Ind. Eng. Chem. Res. 1993,32, 743-746
743
Hindered Diffusion in Slit Pores: An Analytical Result Yashodhara Pawar and John L. Anderson’ Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
An analytical result for the hindered diffusional mobility of a spherical particle in a slit pore is derived using an asymptotic matching technique and published expressions for the enhanced viscous drag on the particle caused by the flat walls. A hard sphere-hard wall potential is assumed for the interaction between the particle and pore wall. The mobility is presented as a regular expansion in the parameter X and is correct to O(X3), where X is the ratio of particle radius to half-width of the pore. Our result agrees quantitatively with published numerical evaluations of the mobility up t o X 0.5. The derivation is sufficiently general that i t allows for inclusion of nonsteric interactions between the particle and the pore wall.
-
Introduction The apparent diffusion coefficient of molecules in liquidfilled pores of comparable dimension is reduced from the value in free solution. Hindered diffusion through small pores is a factor in transport in microporous media, for example, dialysis of proteins and other macromolecules across blood capillaries and diffusion of reactants into microporous catalysts. A review of progress made in theoretical and experimental studies of hindered diffusion is given by Deen (1987). While there is basic understanding of the mechanism behind hindered diffusion provided by the pore walls, at present there is only one equation relating the diffusion coefficient (D(P))to the ratio of molecule-to-pore radius (A). The following expression was derived by Brenner and Gaydos (1977) for a hard-sphere molecule (“particle”) diffusingwithin a long circular cylindrical pore and applies when the particle/pore wall interactions are purely steric:
D(P) 9 1+ gX In X - 1.539X + O(X)] -= (1(1) D, D , is the diffusion coefficient of the particle in bulk solution, and D(P)is the diffusion coefficient within the pore. If z is the coordinate along the pore’s axis, then D(P) is defined as ( Az2)/27 in the terminology of random walks and Brownian motion, where the particle moves a distance Az in time 7 and the brackets denote an ensemble average. The particle samples all radial positions according to its equilibrium partition function. D(P) is the diffusion coefficient when the flux of particles (diffusion rate per area of pore) is expressed in terms of the gradient of the intrapore concentration of particles, that is, the crosssectional average of concentration a t each axial position z. Because solute concentration is determined by the volume available to the centers of the spheres, but defined by the total pore volume, the intrapore concentration is less than the bulk concentration outside the pores by a “partition factor”. To obtain the effective diffusion coefficient based on the equivalent concentration of particles in the bulk solution, one must multiply the right side of eq 1 by the partition factor (1An important limitation of eq 1is that it is only valid to O(X), which probably means that this equation is unreliable above X = 0.1. The reason why Brenner and Gaydos were forced to truncate their result at O(X) is that the wall effect on the mobility of the particle is known only to this order of X for a sphere at an arbitrary radial position inside a circular cylinder. This limitation does
not exist for a slit pore, which is formed by two infinite flat surfaces spaced a distance 2d. In fact, the mobility of a sphere of radius a moving parallel to two flat surfaces can be determined to O(X3),where X = ald, for arbitrary positions of the particle using the hydrodynamic expressions found in the text by Happel and Brenner (1991). In this paper we derive the equation, correct to O(X3), for the diffusion coefficient of spherical particles in a “slit” pore. The derivation follows the approach of Brenner and Gaydos, in which the hydrodynamic effects of the walls are expressed as inner and outer expansions. Our result applies to situations where the particle interacts with the pore wall only by a hard sphere-hard wall potential. Comparison with numerical evaluations of D(P)lD, by Weinbaum (1981) shows our analytical expression to be accurate up to X = 0.5. The matched asymptotictechnique can be extended to account for nonsteric (longer range) interactions, such as those electrostatic in origin, between the particles and the pore wall.
Derivation of 13(P) The geometry is shown in Figure 1. A spherical particle executes Brownian motion between the two infinite flat surfaces that form the “pore”. The half-width of the pore is d and the radius of the particle is a; X = ald. The dimensionlessparameter y indicates the fractional position of the center of the particle from the centerline of the pore; it is related to the distance from the walls by hl=(l-y)d; h z = ( l + y ) d (2) where hl and hz are the distances of the center of the particle from walls “1”and “2”. y = 0 denotes the centerline of the pore, and y = 1is wall 1. The hydrodynamic mobility of the particle is defined as its velocity in the z-direction per unit force. It is a function of position y. A t either wall Cy = f l )the mobility is zero because the walls are stationary and the no-slip condition of the fluid is assumed. We defiie F(y) to be the ratio of the mobility at position y divided by the mobility were the particle outside the pore in an unbounded fluid. The diffusion coefficient is given by the average value of F
s
l-X
D(p’ -- - o
F W exp[-EIkTJ dy
s
(3)
l-X
Dm
exp[-ElkTJ dy
0
* To whom correspondence should be addressed. 0000-5005/93/2632-0743$04.00/0
The upper limit of integration means that the center of 0 1993 American Chemical Society
744 Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993 Wdi
Table I. Values of the Function fln)Defined in Eq 8 fmm the Calculations of Goldman Bt al. (1967).
I
w
f(w)
111
fhll
10.0677 3.1622 2.3524 1.5431 1.1276
5.2843 X 1 V O.rn8999 0.0059004 0.031535 0.12352
1.0453 1.005004 1.003202 1.00
0.19360 0.29933 0.31449 0.5625
w s 1.1: 1 + expt1.80359(w - 111 + 15” -9 + -wa 8 In(-1) 16 0.319031(w - 1)o.’682 W>l.l:
fCw, =
Figure 1. Spherical particle of radiua (I in a slit pore of spacing 2d. A = old; w = hlo; y = 9ld.
molecule cannot approach the pore wall closer than ita radius; this is called the “hard spherehard wall” potential. E@) is the long-range potentialenergy between the particle and the walls. In sterically dominated situations, as assumed here, the hard sphere/hard wall interaction dominates and E .= 0 for all y. The effective diffusion coefficient (Den),which is based on concentration driving forces outside the pore and is often used in modeling membrane transport (Anderson and Quinn, 1974; Baltus and Anderson, 1983; Deen, 1987; Kathawalla and Anderson, 1988), is given by the following:
D , = (1- X)D‘p’ (4) (1- A) is the partition factor for the parallel slit geometry. Setting E = 0 in eq 3 gives the following for the stericdominated case: D‘P’ 1-A -= (1- X)-’S F@) dy (5) D, O The task is to develop an expression for F that is valid over the total range of integration. To do this, we divide the pore into an ‘outer” region comprising all values of h >>a (or equivalently, 1-y >> A), and an ‘inner region” for which h O(a)(i.e., 1- y OW), where h equals hl and is related t o y by eq 2. The mobility in the outer region can be expressed as an expansion in X: EI”’=l-[A(l-y)~’IX+[E(l-y)3 ]A 3 -
-
-
[C(1 -y)“lX5 + ... (6)
where A, E, and C are integrals involving y hut are independent of X (Happel and Brenner (1991). p 325). [Note that the integral for E given by Happel and Brenner contains the following typographical error: the (1- h)2 prefactor should be (1- hP.1 For the inner region we use thedimensionlessvariablew = (1-y)/h;theparticlemakes contact with the wall when w = 1. Only the hydrodynamic effect of one wall is important in this region. Solutions for asymptotic values of the dimensionless mobility for w and w 1are given by Faxen (1923) and Goldman et al. (1961), respectively:
-- -
p)= 1- (9/16)w-’+
(1/8)w3(1/16)wJ
(45/256)w4O(w4) as w
+
--
(7a)
In view of these asymptotic forms, it is convenient to write
F” as
1- (9/16)w-’ + (1/8)w3 -f(w) (8) where the function f(w) can be found from tabulated
p’
45 256
f ( w ) = -w4
+ L16w 4 - 0.22206w4 + 0.205216~~~
The above equations are ‘best fits” to the tabulated values. Table 11. Values of the Functions Y
w )and E&)*
G/U-v )
H l ( 1 - v)3
0 0.05 0.1 0.15
0.44162 0.292956 0.414016 0.341689 0.387217 0.261031 0.360933 0.248241 0.2 0.334946 0.236865 0.25 0.309101 0.226510 0.3 0.283299 0.216848 0.35 0.257488 0.207603 0.45 0.205891 0.189373 0.5 0.18025 0.18wOo 0.55 0.1549 0.169964 0.6 0.130042 0.102969 0.65 0.105949 0.147405 0.7 0.0829567 0.134259 0.75 0.061492 0.11936 0.8 0.042075 nm22s 0.85 0.0253533 0.0823704 0.9 o.oizn8 0.059 0.95 0.00268 0.032 1.0 0.0 0.0 Gl(1-y) =0.106195(1-y) +0.659475(1-y)2-0.329351(1-y)3 Hl(1 -yP = 0.622201(1 -y) - 0.798834(1 - Y ) + ~ 0.495588(1 -y)3 a
The above equations are “best tits” to the tabulated values.
solutions of the Stokes flow problem for a sphere moving parallel to a single flat surface (Goldman et al., 1967). From the boundary condition requiringzero mobility when the sphere touches the surface, we know that f - 9/16 as w 1. From eq la we see that f w4 as w Values o f f are tabulated in Table I. Because the inner and outer expressions for F must match, we know that A 9/16 and E 1/8in the limit y 1,making eqs 6 and 8 equivalent. It is convenient in the matching of the two regions to reexpress the mobility in the outer region as
-
-
-
-
Po’= 1- [(9/16) + G@)l(l -y)-’X
+
+
--.
...
- (9) [(1/8) H@)l(l where the functions GCy) and H@)are obtained from the coefficientsA and E which appear in eq 6;these functions are tabulated in Table 11. The integrations required for A and E were evaluated numerically by Mathematica. To evaluate the integral in eq 5, we must form a composite expression for F that is valid over the entire range of integration. The usual method of matched asymptotic expansions is used to obtain
F )= P ( W ) + P C y ) - (EI”)))-”
(10) wherethesubscripty- w meansthey variableis replaced hy 1- Aw and the asymptotic limit w 0 is taken. Terms
-
Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993 745
+
= 1- 1.004X + 0.418h3+ 0.210X4 - 0.169X5 O(X6) (15) centerline (eq 15)
O u r result (eq 14)
0
0
0.2
0.6
0.4
0.8
1
h
Figure 2. Comparison of o w result (eq 14) with the numerical calculations of Weinbaum (1981) (symbols) and the centerline approximation (eq 15).
of equal order in X are collected to form the following composite solution that is valid over the total range of integration:
F'"' = [l- (9/16)~-'+ ( l / 8 ) ~--f(w)] ~ - G(y)(l -y)-lX + H(y)(l- y)-3X3 (11) which is correct to O(X3). The mean translational diffusion coefficient of a particle within the pore is determined to O(h3)by substituting eq 11 into eq 5. The integration is split into two parts to allow for integration over the w and y variables separately. We first integrate the function of w within the square brackets of eq 11: (1- X)-'X~,l'x[l (1- X)-'[(l-
+ ( 1 / 8 ) ~-f(w)l - ~ dw = A) + (9/16)X In (1)+ (1/16)(X - X3) -
- (9/16)w-'
0.0655X + XJlyAf(w) dwl (12)
-
The coefficient 0.0655 equals the integral off over w = 1 03 (Brenner and Gaydos, 1977). Because f is O(w-9 when w >> 1,the integral of f(w) in eq 12is O(h4)and hence negligible here. The integrations of the terms containing G and H in eq 11 must be done numerically. For this purpose we have approximated G/(1- y ) and Hl(1- yI3 by cubic polynomials, as given in Table 11. These were then integrated over y to give the following results for the integrals of the second and third terms in eq 11: (1- X)-'[-O.l90583X
+ 0.221817X31
(13)
The final result for the diffusion coefficient is obtained by adding the right side of eq 12 to eq 13:
D(P)
-= (1- X)-'[l D,
+ (9/16)XIn (A) - 1.19358X + 0.159317X31 (14)
where the term in square brackets has an error of O(X4). Discussion Using numerical techniques for solving the Stokes hydrodynamic problem of a sphere translating parallel between two flat surfaces, Ganatos et al. (1980) evaluated F a t various positions in the range 0
