4050
Langmuir 1996, 12, 4050-4052
A Transition in the Diffusivity of Adsorbed Fluids through Micropores
D0 )
Department of Chemistry, Imperial College of Science, Technology and Medicine, London SW7 2AY, U.K.
c
(∂µ∂x) ) (∂p∂x)
(3)
and the viscous part of the flux is then
Nicholas Quirke‡ Department of Chemistry, University of Wales at Bangor, Bangor, Gwynedd LL57 2UW, U.K. Received February 27, 1996. In Final Form: May 2, 1996
When a fluid is transported through a porous material, two modes of transport are possible: diffusive flow and viscous (or cooperative) flow. An abiding question is the relative importance of these two modes under different conditions. Some insight can be gained using the general formulation of transport in membranes, developed by Mason and co-workers,1 based on the statistical mechanical theories of Kirkwood et al.2 For isothermal steady state flow, in the absence of external fields, the flux in the x-direction (molecules per unit area per unit time) is given by the expression
cD0 ∂µ cB0 ∂p kT ∂x η ∂x
( )
(2)
for a molecule of hard sphere diameter σ. The GibbsDuhem equation can be expressed as3
David Nicholson* and Roger Cracknell†
J)-
kT 3πησ
( )
(1)
The first term in eq 1, the diffusive flux JD, contains the diffusion coefficient, D0, and the molecular concentration, c(x), inside the pore space; it is driven by the chemical potential gradient ∂µ/∂x. The second term, the viscous flux, JV, is driven by the pressure gradient and contains the viscosity coefficient η and a geometrical term B0. In a slit of internal width H′, sufficiently large that the Navier Stokes equation is valid, B0 would be given by (H′)2/12, and p here stands for the mean value of the transverse (xx or yy) component of the pressure tensor. When the fluid is gaseous, eq 1 transforms into the well-known slip flow equation for gases, and the viscous component becomes increasingly important as pressure is increased. In gases at low pressure, when the mean free path of the molecules exceeds the pore width, the diffusion term is dominant and reduces to the Knudsen limit at zero pressure. Adsorption complicates this picture because gas molecules concentrate in regions of low potential energy. Consequently the fluid is nonuniform because of the external adsorption field, as well as the concentration gradient, and the flux differs from that expected in a gaseous fluid that is not subject to adsorption forces. At low temperature, the total flux exceeds the gaseous flux and the extra flux is often referred to as surface flow. In adsorbed fluids at higher temperatures, and especially when the bulk fluid is supercritical, there is still nonuniformity due to the adsorption field, but the surface flow picture is no longer physically realistic. The relative importance of the contributions from diffusive flux and viscous flux is expected to depend on pore size and pressure as well as on temperature. A simple analysis can be given if we assume that the Stokes-Einstein equation is valid so that the diffusion coefficient is related to viscosity by * To whom correspondence should be addressed. E-mail: dn@i c.ac.uk. † Present address: Shell Thornton Research Centre, Chester CH1 3SH, U.K. E-mail:
[email protected] ‡ E-mail:
[email protected]. (1) Mason, E. A.; Viehland, L. A. J. Chem. Phys. 1978, 68, 35623573. (2) Kirkwood, J. G. J. Chem. Phys. 1946, 14, 180-201.
S0743-7463(96)00179-5 CCC: $12.00
JV )
c2B0 ∂µ η ∂x
( )
(4)
Combining eqs 2 and 4 gives the ratio of viscous to diffusive flux from eq 1 as
JV ) 3πσB0c JD
(5)
The last expression shows that the viscous term rapidly becomes the most important mode of transport when pore width is increased, since B0 increases as the square of the pore width. Conversely diffusive flow is the most important transport mode in very small pores. In a pore of given size the viscous mode is expected to supersede the diffusive mode as the density of the adsorbate increases. For example a recent study,4 using NEMD in slit pores having a width of about 26σ, revealed almost perfect Poiseuille profiles at normal liquid densities. Conversely it is widely held that diffusive transport is the important mechanism in zeolitic pores.5 Equation 5 cannot be regarded as better than a crude guide to the nature of transport flux in micropores, since it is to be expected that the Navier-Stokes expression for B0 will break down as pores get smaller6 and also since the Stokes-Einstein equation is derived from macroscopic hydrodynamic considerations. This paper presents the results of computer simulation studies of a confined fluid at bulk supercritical temperature in a narrow slit pore. It is shown that, although the general trend in flux as a function of concentration is in keeping with that expected from the foregoing arguments, the changeover from a predominantly diffusive to a predominantly viscous mode of transport can occur over a very small range of concentration and appears as a rather steep transition. To our knowledge, such transitions have not been previously reported. Simulation Studies The methane adsorbate was modeled as spherical LennardJones particles with the parameters /k ) 148.2 K and σ ) 0.3812 nm. The temperature of 296.2 K (kT/ ) 2.0) is above the critical temperature for the bulk fluid. The walls of the model slit pore were represented as a graphite continuum, which interacts with the adsorbate through a 10-4-3 potential with the standard parameters;6 cross parameters were calculated from the LorentzBerthelot rules. The adsorbent field in the pore was calculated as the sum of the interaction from opposing graphitic surfaces, which were separated by a distance of H ) 0.953 nm (2.5σ), where H is the “physical” width of the pore and is greater than the internal width H′, accessible to the diffusing molecules. Self-diffusion coefficients, Ds, were estimated from the limiting slope of the mean square displacement in directions parallel to (3) Cracknell, R. F.; Nicholson, D. Adsorption 1995, 1, 7-16. (4) Todd, B. D.; Evans, D. J.; Daivis, P. J. Phys. Rev. E 1995, 52, 1627-1638. (5) Ka¨rger, J.; Ruthven, D. M. Diffusion in zeolites and other microporous solids; Wiley: New York, 1992. (6) Bitsanis, I.; Somers, S. A.; Davis, H. T.; Tirrell, M. J. Chem. Phys. 1990, 93, 3427-3431. (7) Steele, W. A. The interaction of gases with solid surfaces; Pergamon: Oxford, 1974.
© 1996 American Chemical Society
Notes
Langmuir, Vol. 12, No. 16, 1996 4051
Figure 2. Simulated isotherm for methane in a slit-shaped graphitic pore: H ) 0.953 nm; T ) 296.2 K. In the inset curve, fugacity is plotted on a logarithmic scale.
Figure 1. Diffusion coefficients from simulations at 296.2 K plotted against adsorbate density (number of molecules per “physical” volume of pore space) for spherical methane molecules in a slit-shaped graphitic pore with H ) 0.953 nm: open circles, self-diffusion coefficients from EMD simulations; closed circles, Darken transport diffusion coefficients calculated from eq 10; triangles, mean effective diffusion coefficients from NEMD. Some points for Dtrans at low density have been omitted for clarity. See also refs 8 and 16. the pore walls and calculated over a range of adsorbate densities, using equilibrium molecular dynamics simulations.8 This diffusion coefficient does not equate to D0 in eq 1, which can, in principle, be found from the correlation of the streaming velocity u:
D0 )
∫ 〈u(t) u(0)〉 dt
N 2
∞
0
(6)
D0 includes cross-correlations between the molecular velocities, as can be seen from the relationship between u and v N
∑v u)
i
i
(7)
N
In practice, as pointed out by Maginn et al.,9 the streaming velocity correlation has a very long time tail, and indeed in the present model system it appears to oscillate indefinitely. Further investigation of this property will be reported in later work. A transport diffusion coefficient, Dtrans, which does not include cross-correlations, can be calculated from the self-diffusion coefficient. Dtrans is defined by a Fick equation;4 from eq 1, we can write
cDs
(∂x∂c) ) - kT (∂µ∂x)
JD ) -Dtrans
(8)
whence
∂ ln f Dtrans ) Ds ∂ ln c
(
)
(9)
where f is the fugacity of the gaseous adsorptive phase in equilibrium with the adsorbate (approximately equal to the gas phase pressure pg under the conditions used). We refer to this (8) Cracknell, R. F.; Nicholson, D.; Gubbins, K. E. J. Chem. Soc., Faraday Trans. 1995, 91, 1377-1383. (9) Maginn, E. J.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1993, 97, 4173-4181. (10) Cracknell, R. F.; Nicholson, D.; Quirke, N. Phys. Rev. Lett. 1995, 74, 2463-2466.
diffusion coefficient as the Darken diffusion coefficient.5 It is to be anticipated that cross-correlations become less important as density decreases. In order to calculate Dtrans from eq 9, the slopes of the isotherm are needed at different adsorbate concentrations. These were found by fitting the simulated isotherm.8 Our NEMD method has been described in detail elsewhere.10 Related methods have been reported earlier11,12 and also very recently in application to the direct simulation of tracer diffusion13 and to hard sphere diffusion in a model porous medium of silica microspheres.14 In our method, the simulation box is divided into three sections in the direction of flow. The two end sections are maintained at different constant chemical potentials by GCMC (grand canonical Monte Carlo). MD is carried out in a small fraction of the simulation steps. The density gradients ranged between 7 × 10-4 and 3 × 10-4 in reduced units and were chosen so as to ensure that the flux could be obtained with acceptable accuracy. In earlier work10 two of the points shown in Figure 1 (one on either side of the transition) were studied in greater detail. Several runs were made at the same number density, each with a different gradient, and it was shown that the same transport coefficient resulted for each gradient. A bootstrap mechanism was used to find the streaming velocities in the end regions from the flux, and these were added to the Maxwell velocities of newly created particles. A diffuse reflection condition was imposed at the pore walls8,10 such that the x- and y-components of the molecular velocity were randomized after the z-component of a molecular trajectory had been reversed in the repulsive part of the molecule wall potential. The temperature was kept constant using a Gaussian isokinetic technique.15 A typical simulation was run for 1.5 × 106 time steps together with a total of approximately 5 × 107 stochastic trials. Flux was calculated by counting creation and destruction in the end control volumes, after rejection of the first 3 × 105 time steps. From this we can estimate an overall or “effective” diffusion coefficient via the Fick’s equation:
J ) -Deff
(∂x∂c)
(10)
Deff cannot be obtained directly from simulations, since eq 10 cannot be integrated without prior knowledge of the concentration dependence of Deff, and the NEMD method uses a finite concentration gradient, ∆c/l, where l is the length of the pore. (11) Cielinski, M. M. M.S. Thesis, University of Maine, Orono, 1985. Cielinski, M. M.; Quirke, N. Unpublished. Quirke, N. Mol. Simul. 1996, 16, 193-199. (12) Sun, M.; Ebner,C. Phys. Rev. A 1992, 46, 4813-4818. (13) Heffelfinger, G. S.; van Swol, F. J. Chem. Phys. 1994, 100, 75487552. (14) MacElroy, D. M. J. Chem. Phys. 1994, 101, 5274-5280. (15) Evans, D. J.; Morriss, G. P. Statistical Mechanics of NonEquilibrium Liquids; Academic Press: San Diego, CA, 1990.
4052 Langmuir, Vol. 12, No. 16, 1996
Notes
Figure 3. Fluctuation properties (see text) plotted against reciprocal density. The squares are proportional to the reciprocal of the reduced compressibility (number fluctuation); the circles and triangles are fluctuations in neighbor separations with one neighbor either in the wall (|z| > σ/2) or in the central region of the pore, respectively. The central plane is through z ) 0. Instead, we report the closely related property D h eff, defined by
J)D h eff
(∆cl)
(11)
D h eff is a mean diffusion coefficient and approaches Deff when ∆c is small compared to the mean concentration.
Results and Discussion Figure 1 shows the transport properties plotted against mean (physical) density F*. The self-diffusion coefficient decreases with concentration, whilst the Darken transport diffusion coefficient passes through a maximum. These phenomena have been reported before and discussed elsewhere.8,16 The effective diffusion coefficient is greater than the Darken transport coefficient over the whole range of densities studied, as expected from eq 1. However the unexpected feature is the very rapid increase in D h eff in the density region between 0.21 and 0.25. This can be interpreted as a sudden onset of cooperative flow once a critical density has been reached. Above this transition the effective diffusion coefficient gradually declines as the fluid densifies further. This decrease would be expected in a bulk dense fluid, since the viscosity coefficient increases with density. Figure 2 shows the adsorption isotherm calculated by standard GCMC simulation. There is no indication of a transition in the region corresponding to the transition in D h eff, although it is clear that the latter occurs where the isotherm slope begins to decrease rapidly. Figure 3 shows the variation in fluctuation properties as a function of reciprocal density. The fluctuations are defined by [〈g2〉/ 〈g〉2 - 1] and include the number fluctuation (g ) N), which is proportional to compressibility, and the fluctuation in pair separation (g ) rij), with one member of the pair in the central and wall parts of the pore, respectively. There (16) Cracknell, R. F.; Nicholson, D. Proceedings of the Fifth International Conference on Fundamentals of Adsorption, Monterey, 1995; LeVan, M. D., Ed.; in press.
Figure 4. Singlet molecular distributions. The bottom panel shows the molecular distributions in the central region. Mean reduced densities are indicated on the graphs.
is a small but distinct discontinuity in compressibility through the transition region and a more distinct break in the separation fluctuations. The reason for this is made clear in Figure 4, where we show singlet density profiles through the transition region. These profiles demonstrate that the adsorbate initially accumulates near the wall, where there is a shallow minimum in the potential.8,16 The transition occurs after the density increases in the central part of the pore-momentum transfer across the xz plane and then becomes much more efficient. In the vicinity of the transition there is not much alteration of density in this central region. As the mean density increases, the fluid in the wall region becomes more disordered, as indicated by the broadening maxima; beyond the transition intermolecular encounters are increasingly repulsive, and consequently viscosity increases, causing a decline in the flux. We are not aware of previous instances of the transition phenomenon we report. At the present time we are unable to predict whether such phenomena are widespread or restricted to a small region of the relevant phase diagram and to specific geometries. More extensive studies of related systems are in progress. LA960179K
