Diffusion in Pore Networks: Effective Self-Diffusivity and the Concept of

Article pubs.acs.org/JPCC

Diffusion in Pore Networks: Effective Self-Diffusivity and the Concept of Tortuosity Mauricio Rincon Bonilla and Suresh K. Bhatia* School of Chemical Engineering, The University of Queensland Brisbane, QLD 4072, Australia ABSTRACT: Molecular transport in confined spaces plays a fundamental role in well-established and emerging technologies such as catalysis, gas separation, electrochemical energy storage, and nanofluidics. In all these applications, fluids penetrate the voids of porous materials often having a complex morphology. This demands a deep understanding of how structural variables such as pore size distribution and pore connectivity affect transport in order to design or optimize processes and devices. In previous work from this laboratory, the effect of these structural variables had been studied at infinite dilution. Here, we consider nonvanishing densities and investigate the influence of adsorbate density on the effective diffusivity and tortuosity. At finite densities the transport and self-diffusivity are not equivalent at the single pore level, leading to differing effective diffusion coefficients and tortuosities. The tortuosity has previously been shown to be governed not only by the pore network topology, but also by temperature and the gas species. Here, we show it to vary significantly with adsorbate density and the diffusion mode (i.e., collective or self), which has important repercussions on modeling and interpretation of experimental data. While the effective self-diffusivity of supercritical gases is found to qualitatively behave similar to that estimated in single pores, subcritical fluids present widely different behavior, with a maximum at low adsorbate densities when the coexisting population of small and large mesopores is significant, consistent with experimental evidence. We discuss under what conditions a monotonically decreasing single-pore self-diffusivity can lead to the appearance of such maximum in a network. where kB is the Boltzmann constant, T is temperature, and ρ̂ is the mean density of the adsorbed fluid. D0 is commonly known as the “corrected diffusivity” to distinguish it from the Fickian diffusivity DT for which the density gradient is taken as the driving force.14 The second transport mode is the self or tracer diffusion, representing the displacement of labeled but otherwise identical molecules within the fluid at uniform chemical potential (i.e., at equilibrium) under the action of the surrounding molecules and the pore wall. The self-diffusion coefficient Ds is defined by a law similar to that in eq 1, with the only difference that the flux, chemical potential gradient, and density correspond to those of the labeled molecules, indicated herein by the superscript l

1. INTRODUCTION The transport of fluids in confined spaces has attracted much attention for over a century due to its crucial role in several conventional and emerging technologies in the areas of catalysis, adsorptive- and membrane- based separations, and electrochemical energy storage.1,2 The explosive development of novel nanoporous materials such as carbon nanotubes,3 periodic mesoporous silicas4 and metal−organic frameworks5 with ordered structures and simple and ideal geometries has encouraged much fundamental research in the field, allowing for the first time the unequivocal validation of adsorption and transport theories free from nonidealities of pore geometry and connectivity.6 However, disordered nanomaterials such as carbide-derived carbons7,8 and disordered silicas9 continue to be of great practical importance, and substantial effort has been put in understanding how the material structure affects the motion of guest molecules.10−12 Two modes of transport occur within confined spaces when considering a pure fluid.13 One of these is transport diffusion, representing the motion of the center of mass of the fluid under the action of a chemical potential gradient −∇μ. This mode is characterized by the collective diffusivity D0, which relates the molecular flux j and the driving force −∇μ through the phenomenological relation j=

D0ρ ̂ ( −∇μ) kBT

jl =

(2)

An important difference between transport and self-diffusivity is that, while the former tends to increase with density, the latter tends to decrease with density, at least at the single pore level.15−18 However, experimental evidence obtained through pulse field gradient nuclear magnetic resonance (PFG-NMR)19 has shown that material heterogeneity may lead to the presence of a maximum in the self-diffusion coefficient for subcritical Received: July 17, 2012 Revised: January 15, 2013 Published: January 19, 2013

(1) © 2013 American Chemical Society

Dsρl̂ ( −∇μl ) kBT

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dx.doi.org/10.1021/jp3070954 | J. Phys. Chem. C 2013, 117, 3343−3357

The Journal of Physical Chemistry C

Article

fluids as density increases. Such heterogeneity may be interpreted as the presence of a pore size distribution (PSD) in which the relative contribution of large pores with high selfdiffusivity increases as density increases. In particular, for condensable fluids at pressures below saturation, there would be a large difference in self-diffusivity between narrow pores filled with liquid and wide pores predominantly filled with gas, leading to the encountered maximum. Whereas this is essentially the scenario described by Valiulllin et al.,12 the model employed by these authors assumes the self-diffusivity at the single pore level to be a weighted average between the selfdiffusivity of the fluid in the adsorbed phase near the wall and the self-diffusivity of the gas phase in the pore center, leading to a maximum in the single pore self-diffusivity that seems to contradict the simulation evidence. Nevertheless, current experimental techniques do not have the resolution to unequivocally obtain the self-diffusivity of small molecules in a single nanopore, which leaves the phenomenon of transport at this scale somewhat speculative. Moreover, available simulation studies rarely consider pores larger than a diameter of ∼5 nm for guest molecules of ∼0.3 to 0.7 nm effective diameter over a wide range of loadings. However, it is only above this pore size and at sufficiently high loadings that fully distinguishable gas and adsorbed phases could probably be observed within a single pore. It is precisely the exchange between these phases12 that could create the conditions for a maximum to exist in the self-diffusion versus loading curve in a single pore. In this work, we use the distributed friction model (DFM) developed in this laboratory18,20 as an alternative to molecular dynamics (MD) to obtain the self-and transport diffusivities of LJ fluids in single pores when diffuse reflection occurs at the pore walls. The DFM has been shown to be very accurate to predict the diffusion coefficient of Lennard-Jones (LJ) fluids in nanoscale pores (i.e., when at least two molecules can be accommodated in the pore cross-section) under conditions of diffuse reflection, and it is several orders of magnitude faster than conventional MD simulations. The model requires the density profile within the pore, which corresponds to the equilibrium density profile even during transport according to recent results.21 The density profile is determined here through the fundamental measure prescription of classical density functional theory (DFT),22,23 although grand-canonical Monte Carlo (GCMC) simulation would also be suitable to this end. To investigate the influence of PSD and pore connectivity on the overall diffusion behavior in a pore network, a proper averaging technique combining the contribution of virtually an infinite number of pores with different sizes and orientations must be employed. We use a hybrid theory called effective medium-correlated random walk theory (EMT-CRWT), which provides excellent agreement with pore network simulations11,24 and experimental data.25 Using the theoretical techniques mentioned above, we investigate the effective diffusion coefficient of CH4, CF4, H2, CCl4, and hexane in pore networks with known PSD and connectivity. Special emphasis is made on the effective selfdiffusivity because of the apparent disagreement between simulation and experimental measurements. In particular, it is found that a unimodal continuous distribution does not produce a maximum in the self-diffusivity versus loading curve if monotonic decrease in the self-diffusivity at the single pore level occurs. However, if some degree of microporosity is randomly scattered into a mesoporous network (i.e., the sample

has a bimodal micromeso distribution), then a maximum occurs at small loadings provided the guest fluid is under subcritical conditions. The apparent tortuosity, that is, the tortuosity that would be experimentally measured from permeation (transport diffusion) or tracer exchange (self-diffusion) experiments, varies with adsorbate density and depends on the transport mode. For self-diffusion at high loadings the apparent tortuosity tends to be purely structure-dependent because molecular displacements take place just as in the bulk phase.

2. MATHEMATICAL MODEL 2.1. Collective and Self-Diffusion in a Cylindrical Pore. In the DFM, the collective diffusivity is derived from the solution of the microscopic momentum balance for an n-species fluid in a cylindrical pore of radius rp20,26 n xixj( vi̅ − vj̅ ) 1 d ⎛ d vi̅ ⎞ ⎜rηi ⎟ = ρi (r )kBT ∑ Dij r dr ⎝ dr ⎠ j=1

+ ξiρi (r ) vi̅ a(r − r0i)

(3)

Here r represents the radial coordinate, ρi(r) is the local density of component i at position r, and vi̅ is the average axial component of the streaming velocity of the molecules of i. ρ(r) = Σni=1 ρi(r) is the mixture density, ηi is the partial viscosity of i, and Dij is the mutual diffusion coefficient of species i and j. The partial viscosity can be estimated from the mixture viscosity η through ηi = ωiη, where ωi corresponds to the mass fraction of i. Furthermore, Dij and η are evaluated at a coarse-grained density {ρ̃i(r)} of the mixture components, estimated by local averaging of the density profile over a sphere of diameter σff,i (the LJ collision diameter of the fluid particles of i). This is the local averaged density model (LADM), first suggested by Bitsanis et al.,27 which has been successful in allowing the application of the Newtonian shear stress model for nonuniform fluids and has extensively been employed in this group.18,21,26,28 The last term in the right-hand side of eq 3 represents the rate of momentum loss due to the interaction of the molecules with the wall in the repulsive region of the solid− fluid potential r0i < r < rp, where r0i represents the position of the potential minimum. ξi is a density-independent friction coefficient, solely determined by the nature of the molecule− wall interactions, and a(r − r0) is the Heaviside function having the value of unity for r > r0 and zero otherwise. Equation 3 can be solved for the streaming velocity profile subject to the boundary conditions dvi̅ /dr = 0 at r = 0 and r = rp. For a single component, the solution for the velocity profile provides the collective diffusion coefficient (also termed corrected diffusion coefficient) as D0 =

2kBT ρ (̂ −∇μ)r p2

∫0

rp

ρ ( r ) v ̅ (r )r d r

(4)

where ρ̂(rp) is the mean density in the nanopore, given by ρ̂(rp) = (2/r2p) ∫ r0p ρ(r)r dr. In recent work, the self-diffusion coefficient for a fluid confined in a cylindrical pore of radius rp has also been obtained from the above DFM by considering the color diffusion problem.18 Considering two identical species that differ only in color, eq 3 can be rewritten as18 3344



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k Tρ(r ) v ̅ l(r ) dμ l 1 d ⎛ dv ̅ l ⎞ ⎟ = ρ (r ) ⎜rη + B dz r dr ⎝ dr ⎠ D11 + ξρ(r ) v ̅ l(r )a(r − r0)

has correctly predicted the trends in the variation of apparent tortuosity with temperature in mesoporous glass membranes for several light gases.25 For calculation of the density profiles, we use the fundamental measure prescription of DFT,23,33,34 which offers fast and reliable results for LJ fluids in pores of ideal geometry. 2.2. Hybrid Effective Medium Theory. In effective medium theory, the actual pore network, comprising a distribution of pore conductances, is replaced by an equivalent network of pores with identical conductances λe, in such a way that both the original and uniform networks have, on average, the same resistance to flow. The equivalent conductance λe has been shown to be the solution of30,31

(5)

where D11 is the bulk self-diffusion coefficient, evaluated at the coarse-grained density given by the LADM, and ρ(r) is the (total) density profile. Superindex l corresponds to the “labeled” or tracer molecules, for which the momentum balance can be solved under the aforementioned boundary conditions to provide the following expression for the self-diffusion coefficient18 Ds(ρ )̂ =

2kBT rp2ρ ̂

∫0

rp

⎛ v l (r ) ⎞ ρ (r )⎜ ̅ l ⎟r d r ⎝ ( −∇μ ) ⎠

∫0

(6)

For the self-and transport diffusion models to be complete, a method for the calculation of the friction coefficient ξ must be provided. It has been shown20,26 that ξ is related to the lowdensity diffusivity DLD 0 through kBT ∫ r e−ϕfs(r) dr 0

rp

D0LD ∫ r e−ϕfs(r) dr

λ=

(7)

r0

where ϕfs is the 1-D fluid−solid interaction potential. The lowdensity diffusivity can be calculated through MD simulations or, for spherical LJ molecules, diffusely reflecting off the wall through the oscillator model developed in this laboratory,28,29 which provides a fast and accurate method for the estimation of DLD while accounting for the solid−fluid interactions. The 0 theory solves the Newtonian equations of motion of a fluid particle moving in a cylindrical pore under the influence of the fluid−solid interaction and leads to the expression for the lowdensity diffusion coefficient D0LD

2 = πmQ

∫0

∞

∫0

∞

e−βϕfs(r) dr

2

e−βpθ /2m dpθ

∫r

rc1

c0

−1

∞ 0

∫0

∞

e

−βpr2 /2m

jl =

(8)

−βϕfs(r)

jl =

⎡ pr (r′, r , pr , pθ ) = ⎢2m(ϕfs(r ) − ϕfs(r′)) ⎢⎣ ⎤ ⎛ pθ ⎞2 ⎡ ⎛ r ⎞2 ⎤ 2 ⎟ ⎢1 − ⎜ ⎟ ⎥ + p ⎥ r ⎥ ⎝ r′ ⎠ ⎦ ⎝r⎠⎣ ⎦

(10)

̂ s πrp2xρ D lkBT

(11)

ε(N − 1)λe⟨l 2⟩ 3π (N + 1)⟨rp2l⟩

( −∇μl ) (12)

for molecules of a chosen color tagged with the superscript l. Here ε is the material porosity, and the factor (N − 1)/(N + 1) accounts for a one-step correlation in the trajectory of molecules.10.This correlation arises from the possibility that a molecule returns into a pore that it has just traversed, as it meanders between pore intersections. We define the effective network self-diffusion coefficient Deff s by

where β = (kBT) and Q = ∫ e r dr. Here pr(r′,r,pr,pθ) is the radial momentum at position r′of a particle having angular momentum pθ and radial momentum pr at radial position r, following

+

f (rp) drp = 0

where x is the mol fraction of labeled molecules and l is the pore length, assumed to be significantly larger than rp so the diffusivity is independent of this dimension. Upon use of the smooth field approximation, the net flux through the effective network can be estimated as24

dpr

dr ′ pr (r′, r , pr , pθ )

[λ(rp) + (N /2 + 1)λe]

where N represents the coordination number or connectivity, that is, the average number of pores meeting at an intersection, and f(rp) is the number distribution of pore radii. For a pore of radius rp, the conductance is given by

rp

ξ=

[λ(rp) − λe]

∞

Dseff xρ ̅ ( −∇μl ) kBT

(13)

Here x represents the mole fraction of the tagged molecules and ρ̅ is the total adsorbed density, which is the ratio between the number of moles adsorbed in the sample and the void volume

⎜

∞

(9)

ρ̅ =

Here rc0 (r,pr,pθ) and rc1 (r,pr,pθ) are the radial bounds of the trajectory extracted from the solution of pr (r′,r,pr,pθ) = 0 Equations 3−9 constitute the models for collective and selfdiffusion in a nanopore. To predict the effective diffusivity in a pore network following a known PSD, we must take an appropriate average of the fluxes over the pores. We use the EMT-CRWT24 that merges the effective medium theory for random circuits from Kirkpatrick30 (later combined with the smooth field approximation by Burganos and Sotirchos31) with the correlated random walk theory of Bhatia.10,32 The theory yields excellent agreement with simulation in both its single component version24 and its multicomponent extension11 and

∫0 ρ ̂(rp)rp2f (rp) drp ∞

∫0 rp2f (rp) drp

(14)

Upon combining eqs 12 and 13, we obtain Dseff =

εkBTλe(N − 1)⟨l 2⟩ 3π (N + 1)⟨rp2l⟩xρ ̅

(15)

Because the labeled and unlabeled molecules are identical, x is independent of pore radius and eq 10 yields a value of λe that is proportional to x. Equation 15 then leads to an effective selfdiffusivity that is independent of x. For the effective transport diffusivity, an analogous result is obtained 3345


The Journal of Physical Chemistry C D0eff =

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εkBTλe0(N − 1)⟨l 2⟩ 3π (N +

1)⟨rp2l⟩ρ ̅

diffusivities are significantly different and so are the corresponding apparent tortuosities. The LJ interaction parameters for the fluids are summarized in Table 1, whereas

(16)

where the effective transport conductance λ0e is the solution to eq 10 when λ corresponds to the collective transport conductance, λ(rp) = πr2pD0/(lkBT). If the pore length distribution follows that arising from points randomly connected to their N nearest neighbors in a random order, we have found in a previous publication11 that ⟨l2⟩/⟨l⟩2 ≈ (0.001111N2 + 0.043333N + 0.910). By selecting a mean pore length ⟨l⟩ to calculate the conductance and using the previous correlation to correct for pore length dispersion, eqs 15 and 16 produce estimates of the effective diffusivity that agree well with simulation.11 It is important to keep in mind that for self-diffusion the adsorbed fluid is at equilibrium with the bulk gas surrounding the sample and, consequently, the chemical potential is uniform throughout. The chemical potential gradient in eq 13 corresponds to that of the labeled molecules only; for that reason, permeation experiments for the measurement of the effective self-diffusivity require special methods such as tracer exchange or PFG-NMR.12,13 In the case of tracer exchange, the sample is subjected to a step change in the concentration of labeled species (e.g., a radio isotope) at the external surface and the progress toward equilibrium is followed by monitoring the concentration of labeled species in the bulk phase, keeping the total density constant at all times. A novel technique based on interference microscopy (IFM) and IR microimaging35,36 allows for the observation of the transient concentration profiles within the sample, from which the self-diffusion coefficient can be extracted without the disturbing effect of external resistances. In general, it is common practice to define an apparent tortuosity in such a way that the flux of labeled molecules is related to the diffusivity at a single mean pore size rp̅ .37−41 εDs( rp̅ )xρ ̅ j= ( −∇μl ) γappkBT (17)

Table 1. Lennard-Jones Parameters of the Investigated Fluids εff/kB (K)

reference

0.381 0.4662 0.2915 0.5927 0.6182

148.2 134.0 38.0 322.7 297.1

25 25 25 25 49

3. RESULTS AND DISCUSSION Equations 15−18 provide the apparent tortuosity and diffusivity of a fluid in a porous material composed of randomly interconnected pores of arbitrary size having a uniform coordination number N at each node. Here we investigate the effect of structural parameters and adsorbate density on the diffusion of several gases in mesoporous networks, with the single pore diffusivities given by the diffusion model presented above and described in detail elsewhere.18,20 To this end, we consider the Rayleigh distribution of pore sizes fu (rp , r0 , rm) =

(rp − r0) 2

(rm − r0)

2

e−(rp− r0)

/2(rm − r0)2

; r0 ≤ rp

(19)

where f u(rp) represents the number distribution of pore radius, r0 is the minimum pore radius, and rm is the modal pore radius. The standard deviation of this distribution is readily seen to be s = 0.7024(rm − r0)

(20) 11,25,45

This distribution has been considered in previous work because it provides a qualitatively good representation of the PSD for a host of mesoporous solids. We have also considered the case of a bimodal number distribution f b(rp) consisting of a linear combination of two Rayleigh number distributions f u1 and f u2, with modal radii rm1 and rm2 and minimum radii r01 and r02, respectively

3π (N + 1)Ds( rp̅ )⟨rp2l⟩xρ ̅ kBT (N − 1)λe

σff (nm)

the solid is taken to be silica with the interaction assumed to be dominated by a surface layer of oxygen ions, having potential parameters ε00/kB = 492.7 K and σ00 = 0.28 nm, with ρs = 10.47 nm2 based on the results of Neimark et al.42 The fluid−solid interaction parameters are obtained through the Lorentz− Berthelot rules, that is, εfs = (εff·εoo)1/2 and σfs = (σ00 + σff)/2. For CH4, CF4, and CCl4, the bulk self-diffusivities and viscosities were obtained through MD simulations, with simple correlations reported in a previous work.18 For the other gases, we use correlations for LJ fluids available elsewhere.43,44

where γapp is an “apparent tortuosity” coefficient, corresponding to the tortuosity that would be extracted after fitting the experimental results through eq 17. For self-diffusion, Ds (rp̅ ) is usually identified with the bulk self-diffusivity Dsbulk,39−41 whereas for transport, Ds (rp̅ ) is replaced by the collective diffusivity estimated at a characteristic pore radius,36−38 often assumed as twice the pore volume to surface area ratio, with the mole fraction x taken as unity. The apparent tortuosity will not only be affected by the orientation of the pores but also carry the effect of the PSD. Equating the fluxes in eqs 13 and 17, with Deff s following eq 15, yields a theoretical prediction for the apparent tortuosity γapp =

fluid CH4 CF4 H2 CCl4 C6H12

fb (rp) = (1 − α)fu1 + αfu2 (18)

(21)

The mixing parameter α is related to the porosity ratio of the distributions, ε1/ε2, by

For collective diffusion, Ds and λe in eqs 17 and 18 should be replaced by D0 and λ0e , with x taken as unity. In previous work,25 we explored the variation of this theoretical apparent tortuosity with PSD parameters, temperature, and gaseous species in DDR zeolites and disordered mesoporous glass membranes at low densities, where self and collective diffusivities are approximately equal. Here we explore such variations in a range of densities for which the self and transport

ε1⟨rp2⟩ f 1 u2 =1+ α ε2⟨rp2⟩ f

u1

(22)

where ⟨·⟩f ui refers to the average according to f ui. This distribution is representative of materials with both micro- and mesopores thoroughly interconnected, which makes it helpful 3346



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ing s essentially increases the weight of the narrow, poorly conducting pores and tends to decrease the self-diffusion coefficient. The collective diffusivity follows the Hagen− Poiseuille formula for very wide pores at moderate to high densities, increasing approximately with rp2. As the density decreases in these pores, the collective diffusivity tends to the Knudsen diffusivity DK46

to understand the effect of randomly distributed defects acting as narrow channels between meso- and macropores. 3.1. Unimodal Distribution: Supercritical Fluids. Figure 1 depicts the variation of the transport (a) and self-diffusion (b)

DK =

8kBT 2 rp 3 πm

(23)

increasing linearly with rp. Consequently, the poorly conductive pores added as s is increased cannot counteract the effect of the wide and highly conductive pores that are also included, leading to the increase in Deff 0 with increasing s that is observed at all eff densities. It is well known that Deff 0 should approach Ds as the 13 density is lowered. This is clearly seen when comparing Figure 1a and Figure 1b, explaining why at low density ( 0.1 until ρ̅ ≈ 5 to 6 nm−3 and subsequently remains nearly constant. A similar situation was observed in a previous work45 for several light gases in a macroporous alumina support, which was explained by the increasing influence of viscous flow at high densities. The case is similar here: because the viscous diffusivity tends to be proportional to density in large mesopores and the adsorbate densities in large individual pores become increasingly similar, the effect of ρ̅ in the numerator and the denominator (embedded in λe) of eq 18 roughly cancels out, producing an almost constant tortuosity at high densities. As density decreases, the larger pores become increasingly dominated by the wall-mediated diffusion given by the oscillator model in eq 8 and approach the Knudsen limit in eq 23 for rp > ∼20σff,46 which produces a short-circuiting effect that increases γ0app. When s = 0.1, γ0app slightly increases with ρ̅, although the maximum increase it experiences from its value at ρ̅ = 0 is just over 0.5%; consequently, it is reasonable to assume that, at least for this coordination number and modal pore radius, γ0app is fully determined by geometric factors when s ≤ 0.1. Variations in the coordination number, temperature and mean pore size can yield a rich variety of behavior of γ0app and γsapp. An example of this is depicted in Figure 4a−d, in which the variation of apparent tortuosity with adsorbate density and pore

size dispersion in a network following a Rayleigh distribution of rp is shown at several modal pore sizes, coordination numbers, and temperatures. Whereas the way in which γsapp changes with adsorbate density is similar to that depicted in Figure 2a for large modal pore sizes at most coordination numbers, it is not the case when N = 3, as observed in Figure 4a. Here γsapp exhibits a similar behavior to that shown in Figure 2a at low densities but continues to decrease after the maximum occurring at ρ̅ ≈ 1.5 m−3 to become approximately constant at high densities. Note that γsapp (n = 6) < γsapp (n = 3) at all ρ̅, which is expected because the higher the coordination number the lower the chance for a molecule to retrace its path at an intersection. The decrease in tortuosity with density after ρ̅ ≈ 1.5 m−3 can be explained by noting that at high coordination numbers it is more likely to have both narrow and wide pores meeting at each intersection, whereas for small coordination numbers there is a higher chance to have just narrow (or just wide) pores at a given intersection. Put another way, the network turns closer to the limiting case of an arrangement of pores in series (N = 2), providing narrow pores with low conductance a higher weight in determining the effective conductance. Furthermore, when the adsorbate density increases from moderate to high values, the conductance in small pores decreases at a considerably lower rate than it does in wide pores, as inferred from Figure 3. Because narrow pores are now having a more significant role in the average diffusion behavior, λ e tends to decrease slower than λ*, and, consequently, the apparent tortuosity tends to decrease. Figure 4b depicts the variation of γsapp with ρ̅ when N = 6 at a considerably lower modal pore radius, rm = 0.8 nm. The general behavior resembles that in Figure 2a, although significantly 3349



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as high as that in a 6 nm diameter pore, whereas at 300 K and the same bulk density it is just twice as high. The ratio between the self-diffusion coefficients of the 6 and 1.6 nm pores is two at 200 and five at 300 K. Therefore, the difference in the conductance between small and large pores is considerably more pronounced at 200 K than it is at 300 K, which explains the increase in the intensity of the apparent tortuosity maximum as temperature is lowered. Furthermore, a small bulk density produces a considerably higher ρ̅ at 200 K than it does at 300 K or 400 K, explaining the shift toward the left in Figure 4c as T decreases. Bhatia25 predicted that at infinite dilution the apparent tortuosity increases with temperature. Figure 4c shows that as density increases this behavior no longer holds. However, at high density, when self-diffusion is molecularly dominated in most pores, the tortuosity tends to become purely geometric and independent of both density and temperature. Figure 4d shows the variation of γ0app with ρ̅ for N = 3 and rm = 5 nm. Similar to the previous case, the trend of the curve shifts remarkably compared with that observed in Figure 2b by only changing the coordination number. The explanation is essentially the same as before: when the adsorbate density increases from moderate to high values the conductance in small pores increases at a considerably lower rate than it does in wide pores. Because narrow pores have a greater impact at N = 3 than they have at, for example, N = 6, λe tends to increase slower than λ* and the apparent tortuosity will now increase. The self-diffusivity and apparent tortuosity of other noncondensable gases is affected by the density of the adsorbed phase in a somewhat similar manner, as depicted before for CH4. Figure 6 illustrates the variation of Deff s with ρ̅ for (a) CF4

displaced toward higher densities. The difference between large and narrow pores is less pronounced as 99% of the pores have a radius below 2.5 nm when s = 0.5 and below 2.0 nm when s = 0.3. In this scenario, higher densities are required for the smallest pores (those below rm) to be significantly more controlled by fluid−fluid interactions than the largest pores (those above rm). The maximum in the conductance occurs at a higher density because the self-diffusion coefficient drops at a slower rate for smaller pore sizes, as depicted in Figure 3b. Interestingly, the tortuosity can be lower at low adsorbate densities than the value corresponding to a uniform pore network with N = 6 (γ = 4.2). This had previously been observed by Bhatia25 when applying the EMT-CRWT to the analysis of transport of light gases in silica pore networks at infinite dilution. Figure 5 depicts the variation of pore

Figure 5. Pore size dependence of the conductance of various gases at 300 K in silica pores. The adsorbate is in equilibrium with the bulk gas at a density of 10−9 nm−3.

conductance with pore radius for three gases, H2, CH4, and CF4, at 300 K and a constant bulk density ρb = 10−9 nm−3, that is, in the Henry law region, where ρ̂ = Kρb where K is the equilibrium constant. Under these conditions, the self and transport diffusion coefficients are approximately equivalent and can be estimated through the oscillator model. The local maximum observed at small pore diameters when the pore can accommodate one single molecule is due to two factors: the most important one is the occurrence of a maximum in the equilibrium constant as a consequence of the merging of the two local minima in the fluid−solid potential, strengthening adsorption. Second, the so-called “levitation effect”, which refers to a local maximum in the diffusivity as a result of a local minimum in the wall-particle collision frequency, takes place.47,48 Therefore, while λ* in eq 26 is determined by the diffusivity at the mean pore size, λe includes the effect of both larger and narrower pores with higher conductance, which makes the ratio λ*/λe smaller than unity and hence reduces γsapp = γλ*/λe below its corresponding value for a uniform network. Figure 4c depicts the variation of γsapp with ρ̅ for methane in a pore network with N = 6 and rm = 2.5 nm at three different temperatures, 200, 300, and 400 K. Decreasing the temperature has the effect of both increasing the intensity of the tortuosity maximum and displacing it toward higher adsorbate densities. This behavior can be explained by noting that adsorption strengthens as temperature is lowered, increasing the difference between the adsorbate densities in narrow and wide pores at low bulk densities. For example, at 200 K and a bulk density of 0.04 bar, the density within a 1.6 nm diameter pore is ten times

Figure 6. Effect of standard deviation in pore size distribution on variation of the effective self-diffusion coefficient with adsorbate density for (a) CF4 and (b) H2 at 300 K in a disordered silica pore network following a Rayleigh pore size distribution. 3350



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with modal radii 1.0 and 0.7 nm, respectively, at several coordination numbers. The shape of the tortuosity curve for CF4 resembles that of CH4 in Figures 2a and 4b, although a constant tortuosity is never reached for the investigated adsorbate densities. This is explained by the fact that at rm = 1.0 nm the self-diffusivity is strongly correlated with the pore radius in most of the pores at all densities, which means the diffusion coefficients embedded in the factors λ* and λe never cancel out. Furthermore, γsapp (s > 0) is below γsapp (s = 0) at low densities. A similar behavior was observed for CH4 in Figure 4b, and the reason for this lies again in Figure 5, where it is clear that at rm = 1.0 nm adding smaller and larger pores will produce a rapid increase in λe as a consequence of the local maximum in the pore conductance. According to the same Figure, for H2, the pore conductance does not exhibit a pronounced local maximum in molecularly sized pores due to weak adsorption, which leads γsapp (s > 0) to be larger than γsapp (s = 0) at all densities. Moreover, the tortuosity is quite insensitive to the adsorbate density, with a very mild decrease in γsapp with ρ̅ for N = 3 and a slight increase for N = 6. This can be attributed to the fact that for a weakly adsorbed species such as hydrogen shortcircuiting effects arising from large differences in density between large and narrow pores are considerably less pronounced. 3.2. Unimodal Distribution: Subcritical Fluids. A more interesting behavior arises when condensable gases are considered. Figure 8 depicts the variation of the effective self-

and (b) H2 at 300 K in a network following a Rayleigh distribution of pore radii with the parameters indicated in the Figure. Following the previous discussion, it is not unexpected that increasing s leads to a decrease in the self-diffusivity at high adsorbate density. However, an apparently more complicated behavior is displayed by H2 as density decreases, with −3 Deff s increasing with s for ρ̅ < 5 nm . This behavior is equivalent to what occurred for CH4 at very low adsorbate densities in Figure 1a: as ρ̅ decreases, the contribution of large pores becomes more significant because the fluid inside them has a considerably lower density and, consequently, a much higher diffusivity compared with the fluid in narrow pores. Therefore, increasing s at low ρ̅ adds large pores whose contribution outweighs that of the narrow pores that are also added. The eff shift between an s-increasing Deff s and an s-decreasing Ds occurs at a much higher adsorbate density for hydrogen than it does for methane as a consequence of the significant difference between their molecular diameters and the strength of their intermolecular interactions. As a matter of fact, “bulk-like” behavior in large pores giving rise to an s-decreasing Deff s is only possible for H2 when there is a substantial number of particles in a unit volume due to the small collision diameter of H2 molecules (σf = 0.2915 nm) and their weak intermolecular interaction (εf = 38 K). This explanation is further corroborated by Figure 6a, where the above shift in the variation of Deff s with s is observed at a low density of 0.5 nm−3, which is because of the considerably larger size and fluid−fluid interaction strength of CF4 relative to H2. Whereas the self-diffusivity of both CF4 and H2 does not exhibit a peculiarly different variation with ρ̅ and s when compared with CH4, the way tortuosity changes with these parameters for H2 is indeed quite different. Figure 7 depicts the variation of γsapp with ρ̅ for (a) CF4 and (b) H2 in a pore network following a Rayleigh distribution of pore sizes

Figure 8. Variation of normalized effective self-diffusivity with bulk pressure for several gases at 300 K in a disordered silica pore network following a Rayleigh distribution of pore sizes with N = 4, s = 0.5, and rm = 5σff. The normalization parameters are: H2: P* = 300 bar, D* = 25.3 × 10−9 m2 s−1; CH4: P* =250 bar, D* = 9.4 × 10−9 m2 s−1; CF4: P* = 30 bar, D* = 3.1 × 10−9 m2 s−1; CCl4: P* = 0.2 bar, D* = 2.5 × 10−10 m2 s−1; cyclo-C6H12: P* = 0.2 bar; D* = 1.9 × 10−10 m2 s−1.

diffusivity with bulk pressure (P) for several gases at 300 K in a disordered silica pore network following a Rayleigh distribution of pore sizes, with N = 4 and rm = 5σff. Of these gases, CCl4 and cyclohexane are subcritical at room temperature and, consequently, capillary condensation occurs. The values of P and Deff s are normalized to allow representation of the data in a single plot, with the normalization parameters indicated in the Figure caption. It must be noted that cyclohexane is a large molecule with a complex geometry that cannot be exactly captured by the one-site LJ potential model used here, with the LJ parameters extracted from viscosity data, as reported by Hirschfelder et al.49 The results depicted here regarding this molecule are consequently not expected to be quantitatively exact but meant to offer a qualitative view of how subcritical fluids diffuse in pore networks. For both CCl4 and cyclohexane,

Figure 7. Variation of γsapp with ρ̅ for (a) CF4 and (b) H2 at 300 K in a disordered silica pore network following a Rayleigh pore size distribution with coordination number as indicated in the plot. 3351



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the curve is significantly different from that corresponding to supercritical gases like H2, CF4, and CH4 and qualitatively resembles that obtained by Valiullin et al.50 for self-diffusion of cyclohexane in MCM-41 agglomerates with a mixture of macro and mesopores through PFG-NMR. The adsorption isotherms of CCl4 and cyclohexane in the considered pore networks obtained using DFT are depicted in Figure 9; the normalization

Figure 10. Variation of self-diffusion tortuosity for CCl4 at 300 K in a disordered silica pore network following a Rayleigh distribution of pore sizes with the parameters indicated in the Figure.

Figure 9. Adsorption isotherms of CCl4 and C6H12 at 300 K in a disordered silica pore network following a Rayleigh distribution of pore sizes with N = 4, s = 0.5, and rm = 5σff.

pressure of these species corresponds to their saturation pressure, that is, P0 = P*. The adsorption isotherms reveal that condensation progressively occurs within the network pores, which are mostly below rp ≈ 20σff, up to a relative pressure of 0.4 for cyclohexane and 0.3 for CCl4. From P/P* = 0 to this threshold pressure, Deff s decreases quickly as the fraction of pores filled with poorly conductive condensate increases. After this point, Deff s reaches a minimum and increases slightly with increasing relative pressure, particularly in the case of CCl4. According to Valiullin et al.50 this increase can be explained by the presence of gas-filled macropores in which the increase in density arising from an increasing pressure leads to a net increase of the fraction of molecules adsorbed in gas form and, consequently, of the effective diffusivity. Although there are no macropores in the network considered here the very mild increase observed in Figure 8 (as opposed to the significant increase reported by Valiullin et al.50) is probably the effect of a few pores large enough (rp > 30σff), in which condensation to occurs near P0, producing this way, in a more subtle manner, 50 the slight increase in Deff s described by Valiullin et al. The tortuosity of subcritical gases displays a considerable variation with bulk pressure, as depicted for CCl4 in Figure 10. The maximum at P/P0 ≈ 0.15 when s = 0.4 and 0.24 reflects the short-circuiting effects from large pores, arising from slow diffusion in nanopores filled with closed-packed fluid and condensation in small mesopores. Such short-circuits are weakened as condensation progresses with increasing bulk pressure because the difference between conductances of pores with various sizes is reduced. Finally, a sudden increase in tortuosity takes place shortly after condensation has occurred in the majority of pores, as a consequence of the remaining large pores still filled with gas that gain weight due to the increasing adsorbate density, as discussed before. When condensation has completed at all pore sizes, the tortuosity stabilizes and reaches a nearly constant value. 3.3. Bimodal Distribution. Figure 11 depicts the (a) number distribution function f (rp) and (b) porosity

Figure 11. (a) Number distribution function f (rp) and (b) porosity distribution function ε(rp) derived from eq 21 for a bimodal distribution with rm1 = 4 nm, rm2 = 1 nm, s1 = 0.14, and s2 = 0.3 and several values of the porosity ratio ε1/ε2.

distribution function ε (rp) derived from f (rp), as given by eq 21 for a bimodal distribution with modal pore radii rm1 = 4 nm and rm2 = 1 nm, standard deviations s1 = 0.14 and s2 = 0.3, and several values of the porosity ratio, ε1/ε2. From the Figure, it is clear that a small amount of microporosity, from a volumetric standpoint, may correspond to a significant number of micropores (assuming, clearly, uniform pore lengths). For example, whereas ε1/ε2 = 75 represents a small degree of microporosity in a volume distribution, it is quite significant when represented in a number distribution. Moreover, because it is the number distribution that ultimately determines the effective conductance in eq 10, a relatively small microporous volume can have a very significant effect on the effective selfdiffusivity. Figure 12 shows the variation of Deff s with adsorbate density for CCl4 at 300 K in a network following the bimodal 3352



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Figure 12. Variation of Dseff with adsorbate density for CCl4 at 300 K in a network following a bimodal distribution of pore radii with several coordination numbers. The volume ratio between mesopores and micropores, ε1/ε2, is (a) 1865, (b) 358, (c) 75, (d) 44, (e) 18 and (f) 2.

distribution given in eq 21. The porosity ratio between mesopores and micropores is (a) 1865, (b) 358, (c) 75, (d) 44, (e) 18, and (f) 2, whereas the modal pore radii and standard deviations are kept as in Figure 11. For cases (a) and (f), there is a negligible amount of either micropores or mesopores, yielding a value of Deff s that varies with ρ̅ in a similar manner as in the unimodal distributions considered in the previous section. In all of the other cases, a maximum occurs at smalls adsorbate densities, intensifying its relative magnitude for ε1/ε2 = 44 and 18, where the number of micro and mesopores is similar. It is interesting to see that at a high porosity ratio such as ε1/ε2 = 358 the maximum in Deff s is still of considerable magnitude, even though the porosity distribution in Figure 11b shows a nearly insignificant microporous volume. The reason for the occurrence of the maximum is as follows: consider the simple case of large N (the pores are thoroughly connected), for which eq 10 reduces to the classical smooth field approximation51 λe =

∫0

or Dseff = =

∫0

∫0

∞

rp2Ds(rp)ρ (̂ rp)f (rp) drp

∞

Ds(rp)f *(rp) drp

(28)

where f*(rp) is a probability density function defined as f *(rp) =

ρ ̂(rp)rp2f (rp) ρ ̅ ⟨rp2⟩

(29)

Therefore, the contribution of pores with radius rp in the value of Deff s is strongly influenced by the fraction of molecules adsorbed within the pores of this size, p(rp) = ρ̂(rp)r2p/(ρ̅⟨r2p⟩). Whereas for supercritical gases the fraction p remains roughly proportional to the pore volume, this is not the case for subcritical fluids. As a matter of fact, if a significant amount of micropores and small mesopores exists in which the density reaches the condensate density quickly, then an increase in the bulk pressure (and hence of ρ̅) will keep the value of ρ̂ within

∞

λ(rp)f (rp) drp

ε 3

ε 3⟨rp2⟩ρ ̅

(27) 3353



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these narrow pores nearly constant. ρ̂ will increase rapidly with bulk pressure in the remaining pores, which still contain gaseous adsorbate, increasing their corresponding values of p. Consequently, the weight of large, highly conducting pores is increased as ρ̅ increases, producing the maximum observed in those plots in which the population of narrow pores is sufficiently large. As ρ̅ continues to increase, condensation occurs in medium-sized and large mesopores, that is, in those whose population is given by the distribution f u1(rp). From this point, the narrow pores given by f u2(rp) become less relevant and Deff s decays in the typical way observed for a unimodal distribution. Figure 13 depicts f *(rp) at several adsorbate

Figure 14. Variation of self-diffusivity with pore filling fraction for the curves in Figure 12a−c at coordination number N = 3.

comparing Figures 12 and 14. In Figure 12, it is clear that the maximum occurs at a similar adsorbate density in all cases. Down to a certain value of ε1/ε2, the number of small pores needed to be filled to reach a given adsorbate density increases because the adsorbate density is a volume average where wide pores have a larger weight than narrow ones. Obviously, when ε1/ε2 decreases to a given threshold, the small pores dominate and the location of the maximum starts to recede until, as seen in Figure 12f, it vanishes. Figure 15 depicts a similar behavior for the case of cyclohexane in a bimodal network with rm1 = 4 nm and rm2 =

Figure 13. Distribution function f *(rp) at several adsorbate densities for a bimodal number distribution f(rp) with the form of eq 21 and rm1 = 4 nm, rm2 = 1 nm, r01 = 2.2 nm, r02 = 0.8 nm, and ε1/ε1 = 44.

densities for a bimodal number distribution f b(rp) with the parameters employed in Figure 12d. As the adsorbate density increases, the weight of pores above rp = 2 nm increases, as previously discussed. However, the plot indicates that such increase in their contribution slows down when ρ̅ > 0.2 nm−3, explaining why further points of maxima are not observed, but instead just the normal monotonic decrease in Deff s occurs. Equation 28 can also be written as ε Dseff = ⟨pDs(rp)⟩f (30) 3 which resembles the multiregion diffusion model from Kärger and coworkers, lucidly summarized in a recent review.52 However, eq 30 does not take into account the pore network connectivity N, which, despite having apparently little influence in the general shape of the curves in Figure 12, has an obvious effect on the magnitude of Deff s . Figure 14 depicts the results in Figure 12a,b,d when N = 3 as a function of pore filling fraction, defined here as the number fraction of pores in which condensation (or saturation in the case of pores below two molecular diameters) has already occurred. According to the plot, the maximum in self-diffusivity occurs at a higher pore filling fraction as the meso- to micropores volume ratio, ε1/ε2, decreases. This can be rationalized by noting that increasing the number of narrow pores with respect to that of wide pores requires a higher pore filling fraction for condensation to start occurring in the wide pores. Because it is the increasing contribution to the overall diffusion of wide pores at low densities that produces the initial increase in self-diffusion, it is reasonable to expect that if there are more narrow pores then this increase in contribution will be sustained for a wider range of pore filling fraction. A different perspective arises from

Figure 15. Variation of Dseff with adsorbate density for cyclohexane at 300 K in networks of various coordination number.

1 nm, standard deviations s1 = 0.14 and s2 = 0.3 and ε1/ε2 = 70. The curve is qualitatively similar to that obtained by Naumov et al.53 for self-diffusion of cyclohexane in Vycor porous glass with a mean diameter of 6 nm, illustrated in Figure 16. However, the PSD of the sample was not provided, which makes it difficult to establish if the effect of microporosity was at least partially responsible. Furthermore, hysteresis was observed in the Naumov et al.53 measurements, which was attributed to interplay between cavitation and pore-blocking. Such effects cannot be observed in the present model, where no blocking or cavitation is considered. However, the qualitative similarity between the plots is remarkable. It should also be noted that the pronounced maximum experimentally observed by Valiullin et al.19 for self-diffusion of cyclohexane in stacked silicon wafers with rp̅ ≈ 5 nm could also be influenced by the anisotropy of the medium, which produced a preferential diffusion direction. 3354



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influenced by loading and structural network parameters. Special emphasis was made on the effective self-diffusivity due to the apparent disagreement observed between simulations at the single pore level and experimental measurements. Indeed, MD simulations at the single pore level suggest an almost monotonically decreasing self-diffusivity with increasing loading, whereas experimental measurements through PFGNMR reveal the possible existence of a maximum when the guest fluid is subcritical, presumably due to networking effects (heterogeneity). It was found that a unimodal continuous distribution of pore radii does not produce a maximum in the self-diffusivity versus loading curve at any coordination number if a monotonic decrease in the self-diffusivity at the single pore level occurs. However, if some degree of microporosity exists within a mesoporous network (i.e., the sample has a bimodal micromeso distribution), then a maximum occurs at small loadings provided the guest fluid is under subcritical conditions. Nevertheless, PFG-NMR studies have been usually performed with species that can hardly been considered single-site LJ fluids such as methanol, acetone, or cyclohexane,12,19,35 and little information is available from MD simulations with polar or multisite LJ fluids in pores large enough for two distinguishable phases, an adsorbed one close to the wall and a bulk-like one in the pore center, to form and produce the exchange that could theoretically lead to a maximum in the self-diffusivity versus loading curve at the single pore level. On the other hand, the collective and self-diffusion tortuosities were found to be significantly different, suggesting that experimental determination of this parameter must take into account the transport mode to be studied. The adsorbate density was seen to strongly affect the apparent tortuosity for the two transport modes, although this influence tends to vanish at high densities when the pores are wide enough for bulk-like (in the case of selfdiffusion) or pipe-like (in the case of transport diffusion) to occur. It is clear that the apparent tortuosities obtained from permeation experiments on a given sample cannot be directly used to interpret tracer-exchange experiments and vice versa. The fundamental reason for this difference lies in the fact that the solid structure has a very different role in both processes, except for the low-density case in which the two transport modes coincide. Indeed, as density increases for the adsorbate in equilibrium with its surroundings (the case of self-diffusion) the solid tends to progressively act more like “excluded” space for the molecules that would otherwise diffuse just as in the bulk phase. The solid is not only bounding the fluid in the collective diffusion case but also generating the friction that allows the system to reach a steady-state under the action of an external force. In other words, the very occurrence of collective diffusion is tied to the presence of the solid. From our calculations, it is evident that the apparent tortuosity in nanoporous materials displays a complex behavior dependent on process conditions, adsorbate species, and transport mechanism. The latter aspect has an important implication: probing the structure of a material through transport diffusion and self-diffusion experiments may yield different results when adsorption is significant and a PSD exists (as it is most often the case). Experimental studies where both diffusion mechanisms are used to probe pore topology on the same sample are, to the best of our knowledge, not available. Research in this direction would be of great interest. From our current estimations, at high density the apparent tortuosity is less sensitive to loading because both self and

Figure 16. Variation of effective diffusivity Deff s for cyclohexane with loading in Vycor porous glass (rp̅ = 3 nm) obtained through PFGNMR at 279 K. The relative amount adsorbed represents the ratio between the amount adsorbed at a given pressure and the adsorbed amount at saturation pressure. The open and filled circles symbolize the experimental data in the adsorption and the desorption branch, respectively. The hysteresis suggests a history-dependent diffusivity. Adapted with permission from Naumov et al.53

Such anisotropy is not considered in the present theory. Finally, we note that the maximum is either nonexistent or very mild for supercritical fluids in materials having this type of PSD, as extensively confirmed by our calculations, due to the fact that the difference of adsorbate densities between pores of different sizes is significantly less pronounced than it is for the case of subcritical fluids. The microporosity introduced in the pore network can also be understood as surface heterogeneity uniformly distributed in meso- and macropores, manifesting in the form of, for example, surface roughness, which generates surface curvature where the adsorption field is stronger, or pore constrictions. This could explain the maximum in self-diffusion observed in Figure 15 despite the fact that the silicon sample had a unimodal PSD. It is worth noting that agreement between theory and experiment in Figures 15 and 16 is qualitative only, and refinements in the calculations need to be made to reach better quantitative agreement. We have assumed the adsorbate (cyclohexane) to be spherical, a gross approximation for a molecule whose geometry is better described as disk-like. The experimental data in Figure 16 were obtained from a system that is not truly isotropic because the solid framework is composed of silicon wafers stacked together. Further development of the averaging method for systems that are anisotropic in one direction is required. Finally, real structures generally have defects that are difficult to characterize and are not captured in fully predictive models.

4. CONCLUDING REMARKS The effective diffusion coefficient of pore networks with a given PSD and connectivity has been estimated through the EMTCRWT using the DFM20,26 at the single pore level to obtain both the transport and self-diffusion coefficients. The DFM has been thoroughly tested for LJ fluids against MD simulation in pores of several sizes and at a wide range of loadings.18,20,26 Furthermore, the EMT-CRWT has repeatedly proven to be successful in predicting simulation and experimental data,24,25 which makes it a reliable averaging technique. For these reasons, we believe that the current calculations, although theoretical, provide valuable insight into how effective diffusivities and apparent tortuosities of pore networks are 3355



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(9) Sharma, R. K.; Rana, B. S.; Varma, D.; Kumar, R.; Tiwari, R.; Jha, M. K.; Sinha, A. K. Microporous Mesoporous Mater. 2012, 155, 177− 185. (10) Bhatia, S. K. Chem. Eng. Sci. 1986, 41, 1311−1324. (11) Bonilla, M. R.; Bhatia, S. K. Langmuir 2011, 28, 517−533. (12) Valiullin, R.; Karger, J.; Glaser, R. Phys. Chem. Chem. Phys. 2009, 11, 2833−2853. (13) Kärger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; John Wiley and Sons: New York, 1992. (14) Gubbins, K. E.; Liu, Y.-C.; Moore, J. D.; Palmer, J. C. Phys. Chem. Chem. Phys. 2011, 13, 58−85. (15) Nicholson, D.; Travis, K. Molecular Simulation of Transport in a Single Micropore. In Recent Advances in Gas Separation by Microporous Membranes; Kanellopoulos, N., Ed.; Elsevier: Amsterdam, 2000. (16) Krishna, R.; van Baten, J. M. Microporous Mesoporous Mater. 2011, 138, 228−234. (17) Krishna, R.; van Baten, J. M. Chem. Eng. Sci. 2012, 69, 684−688. (18) Bhatia, S. K.; Nicholson, D. J. Phys. Chem. B 2011, 115, 11700− 11711. (19) Valiullin, R.; Kortunov, P.; Karger, J.; Timoshenko, V. J. Chem. Phys. 2004, 120, 11804−11814. (20) Bhatia, S. K.; Nicholson, D. J. Chem. Phys. 2008, 129, 164709− 164712. (21) Bhatia, S. K.; Nicholson, D. Phys. Rev. Lett. 2003, 90, 016105-1− 016105-4. (22) Malijevsky, A. J. Chem. Phys. 2007, 126, 134710-1−134710-0. (23) Rosenfeld, Y. J. Phys.: Condens. Matter 1996, 8, 9289−9292. (24) Deepak, P. D.; Bhatia, S. K. Chem. Eng. Sci. 1994, 49, 245−257. (25) Bhatia, S. K. Langmuir 2010, 26, 8373−8385. (26) Bhatia, S. K.; Nicholson, D. Phys. Rev. Lett. 2008, 100, 236103. (27) Bitsanis, I.; Magda, J. J.; Tirrell, M.; Davis, H. T. J. Chem. Phys. 1987, 87, 1733−1750. (28) Bhatia, S. K.; Jepps, O.; Nicholson, D. J. Chem. Phys. 2004, 120, 4472−4485. (29) Jepps, O. G.; Bhatia, S. K.; Searles, D. J. Phys. Rev. Lett. 2003, 91, 126102. (30) Kirkpatrick, S. Rev. Mod. Phys. 1973, 45, 574−588. (31) Burganos, V. N.; Sotirchos, S. V. AIChE J. 1987, 33, 1678−1689. (32) Bhatia, S. K. Proc. R. Soc. London, Ser. A 1994, 446, 15−37. (33) Roth, R.; Evans, R.; Lang, A.; Kahl, G. J.Phys.: Condens. Matter 2002, 14, 12063−12078. (34) Yu, Y.-X.; Wu, J. J. Chem. Phys. 2002, 117, 10156−10164. (35) Kärger, J.; Chmelik, C.; Heinke, L.; Valiullin, R. Chem. Ing. Tech. 2010, 82, 779−804. (36) Chmelik, C.; Kärger, J. Adsorption 2010, 16, 515−523. (37) Wang, C. T.; Smith, J. M. AIChE J. 1983, 29, 132−136. (38) Dozier, W. D.; Drake, J. M.; Klafter, J. Phys. Rev. Lett. 1986, 56, 197−200. (39) Geier, O.; Vasenkov, S.; Karger, J. J. Chem. Phys. 2002, 117, 1935−1938. (40) Vasenkov, S.; Geir, O.; Karger, J. Eur. Phys. J. E 2003, 12, 35− 38. (41) D’Agostino, C.; Mitchell, J.; Gladden, L. F.; Mantle, M. D. J. Phys. Chem. C 2012, 116, 8975−8982. (42) Neimark, A. V.; Ravikovitch, P. I.; Grün, M.; Schüth, F.; Unger, K. K. J. Colloid Interface Sci. 1998, 207, 159−169. (43) Galliéro, G.; Boned, C.; Baylaucq, A. Ind. Eng. Chem. Res. 2005, 44, 6963−6972. (44) Reis, R. A.; Nobrega, R.; Oliveira, J. V.; Tavares, F. W. Chem. Eng. Sci. 2005, 60, 4581−4592. (45) Gao, X.; Bonilla, M. R.; Costa, J. C. D. d.; Bhatia, S. K. J. Membr. Sci. 2012, 409−410, 24−33. (46) Bonilla, M. R.; Bhatia, S. K. J. Membr. Sci. 2011, 382, 339−349. (47) Yashonath, S.; Santikary, P. J. Phys. Chem. 1994, 98, 6368−6376. (48) Anil Kumar, A. V.; Bhatia, S. K. J.Phys. Chem. B 2006, 110, 3109−3113. (49) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954.

transport diffusivities are more strongly influenced by intermolecular interactions than wall−fluid interactions. For mesoporous materials, this is a limit in which an apparent tortuosity reflecting pore connectivity and pore size dispersion can be obtained. Therefore, if the major interest is obtaining insight of the material pore topology, then one good option is to choose weakly adsorbed fluids (e.g., alkanes instead of hydrocarbons with functional groups) and select pressures high enough to guarantee that viscous diffusion dominates over wallmediated diffusion. An alternative option is selecting low pressures and temperatures high enough for wall-mediated diffusion (i.e., negligible adsorption) to dominate in the widest pores. Nevertheless, Figure 2 shows that a small increase in adsorbate density at the lowest adsorbate densities may lead to significant tortuosity variations. Clearly, this does not matter if the pores are large enough to consider adsorption negligible. In such case, however, it is still desirable to have one single diffusion mechanism (i.e., viscous or Knudsen) to dominate transport, as the combination of both has been shown to lead to variations in apparent tortuosity with adsorbate density even in macroporous materials.45 Still, the self and collective diffusion lead to different tortuosities. The reason for this is the existence of a PSD because the self-and collective diffusivity do not increase proportionally with variations in the pore size. In this sense, the tortuosity must be read as a useful “comparative” measure of connectivity and pore size dispersion between, for example, two samples of the same material synthesized through different methods, as long as the same methodology to measure it is used. Notice that apparent tortuosity varies with temperature mostly because adsorption is very sensitive to temperature. Comparative measurements must clearly be carried out at the same temperature.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research has been supported by a grant from the Australian Research Council under the Discovery Scheme. S.K.B. acknowledges an Australian Professorial Fellowship from the Australian Researches Council. We thank the High Performance Computing Unit of the University of Queensland for access to supercomputing facilities.

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REFERENCES

(1) Jackson, R. Transport in Porous Catalysts; Elsevier Scientific: Amsterdam, 1977. (2) Wesselingh, J. A.; Krishna, R. Mass Transfer in Multicomponent Mixtures; Delft University Press: Delft, The Netherlands, 2000. (3) Iijima, S.; Yudasaka, M.; Nihey, F. NEC Tech. J. 2007, 2, 52−56. (4) Selvam, P.; Bhatia, S. K.; Sonwane, C. G. Ind. Eng. Chem. Res. 2001, 40, 3237−3261. (5) Davis, M. E. Nature 2002, 417, 813−821. (6) Bae, J.-S.; Nguyen, T. X.; Bhatia, S. K. J. Phys. Chem. C 2009, 114, 1046−1056. (7) Bonilla, M. R. n.; Bae, J.-S.; Nguyen, T. X.; Bhatia, S. K. J. Phys. Chem. C 2010, 114, 16562−16575. (8) Nguyen, T. X.; Bae, J. S.; Bhatia, S. K. Langmuir 2009, 25, 2121− 2132. 3356



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(50) Valiullin, R.; Dvoyashkin, M.; Kortunov, P.; Krause, C.; Karger, J. J. Chem. Phys. 2007, 126, 054705-1−054705-6. (51) Feng, C.; Stewart, W. E. Ind. Eng. Chem. Fund. 1973, 12, 143− 147. (52) Valiullin, R.; Kärger, J. Chem. Ing. Tech. 2011, 83, 166−176. (53) Naumov, S.; Valiullin, R.; Monson, P. A.; Kärger, J. r. Langmuir 2008, 24, 6429−6432.

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Diffusion in Pore Networks: Effective Self-Diffusivity and the Concept of

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