Langmuir 2005, 21, 229-239
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Effects of the Juxtaposition of Carbonaceous Slit Pores on the Overall Transport Behavior of Adsorbed Fluids Owen G. Jepps and Suresh K. Bhatia* Division of Chemical Engineering, The University of Queensland, Brisbane QLD 4072, Australia
Debra J. Searles School of Science, Griffith University, Brisbane QLD 4111, Australia Received August 11, 2004. In Final Form: October 8, 2004 It is a common approximation in the modeling of adsorption in microporous carbons to treat the pores as slit pores, whose walls are considered to consist of an infinite number of graphitic layers. In practice, such an approximation is appropriate as long as the number of graphitic layers in the wall is greater than three. However, it is understood that pore walls in microporous carbons commonly consist of three or fewer layers. As well as affecting the solid-fluid interaction within a pore, such narrow walls permit the interaction of fluid molecules through the wall, with consequences for the adsorption characteristics. We consider the effect that a distributed pore-wall thickness model can have on transport properties. At low density we find that the only significant deviation in the transport properties from the infinite pore-wall thickness model occurs in pores with single-layer walls. For a model of activated carbons with a distribution of pore widths and pore-wall thicknesses, the transport properties are generally insensitive to the effects of finite walls, in terms of both the solid-fluid interaction within a pore and fluid-fluid interaction through the pore walls.
I. Introduction Understanding the dynamics of fluid transport through microporous media is of great importance for the development of a variety of new applications.1 There are many factors that are important in determining this transport behavior, related to the fluid properties, the solid properties, and the nature of their interaction. Many microporous systems of current interest, such as zeolites, are strongly regular in nature; by contrast, the myriad structures that microporous carbons can adopt present one of the greater difficulties in studying transport in such carbonaceous systems. This is problematic in the development of theories describing microporous transport, because solids with even relatively simple topological structures can require complex calculations to simulate transport data. Consequently, analyzing the transport properties of microporous carbons requires simplifying assumptions about the underlying geometry of the pores themselves, as well as about their connectivity. In this context, qualitative observations of the properties of microporous carbons determined with such models fulfill an important role in helping to identify which aspects of the structures are significant to the transport behavior and to refine our models accordingly. In the case of activated carbons, the porous system is commonly represented using the slit-pore model.2-4 Pores are considered to be slit-shaped, and pore walls are deemed * To whom correspondence may be addressed. E-mail: sureshb@ cheque.uq.edu.au. (1) Bandosz, T. J.; Biggs, M. J.; Gubbins, K. E.; Hattori, Y.; Ilyama, T.; Kaneko, K.; Pikunic, J.; Thomson, K. T. Chem. Phys. Carbon 2003, 28, 41. (2) Aukett, P. N.; Quirke, N.; Riddiford, S.; Tennison, S. R. Carbon 1992, 30, 913. (3) Maddox, M.; Ulberg, D.; Gubbins, K. E. Fluid Phase Equilib. 1995, 104, 145. (4) Chen, X. S.; McEnaney, B.; Mays, T. J.; Alcaniz-Monge, J.; CazorlaAmoros, D.; Linares-Solano, A. Carbon 1997, 35, 1251.
to consist of graphitic planes. One of the simplifying assumptions often made in studies of slit-pore transport is that these walls are semi-infinite; that is, both walls consist of an infinite number of graphitic layers. In this case, the Steele 10-4-3 potential5 is usually used to represent the interaction between fluid molecules and the pore wall. Models for microporous carbons have been developed by considering domains of parallel pores, with a distribution of pore widths, and then introducing factors into this model to represent the topological structure connecting these domains.1 Recently, researchers have begun to consider alternative models for the pore walls, in light of experimental evidence6,7 and theoretical mass balance considerations that suggest relatively thin walls.8 The effect of varying the pore-wall thickness on the adsorption properties has already been studied.9,10 Furthermore, recent work based on theoretical considerations has suggested the consideration of a distribution of pore-wall thickness in activated carbons;8 this proposal has been subsequently validated with experimental data.11,12 These results indicate a relatively high proportion of single- or double-layer walls, and their existence leads to a significant departure from the characteristics observed under the assumption of semi-infinite walls. The reason for this departure is primarily the dependence of the solidfluid interaction on the pore-wall thickness, affecting the adsorption energies. Further, it is possible for molecules (5) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974 (6) Fryer, J. R. Carbon 1981, 18, 397. (7) Marsh, H.; Crawford, D. Carbon 1982, 20, 419. (8) Bhatia, S. K. Langmuir 2002, 18, 6845. (9) Mays, T. J. Fundamentals of Adsorption 5; Kluwer Academic Publishers: Boston, 1996. (10) Ravikovitch, P. I. Presented at Carbon 01, International Conference on Carbon; Lexington, July 14-19, 2001. (11) Nguyen, T. X.; Bhatia, S. K. Langmuir 2004, 20, 3532. (12) Nguyen, T. X.; Bhatia, S. K J. Phys. Chem. B 2004, 108, 14032.
10.1021/la047984g CCC: $30.25 © 2005 American Chemical Society Published on Web 12/07/2004
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to interact with one another through the pore walls in the single-layer case, further influencing the adsorption isotherm. In this work, we consider the implication of these findings on the transport behavior of molecules in micropores. First of all, we adapt a recently developed model of transport at low densities in symmetric slit pores13,14 to estimate transport properties in the low-density limit. To apply this theory, we extend it to the case of an asymmetric slit pore in section II-A (with details in the appendix). In section II-B, we consider a simple model of activated carbon, as a juxtaposition of slit pores with a given distribution of pore widths and pore-wall thicknesses, and derive an expression for the transport diffusion coefficient for this model. Apart from estimating transport coefficients through the low-density transport theory, we also use Monte Carlo and molecular dynamics techniques to examine the adsorption and transport properties of these systems (respectively). Details of our application of these simulation techniques are given in section III. In Section IV we compare our theory with the simulation results. Starting with the case of a fluid in the Henry’s law region, we consider the effect of a distributed wallthickness model on transport for a single pore (section IV-A) and then for the activated carbon model which juxtaposes such pores (section IV-B). In both cases, we compare results with the semi-infinite pore-wall (infinite pore-wall thickness, IPWT) model. These results are obtained in the low-density limit, where fluid densities are insufficient to lead to significant interactions either within a single pore or through the pore walls. Therefore, in section IV-C we consider the effect of pore-pore correlations on the overall transport. Interestingly, in all cases, we observed that the effects of including the finite wall thicknesses in our model do not lead to significant changes in the transport properties, in contrast to the changes observed in the adsorption behavior. II. Theory A. Low-Density Single-Pore Transport Model. In ref 13, the authors developed a theory for the transport of molecules along pores with a slit or cylindrical geometry, where the solid-fluid interaction was symmetric across the pore. While the basic premise of our model remains the same, in the current work we must consider an extension where the solid-fluid interaction across the slit pore is no longer symmetric. The solid-fluid interaction in our system is given by a conservative interaction potential V(x) across the pore width and a boundary condition to permit momentum exchange in the yz plane (parallel to the pore walls). V(x) is determined using a Steele 10-4 potential5 φ(x) to represent the interaction between a single graphitic layer and a fluid molecule:
[ ( ) ( )]
φ(x) ) 2πFnsfσsf2
2 σsf 5 x
10
-
σsf x
4
(1)
where Fn represents the surface density of the wall atoms, sf and σsf represent the (LJ) energy and length-scale parameters for the solid-fluid interaction, and x represents the normal separation between the fluid molecule and the graphitic plane. Given that each pore is bounded (13) Jepps, O. G.; Bhatia, S. K.; Searles, D. J. Phys. Rev. Lett. 2003, 91, 126102. (14) Jepps, O. G.; Bhatia, S. K.; Searles, D. J. J. Chem. Phys. 2004, 120, 5396.
Figure 1. Range of oscillation of a molecule is determined by the solid-fluid interaction potential V(x) and by Ex. In parts a and b, there is only one interval of oscillation for the energy Ex, bounded by x(. In part b, this can only occur if Ex is sufficiently large. Otherwise in part c, the molecule cannot cross the central maximum and must oscillate in one of two regions. In the double-minimum pore, we define a bounded region between the potential minima xL0 and xR0, corresponding to the vertically striped region. We also define a repulsive region, corresponding to the dotted regions.
by two walls made up of graphitic layers, the potential across the pore takes one of two general shapes, a singleminimum potential (“U” shape) in narrower pores and a double-minimum potential (“W” shape) in wider pores (see Figure 1). The fluid molecule in the potential field V(x) alone would only exchange momentum normal to the pore walls and not parallel with them. In a real system, fluctuations in the yz momentum arise from the exchange of yz momentum during interactions with the boundary, due to surface roughness. In our system, we model this exchange using diffuse boundary conditions:15 at each boundary reflection, a molecule’s momentum is randomly reoriented in the yz plane. In the appendix, we determine the transport diffusion coefficient D0 of a fluid at equilibrium in an asymmetric pore, in the limit of low density, where fluid-fluid interactions are negligible. We find, in analogy with the result of ref 13, that the transport diffusion is given by
D0 )
kT 〈τ〉 2m
(2)
for a system of fluid molecules, each of mass m, at temperature T. In eq 2, k denotes Boltzmann’s constant, and 〈τ〉 denotes the mean time between diffuse boundary reflections, averaged over a distribution consistent with (15) Cracknell, R. F.; Nicholson, D.; Quirke, N. Phys. Rev. Lett. 1995, 74, 2463.
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a system at constant temperature (i.e., a canonical distribution of molecular energies). The molecular dynamics across the pore is separable, so that the time between diffuse reflections is completely determined by V(x) and can be calculated for a molecule from the relation
dτ ) xm/2[Ex - V(x)]-1/2 dx
2 - R kT 〈τ〉 R 2m
tf∞
〈(
)〉
∑iNiri(t) 2 ∑iNi 1 〈∑iNi2ri2(t)〉 ) lim tf∞ 2Nt
N〈r2(t)〉 N ) lim tf∞ 2t 2t
(3)
where Ex(x, px) ) px2/2m + V(x), by integration along the appropriate path (see the appendix). Finally, it is interesting to extend the theory to the case of partially diffuse boundary conditions, as was done in ref 14. We consider the style of the partially diffuse boundary condition introduced by Maxwell to understand slip flow.16 Under this boundary condition, reflections are either totally diffuse or totally specular. An accommodation coefficient R describes the mean ratio of diffuse reflections to total (diffuse and specular) reflections that take place. Alternatively, it describes the probability that a diffuse reflection is observed at the end of an oscillation. In the appendix, we determine the transport diffusion coefficient for this more general case
D0 )
D0 ) lim
)
1 N
∑iNiD0i(t)
(5)
For a system with pore-width and pore-wall thickness distributions f(H) and p(l), the transport diffusion coefficient becomes
D0(µ) )
∫Fˆ (l, m, H, µ) HYZD (l, m, H, µ) f(H) dH ∑ p(l) p(m) ∫Fˆ (l, m, H, µ) HYZf(H) dH
∑
l,mp(l)
p(m)
0
l,m
(6) To produce a Fickian transport coefficient, we multiply D0(µ) by the Darken factor. As discussed above, we must also introduce a correction due to the anisotropy of our model. These combined corrections give a Fickian diffusion transport coefficient of
(4)
which reduces to eq 2 in the case of totally diffuse reflections (R ) 1). Thus, the accommodation coefficient contributes a multiplicative factor (2 - R)/R, once again in agreement with the result derived previously for the nonequilibrium system.14 B. Activated Carbon Transport Model. We now consider an expression for the overall transport coefficient observed in a system of activated carbon micropores. Adsorption properties are commonly modeled by considering the microporous network to be at equilibrium with a surrounding gas at a given bulk fugacity µ. Our model8 consists of a domain of parallel, juxtaposed pores, where each pore wall serves as the boundary between two (neighboring) pores. The center-to-center pore widths H are governed by a probability distribution f(H), and the pore-wall thicknesses l (the integer number of graphitic layers comprising the wall) by a distribution p(l). We consider the pore-wall thicknesses on either side of a pore to be statistically independent. While such a model may provide a useful description on the local scale, it fails to take into account the topology of the larger-scale pore structure. To accommodate this, we introduce a simple tortuosity factor δ, set at δ ) 3 to correct for the anisotropic nature of this model (because all walls have their normal vector in the x direction only). While this is an extremely simple way of incorporating complex geometric and topological information, we anticipate that much can still be learned about the nature of microporous transport in activated carbons from such a model. If each pore in the domain has the same breadth, Y, and length, Z, then the number of molecules Ni in each pore is given by Fˆ iHiYZ, where Fˆ i denotes the mean pore density and the total number of molecules is given by N ) ∑iFˆ iHiYZ ) YZ ∑iFˆ iHi. To determine the transport coefficient of this system, we write the center of mass of the molecules in the system, in terms of the centers of mass in the individual pores: (16) Maxwell, J. C. Scientific Papers; Cambridge University Press: Cambridge, 1890; Vol. II, p 708.
Dt0(Fˆ ) )
1 ∂µ × δkBT ∂ ln Fˆ
∑l,mp(l) p(m) ∫Fˆ (l, m, H, µ) HD0(l, m, H, µ) f(H) dH ∑l,mp(l) p(m) ∫Fˆ (l, m, H, µ) Hf(H) dH (7) In the low-density limit, this expression can be simplified, because in that limit the transport coefficient D0(l, m, H, µ) is independent of µ, and the Darken factor is unity. Furthermore, the pore densities at low densities correspond to those predicted by statistical mechanics:
Fˆ (l, m, H)H )
eµ/kBT kBT
H/2 -V(x,l,m,H)/k T e dx ∫-H/2 B
(8)
so that the low-density limit transport coefficient is given by
Dt0LD )
∑
∫D (l, m, H) f(H)∫ δ∑ p(l) p(m) ∫f(H)∫ e
l,mp(l)
p(m)
0
H/2
l,m
H/2
-H/2
e-V(x,l,m,H)/kT dx dH
-V(x,l,m,H)/kT
-H/2
dx dH
(9) III. Simulation Details In this paper, we apply the theory developed in the previous section to the transport of methane and carbon tetrachloride in carbon slit micropores, by comparing the predictions of the theory with results from equilibrium molecular dynamics (EMD) simulations. All the pores considered were bounded by walls consisting of graphitic layers, with pore widths in the range 0.5 < H e 2.0 nm. In the simulations, fluid molecules were represented as spherically symmetric particles, interacting with one another through a Lennard-Jones (LJ) 12-6 potential. The interaction between each graphitic layer and the fluid molecules was represented by a Steele 10-4 potential, with LJ parameters defined via the Lorentz-Berthelot combining rules. The following LJ parameters were used: for
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methane, f/kB ) 148.1 K and σf ) 0.381 nm; for carbon tetrachloride, f/kB ) 322.7 K and σf ) 0.5947 nm; and for carbon, s/kB ) 28.0 K and σs ) 0.340 nm. Fluid-fluid interactions were cut off at a radius ≈4σf. Initial configurations were generated using grand canonical Monte Carlo (GCMC) simulations. System dimensions were chosen to ensure that the mean number of molecules produced was approximately 500, and the simulations were run for the order of 108 Monte Carlo steps, to ensure sufficient convergence of the density profiles. Values of D0 were determined at a constant number of molecules, constant volume, and constant temperature using EMD. To control the system temperature, a Gaussian thermostat was introduced,17 with equations of motion
r3 i ) pi/m
(
p3 i ) -
)
∂Φ + R(t)pi′ ∂ri
(10)
where ri and pi represent the positions and momenta of the fluid molecules, Φ is the combined solid-fluid and fluid-fluid interaction potential, and R(t) is the thermostat multiplier, acting on the peculiar momenta pi′sthe peculiar momentum being that part of the instantaneous momentum that is additional to the average streaming momentum. These equations were solved using a 5th order Gear predictor-corrector integrator, with a time step of 2 fs. To permit momentum exchange between the solid and the fluid parallel to the walls, diffuse boundary conditions were applied as described in ref 15. Data were obtained by averaging results from runs of length 5-10 × 106 time steps, each beginning from distinct initial conditions. For each run, averages were collected only after the system had been allowed to relax to equilibrium or the steady state (over approximately 500 000 time steps). Values of D0 were determined using the GreenKubo relation:
D0 )
1 N
lim tf∞
∫0t 〈∑∑vzi(0)vzj (ξ)〉 dξ i
Figure 2. Diffusion coefficient as a function of pore width, for methane transport at 298 K in the low-density limit, and for pore walls of various thicknesses. Data are collected by wall thickness: the 1-1 pore is used as a reference in all four graphs. Lines represent data obtained from the theory; symbols represent data obtained from simulation (and appearing in ref 14).
(11)
j
for molecular velocities vzi(t). IV. Results and Discussion A. Low-Density Dependence (Single Pore). We begin by considering transport at low density along a single pore of width H, with walls consisting of l graphitic layers on one side and m graphitic layers on the opposite side. For convenience, we assume that l < m and hereafter refer to such a pore as an l-m pore of width H. We investigated the variation of the transport diffusion coefficient D0 with varying H, l, and m at various temperatures in the range 250-500 K, in the low-density limit. Values for the transport coefficient were obtained using the low-density transport model outlined in section II-A. Figures 2 and 3 depict the variation of D0 with pore width for methane and carbon tetrachloride adsorbates at 298 K, for the various combinations of pore-wall thicknesses considered. The results are grouped according to the number of layers in the thinner wall (i.e., by l), and the 1-1 pore results are included in each graph as a reference. We show results at 298 K only; while the pore-width(17) Evans, D. J.; Morriss, G. P. Statistical Mechanics of Nonequilibrium Liquids; Academic Press: London, 1990.
Figure 3. Diffusion coefficient as a function of pore width, for carbon tetrachloride transport at 298 K in the low-density limit and for pore walls of various thicknesses. Data are collected by wall thickness: the 1-1 pore is used as a reference in all four graphs. Lines represent data obtained from the theory; symbols represent data obtained from simulation.
dependent behavior varies with temperature, the relative behaviors due to the variation in pore-wall thickness, for a given pore width, change considerably less with temperature. We show results at a lower end of the temperature range observed, where the effects due to pore-wall thickness are somewhat stronger. While the low-density theory has previously been validated against methane transport in the semi-infinite pore-wall model at 298 K,13,14 here we provide values of the transport coefficient determined by EMD simulation for carbon tetrachloride transport as well. These data also appear in the figures and show excellent agreement between the theory and the simulation. The results for the two different adsorbates reveal some common features. At narrow pore widths, the transport coefficient is essentially independent of the pore-wall thickness. The pore-wall thickness dependence begins to take effect in the vicinity of the values of H at which the solid-fluid interaction changes from a single to a double minimum (denoted HC: HC = 0.83 nm for methane, HC =
Juxtaposition of Carbonaceous Slit Pores
Figure 4. Solid-fluid interaction potentials for methane and carbon tetrachloride at 298 K. Potentials are shown for pores with both walls comprising five graphitic layers, for various pore widths.
1.08 nm for carbon tetrachloride). This point also appears to serve as a local maximum in the transport coefficient: as the pore width continues to increase, the value of D0 decreases to a local minimum, before increasing again toward infinity (this behavior is less obvious in the case of carbon tetrachloride but is consistent with the theory, as outlined below). This nonmonotonic behavior may appear counter-intuitive and is directly related to the variation in V(x) as the pore width increases and the consequent effect on the mean oscillation time 〈τ〉. For both adsorbates, there is little distinction in the transport behavior once at least one wall has two or more graphitic layers. Our results indicate that systems where at least one wall has more than one layer are equivalent to the semi-infinite wall, for the purposes of transport behavior. We interpret these features as follows: while the adsorption isotherm depends on the absolute values of the solid-fluid interaction energy in the pore, the transport behavior depends on the dynamical properties and, therefore, on the forces, or gradients of the solidfluid interaction energy across the pore. In the narrower pores, the energy gradients are overwhelmingly dominated by the innermost graphitic layers in the pore walls so that the transport coefficient is essentially insensitive to the outer layers. In the vicinity of the pore width marking the single/double minimum transition, the gradient of V(x) is low and slow-varying about the minimum, as the higherorder derivatives are also close to zero near the minimum (see Figure 4). The slope of V(x) is, therefore, more sensitive to the influence of the graphitic layers beyond the first in either wall and remains so for wider pores. As the pore width increases beyond the single/double minimum transition width HC, the local maximum acts as a barrier, reducing the mean oscillation time and, therefore, the transport diffusion (see Figure 4). This effect is more pronounced for carbon tetrachloride transport, as a result of the stronger retarding forces induced by the emerging local maximum. However, once the pore becomes sufficiently wide that the influence of one wall on the potential at the opposite potential minimum becomes negligible, the shapes of the local minima remain essentially constant and the mean oscillation time will once again begin to increase. At this stage, the crossing time of molecules trapped about one minima or the other will
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Figure 5. Solid-fluid potential in an asymmetric pore with pore width (a) less than or (b) greater than the single/double minimum transition pore width. In part a, the molecule with energy Ex undergoes two diffuse reflections per oscillation. In part b, the appearance of a bounded region between the two local minima causes the same molecule to undergo a single diffuse reflection per oscillation. This leads to a discontinuity in D0, as a function of pore width H.
be unaffected by the increase in H, but the crossing time of molecules traversing the entire pore will gradually increase. The fraction of traversing molecules can be related to the difference in energies between the central maxima and local minima via statistical mechanics; consequently, this eventual increase in D0 is more pronounced for methane transport than for carbon tetrachloride. This is consistent with the results of Figures 2 and 3. The choice of diffuse boundary conditions also leads to a discontinuity about H ) HC in the transport coefficient D0, as a function of pore width H, for asymmetric pores. This discontinuity is not present for symmetric pores, and the continuous curves for the symmetric pores serve as upper and lower bounds for the asymmetric results (e.g., the 1-2 pore curve lies between the 1-1 curve and the 2-2 curve). This phenomenon can be understood if we consider the solid-fluid interaction potential at pore widths on either side of the single/double minimum transition pore width HC, for a given asymmetric pore. In Figure 5a, at pore widths less than the transition width, the entire pore corresponds to a repulsive region. All oscillatory trajectories consist of diffuse reflections at both ends of the oscillation, including the oscillation marked as a dotted line in Figure 5a. In Figure 5b, at pore widths greater than the transition width, a bounded region appears as a result of the creation of a second local minimum in the solid-fluid potential. Molecules that oscillate with a turning point in the bounded region now undergo a diffuse reflection at one end only of the oscillation, including the molecule whose oscillation is marked as a dotted line in Figure 5b. As the pore width increases, the oscillation period of the molecule of energy Ex changes continuously; however, the oscillation contains two diffuse reflections if H < HC but only one diffuse reflection when H > HC so that the contribution from molecules with turning points in the bounded region to the overall transport doubles as the pore width increases beyond the transition from a single to a double minimum. This leads to the discontinuity described above. We note that, in the symmetric case, the central minimum “splits” continuously into two distinct minima: it is the sudden appearance of a bounded region of nonzero width that causes the discontinuity in D0 as a function of pore width H. The steepness of the D0 versus H curves at small pore widths in Figures 2 and 3 also explains the molecular
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sieving capability of carbons having an open pore width of about 4.5 Å. Such carbons are used in kinetic separations, for example, of oxygen and nitrogen from air. Figures 2 and 3 clearly show that at a center-to-center pore width of about 8 Å (or open pore width of about 4.6 Å, given the carbon diameter of 3.4 Å) the diffusivity is very highly sensitive to the pore width. In such a case small differences in molecular size can lead to large differences in diffusivity, making possible kinetic or diffusional separation from binary mixtures. B. Low-Density Dependence (Activated Carbon Model). Using the transport coefficients determined from the single-pore transport model, we determine a transport coefficient for an activated carbon described by our model of section II-B. To determine the transport coefficient, we require expressions for f(H) and p(l). We assume that the pore-size distribution takes the form8
f(H) )
q(qH)γe-qH Γ(γ + 1)
Figure 6. Temperature dependence of the transport coefficient for the model of activated carbon. The pore-wall thickness distribution is represented using three models: the DPWT, IPWT, and SLPWT models.
(12)
for parameters q and γ, where
Γ(x) )
∫0∞ tx-1e-t dt
(13)
is the usual Gamma function. The quantity q/Γ(γ + 1) in eq 12, therefore, serves as a normalization factor. The parameters q and γ are set at q ) 4.5 and γ ) 46.0, in accord with published values for Norit activated carbon.18 We assume the pore-wall thicknesses are described by a Poisson distribution
p(l) )
a(l-1)e-a (l - 1)!
(14)
where e-a serves to normalize the distribution. The distribution is a Poisson distribution for the variable (l - 1): we exclude the possibility that l ) 0 because each wall must comprise at least one layer. For the Poisson distribution, the parameter a ) 〈l - 1〉, the mean porewall thickness. To contrast the effect of a distribution of pore-wall thicknesses with the semi-infinite wall model, we set a ) 0.5, which yields p(1) ≈ 60% and p(2) ≈ 30%; these figures agree with previous studies of the pore-wall thickness distribution for activated carbons.11 These distributions for f(H) and p(l) are used to describe the physical structure of the microporous material, independent of the adsorbed fluid. The transport coefficients for the individual pores are taken from the theory outlined in section II-A and are combined using the model developed in section II-B. We calculated the overall transport coefficient determined using the activated carbon model, for methane and carbon tetrachloride adsorbates, at temperatures in the range 250-500 K. For each choice of temperature and adsorbate, we calculated three transport coefficients, using the same pore size distribution but different pore-wall thickness distributions. The first is the diffusion coefficient obtained using the distributed pore-wall thickness (DPWT) model. We note that the properties of an l-m pore, l e m, do not vary significantly once m > 5swe, therefore, assume that the properties of an l-m pore, m > 5, are the same as those of an l-5 pore (or a 5-5 pore, if 5 < l e m). The second diffusion coefficient is obtained using a model where all walls consist of five graphitic layers, which for both adsorbates is effectively equivalent to an IPWT model, (18) Ding, L. P.; Bhatia, S. K. AIChE J 2003, 49, 883.
Figure 7. Occupation distribution (normalized) as a function of the pore width, for the DPWT (solid line), IPWT (dashed line), and SLPWT (dotted line) models. The relative occupations are largely independent of the pore-wall thickness model used.
the prevalent pore-wall thickness model used in the literature. The third diffusion coefficient is obtained using a model where all walls consist of a single graphitic layer. From the previous section, we note that the 1-1 pores admit the fastest transport so that the single-layer porewall thickness (SLPWT) model provides an upper bound for the transport coefficient, for a given pore size distribution. In Figure 6, we show the temperature dependence of these overall transport coefficients, using the three porewall models. We observe that the discrepancy between the distributed and semi-infinite wall models lies in the range 3-5%, from which we deduce that the assumption of semi-infinite walls does not significantly affect the calculation of the overall transport coefficient. This result can be understood when we recall that the overall contribution of an individual pore to the overall transport is weighted by two pore-width-dependent functions: one [f(H)] pertains to the physical structure of the porous material, describing the probability of finding a H/2 F(x) dx] pore of a particular width; the other [∫-H/2 pertains to the solid-fluid interaction, describing the relative occupancy of such a pore. The distribution f(H) is identical for each model, because it represents the physical nature of the pore, whereas the occupancy depends on the pore-wall thicknesses and pore width, as well as the fugacity of the environment surrounding the porous system. In Figure 7, we depict the normalized distribution of H/2 F(x) dx, as a function of pore width, the occupation, ∫-H/2
Juxtaposition of Carbonaceous Slit Pores
Figure 8. Pore-width distribution (dotted line), relative occupation as a function of pore width for the DPWT model (dashed line), and their combined distribution (solid line) for methane and for carbon tetrachloride at 298 and 500 K. The combined distribution gives the weighting for D0(H), used to determine the overall transport coefficient.
for the three models. Interestingly, the distribution is not strongly dependent on the model chosen, indicating that, while the actual densities may be affected by the porewall thickness model, the relative occupations are much less sensitive. Because the physical structure of the porous material is the same for each model, the combined structural and occupation weights are similarly insensitive to the pore-wall thickness model. In Figure 8, we show individual and combined effects of the structural and occupation weights for four systems, corresponding to the lower and higher ends of the temperature range considered for methane and carbon tetrachloride. From the combined distribution, the range of pore widths that contribute significantly to the overall transport coefficient can be understood. At 300 K, the significant range of pore widths for methane transport is in the 7-8 Å range, while at 500 K the range widens to 7-14 Å. Significant pore-wall thickness effects are only observed for pores wider than 9 Å for methane and are only significantly different from the semiinfinite model in the case of the 1-1 pore. For our DPWT model, such pores make up about p(1)2 ≈ 36%; all other contributions are effectively equivalent to the semi-infinite pore wall. This, combined with the low relative discrepancy between values of D0 between the 1-1 and 5-5 pores, helps to explain the small difference between the DPWT distributed and semi-infinite pore-wall thickness models. For carbon tetrachloride, the significant range of pore widths remains in the 9-11 Å range. In this range of pore widths, the pore-wall thickness has only a minor effect on the transport. C. Pore-Pore Correlations. Up to this point, our results have assumed that there is no interaction of molecules through the graphitic layers separating the individual pores. This is a reasonable assumption in the context of semi-infinite pore walls. However, the existence of walls as narrow as a single graphitic layer admits the possibility that molecules could indeed interact with molecules in neighboring pores. Such interactions can affect the adsorption of large molecules in carbon micropores.8 At this point, we investigate whether fluid interactions through the pore walls perturb transport. To examine the effect of pore-pore correlations on the transport behavior, we consider a system of two juxtaposed pores (sharing a common pore wall). We choose the pore
Langmuir, Vol. 21, No. 1, 2005 235
sizes and pore-wall thicknesses of these adjoining pores in such a way as to maximize the possible effect of interactions through the pore wall. To achieve this, we require the following conditions: that molecules in neighboring pores can get close enough together to interact significantly, that there is a region of sufficiently high fluid density for the intermolecular force in the neighboring pore to be significant, and that there is a region of sufficiently low fluid density so that the intermolecular forces from the neighboring pore are not dominated by local intermolecular interactions or the local solid-fluid interaction. These last two conditions may appear contradictory; however, by an appropriate choice of the test system, all three conditions can be met. Consequently, we consider a system with one pore of width H ) 2σsf, the other of arbitrary width, and where all three wall layers consist of a single graphitic layer (hereafter described as a 1-1-1 system). The above system meets the three conditions that we have set. The first condition is met by the choice of singlelayer walls. The single-layer central wall minimizes the separation between molecules on either side. Furthermore, if one of the outer walls has more than one layer, then it is energetically more favorable for molecules to move away from the center toward this outer wall. The choice of single-layer outer walls avoids this situation. Given that σf/σsf is greater for carbon tetrachloride than for methane, we choose the former as our test system adsorbatessuch a choice allows for more significant fluid-fluid interactions for fluid molecules separated by the central wall. The pore widths are chosen to meet the second and third conditions. Because both pores are in equilibrium with the same external chemical potential, we aim to choose pore widths such that one side is at high density while the other is at low density. The high density of molecules on one side produces a significant potential that has the capacity to influence the dynamics of molecules in the neighboring low-density pore. In the Henry’s law limit, the mean pore densities can be estimated via eq 8, where the solid-fluid pore potential is determined as the sum of Steele 10-4 potentials for each graphitic layer from either wall. While this integral does not have a convenient closed form, the maximum of Fˆ (H), as a function of H, is well approximated at H ) 2σsf ) 9.35 Å (the pore width with the lowest potential minimum). This pore size is also a good approximation of the pore width of maximum density, for fugacities beyond the Henry’s law regime. We choose a pore width of 20 Å for the neighboring poresat the same fugacity, the density of molecules in this pore is several orders of magnitude less as Henry’s law regime fugacities. To estimate the effects of the pore-pore interaction, we compare the adsorption and transport properties of the 1-1-1 system with the properties of isolated 1-1 pores of widths 9.35 and 20 Å. We performed GCMC simulations for these systems, to determine the adsorption isotherms for carbon tetrachloride depicted in Figure 9. For the 1-1-1 system, we also show the individual isotherms for the 9.35 and 20 Å pores comprising the 1-1-1 system. As expected, at very low densities in both pores, the isotherms for the juxtaposed pores match those of the isolated pores, in the absence of significant interaction through the pore walls. In the Henry’s law limit, the density in the 20 Å pore is much less than the density in the 9.35 Å pore. Thus, once the adsorbate in the 9.35 Å pore reaches a mean pore density greater than 0.01 nm-3, the potential energy generated by these molecules is sufficient to change the mean pore density in the neighboring 20 Å pore. In Figure 10a, we see the influence of
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Figure 9. Adsorption isotherms for carbon tetrachloride at 298 K for the 20 Å pore of the 1-1-1 system; the 9.35 Å pore of the 1-1-1 system; the whole 1-1-1 system; the isolated 1-1 pore of width 20 Å; and the isolated 1-1 pore with 9.35 Å.
Figure 10. Density profile for carbon tetrachloride at 298 K in the 1-1-1 system: (a) at low density, transport is essentially through the narrow pore; (b) as the density increases in the narrow pore, intermolecular interactions through the central wall continue to affect the density profile in the wider pore; and (c) as the density increases further, the interactions through the wall are dominated by local fluid-fluid and solid-fluid interactions. The solid vertical bars denote the position of the pore walls.
the molecules in the 9.35 Å pore on the density profile of molecules in the 20 Å pore: molecules tend to occupy the side of the 20 Å pore nearest the 9.35 Å pore, as a result of the potential field generated. As the pore density increases, the effect on the profile remains (Figure 10b), and the discrepancy between the juxtaposed 20 Å pore and the isolated 20 Å pore increases (Figure 9). However, as the density continues to increase, the juxtaposed 20 Å pore reaches sufficiently high densities that the fluidfluid interactions within the pore and the solid-fluid interaction dominate the influence of the neighboring pore. At this stage (Figure 10c), the asymmetry observed at low densities has disappeared, and there is no longer any discrepancy between the isolated and juxtaposed pores. We note that, at all fugacities, there is no discernible
Jepps et al.
Figure 11. Transport coefficients for carbon tetrachloride at 298 K in the 1-1 pore of width 9.35 Å, in the 1-1 pore with 20 Å, in the 1-1-1 system, and in the wider pore of the 1-1-1 system. Lines are to guide the eye only.
difference between the isotherm of the juxtaposed and that of the isolated 9.35 Å pore: the density of molecules in the neighboring 20 Å pore is insufficient to significantly perturb the density-dependent behavior. At this point we consider the implications of the above results for the transport diffusion of the 1-1-1 system. At low densities, it is apparent that essentially all transport takes place in the 9.35 Å pore (indeed, this is the same phenomenon that we observed in the previous section, where the transport coefficient weights were biased toward pores in this range of widths) so that the low-density transport coefficient will be equal to that of the 9.35 Å pore. Conversely, at high density, the pore-pore correlation is negligible, so we expect the overall transport coefficient simply to be the weighted average of the isolated pore transport coefficients at high density. We performed EMD simulations to measure the transport diffusion coefficient for carbon tetrachloride in the 1-1-1 system and the isolated 1-1 pores. The results of these simulations appear in Figure 11. We observe that the low-density transport coefficient for the 1-1-1 system agrees with that of the isolated 9.35 Å pore, as anticipated. Furthermore, we note that the diffusion coefficient for the isolated 9.35 Å pore is essentially constant over the range of densities examined, because the shape of the density profile does not vary with density. This phenomenon has been observed previously in slit pores14 and in single-file transport in cylindrical pores.13,19 In Figure 11, we also show an estimate of the transport coefficient, based on the transport coefficient that one would expect if there were no correlation between the two pores (as in eq 6):
D0(µ) )
∑iFˆ (Hi, µ) HiD0(Hi, µ) ∑iFˆ (Hi, µ)Hi
(15)
We find excellent agreement between the simulation values and the estimate from eq 15, up to a density of about 2 nm-3. Interestingly, this range of densities includes the region where significant pore-pore correlation induces the greatest relative change in the mean fluid density in the 20 Å pore. At higher densities, the agreement between the simulation values and the estimate is still good, although not as close as at lower densities. It is possible that this reflects the greater difficulties in obtaining (19) Bhatia, S. K.; Jepps, O. G.; Nicholson, D. J. Chem. Phys. 2004, 120, 4472.
Juxtaposition of Carbonaceous Slit Pores
accurate values of the transport coefficient at high densities (as reflected in the larger error bars). We also note that, while we have used data obtained at the same bulk fugacities for the simulation values and the estimates, the resulting densities do not agree; this is expected, as a result of the pore-pore correlation. However, agreement in the density dependence of the transport coefficient between the two sets of data indicates that the transport coefficient in the juxtaposed 20 Å pore is well approximated by the transport coefficient of the isolated 20 Å pore at the same density (because the transport coefficient in the 9.35 Å pore remains essentially constant). We therefore conclude, from these results, that while pore-pore correlations may affect the adsorption behavior of a system, the transport behavior is much less significantly affected and can be well approximated by the transport in the isolated pore at the same density. Conclusion Theories of microporous carbons are commonly based on a slit-pore model, where localized domains of parallel slit-shaped pores are bounded by walls which are treated as semi-infinite, for the purposes of determining molecular interactions. Such an approximation effectively assumes that pore walls contain at least four graphitic layers; however, evidence suggests that a significant proportion of pore walls consist of three or fewer layers. This variation in pore-wall thickness has two consequences for the nature of molecular interactions in such systems: the solid-fluid potential depends on the number of layers in the pore walls, and the existence of narrow pore walls admits the possibility that fluid molecules in neighboring pores can interact with one another. Such effects have already been observed to influence the adsorption properties of microporous carbon models, and it is, therefore, of interest to examine their effect on transport. At low density, we observe that the transport coefficient is indeed dependent on the pore-wall thicknesses, beyond a critical pore width dependent on the transition from a single-minimum potential to a double-minimum potential. The pores bounded by single graphitic layers appear to give the fastest transport for a given width at low density. However, we find that once both walls consist of at least two graphitic layers, the deviation of the transport coefficient from the semi-infinite wall model is small. The discontinuity obtained in the variation of the transport coefficient with pore width in the asymmetric pores highlights a difficulty in implementing diffuse boundary conditions. The nature of the diffuse boundary condition is to randomly reorient the reflected molecule’s momentum due to a collision at the boundary. In particular, a single wall is considered responsible for a given reflection, and each reflection is treated equally. These last two conditions are difficult to reconcile with the physical behavior of molecules in systems with continuous solid-fluid interactions. These difficulties manifest themselves most obviously in the asymmetric pores, where the sudden emergence of a second potential minimum leads to discontinuous changes in our interpretation of the dynamics of oscillating molecules (as demonstrated in Figure 5). Such difficulties of interpretation also exist for the symmetric case, where each reflection is treated equally, independent of how far from the wall it takes place. To overcome these problems, it would be necessary to introduce a continuous “smoothing” of the reflection condition to recognize the “relative” contribution to the solid-fluid interaction from either wall. The development of an appropriate smoothing function would be the subject of further work.
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We find a small (3-5%) difference between the transport coefficient calculated from a simple model for activated carbons, whether the pore-wall thicknesses are considered all semi-infinite, or varying with a probability distribution in accordance with the current understanding of activated carbon structures. Despite such a choice for the pore-wall thickness distribution, a significant contribution to the transport coefficient comes from pores with two or more layers in both walls. Furthermore, over the range of pore widths considered by our activated carbon model, a significant contribution to the overall transport coefficient comes from the range of pore widths which exhibit virtually no dependence on the pore-wall thicknesses. The level of insensitivity to the pore-wall thickness distribution evident from our results suggests that, at low density, one can safely consider the pores to be semi-infinite, for the purposes of determining the transport behavior. We also consider the effect on the transport behavior of interactions through the pore wall. We consider the interaction through the pore wall to be of the same nature (LJ) as fluid-fluid interactions in the same pore. While this assumption ignores possible screening effects due to the presence of the carbon wall, we anticipate that our results should qualitatively indicate the extent of any effect such interactions can have on the overall transport. We have chosen a test system that enhances the effect of porepore interactionssthe central wall is of minimal thickness to promote the strength of pore-pore interactions, and the relative densities of molecules on either side of the pore wall are such that molecules on the denser side generate a potential field that is significant in comparison with fluid-fluid and solid-fluid interactions on the less dense side. We find, for this system, that the transport behavior is well approximated if we assume that the transport coefficient in a pore is given by the transport coefficient of the isolated pore at the same density. Because this system was chosen for the strength of the pore-pore correlation, we expect that such correlations will be weaker in other systems. Consequently, while pore-pore correlations may shift the fluid density in a pore at a given fugacity, this information alone appears sufficient to predict any corrections to the overall transport. We note that the range of pore widths with significant fluid densities is such that, for activated carbons, the overall contribution of pore-pore correlations is expected to be weak. In this paper, we have only considered systems with totally diffuse reflections. Currently, there is much interest in systems whose solid-fluid interactions can be characterized as almost specular, where the accommodation coefficient R is small. The extension of our results to such systems is not trivial; however, we make the following remarks. In the low-density limit, the effect of the accommodation coefficient on the transport coefficient is transparently given by the leading coefficient (2 - R)/R so that the relative transport behavior at low density will be unchanged in neighboring pores of the same R. At higher densities interpore interactions may serve to increase R and significantly reduce the transport coefficient because of the high sensitivity of the factor (2 - R)/R to changes in R when R is very small (i.e., reflection is nearly specular). Acknowledgment. Financial support of this research by the Australian Research Council through the Discovery Grants scheme is gratefully acknowledged. We are also grateful to the Australian Partnership for Advanced Computing, for access to the supercomputers at the National Facility.
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Appendix The transport diffusion coefficient was determined in ref 13 using a nonequilibrium formulation, considering the transport of a fluid under the influence of a constant external force F. Here we determine the transport diffusion coefficient of a low-density fluid at equilibrium in an asymmetric slit micropore. The transport diffusion coefficient for an equilibrium fluid of N identical particles can be calculated from the center-of-mass velocity autocorrelation function:20,21
N〈rz2(t)〉 ) N lim tf∞ tf∞ 2t
D0 ) lim
)
1 N
lim tf∞
∫0t〈vz(0) vz(ξ)〉 dξ
∫0 〈∑∑vzi(0) vzj(ξ)〉 dξ t
i
H)
1
t 〈∑vzi(0) vzi(ξ)〉 dξ + ∫ 0 N tf∞
lim
i
1 N
lim tf∞
∫0t〈∑∑vzi(0) vzj(ξ)〉 dξ
px2 + py2 + pz2 + V(x) 2m
(18)
(16)
j
where rz(t) represents the position of the fluid center of mass at time t, vz(t) its velocity, and vzi(t) the velocity of the ith particle. Consider, therefore, a fluid at equilibrium in a slit pore at temperature T, with pore walls normal to the x axis. If the center-to-center pore width (or size) is given by H, then without loss of generality we set x ) 0 at the midpoint between the innermost graphitic layers of both walls; these innermost layers, therefore, lie at x ) (H/2. At low densities, we can neglect intermolecular interactions, considering only the solid-fluid interaction. We can split the transport coefficient eq 16 into two parts:
D0 )
(diffuse plus specular) reflections is represented by the accommodation coefficient R so that R represents the probability of observing a diffuse reflection at a boundary reflection. Consequently, the probability of l consecutive specular reflections followed by a diffuse reflection is R(1 - R)l. We note that totally diffuse boundary conditions, where a molecule’s momentum is randomly reoriented at every boundary reflection, corresponds to the case R ) 1. In the absence of intermolecular interactions, the dynamics of a fluid molecule can be determined using the Hamiltonian
(17)
i j*i
In the low-density regime, correlations between the velocities of different molecules disappear, and this second term on the right side of eq 17 (sometimes denoted Dξ20) goes to 0. Therefore, the transport diffusion is given by the first term only, which is equivalent to the selfdiffusivity, DS. As outlined in section II-A, the solid-fluid interaction is given by a conservative interaction V(x), consisting of a sum of Steele 10-4 contributions, combined with a diffuse boundary condition. In narrow pores, V(x) has a single potential minimum (Figure 1a), whereas in wider pores V(x) has two local minima on either side of a “central” maximum which, in the asymmetric pore, need not be at the pore center (Figure 1b,c). As in the case of the symmetric pores, we describe two regions: a repulsive region beyond the two local minima and a bounded region between the local minima. In the narrow pores, the entire pore corresponds to the repulsive region. To model the effects of surface roughness, we permit yz momentum exchange through the inclusion of partially diffuse boundary conditions.15 Under these conditions, boundary reflections are either totally diffuse (a molecule’s momentum is randomly reoriented in the yz plane) or totally specular (a molecule’s momentum is preserved in the yz plane). The mean ratio of diffuse reflections to total (20) Nicholson, D.; Travis, K. P. Molecular Simulation of Transport in a Single Micropore. In Recent Advances in Gas Separation by Microporous Membranes; Kanellopoulos, N., Ed.; Elsevier: Amsterdam, 2000. (21) Maginn, E. J.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1993, 97, 4173.
The momenta py and pz are, thus, constants of the motion between the diffuse reflections at the wall, where the reorientation of the yz momentum on reflection introduces a discontinuity in py and pz. Ex(x, px) ) px2/2m + V(x) is also a constant of the motion, independent of the diffuse reflections at the wall. The motion in the x direction will, therefore, be periodic, and for a given trajectory we denote the time between diffuse reflections by τ [)τ(x, px)]. We will consider how to evaluate τ(x, px) shortly; at this point, we merely note that τ is constant along a given trajectory. The motion of a fluid molecule parallel to the yz plane is characterized by a random walk process: a molecule will maintain yz momentum p1 for l1 steps, as a result of specular reflection, before undergoing a diffuse reflection. It will then proceed with yz momentum p2 for l2 steps before its next diffuse reflection and so on. To determine D0, we must consider the possible trajectories that a molecule can follow and calculate the mean square displacement in the long-time limit. The diffusion coefficient for this two-dimensional random walk is given by
D0 ) DS ) lim
〈r2(t)〉pi,li,τ
(19)
4t
tf∞
where the average is taken over the pi, li, and τ. After n diffuse reflections, the displacement of a molecule with this sequence of momenta is given by
r(Lτ) )
l1p1τ
+
l2p2τ
m
m
+ ... +
lnpnτ m
n
where
li ) L ∑ i)1
(20)
Because the pi are uncorrelated,
〈r2(Lτ)〉pi,li )
〈(
〈∑( ) 〉 n
)
)〉
lnpnτ l1p1τ l2p2τ + + ... + m m m
i)1
lipiτ m
2
pi,li
)
2
2kTτ2 n 2 〈 li 〉l m i)1 i
∑
pi,li
(21)
We can rewrite this in terms of the specular intervals. If, along a given trajectory, ml represents the number of specular intervals of length l, the sum in angled brackets can be rearranged thus
〈r2(Lτ)〉pi,li )
2kTτ2 L 〈 mll2〉li m l)1
∑
(22)
Juxtaposition of Carbonaceous Slit Pores
In the limit t f ∞,
tf∞
m
Lf∞
D0 )
〈 〉 L
2kTτ
lim 〈r2(t)〉pi,li ) lim
Langmuir, Vol. 21, No. 1, 2005 239
∑ l)1
ml l2 t + O(tλ), λ < 1 L li (23)
H/2 τ(x, px) e-E (x, px)/kT dx dpx ∫-∞∞ ∫-H/2 H/2 -E e (x, px)/kT dx dpx ∫-∞∞ ∫-H/2 x
x
)
However, by the law of large numbers, we have22
ml L ml L
L
kT × (2 -R R)2m H/2 τ(x, Ex)xm/2[E - V(x)]-1/2 dx dEx ∫-∞∞ e-E /kT∫-H/2 H/2 xm/2[E - V(x)]-1/2 dx dEx ∫-∞∞e-E /kT∫-H/2
ml f pl ) R(1 - R)l-1 ∑ l)1
x
) ml /
x
(24)
∞
L
kT × (2 -R R)2m
lml f pl/∑lpl ) ∑ l)1 l)1
) ml /
(29) ∞
lR(1 - R)l-1 ∑ l)1
R(1 - R)l-1/
in the limit L f ∞, for almost every trajectory. Consequently, for molecules of oscillation period τ,
lim 〈r2(t)〉pi,li ) lim tf∞
tf∞
∑l lR(1 - R)l-1
m
tf∞
) lim
( )
l2R(1 - R)l-1 ∑ 2kTτ l
2kTτ 2 - R t + O(tλ) m R
(
)
t + O(tλ)
(25)
At this point, we identify two possible types of oscillation: those with both turning points in the repulsive region (Figure 1a,b) and those with one turning point in the repulsive region and one turning point in the bounded region (Figure 1c). In the former case, the period of oscillation is 2τ, because the molecule undergoes a diffuse reflection at both turning points, and for a given energy Ex there is only one value of τ ) τ(Ex). In the latter case, the period of oscillation is τ, because the molecule only undergoes one diffuse reflection per full oscillation. This case can only occur in the double-minimum potential, with energies Ex lower than the potential at the “central” maximum, denoted EC. It follows that for every Ex < EC there will be two disjoint intervals in which molecules oscillate and, consequently, two values of τ that we denote τL(Ex) and τR(Ex). Thus
Averaging over all oscillation periods,
2 - R 2kT〈τ〉τ t + O(tλ) (26) lim 〈r2(t)〉pi,li,τ ) lim tf∞ tf∞ R m
(
)
and, therefore,
D0 )
kT 〈τ〉 (2 -R R)2m
(27)
Thus, the accommodation coefficient contributes a multiplicative factor (2 - R)/R, with the result
D0 )
kT 〈τ〉 2m
(28)
in the special case of totally diffuse reflections (R ) 1). We note that this result is in agreement with results derived previously for the nonequilibrium system.13,14 As in refs 13 and 14, we assert that fluid-fluid interactions, in the limit of low density, serve to mix the energies of molecules in the system so that, on the average, their distribution corresponds with the canonical distribution and (22) Moran, P. A. P. An Introduction to Probability Theory; Clarendon Press: Oxford, 1968.
∫I [Ex - V(x)]-1/2 dx
τ(Ex) ) κxm/2
(30)
where κ ) 2 if Ex < EC and κ ) 1 if Ex g EC and where I represents the range of x over which the molecule oscillates (either [xL-, xL+], [xR-, xR+], or [x-, x+]). It follows that
D0 )
kT × (2 -R R)2m 2
(Ex) + τR2(Ex) dEx + 2 EC -Ex/kTτL(Ex) + τR(Ex) dEx + e -∞ 2
EC -Ex/kTτL e -∞
∫
∫
∫E∞ e-E /kTτ2(Ex) dEx x
C
∫E∞ e-E /kTτ(Ex) dEx x
C
(31) In the symmetric pore, we can define τ(Ex) ) τL(Ex) ) τR(Ex) for Ex < EC. In this case, τ(Ex) is a continuous function of Ex for all energies and eq 31 reduces to
2 - R kT D0 ) R 2m
(
LA047984G
)
∫-∞∞e-E /kTτ2(Ex) dEx ∫-∞∞e-E /kTτ(Ex) dEx x
x
(32)
