al.,25 invoking binary partial viscosities that are ex-pressed as spatial integrals over terms involving sphericalharmonic expansion coefficients and perturbed pair distribu-tion functions. However, the complexity of their equationshas served as a deterrent to a fuller development of thistheory. As yet there exists no way to estimate the binarypartial viscosities from elementary molecular properties. In-deed, simpler species partial viscosities which sum to givethe mixture shear viscosity were also invoked by the priorBearman–Kirkwood �BK� treatment,22 but there exists noreadily accessible prescription even for their estimation.Moreover, the BK equations are also based on expansionaround the mixture center of mass velocities.

For mixture transport in porous materials perhaps themost commonly used approach is the dusty gas model ofEvans and co-workers,9,10 which was originally based onphenomenological modeling. Later Mason and Viehland26

extended this model to dense phases. They argued that themost fundamental staring point was the Liouville equationand again used the BK equations �derived from the Liouvilleequations� as a point of departure. Their final result for mix-tures included both a Stefan–Maxwell term and a viscousterm, although the manner in which this was incorporatedhas been called into question;23,24 furthermore, the underly-ing assumption of expansion about the mixture center ofmass velocity remains despite the fact that these authors dis-miss this as “not critical.” Their equations were primarilyaimed at membrane transport and this somewhat global viewmeant that adsorbate density gradients and inhomogeneitiesarising from the fluid-solid interaction were overlooked and,for the case of cylindrical capillaries, relies on the parabolicvelocity profile arising from the classical Poiseuille flowmodel.

The approximation of using purely hard sphere interac-tions has been demonstrated by us15,17 to be as much as anorder of magnitude in error in estimating the diffusivity atlow pressures where fluid-solid interactions are dominant.The neglect of dispersive interactions allows the flux to besimply written as the sum of purely viscous and diffusivecomponents, the latter including the binary as well as theKnudsen �or wall mediated� diffusion, but it is not clear ifthis is generally possible. Thus, there exists no clear route tointroduce wall effects in the modeling based on fundamentalprinciples. Finally, the Onsager formalism, with the well-

known reciprocity relations between cross coefficients, isgenerally used only for the diffusive component,27–29 al-though it is readily seen that the total flux in these ap-proaches, expressed in the same formalism, following

ji = �j

�ij�− �� j� �1�

will also obey the reciprocity relations. Here ji is the totalflux of species i, �� j the chemical potential gradient of spe-cies j and �ij the Onsager coefficients.

For narrow nanopores, or micropores �typically havingdiameter �2 nm�, there has been much work reported on thedevelopment of phenomenological models for multicompo-nent transport by Krishna and co-workers,30–34 based on theMaxwell–Stefan approach. These models also overlook thepresence of inhomogeneity due to adsorption forces which,as will be shown, has a strong influence on the equilibriumand transport properties in narrow micropores. Nevertheless,much success in modeling mixture transport in microporouszeolites31,32 and carbon nanotubes34 has been reported, usingphenomenological parameters from pure component trans-port. The prediction of these parameters based on molecularprinciples, as well as the consideration of inhomogeneity inmicropores, remain fundamental challenges. Further, al-though included in the formulation of Krishna andWesselingh,30 the viscous contribution is small in such nar-row pores and the approach requires validation in this re-spect.

Here we develop an approach that overcomes all of theabove limitations, and for the first time present a tractabletheory that is able to handle mixture transport in nonuniformfluids from the nanopore to the mesopore range of confine-ment. The theory is based on a constitutive model for theshear stress on a given component that is based on the strainrate of that component as suggested by Kerkhof andGeboers,23,24 while also representing wall effects in a novelmanner through a species-specific friction coefficient in themomentum balance. The theory is validated with the help ofmolecular dynamics simulations for the H2 /CH4 binary sys-tem in a silica nanopore. A preliminary report on the devel-opment has recently been provided elsewhere.35

II. THEORY

We consider the one-dimensional axial flow of an ncomponent fluid mixture in a cylindrical pore of radius R,measured between centers of diametrically opposite surfacesites on the pore wall. The starting point of the theory is theequation of change for species i

1

r

d

dr�r�i

dv̄i

dr� = �i�r�

d�i

dz+ �t�r�kBT�

j=1

nxixj�v̄i − v̄ j�

Dij

+ �i�iv̄ia�r − roi� �2�

in which d�i /dz represents the axial chemical potential gra-dient of species i, �i�r� is its local number density, xi�r� itsmole fraction, v̄i�r� its local mean axial velocity, �i its partialviscosity, and �t�r� is the total number density. The variableDij represents the well-known binary or mutual

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diffusivity,5,36,37 whose inverse may effectively be viewed asa friction coefficient between species i and j. Further, �i is awall friction coefficient for species i, such that the last termon the right hand side of Eq. �2� represents the rate of mo-mentum loss due to molecule wall collisions in the repulsiveregion of the fluid-solid interaction potential, roi�r�R,where roi represents the location of the minimum of thefluid-solid potential for species i, and, and a�r−roi� is theHeaviside function having the value of unity for r�roi andzero otherwise. As we will show, this friction factor, �i, isreadily obtained from the wall mediated diffusion coefficientat low density.

Equation �2� differs from the conventional models5,9,10,22

in two ways. The first of these relates to the novel consider-ation of the momentum loss due to wall collision through thelast term on the right hand side of Eq. �2�, effectively repre-senting a frictional effect of the pore wall. Molecules movingtoward the wall will lose kinetic energy on moving up thepotential hill in the repulsive region after crossing the mini-mum of the potential, and on changing direction when theradial momentum is zero they will also on the average havelost axial momentum. This effect is modeled here throughthe frictional term representing the rate of axial momentumloss per unit volume. Figure 1 illustrates this zone of frictionin the steep repulsive region beyond the potential minimumlocation.

A second departure from conventional hydrodynamicmodeling relates to the left hand side of Eq. �2�, which inspirit follows the approach of Kerkhof and Geboers23,24 whopropose a stress tensor based on the individual species ve-locity. However, while no simple relation between fluidproperties and the partial viscosity has yet been suggested,we propose that �i=wi�, where � is the shear viscosity ofthe mixture, and wi is the weight fraction of species i in themixture. This result is obtained by requiring that for a homo-geneous fluid the total shear stress on all the components isthat on the mixture as a whole, i.e., �i=1

n �idv̄i /dr=�dv̄ /dr,where v̄ is the mass-average mixture velocity �i=1

n wiv̄i. Fora homogeneous fluid, having uniform wi, this conditionprovides

�i=1

n

��i − wi��dv̄i

dr= 0. �3�

Since �i, wi, and � must be functions only of state variablesand independent of the arbitrary velocity field, dv̄i /dr, Eq.�3� leads to the result �i=wi�, overcoming a major difficultyin the use of the Kerkhof and Geboers23,24 approach.

For the inhomogeneous nanopore fluid the partial viscos-ity of species i as well as the binary �i.e., mutual� diffusivi-ties, Dij, are nonlocal properties, expressed as functions oflocally averaged densities, �̄k, following the LADM

�̄k�r� =6

� f3�

r��f/2�k�r + r��dr� �4�

in which species density, �k, is averaged over a sphere ofradius k /2, where k is its LJ size parameter.14,17,21 Equa-tion �2� may now be formally integrated subject to the zeroshear stress condition for each component at r=0 and at r=R, i.e., dv̄i /dr=0 at these locations. The condition at r=0arises from symmetry, while at r=R �the centerline radius ofthe surface sites of the solid� it results from the absence ofcolliding fluid molecules. Integration of Eq. �2� subject tothese conditions yields, for any species i,

d�i

dz�

0

R

r�i�r�dr + kBT�j=1

n �0

R r�i�r�� j�r��v̄i − v̄ j��t�r�Dij�r�

dr

+ �i�roi

R

r�i�r�v̄i�r�dr = 0, i = 1,2,...n �5�

representing a force balance on this species and, for the ve-locity profile,

v̄i = v̄io +d�i

dz�

0

r dr�

r��i�r���

0

r�r��i�r��dr�

+ kBT�j=1

n �0

r dr�

r��i�r���

0

r� r��i�r��� j�r���t�r��Dij�r��

�v̄i − v̄ j�dr�

+ a�r − roi��i�roi

r dr�

r��i�r���

roi

r�r��i�r��v̄i�r��dr�, �6�

where the chemical potential gradient of any species is con-sidered to be constant over the pore cross section. This fol-lows from our earlier observation,14,17 based on non-equilibrium molecular dynamics �NEMD� studies, thatequilibrium radial density profiles are attained even duringtransport. In addition, the friction coefficient �i is taken to bedensity and position independent. The value of this coeffi-cient may be obtained from a known estimate of the trans-port coefficient at any density, for the pure component case.The low density limit is the most suitable for this, since itisolates wall effects without intrusion of fluid-fluid interac-tions. In particular, we have already developed an exact sta-tistical mechanical theory for the low density transport coef-ficient when wall reflection is diffuse,15 which offers aconvenient route to relate the friction factor to the fluid-solidinteraction potential. In this low density limit Eqs. �5� and�6� combine to yield

FIG. 1. Illustration of potential energy profile and region of friction corre-sponding to the repulsive part of the potential energy curve.

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v̄i = v̄io = −�d�i/dz�0

Rr�i�r�dr

�iroi

R r�i�r�dr. �7�

Equation �7� may be combined with the phenomenologicalmodel defining a transport coefficient Doi

v̄i = −Doi

kBT

d�i

dz�8�

to provide the following relation between �i and the lowdensity diffusion coefficient of species i, Doi

LD,

�i =kBT0

Rre−i�r�/kBTdr

DoiLDroi

R re−i�r�/kBTdr, �9�

where i�r� is the fluid-solid interaction potential field, andwe have used the canonical form for �i�r� in the low densitylimit. Equation �9� now provides an unambiguous route toincorporating wall effects in terms of the low density diffu-

sion coefficients, in our mixture transport model, even in thepresence of dispersive interactions. We note that we do notaddress the question of the momentum accommodationcoefficient38,39 here, assuming this to be embedded within thelow density diffusion coefficient Doi

LD. Thus, the above for-mulation applies even for partially specular reflection, inwhich case the value of Doi

LD may be obtained by moleculardynamics simulation or experiment.

Equations �5� and �6� provide a coupled system of equa-tions that may be solved for the centerline velocities v̄io

andthe velocity profiles v̄i�r�, by writing v̄io=−� j=1

n Aij�d� j /dz�and v̄i�r�=−� j=1n Xij�r��d� j /dz�, and obtain-

ing a coupled system of equations for the coefficients Aij andXij�r�, which may be solved iteratively. To obtain the coupledsystem of equations, we make the above substitutions intoEqs. �5� and �6�, and compare coefficients of d� j /dz, whichprovides

Aij =1

�iroi

R r�i�r�dr��

0

R

r�i�r�dr − kBT�k=1

n �0

R r�i�r��k�r��t�r�Dik�r�

�Xij − Xkj�dr + �ij�i�roi

R

r�i�r�dr�0

r dr�

r���r���0

r�r��i�r��dr�

− kBT�i�k=1

n �roi

R

r�i�r�dr�0

r dr�

r���r���0

r� r��i�r���k�r���t�r��Dik�r��

�Xij − Xkj�dr� − �i2�

roi

R

r�i�r�dr�roi

r dr�

r���r���roi

r�r��i�r��Xij�r��dr��,

i, j = 1,2, . . . ,n , �10�

Xij�r� = Aij − �ij�0

r dr�

r���r���0

r�r��i�r��dr� + kBT�

k=1

n �0

r dr�

r���r���0

r� r��i�r���k�r���t�r��Dik�r��

�Xij − Xkj�dr�

+ �ia�r − roi��roi

r dr�

r���r���roi

r�r��i�r��Xij�r��dr�, i, j = 1,2, . . . ,n . �11�

Equations �10� and �11� constitute the core of the presenttheory, and their solution is readily accomplished by succes-sive substitutions, mixing old and new solutions to promoteconvergence, following

Xij�r� = �Xijnew�r� + �1 − ��Xij

old�r� , �12�

Aij = �Aijnew + �1 − ��Aij

old, �13�

where � is a suitably chosen constant. The use of the phe-nomenological relation in Eq. �1� then leads to the expres-sion

�ij =2

R2�0

R

r�i�r�Xij�r�dr , �14�

where ji is the pore flux of species i. Equation �14� permitsthe estimation of the Onsager coefficients from the solution,Xij�r�, of Eqs. �10� and �11�, thereby enabling their determi-nation from molecular properties of the adsorptives. To thisend it is also necessary to use suitable theories or correlationsexpressing the bulk mixture viscosity and the binary �i.e.,

mutual� diffusivities in terms of the species densities andmolecular properties, which may be combined with theLADM for the inhomogeneous pore fluid. A number of suchcorrelations, specifically developed for LJ as well as molecu-lar fluids are available in the literature.40–45 Here we haveused the method of Galliéro et al.41 for the mixture viscosity,and for the mutual diffusivities we used the method of Reiset al.43 Further, equilibrium density distributions, which arerequired in the theory, may be obtained from either densityfunctional theory or grand canonical Monte Carlo �GCMC�simulation. Here we have used the latter. Equations �9�–�11�and �14�, together with the equilibrium density distributionsand estimation methods for the bulk mixture viscosity andbinary diffusivities, as well as the LADM, then constitute acomplete predictive theory for the species fluxes or binaryOnsager coefficients for mixture transport in a nanopore.

III. SIMULATION

We have investigated here the transport of a binary mix-ture, that of hydrogen and methane at 300 K, in cylindrical

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silica nanopores of radii 0.783 and 1.919 nm, and determinedthe Onsager coefficients based on the above theory as well asby equilibrium molecular dynamics �EMD� simulations. Weconsider the fluid particles to be LJ spheres, and the porewalls to be infinitely thick and amorphous, comprised ofclose packed LJ centers. For methane we use the establishedLJ parameter values f /kB=148.1 K, f =0.381 nm, whilefor hydrogen we use f /kB=38 K, f =0.2915 nm. For theLJ centers in the solid we use s /kB=290 K, s=0.29 nm,which have been obtained from earlier fits of argon adsorp-tion data on MCM-41.14,46 For the mixture viscosity we usethe method of Galliero et al.,41 and for the mutual diffusivitywe used the correlation of Reis et al.,43 both developed forLJ fluid mixtures. To validate the theory we have conductedEMD simulations, as described elsewhere.46,47 In the simula-tions we solve the equations of motion for a mixture, typi-cally having about 500 particles, using a Gaussianthermostat.48 Each EMD simulation is started with an arbi-trary initial configuration having a prespecified density ofeach species, obtained by Monte Carlo simulation. A fifthorder predictor-corrector method is used to solve the equa-tions of motion, using a time step of 0.5 fs, with each runcomprising 30�106 time steps. The cutoff distance is takento be 2.5 nm, and the Lorentz–Berthelot mixing rules areused for the cross-interaction parameters. We use a diffusereflection condition at the pore wall, whereby particles mov-ing toward the wall are diffusely scattered in the osculatingplane upon reversing direction after crossing the potentialminimum location. Nevertheless, we do note here that thetheory developed above is more general, and not restricted tothe diffuse reflection condition, since the wall effect embed-ded in the low density diffusivity may well involve partiallyspecular reflection.

To obtain the Onsager coefficients from the simulationresults we used the autocorrelation of the streaming veloci-ties, and the Green–Kubo relation

�ij =NiNj

kBTVlim�→�

�0

�

� v̄i�0� · v̄ j�t� � dt , �15�

where Ni is the number of molecules of species i, v̄i its axialmean pore velocity, 1

Ni�k=1

Ni dzki /dt, and V is the system vol-

ume.

IV. RESULTS AND DISCUSSION

A. Validation of theory for pure component transport

Initially, studies of pure component transport were con-ducted for the flow of H2 and of CH4 at 300 K and variousadsorbed phase densities, in silica nanopores of diameter inthe range of 0.75 to −3.84 nm. Molecular dynamics simula-tions for the diffusion of the pure species under these condi-tions have previously been reported from this laboratory,47

and their results were used to validate the theory developedabove, simplified for a single species. For the transport of asingle species the present theory provides the coefficient �11,and Eqs. �1� and �8� combine to provide the correcteddiffusivity

Doi��̂i� =kBT

�̂i

�11 �16�

for pure species i at mean pore density �̂i. The recent oscil-lator model from this laboratory14 was used to estimate thelow density transport coefficient for determination of thefriction coefficient via Eq. �9�, and density distributions weredetermined using GCMC simulation. Figure 2 depicts thecomparison between the theoretical and simulation resultsfor the variation of corrected diffusivity of H2 with density,in pores of various diameters. As indicated earlier47 the simu-lation results were considered to be accurate within about5%, based on repeat runs at selected densities. The solid linesin the figure denote the results from the present theory, anddashed lines those from the earlier theory17,47 superimposingthe low density transport coefficient from the exact oscillatormodel theory15 and a viscous flow term, following

Doi��̂� = DoiLD + Doi

vis��̂� . �17�

Here

Doivis��̂� =

2kBT

�̂R2 �o

roi dr

r���̄�r����o

r

r���r��dr��2

�18�

represents the viscous flow contribution, obtained on inte-grating the Navier–Stokes equation with a no-slip boundarycondition at the potential minimum location roi. This super-position of the diffusive and viscous terms, along the lines ofthe dusty gas model,9,10,26,29 is somewhat arbitrary and with-out rigorous theoretical justification in the presence of ad-sorptive forces. As seen in Fig. 2, while the earlier approachdoes reasonably well in comparison with the simulation, thenew theory performs marginally better at all pore sizes. Inparticular, the new theory also predicts the correct behaviorat the narrow pore diameter of 1.05 nm, with the decrease incorrected diffusivity with increase in density. This behavior,not captured in the prior approach, is due to the increasedcollision frequency and consequent momentum loss arisingfrom the presence of only a monolayer on the cylindricalpore surface at this pore size. However, it may be noted thatat this pore width, having only a monolayer, the viscosity

FIG. 2. Variation of transport coefficient with density, for hydrogen at 300 Kin pores of various diameters. Symbols correspond to EMD simulation re-sults, and lines to theory.

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concept, although utilizing the LADM, is unlikely to be ac-curate, leading to the slight underestimation of the diffusivityby the present theory. A recent more accurate theory,18 spe-cifically considering interactions between the molecules onopposite sides of the diameter, performs somewhat better atthis pore size, indicating that the present approach is bettersuited for mesopores, where the intermolecular interactionscan reasonably be considered via the viscosity concept.

Figure 3 depicts the simulation and theoretical results formethane in pores of various diameters, again showing some-what better agreement of the new theory with simulation, incomparison to the previous approach. In this case, due to thelarger size of methane and its stronger intermolecular inter-actions, the new theory starts to deviate from simulation atthe pore diameter of 1.57 nm, while showing generally betteragreement compared to the earlier approach. All of theseresults reinforce the above suggestion that the present theoryis more accurate for mesopores.

It is of interest here to point out that the present theorydoes not support the often utilized no-slip condition, and thatthe pore surface is not well defined in systems having vander Waals interactions. Indeed, Eq. �6� indicates a velocityprofile with a finite value at every point in the region 0�r�R, at any density and driving force �−d�i /dz�. By analogywith a hard sphere system, if we define the pore surface to bethe location of the potential minimum at r=roi, then for thepure component case we obtain the slip velocity

v̄i = v̄io − �−d�i

dz��

0

roi dr�

r��i�r���

0

r�r��i�r��dr�, �19�

While we do not explicitly investigate the velocity profileshere, it is clear that the velocity at the potential minimumsurface will, in general, be finite. Indeed, in the low densitycase, where the present theory provides the same flux as theexact oscillator model theory,15,17 we have already shown thelatter to yield considerable slip at the potential minimumlocation, while reproducing the velocity profile from molecu-lar dynamics simulation. Further, in earlier work46 consider-ing diffuse reflection at this surface the invalidity of the no-slip condition was also noted, and the slip length found to bea strong function of the density at the potential minimum

location with a weak temperature dependence except at veryhigh density.

B. Friction coefficient

Key to the success of the present formulation is the con-sideration of a continuous region of friction, roi�r�R, inthe momentum balance in Eq. �2�, in which molecules mov-ing toward the wall undergo repulsion and lose axial momen-tum on reversing direction. This improves on our earlierpostulate14,46 of hard spherelike collisions at the potentialminimum location, roi, and permits Eq. �2� to be solved overthe entire region 0�r�R, while also avoiding the need toarbitrarily superimpose viscous and wall mediated diffusivecontributions as in Eq. �17�.9,10,17,28,47 Although, in principle,the friction coefficient, �i, will be affected by intermolecularinteractions, and therefore vary with density and position, theresults in Figs. 2 and 3 suggest that this is a secondary effectthat is overshadowed by the wall repulsion. Figure 4 depictsthe variation of this friction coefficient with pore diameterfor H2 at various temperatures, showing an interesting trendhaving both a minimum and maximum with respect to diam-eter. The minimum occurs at a pore diameter of about 0.75nm, which is quantitatively consistent with a theoretical ex-planation from this laboratory49 in which the diffusivity has amaximum at this pore size, following the floating moleculeor levitation effect, first discussed by Derouane et al.50 andsubsequently studied in detail by Yashonath andco-workers51,52 for diffusion in zeolites. This maximum inthe diffusivity, and the minimum in the friction coefficient, isbest explained by considering the expression derivedearlier14

DoiLD =

kBT

m �� �20�

in which the diffusion coefficient at low density is propor-tional to the mean time, ��, spent along a trajectory betweenthe diffuse wall collisions. Equation �20� may be substitutedinto Eq. �9� to obtain

FIG. 3. Variation of transport coefficient with density, for methane at 300 Kin pores of various diameters. Symbols correspond to EMD simulation re-sults, and lines to theory. FIG. 4. Variation of friction factor for hydrogen with pore diameter, at

various temperatures.

164709-6 S. K. Bhatia and D. Nicholson J. Chem. Phys. 129, 164709 �2008�

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�i =m0

Rre−i�r�/kBTdr

��roi

R re−i�r�/kBTdr. �21�

At the pore size of 0.75 nm there is a transition from a singleto a double potential energy well in the pore, as shown inFig. 4, with a single well at smaller pore diameter and adouble well at larger sizes. The width of the potential well inwhich the molecule oscillates, and therefore the low densityoscillation time, ��, is largest at this pore diameter, and thisleads to the minimum in the friction factor at a given tem-perature. With increase in pore diameter the molecule oscil-lates in the narrower potential well near the surface and theoscillation time decreases, leading to an increase in frictionfactor. However, with increasing pore width the depth of thepotential well is reduced, as seen in the inset of Fig. 4, and atsufficiently large pore width the molecule again oscillatesover an increasingly wide region with increase in pore diam-eter. This leads to an increase in oscillation time and themaximum in friction factor seen in Fig. 4.

Even more interesting behavior is observed when theeffect of temperature on the friction factor is investigatedover a wider range of temperature than that covered in Fig. 4.The behavior of the friction factor of H2 with respect totemperature is depicted in Fig. 5, showing a minimum in therange of 130–140 K, at all diameters except the smallest oneof 0.65 nm. This minimum arises because, while the oscilla-tion time �� decreases with increase in temperature becauseof the increase in kinetic energy, the fraction of molecules inthe repulsive region, roi

R re−i�r�/kBTdr /0Rre−i�r�/kBTdr, also in-

creases. Following Eq. �21�, these opposing contributions tothe friction factor then lead to the temperature minimum. Theabsence of the minimum at the smallest diameter examined�0.65 nm� is due to the potential minimum being located atthe pore center in this case, so that the fraction of moleculesin the repulsive region is unity at all temperatures, and thefriction factor then increases with increase in temperaturedue to the decrease in ��.

Figure 6 depicts the variation of friction factor for CH4

with pore diameter at the three temperatures of 150, 300, and450 K. While the behavior at the two higher temperatures issimilar to that for H2, at 150 K a second weak minimum and

maximum in the friction factor also occurs at the lower tem-perature of 150 K, due to the complexities of the interplaybetween the oscillation period, ��, and the fraction ofmolecules in the repulsive region, roi

R re−i�r�/kBTdr /0

Rre−i�r�/kBTdr. Such behavior with respect to pore size mayalso be expected to occur for H2, but at lower temperaturesdue to it being more weakly interacting. As for H2, the levi-tation effect47 related minimum in the friction coefficient isfound at the diameter �0.85 nm for CH4� at which the tran-sition from a single to a double potential well occurs, evidentfrom the inset in Fig. 6. Figure 7 depicts the variation offriction factor for CH4 with temperature at various pore di-ameters, showing a minimum at about 300 K, similar to H2

for which the minimum occurs at about 130–140 K. Interest-ingly, at very low temperature of about 80–150 K, a weakmaximum in the friction factor also occurs, again because ofthe complexity of interplay between the oscillation periodand the fraction of molecules in the repulsive region. Such amaximum appears also for H2 at the pore size of 1.36 nm atabout 50–60 K, as seen in Fig. 5, and may occur for the otherpore sizes at lower temperatures than those studied in Fig. 5.

While illustrating the complexities of the effect of poresize and temperature on the friction factor, the presence ofthe strong minimum at 130–140 K for H2, and about 300 K

FIG. 5. Variation of friction factor for hydrogen with temperature, at variouspore diameters.

FIG. 6. Variation of friction factor for methane with pore diameter, atvarious temperatures.

FIG. 7. Variation of friction factor for methane with temperature, at variouspore diameters.

164709-7 Transport on the nanoscale J. Chem. Phys. 129, 164709 �2008�

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for CH4, in the larger mesopores, indicates that fractionalmomentum dissipation is least and the transport most effi-cient at these temperatures. Thus, there exists a fluid-specificoptimum temperature at which the flow is most efficientfrom a frictional loss perspective. While this conclusion isbased on the diffuse reflection boundary condition adoptedhere, it may be expected to be generally applicable, becauseof the validity of Eq. �9� even for the case of partially specu-lar reflection as discussed earlier. In this regard, we note herethe unusual temperature effects on self-diffusion of methanein carbon nanotubes of various diameter recently reported byJakobtorweihen et al.,53 with a maximum in the low densityself-diffusivity with temperature in some cases. Such a maxi-mum in low density diffusivity with variation in temperaturehas not been found to arise in silica pores or carbon slit poresin our previous studies using our exact low density theory aswell as simulation17,54 utilizing the diffuse wall scatteringcondition, and may be related to the smoother surface ofcarbon nanotubes with nearly specular reflection, or its non-rigid nature as considered by Jakobtorweihen et al. Thus,their maximum in diffusivity is not believed to be related tothe same effect causing the minimum in friction factor foundhere.

C. Transport in binary H2/CH4 mixtures

A key feature of the present approach is the possibility ofmodeling multicomponent transport, based on individualspecies friction factors determined from their low pressurediffusivities. Extension of the earlier approach in Eqs. �17�and �18� along the lines of the dusty gas model, suffers fromthe difficulty that in this approach the viscous flow region,0�r�roi, is species-specific, and therefore ill defined for amixture. The new approach proposed here, integrating theequation of motion over the entire region 0�r�R, over-comes this difficulty. For validation of the new approach,EMD simulations of the binary transport at 300 K in cylin-drical silica nanopores of diameter 3.84 and 1.57 nm wereconducted here, as discussed earlier, for H2 densities of 0.25,1.0, and 4.0 nm−3, and various CH4 densities. Figures8�a�–8�c� depict a comparison between simulation andtheory for the variation of the Onsager coefficients withmethane density for a pore diameter of 3.84 nm, at the threeH2 densities. In this and all subsequent figures, species 1 isH2 and species 2 is CH4. For the simulations four repeat runswith different initial configurations were conducted, and thesymbols represent the average value from these runs, while

FIG. 8. Variation of the Onsager coefficients with methane density, in a cylindrical silica pore of diameter 3.84 nm at 300 K, for �a� a hydrogen density of0.25 nm−3, �b� a hydrogen density of 1 nm−3, and �c� a hydrogen density of 4 nm−3. Symbols represent EMD simulation results, and lines the theoreticalpredictions. �d� Results when the densities are assumed uniform in the theory.

164709-8 S. K. Bhatia and D. Nicholson J. Chem. Phys. 129, 164709 �2008�

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the error bars represent their standard deviation. As for thepure component case, we estimated the species friction fac-tors based on Eq. �9�, using the oscillator model14 to estimatethe low density diffusivities. Further, density distributionsrequired for the theory were obtained by GCMC simulation.Excellent agreement is seen between simulation and theoryfor all four Onsager coefficients, despite their large variation,spanning over 3 decades in magnitude with change in meth-ane density, for all three hydrogen densities. This supportsthe theoretical development considering the shear stress on acomponent to depend on its own strain rate rather than thatof the mixture as a whole. Further, the viscous contributionto the species flux is embedded within the overall Onsagercoefficients, with no requirement for a separate viscous flowterm as is often introduced.28,29,55 At the same time bothsimulation and theory yield �12=�21, as is to be expectedbased on microscopic reversibility of the cross interactions,demonstrating internal consistency of the present develop-ment. We emphasize here that the theoretical results are fullypredictive with no adjustable parameter. We further note thatthe mixture components, H2 and CH4, differ greatly in theirproperties �size and mass�, with the former being much fasterdiffusing, by nearly an order of magnitude,47 and the good

agreement between theory and simulation in Figs. 8�a�–8�c�,despite this difference, provides strong support for thetheory.

The above finding that viscous contributions are embed-ded within the Onsager coefficients for the total flux of anycomponent is consistent with our recent observation frompure component EMD simulations,14 that the transport coef-ficient obtained from the autocorrelation of the streamingvelocity embeds the viscous contribution, despite there beingno net flow. Several earlier studies had claimed that the EMDtransport coefficient represents only the diffusive part, whileviscous effects are captured only in simulations involvingactual flow.55,56 A further interesting feature of the presentresults is the importance of inhomogeneity, incorporated inthe present approach through the density distributions withinthe integrals in Eqs. �10� and �11�. Figure 8�d� depicts thevariation of the Onsager coefficients with CH4 density, for aH2 density of 1.0 nm−3, if the system is taken to be homo-geneous and the mean pore densities of the components usedinstead of their density distributions. In this case, the predic-tions yield larger discrepancy with the simulations, under-scoring the importance of considering heterogeneity.

Since the pore size used above falls in the mesoporerange, studies were also conducted at a smaller pore diameter

FIG. 9. Variation of the Onsager coefficients with methane density, in a cylindrical silica pore of diameter 1.57 nm at 300 K, for �a� a hydrogen density of0.25 nm−3, �b� a hydrogen density of 1 nm−3, and �c� a hydrogen density of 4 nm−3. Symbols represent EMD simulation results, and lines the theoreticalpredictions. �d� Results when the densities are assumed uniform in the theory.

164709-9 Transport on the nanoscale J. Chem. Phys. 129, 164709 �2008�

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of 1.57 nm, lying in the micropore range ��2 nm�. Figures9�a�–9�c� depict the comparison between simulation andtheory for this pore diameter, again showing good agreementdespite the smaller pore size falling in the micropore range.The deviation from theory is somewhat greater than that atthe larger pore size of Fig. 8, particularly for �11, suggestingthat the theory does less well in the micropore region. This isto be expected in view of our recent work18 demonstratingthe importance of packing effects in narrow pores of molecu-lar dimension. The error is larger for �11 because of theeffect of methane on the more weakly interacting hydrogen,and as a result increases with methane density. The impor-tance of considering nonuniformity is further highlighted byFig. 9�d�, showing the large deviation between simulationand theory when the density field is taken to be uniform atthe mean pore density. The approximation of a uniform poredensity is common to existing formulations,9,10,23,24,28–34,55

but is clearly in serious error, particularly for microporeswhere the inhomogeneity in the potential energy profile isstronger. Nevertheless, as indicated earlier, success in inter-preting transport data from simulation or experiment is oftenfound because of the presence of a few empirical phenom-enological parameters.

Given the increase in error on reducing pore diameter afurther study was conducted at an even smaller pore diameterof 1.05 nm, and H2 density of 1.0 nm−3, the results forwhich are depicted in Fig. 10. Further deterioration of thetheory is evident, with an increase in error compared to theresults at 1.57 nm given in Fig. 9�b� for this density, particu-larly for �11 and the Onsager cross coefficients �12 and �21.These results suggest that the theory is best suited for nan-opores larger than the size at which only a monolayer can fit,and this is attributable to the use of a viscosity approach inmodeling intermolecular interactions in the theory. Thus, the1.05 nm diameter pore can fit only a monolayer of CH4,while the corresponding size for H2 is about 0.87 nm. Theratio of molecular size to pore size would appear to be animportant parameter in this regard. At small sizes accommo-dating only a monolayer �i.e., two molecules across a porediameter� viscosity concepts applicable to fluid phases are

inaccurate. Consistent with these arguments, Figs. 2 and 3show that, at the pore diameter of 1.05 nm, the theory ismore satisfactory for H2 as compared to CH4. For the lattercase the recent theory of the authors,18 considering packingeffects and avoiding viscosity concepts, offers a more accu-rate alternative for the single component case but has yet tobe extended for mixtures. Such packing effects are less im-portant at low densities, and Fig. 10 indeed shows betteragreement under such conditions.

One of the important features of the present approach isthe departure from a mixture center of mass based frame ofreference for determining shear stress, to one based on theindividual species mean velocity. The rationale for this is thatwhen the different species have significantly different mobil-ity, the expansion of the individual velocities around the mix-ture center of mass velocity, which forms the basis of exist-ing solutions of the Boltzmann equation, may not readilyconverge, and the ensuing component equations of motiontherefore unreliable.23,24 In the present case where H2 andCH4 have distinctly different low pressure diffusivities by asmuch as an order of magnitude,47 the conventional approachwould appear to be problematic. Figure 11 depicts the varia-tion of the ratio of mean pore velocity of hydrogen to that ofthe mixture center of mass, with methane density, for poresof diameters 3.84 and 1.57 nm, at the three H2 densitiesstudied here. The values of the driving forces, �−d�1 /dz� and�−d�2 /dz� have been taken to be equal for the calculations.It is seen that the ratio can be large, even exceeding 5 underthe conditions investigated, making the expansion around themixture center of mass velocity unviable. At low methanedensities there is a rapid drop in the mixture center of massvelocity with increase in methane density, leading to therapid increase in the ratio to values significantly larger thanunity, because of the distinctly smaller mobility of methane.However, at high densities of both methane and hydrogen theratio is within a factor of 2–3 from unity, and the conven-tional approach probably more viable. Liquid phases ordense fluids under subcritical conditions fall in this category,and the conventional approach may be more suitable underthese conditions. As a confirmation of the failure of the con-ventional approach at supercritical conditions, we attemptedto solve the current problem using the equation of motion

FIG. 10. Variation of the Onsager coefficients with methane density, in acylindrical silica pore of diameter 1.05 nm at 300 K, for a hydrogen densityof 1.0 nm−3. Symbols represent EMD simulation results, and lines the the-oretical predictions.

FIG. 11. Variation of the ratio of mean pore velocity of H2 to that of themixture center of mass, with CH4 density, for pores of diameter �a� 3.84 nmand �b� 1.57 nm, at three H2 densities.

164709-10 S. K. Bhatia and D. Nicholson J. Chem. Phys. 129, 164709 �2008�

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wi1

r

d

dr�r�

dv̄dr� = �i�r�

d�i

dz+ �t�r�kBT�

j=1

nxixj�v̄i − v̄ j�

Dij

+ �i�iv̄ia�r − roi� , �22�

which is based on the mixture center of mass velocity, v̄, asframe of reference. However, the approach gave large devia-tion from simulation results, with even negative Onsagercross coefficients at low density.

V. SUMMARY

The approach proposed here for modeling multicompo-nent transport at the nanoscale has several novel featuresdeparting from the traditional approaches such as the dustygas model.9,10 The first of these is the use of a constitutiveequation for the local shear stress for any component that isbased on the strain rate for that component, as opposed to theusual method employing a mass-averaged velocity basedframe of reference and therefore involving the rate of strainon the mixture. A further novel feature of the approach is theconsideration of a continuous region of wall friction for anyspecies, in the repulsive region of the fluid-solid potential forthat species, incorporating a species-specific friction coeffi-cient. This permits a rigorous route for incorporating walleffects through the low density diffusion coefficient, over-coming limitations of earlier analyses lumping wall frictionat the location of the minimum of the fluid-solid potential forany species. A third key feature of the method is the consid-eration of nonuniformity, utilizing the detailed equilibriumdensity profile for any species, as opposed to the use of auniform density at the mean value, that is common to exist-ing formulations. Comparison of results for H2 /CH4 mixturetransport with molecular dynamics simulations, for variousnanopore sizes in silica, has shown that the theory can accu-rately predict Onsager coefficients in mesopores and largemicropores that can accommodate more than a monolayer onthe surface. While validated here for a LJ fluid mixture underdiffuse reflection conditions, the method should be extend-able also to more complex molecular fluids as well as forpartially specular reflection, provided the species low pres-sure diffusivities are independently known from experimentor simulation. For LJ fluids an exact value of the low densitytransport coefficient is available for the case of diffuse wallreflection, through the oscillator model developed in thislaboratory,15 and the theory is fully predictive. Our calcula-tions have also shown that the use of a uniform density canlead to serious predictive errors, and the error is larger forsmaller nanopores. In addition, the conventional use of amixture center of mass frame of reference, involving mixturestrain rate, yielded unsatisfactory results with even negativeOnsager cross coefficients. It is anticipated that the methodwill have important applications in modeling transport in ca-talysis, membrane, and adsorptive separations, as well as inadsorbed nanoscale films. Further, extension to nanolubrica-tion is also an avenue for further development of the theory.

ACKNOWLEDGMENTS

Support of the Australian Research Council through agrant under the Discovery Scheme is gratefully acknowl-edged.

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