theoretical analysis of pore size distribution effects on membrane transport

Journal of Membrane Science, 82 (1993) 211-227 Elsevier Science Publishers B.V., Amsterdam

211

Theoretical analysis of pore size distribution effects on membrane transport

Seiichi Mochizuki and Andrew L. Zydney Department of Chemical Engineering, University of Delaware, Newark, DE 19716 (USA)

(Received November 3,1992; accepted in revised form March 16,1993)

Abstract

The membrane selectivity can be critically affected by the pore size distribution. We examined the effect of different pore size distributions on the asymptotic membrane sieving coefficient, the hindered solute diffusivity, and the membrane hydraulic permeability by averaging the solute and solvent transport rates over specific pore size distributions. Although the calculated sieving coefficients and hindered diffusivities both increased significantly with an increase in the breadth of the distribution, the relationship between these two membrane transport parameters was relatively unaffected by the pore size distribution, allowing for reasonably accurate predictions of the hindered diffusivity from sieving data (or vice versa). These detailed calculations were also compared with predictions of a recently developed analytical model which implicitly accounts for the pore size distribution by evaluating the effective solute to pore size ratio using an expression for the solute partition coefficient in a random porous media, Model predictions were in good agreement with the detailed integral results for a membrane with a log-normal distribution with geometric standard deviation of around two, which is consistent with the observed pore size distribution for many ultrafiltration membranes. Model calculations also examined the effects of protein adsorption on membrane transport, with very different behavior seen for different adsorption mechanisms.

Key words: microporous and porous membranes; theory; ultrafiltration; pore size distribution

Introduction

One of the critical factors governing the selectivity of commercial ultrafiltration membranes is the wide pore size distribution pres- ent in currently available asymmetric membranes. Direct measurements of the pore size distribution by field emission scanning electron microscopy [ 11, transmission electron

Correspondence to: Andrew L. Zydney, Department of Chemical Engineering, University of Delaware, Newark, DE 19716 (USA). (phone: 302-831-2399).

0376-7388/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

microscopy [ 11, and atomic force microscopy [ 21 indicate that the pore size can vary by well over a factor of ten. For example, scanning emission electron micrographs of a Millipore PTHK polysulfone membrane with 100,000 (100K) molecular weight cut-off showed pores ranging from about 1.5 to 45.9 nm [ 11, with very different pore size distributions seen for membranes from different manufacturers even for the same base polymer.

Most of the initial work on the transport characteristics of membranes with a pore size distribution focused on the breadth of the siev-

212 Seiichi Mochizuki and Andrew L. Zydney /J. Membrane Sci. 82 (1993) 211-227

ing curve, i.e. the steepness of a plot of sieving (or rejection) coefficient as a function of solute size. Cooper and van Derveer [3] showed that plots of the dextran rejection coefficient versus the dextran molecular weight for several different polysulfone membranes were essentially linear on log-probability paper, suggesting that the dependence of the rejection coefficient on solute radius could be well-described using a log-normal distribution. Michaels [ 41 ex- tended this analysis to a range of synthetic and biological membranes, with the pore size distribution of these membranes described, at least qualitatively, by a mean solute size (determined from the size of the solute with sieving coefficient of S, = 0.5) and the geometric standard deviation GSD* (determined experimentally from the ratio of the solute size at S, = 0.159 to that at S, = 0.5). The sharpness of the sieving curve was thus inversely related to the geometric standard deviation, with calculated values of GSD* ranging from about 1.2 for biological membranes to as much as 1.7 for synthetic membranes.

Horiuchi et al. [5 ] used an analysis similar to that employed by Michaels [4] to describe the sieving characteristics of large pore plasma filtration membranes (ethylene/vinyl chloride copolymer and cellulose acetate) and found GSD* values ranging from 1.67 to 2.26, substantially larger than those reported by Mi- chaels [4]. Youm and Kim [6] obtained data for the intrinsic rejection coefficients for dex- trans and polyethylene glycols through a 3K molecular weight cut-off polysulfone membrane which showed GSD*s ranging from 1.73 to 2.86 depending on the applied pressure and bulk solute concentration. This dependence on pressure and concentration was attributed to solute deformation and pore blockage effects, although there is no independent evidence for either of these phenomena.

Leypoldt [ 71 theoretically evaluated the sieving coefficients for membranes with speci-

fied distributions of several discrete cylindrical pore sizes. The sieving coefficient for each pore size was evaluated using available hydrodynamic models, with the sieving coefficient for the membrane determined by averaging over the particular pore size distribution. These calculations provide some general insights into the effects of the mean pore size, the variance, and the skewness of the distribution on solute sieving. In addition, Leypoldt concluded that it was not in general possible to determine the actual pore size distribution from experimental data for solute sieving as a function of the solute molecular weight due to the non-uniqueness of the calculated sieving coefficients on the assumed pore size distribution.

Wendt and Klein [8] evaluated the sieving coefficients for several monomodal and bimo- da1 pore size distributions. The calculated sieving coefficients for the different distributions tended to collapse to a single curve when plotted as a function of the dimensionless solute radius, with the non-dimensionalization performed using the radius of the solute with S, = 0.5 as suggested by Michaels [ 41. Signif- icant differences in the results appeared only for very large solutes (with S, < 0.05), and Wendt and Klein suggested that data in this regime could, at least in principle, be used to distinguish between different pore size distributions.

Kassotis et al. [ 91 developed an approach to evaluate the detailed membrane pore size distribution, expressed as a discrete distribution with the number of pore sizes equal to the number of sieving measurements, from data for dextran sieving as a function of dextran molecular weight. However, this approach requires the accurate evaluation of the size of the smallest dextran with 100 % rejection, a value which can be very difficult to determine experimentally due to the presence of a small number of very large pores in many commercial membranes.

Seiichi Mochizuki and Andrew L. Zydney /J. M~tnbtme Sci. 32 (W93) 211-2.27 213

Aimar et al. [lo] calculated the rejection characteristics for membranes with a log-normal pore size distribution. Sieving data for clean and BSA-fouled polysulfone IRIS membranes were well described using this log-normal distribution with the geometric standard deviations (GSD) ranging from 1.35 to about 1.6 [ 111. BSA adsorption caused a significant reduction in the maximum pore size as well as an apparent increase in the breadth of the distribution. These effects were even more pronounced for adsorption of the smaller proteins ovalbumin and a-la&albumin. Some care must be taken in interpreting these experimental results and fitted parameters since the intrinsic membrane sieving coefficients were evaluated by extrapolation of data for the observed rejection coefficient to zero applied pressure, which ignores the potential contribution of solute diffusion to the observed sieving behavior [ 121.

Robertson and Zydney [ 13 ] and Opong and Zydney [ 141 proposed an alternative approach to including the effects of a membrane pore size distribution on both solute diffusion and sieving. An effective solute to pore size ratio for the membrane was evaluated in terms of the solute partition coefficient as

&f=l-fi (1)

based on results for spherical solutes in cylindrical pores. The solute partition coefficient $ was then evaluated using the model developed by Giddings et al. [ 151 for the partitioning of a rigid solute of arbitrary geometry in an isotropic porous network formed by a random ar- rangement of parallel planes:

@=exp( -R*/s) (2)

where R* is the mean projected solute radius (R* = R, for a sphere) and s is the ratio of the pore volume to pore surface area. This model implicitly accounts for the presence of a pore size distribution through the evaluation of @ by eqn. (2). The sieving coefficients and hindered

diffusion coefficients were then evaluated using available hydrodynamic models for spherical solutes in cylindrical pores with the effective solute to pore size ratio given by eqn. ( 1) . The ratio of the pore volume to surface area can be evaluated by comparison of model predictions with experimental sieving or diffusion data. Alternatively, a priori predictions of the solute transport coefficients can be obtained by relating s to the membrane hydraulic permeability (L,) using the Kozeny-Carman equation for flow in a porous medium as

S= 26,& 1’2 ( > E

(3)

with 8, the membrane thickness and t the porosity. This model was shown to be in relatively good agreement with data for both bovine serum albumin [14] and dextran [12] transport through asymmetric polyethersulfone ultrafiltration membranes (Filtron Technology Corp.); however, the generality of this model to other membranes with different pore size distributions has yet to be demonstrated.

Although these studies provide some insights into the effects of a pore size distribution on membrane sieving, a general understanding of these pore size distribution effects on both solute and solvent transport, including the possible effects of adsorption on solute transport, is currently lacking. The objectives of this study were (1) to examine the effects of several different pore size distributions on the calculated values of not only the asymptotic sieving coefficient, but also the solute hindered diffusivity and the membrane hydraulic permeability, (2) to compare these results with predictions of the analytical model developed by Robertson and Zydney [ 131 and Opong and Zydney [ 141 for solute transport through membranes with a random pore size distribution, and (3) to examine theoretically the possible effects of protein adsorption on these membrane transport parameters using two relatively simple, but very

214 Seiichi Mochizuki and Andrew L. Zydney / J. Membrane Sci. 82 (1993) 211-227

different, physical models for protein adsorp- transport is governed by convection and eqn. tion in the porous membrane. (5) reduces to

Theoretical development N, = #Kc VC, (7)

The quantity @Kc is often referred to as the

The local solute flux through a membrane pore (N,) is given by the sum of the convective and diffusive contributions [ 12,141:

asymptotic sieving coefficient (S,), which is also equal to one minus the intrinsic membrane reflection coefficient.

N s= c KVC KD 3 s- d 03 dz (4)

In the diffusion-controlled regime, i.e. where V-to, the solute flux (N,) becomes

where C, is the radially-averaged solute concentration in the pore, V is the radially-averaged solution velocity in the pore, and D, is the free solution Brownian motion diffusivity. The coefficients Kc and Kd represent the hindrance factors for convective and diffusive transport, respectively, and are functions of the solute to pore size ratio and any long-range (e.g. electro- static) interactions between the solute and pore walls.

N ‘KdD”(C C) 8=- 8, w-f (9)

with the quantity @K,D, equal to the effective solute diffusivity for the membrane.

In membrane filtration, the actual sieving coefficient is evaluated by setting the filtrate concentration equal to the ratio of the solute to solvent flux through the membrane:

For isotropic pores (i.e. pores with properties which are independent of z), eqn. (4) can be integrated over the membrane (of thickness 6,) to give

Cf+?

Substitution of eqn. (9) into eqn. (5) gives, upon rearrangement:

where C, is the solute concentration at the upper surface of the membrane (z= 0), Cf is the filtrate concentration, and $ is the solute equilibrium partition coefficient:

C,(z=O) Cg(z=6,) #= C = C (9)

W f

Equation (5) should also be valid for an asymmetric membrane, with all of the membrane properties (Kc, Kd, @, and &, ) evaluated for the membrane skin layer.

In the limit of very high filtrate flux, solute

The actual sieving coefficient is thus equal to S, in the limit V-+m as expected. At very low velocity, the actual sieving coefficient ap- proaches one since the diffusive flux tends to equalize the solute concentrations on the two sides of the membrane.

Deen [17 ] has recently reviewed the pre- vious theoretical work on the hindrance factors Kc and K& For dilute solutions of spherical solutes in cylindrical pores with no long-range interactions, the hindrance factors are unique functions of the ratio of the solute (23,) to pore (r) radius:

Seiichi Mochizuki and Andrew L. Sydney / J. Membrane Sci. 82 (1993) 21 l-227 215

A=R,/r (11)

Bungay and Brenner [ 181 have developed analytical expressions for Kc and Kd which are valid for all il using matched asymptotic expansions for both small and close-fitting spheres yielding

Kc = @-@K 2Kt

K=!F d Kt

(12)

(13)

where the equilibrium partition coefficient is simply

@=(1-A.)” (14)

The hydrodynamic functions Ic, and Kt are both expressed as expansions in 1:

(15)

with the coefficients a, and b, given in Table 1. The calculated values of S, and qbKd are in good agreement with more rigorous analyses performed by explicitly averaging over the radial direction. Note that for this type of cylindrical

TABLE 1

Expansion coeffkients for hydrodynamic functions Kt and K,ineqn. (15)

Subscript n a, b,

1 - 73/60 7160 2 77,293/50,400 - 2,227/50,400 3 - 22.5083 4.0180 4 -5.6117 - 3.9788 5 - 0.3363 - 1.9215 6 - 1.216 4.392 7 1.647 5.006

pore, the radially-averaged solution velocity (V) is directly related to the applied pressure drop (AP) and the membrane pore radius using Poiseuille’s law as:

V r2AP

=&Iig W-3)

where p is the solvent viscosity. Equations (4)-( 16) are all valid for a mem-

brane with pores of a given radius r. The average solute and solvent flux through a membrane with a known pore size distribution can be determined by integrating the above expressions for N, (eqns. 5,7, or 8) and V (eqn. 16) over the pore size distribution:

co

s N,n(r)m’dr

Ns=ooo

I n(r)m’dr

0

(17)

co

I Vn(r)zr’dr

Q= “,

I n(r)m2dr

0

(18)

In the analysis presented in this paper, we employed two different probability functions for the pore size distribution: a log-normal distribution

n(r) =n,exp[ -(‘“l~~~-‘)3

and a truncated Gaussian distribution

n(r) =n,exp [ (r;oY2)2] -

(19)

(20)

with eqns. (19) and (20) both valid only for r> 0. R, is the pore radius and no the number of pores at the maximum in the distribution functions, while q is the geometric standard deviation for the log-normal distribution and


0 I 2 3 4 5

Dimensionless Pore Radius, r/Flm

Fig. 1. Typical pore size distributions for the log-normal (top panel) and truncated Gaussian (bottom panel ) probability distribution functions.

a, is the classical standard deviation for the Gaussian (normal) distribution. Typical pore size distributions for the log-normal (top panel) and truncated Gaussian (bottom panel) functions are shown in Fig. 1 for a range of standard deviations (0, and a2 ) .

Average transport properties

The average membrane permeability is evaluated experimentally from the ratio of the volumetric filtrate flux to the applied transmembrane pressure drop as:

qlv $=-& (21)

where c is the membrane porosity. &, can thus be evaluated from the pore size distribution using eqns. (16) and (18) as:

co

s n(r)?%

&=&: m

s n(r)r2dr

0

(22)

The average asymptotic sieving coefficient is defined experimentally as

NB = s, Vc, (23)

with the solute flux evaluated in the limit of very high velocity. Substitution of eqns. (7), (16), (17), and (18) into eqn. (23) yields:

co

I $Kcn(r)r4dr

s O oo= co (24)

I n(r)r4dr

0

where @ and Kc are functions of r (and R,) as given by eqns. (12)-(15).

The average effective diffusion coefficient ($Kd) is defined experimentally analogous to the expression for the solute flux through a single pore in the diffusion-controlled regime (eqn. 8):

IV @KdDm(C C) *=- s, w-f

(25)

Substitution of the definition of Ii!* (eqn. 17)) along with eqn. (8)) into eqn. (25) thus yields:

00

__i #Kdn(r)r2dr

#Kd=‘oo (26)

I n(r)r2dr

0

Seiichi Mochizuki and Andrew L. Zydney /J. Membrane Sci. 82 (1993) 21 l-227 217

where # and Kd are again functions of r and R,. The average value of the actual sieving coef-

ficient in the flux-dependent regime cannot be evaluated explicitly. Instead, the expression for N, (eqn. 5) is substituted into eqn. (9) which is then integrated over the pore size distribution yielding (upon some rearrangement) :

co

I l-$F

($K,) V*(r)7cr%r= 0

s V$n(r)nr%r (27) 0 w

where

(28)

with the velocity V given by eqn. (16 ). The average value of the actual sieving coefficient is then evaluated implicitly from eqn. (27) as:

g+ (29) w

which is analogous to the definition used in eqn. (10). The actual sieving coefficient in eqn. (27) was evaluated iteratively for a given solute radius and pore size distribution. All integrals were evaluated numerically using an Euler in- tegration scheme with a variable spacing in the pore radius (Ar) to insure effective conver- gence of the sums.

Results and analysis

The effects of the different pore size distributions on the asymptotic sieving coefficients &, are shown in the top and bottom panels of Fig. 2 for the log-normal and truncated Gaus- sian distributions, respectively. In each case, the data have been plotted as a function of the solute radius normalized by the pore size at the

0 1 2 3 4 5

Dimensionless Solute Radius, R./Sri

Fig. 2. Effect of log-normal (top panel) and truncated Gaussian (bottom panel) pore size distributions on the asymptotic membrane sieving coefficient.

maximum in the distribution function (RJR,). The & values increase with increasing o, and a, due to the large contribution from the solute flux through the largest pores in the distributions. For example, $, for &J&=0.8 increased from 0.05 for a uniform pore size distribution to 0.33 for al= 1.5 and to 0.98 for o, = 3.0. The results for the truncated Gaussian with rs,/R,= 0.6 are similar to those for the log- normal distribution with a, = 1.5, even though there are significant differences in the details of these two pore size distributions (Fig. 1). The variation in SW seen with the log-normal dis-

218 Seiichi Mochizuki and Andrew L. Zydney / J. Membrane Sci. 82 (1993) 21 l-227

tributions for ~~~2.0 were much larger than those for the truncated Gaussians due to the greater number of very large pores for the log- normal distribution with large ol as seen in Fig. 1.

The results shown in Fig. 2 have been replot- ted in Fig. 3 with the solute radius normalized by the radius at &_,=0.5 (R,*) for each distribution as suggested by Michaels [ 41. Although the sieving coefficients for the different distributions for small RJR,* are fairly similar, there are very large discrepancies for RJR: > 1.5, corresponding to s, ~0.1, with the largest sieving coefficients obtained for the log-normal distribution with o, = 3.0. This behavior is similar to that reported previously by Wendt and Klein [8] using an exponential-form for the pore size distribution. Note that the geometric standard deviations in the actual pore size distribution used to generate the curves in Fig. 3 are very different from the geometric standard deviations in the calculated sieving coefficients. These latter geometric standard deviations (GSD* ), evaluated as described by Michaels [4] from the ratio of the solute radius at s,=O.159 to that at L&,=0.5, vary from 1.7

0 1 2 3 4 5

Dimarsionless Solute Radius, FWG

Fig. 3. Asymptotic sieving coefficients for the log-normal (solid curves) and truncated Gaussian (dashed curves) pore size distributions plotted as a function of the solute radius normalized by the radius at & = 0.5.

for the uniform distribution to 1.85 for al = 1.5 and to 2.57 for al = 3.0. Note that the GSD* for the uniform pore size distribution is significantly greater than unity due to the increase in the steric exclusion effects with increasing solute size in the uniform cylindrical pores.

The corresponding calculations for the hindered diffusivity are shown in the top and bottom panels of Fig. 4. The @Kd values at a given value of RJR, are all uniformly smaller than the corresponding S, values, reflecting the much greater hydrodynamic hindrance to diffusion (Kd) than convection (Kc). The results for the Gaussian with a,/R,=O.6 are again

0 1 2 3 4 5 Dimensionless Solute Radius. R&II

Fig. 4. Effect of log-normal (top panel) and truncated Gaussian (bottom panel) pore size distributions on the hindered solute diffusion coefficient.

Seiichi Mochizuki and Andrew L. Zydney /J. Membrane Sci. 82 (1993) 21 l-227 219

similar to those for the log-normal with a, = 1.5. The @Kd values increase substantially with increasing a, or a,, with $& for R,/R,=O.B increasing from 0.0005 for a uniform pore size distribution to 0.03 for al = 1.5 and to 0.52 for a, = 3.0. This last value is three orders of magnitude larger than that for the uniform pore size distribution at the same RJR,.

The importance of the flux dependence of the actual sieving coefficients in the proper inter- pretation of experimental sieving data has been discussed in several recent investigations [12,14]. In both of these studies, the best fit values of the average membrane transport properties ( Sm and $Kd ) were determined from a plot of S, vs. Y similar to that shown in Fig. 5. This approach implicitly assumed that the calculated values of the actual sieving coefficient were described by eqn. (10) with V replaced by Qand with S, and #Kd given by their average values (as determined from the behavior in the limits of the convection and diffusion controlled regimes). The calculated values of

Filtration Velocity, P (dsd

Fig. 5. Comparison between the average value of the actual membrane sieving coefficient (open symbols) and the sieving coefficient evaluad from eqn. (10) using the average values of .‘?, and $Kd (solid curves) for several different size solutes in a membrane with R,=50 A and a, = 2.0.

S,, as given by eqns. (27) and (29), are shown as the open symbols in Fig. 5 for R,= 50, 100, 150, and 200 A for a membrane with a log-normal pore size distribution with R,=50 A and o, = 2.0. These calculated values are compared with those given by eqn. (10) (shown as the solid lines) with S,, +Kd, and V all replaced by their average values as determined from eqns. (24)) (26)) and (18)) respectively. The two calculations are in excellent agreement over the entire range of v, indicating that S, and & for membranes with even broad pore size distributions can be effectively evaluated by fitting experimental data for S, vs P using eqn. (10) as was done by Opong and Zydney [ 141 and Mochizuki and Zydney [ 121. This conclu- sion has also been reached by Knierim et al. [ 191 and more recently by Mason et al. [ 201, with this latter result derived from an analysis of the lower and upper bounds on the coupling constant between diffusive and convective solute transport in membranes with arbitrary pore size distributions.

The calculated values for the hindered diffusivity have been re-plotted as a function of S, for the different pore size distributions in Fig. 6. The results for the different distributions are brought into significantly better agreement when plotted in this manner. This is particularly true for the truncated Gaussian distributions with the qbKd values at a given S, for these distributions being only slightly smaller than those for a membrane with uniform pores over the entire range of S, even when the standard deviation in the distribution was equal to R,. The largest deviation in $Kd occurs at about S,=O.3, with the results for a, = 3.0 about a factor of five smaller than those for the uniform distribution. This decrease in $Kd with increasing q or a2 reflects the greater effect of the largest pores on Sm than on qbKd, which arises from the factor of r4 in eqn. (24) compared to r2 in eqn. (26).

In order to compare these results with those


IO-44 0 0.2 0.4 0.6 0.8 1.0

Asymptotic Sieving Coefficient, S,

Fig. 6. Calculated values of the hindered diffusivity as a function of the asymptotic sieving coefficient for the log- normal (solid curves) and truncated Gaussian (dashed curves) pore size distributions.

for the simple analytical model developed by Robertson and Zydney [ 131 and Opong and Zydney [ 141, the &, and $Kd values in Figs. 2 and 4 have been re-plotted as a function of R,/2s in the top and bottom panels of Fig. 7. The s values for the different pore size distributions were determined from eqn. (3) with the average permeability evaluated from eqn. (22 ) . The &, and $Kd results for the different distributions are brought into closer agreement when plotted in this manner, with the & results for R,/2s < 0.5 (corresponding to s, > 0.4) lying on almost a single curve. For larger values of R,/2s, the calculated values of 9, and $Kd both increase with an increase in the breadth of the pore size distribution, reflecting the different effects of the pore size distribution on sieving, diffusion, and the ratio of the pore volume to surface area (which is related to the permeability through eqn. 3). The predicted values of sm determined using the effective solute to pore size ratio given by eqns. (1) and (2) were in very good agreement with those calculated from the log-normal pore size distribution with a, = 2.0 over the entire range of S, values. Di-

0 I 2 3

Effective solute to Por0 size Ratio, f = R&a

Fig. 7. Variation of the asymptotic sieving coefficient (top panel) and hindered diffusivity (bottom panel) with solute radius (R,) normalized by the specific pore area (s). Solid curves are for the log-normal pore size distributions; dashed curves are for the truncated Gaussians. The bold curve represents the calculated values determined using the partitioning model (eqns. 1 and 2) to evaluate the effective solute to pore size ratio.

rect observations of the pore size distribution in many asymmetric ultrafiltration membranes do in fact show o, values of around 2.0; this is discussed in more detail subsequently. The qbKd values given by this model were somewhat greater than those given by the detailed integrations of the different pore size distributions at small R,, but the results at larger R, were similar in magnitude to those for the log- normal distributions with al between 2 and 3.

Seiichi Mochizuki and Andrew L. Zydney / J. Membrane Sci. 82 (1993) 21 l-227 221

The pore size distribution analysis can also provide important insights into the effects of protein adsorption on membrane transport. Model calculations were performed to examine the effects of protein adsorption on the transport of non-adsorbing solutes, analogous to the use of dextran filtration to evaluate the membrane transport characteristics after BSA adsorption [ 11,161. Calculations were performed using a log-normal pore size distribution with R, = 50 A and al = 1.5 for the clean membrane. Two very different idealized models for protein adsorption were examined: ( 1) blockage of small membrane pores by a large protein, which was simulated by the elimination of all transport through pores with radii r < Rbloek, and (2) constriction of large pores by a monolayer of adsorbed protein, which was simulated by let- ting r= r- Rads for all pores with r> Rads (those pores with r < Rads were simply left unaltered). Protein adsorption in an actual porous membrane would be expected to alter the pore geometry in a much more complicated manner, with some pore blockage possible even for very large pores and some pore constriction possible even for many small pores. However, these two idealized “limiting” cases allow us to obtain important insights into the very different effects of pore constriction and pore blockage on membrane transport.

The effects of pore blockage on the calculated values of both S, and $Kd are shown in Fig. 8 for Rblock=50, 100, 150, and 200 A. Both S, and q5Kd actually increase with increasing R block This somewhat unexpected behavior arises from the fact that the blockage of the small membrane pores causes the “average” pore size of the membrane to increase. Another Way of looking at this phenomenon is that pore blockage causes a much larger reduction in solvent flux than in the solute flux, with &, evaluated from the ratio of the solute to solvent flux at high flow rates (i.e. where convection domi- nates). Likewise, the hindered diffusion coef-

J

._ 0 20 40 60 60 100

soluteRadius,R. (A)

Fig. 8. Effect of pore blockage on the asymptotic sieving coefficient (top panel) and hindered diffusivity (bottom panel) for different values of Rblaelr for a membrane with a log-normal pore size distribution with R,= 50 A and a, = 1.5. Dashed curve represents results for a clean membrane ( Rblock = 0).

ficient is related to the ratio of the total diffusive flux to the available membrane area, with the pore blockage causing a greater reduction in membrane area than in the diffusive flux.

The effects of pore blockage on solute transport and on the membrane porosity, specific pore area (evaluated from eqn. 4)) and permeability are shown more clearly in the top and bottom panels of Fig. 9. S, and @d both increase with increasing Rblock, with the magni-

222 Se&hi Mochizuki and Andrew L. Zydney / J. Membrane Sci. 82 (1993) 211-227

6.0 -

Spcidfic pare area

“0

Fig. 9. Effect of pore blockage on the asymptotic sieving coefficient, hindered diffusivity, and effective diffusivity (top panel) and on the membrane porosity, hydraulic permeability, and ratio of pore volume to surface area (bottom panel) as a function of Rbloct for a membrane with a log-normal pore size distribution with R,=50 A and qz1.5.

tude of this effect being proportionally greater for q3Kd due to its greater dependence on the solute to pore size ratio. In contrast to the results for $Kd, the total rate of diffusive transport across the membrane, which is related to the effective membrane diffusion coefficient ( c@$ ) , decreases with increasing Rblock due to the large reduction in membrane porosity (or membrane pore area) associated with pore blockage. The ratio of the porosity of the pre-

adsorbed membrane to that for the clean membrane:

is shown explicitly as a function of Rblock in the bottom panel of Fig. 9. The porosity and permeability both decrease sharply for R block > 40 A. This effect is more pronounced for the porosity since the pore area is more strongly affected by the elimination of the small pores than is the permeability (the permeability is more heavily weighted by the largest pores in the distribution since the volumetric flow rate SCdeS as r4 While the pore area SC&S as F2).

The specific pore area s, which is proportional to the ratio of the permeability to the porosity, increases with increasing Rblock, an effect which is consistent with the observed increase in both L?, and $Kd seen in Fig. 8. The model developed by Robertson and Zydney [ 131 and Opong and Zydney [ 141 thus correctly predicts the increase in S, and qbKd with increasing Rblock, with the magnitude of the predicted increase in both of these parameters in relatively good qualitative agreement with the rigorous calculations presented in Figs. 8 and 9.

The effects of pore constriction on both SW and $Kd are shown in Fig. 10 for Rads = 25,50, 100, and 150 A. Pore constriction causes a reduction in both S, and #Ke This effect is greatest at an intermediate pore size (in this case Rads ~50 A) due to the significant constriction of a very large number of pores under these conditions. The reduction in S, and @?d is negligible for very small Rads due to the very small degree of pore constriction under these conditions, and it is also negligible for very large Rads due to the small number of pores which are accessible to the proteins (i.e. with F> R,,).

Seiichi Mochizuki and Andrew L. Zydney / J. Membrane Sci. 82 (1993) 211-227 223

0.6

10-4 0 20 40 60 60 100

Solute Radius. R, (A)

0 20 40 60 60 100

RadiusdAdsofbd Protein. Rd. CA)

Fig. 10. Effect of pore constriction on the asymptotic siev- Fig. 11. Effect of pore constriction on the asymptotic sieving coefficient (top panel) and hindered diffusivity (bot- ing coefficient, hindered diffusivity, and effective diffusiv- tom panel) for different extents of pore constriction for a ity (top panel) and on the membrane porosity, hydraulic membr+e with a log-normal pore size distribution with permeability, and ratio of pore volume to surface area (bot- R,= 50 A and a, = 1.5. Dashed curve represents results for tom panel) as a function of Rad. for a membrme with a log- a clean membrane ( Rti = 0 ) . normal pore size distribution with R,=50 A and aI = 1.5.

The results with Rads> 150 A are nearly indis- tinguishable from those for the clean membrane (dashed curve).

The effect of pore constriction on the membrane transport coefficients, the porosity, permeability, and specific pore area are shown in Fig. 11 as an explicit function of the radius of the adsorbed protein. &,, qXd, and qJKd all decrease with increasing Rads for R& < 60 A, but then pass through their minimum values before increasing back towards their values for the

clean membrane at very large Rab This initial reduction in the membrane transport parameters with increasing adsorbed solute size reflects the greater pore constriction caused by the larger solutes. However, as Rads continues to increase, a smaller percentage of the pores are actually constricted by the adsorbed protein since we have assumed that all pores with

r<Rb are inaccessible to the protein. The reduction in $Kd is again greater than that in s, due to its greater dependence on the solute to


pore size ratio, with the largest effect seen with E#& due to the additional reduction in membrane pore area ( E ) .

The effect of pore constriction on the permeability, porosity, and specific pore area (bottom panel of Fig. 11) is similar to that seen for the solute transport parameters, although the minimum values in e, s, and LP are all obtained at a somewhat smaller value of Rads due to the different dependence of these parameters on the pore size. The reduction in permeability is greater than that for the porosity due to the weighting of the permeability by r4 (eqn. 22) compared to only r2 for the porosity. The smallest reduction is seen in the ratio of the pore volume to pore surface area (s ) , which has a scaling that is more nearly linear in the pore radius.

Discussion

The calculations presented in this manu- script confirm many of the conclusions of Ley- poldt [ 71, Wendt and Klein [ 81, and Aimar et al. [lo] on the effects of a membrane pore size distribution on the asymptotic membrane sieving coefficient. The asymptotic sieving coefficient is a strong function of the pore size distribution, although when the results are plotted as a function of RJR,* (with R,* equal to the radius of the solute with S, = 0.5)) the sieving coefficient for relatively small solutes (with sa>O.l) becomes largely independent of the specific pore size distribution [El]. The sieving coefficients for larger solutes (with ,!?m < 0.1) still display a very strong dependence on the breadth of the pore size distribution, which can be of critical importance in the application of size-selective membranes for protein fraction- ation. Note that even under these conditions, the theoretical calculations indicate that the flux-dependence of the actual sieving coefficient is very accurately defined by the values of the average hindered diffusion coefficient and

the average asymptotic sieving coefficient, as has been demonstrated elsewhere [ 19,201.

The current study also extends these theoretical calculations of the asymptotic sieving coefficient to the hindered solute diffusion coefficient, with the hindered diffusivity vary- ing by as much as three orders of magnitude as the breadth of the log-normal pore size distribution increases from aI = 1.0 (corresponding to uniform pores) to cr, = 3.0. Although both S, and @Cd vary dramatically with a, (or cr2), these simulations do indicate that the relationship between the hindered diffusion coefficient and the asymptotic sieving coefficient is relatively insensitive to the pore size distribution (at least for distributions that are not too broad). This means that it is possible to obtain a relatively good initial estimate of the hindered diffusion coefficient from experimental data for the asymptotic sieving coefficient (or vice versa) irrespective of the detailed characteristics of the membrane pore size distribution.

The pore size distribution calculations also demonstrate that the simple analytical model for solute transport developed by Robertson and Zydney [ 131 and Opong and Zydney [ 141 using the partitioning expression of Giddings et al. [ 151 is able to accurately predict both the asymptotic sieving coefficient and the hindered diffusion coefficient for a membrane whose pore size distribution can be characterized by a log- normal distribution function with geometric standard deviation of around 2.0. Although commercial ultrafiltration membranes can clearly have very different pore size distributions, the field emission scanning electron micrographs and transmission electron micrographs obtained by Kim et al. [l] do indicate that the pore size distributions of many of these membranes are similar to those for a log-normal distribution with o, w 2.0. This simple analytical model for solute transport may thus have considerable utility in the characteriza-

Seiichi Mochizuki and Andrew L. Zydney /J. Membrane Sci. 82 (1998) 211-227 225

tion of both solute sieving and diffusion through available asymmetric ultrafiltration membranes, with relatively accurate a priori esti- mates of these solute transport parameters ob- tainable from data simply for the membrane permeability.

Mochizuki and Zydney [ 121 recently evaluated the asymptotic dextran sieving coefficient for polyethersulfone ultrafiltration membranes with a range of molecular weight cut- offs. The sieving coefficients for the 50,000 and 100,000 molecular weight cut-off membranes were found to be in excellent agreement with the predicted values given by the partitioning model (eqns. 1 and 2) with the ratio of the pore volume to surface area evaluated from the hydraulic permeability using eqn. (3). This sug- gests that these 50K and 1OOK membranes have pore size distributions which can be described, at least approximately, by the log-normal function with q x 2.0. In contrast, the sieving data for the 30K membrane were significantly smaller than that predicted by the model, with the opposite behavior seen for the 300K membrane. Although the origin of these discrepancies was uncertain, the calculations presented in Fig. 7 suggest that the breadth of the pore size distribution for the 30K membrane is less than that for a log-normal distribution with a, = 2.0 while that for the 300K membrane is greater than that for the log-normal function. This increase in the breadth of the pore size distribution with increasing membrane molecular weight cut-off is in good agreement with direct measurements of the pore size distribution for different molecular weight cut-off ultrafiltration membranes [ 11. For example, Kim et al. [l] found that the pore size distributions of a PM30 polysulfone 30,000 molecular weight cut-off membrane was approximately that of a log-normal distribution with a, = 1.5, while the pore size distribution of the larger 1OOK membrane was much closer to that of a log-normal with al = 2.0.

Model simulations performed using two very different physical descriptions of protein adsorption, one based on a pore blockage mechanism and one based on a pore constriction mechanism, clearly demonstrate that protein adsorption can have very different effects on membrane transport depending upon the specific mechanism by which adsorption alters the pore structure. In particular, blockage of the small membrane pores actually leads to an increase in the asymptotic sieving coefficients while pore constriction causes a significant reduction in &,. These very different trends can, at least qualitatively, be predicted using the simple analytical model for solute transport developed by Robertson and Zydney [ 131 and Opong and Zydney [ 141, with pore blockage causing an increase in s while pore constriction reduces s. Although protein adsorption in actual porous membranes will be much more complicated than that described by either of these very simple idealized models, these simulations do provide important insights into the adsorption phenomena and its effect on membrane transport. For example, recent data obtained for bovine serum albumin (BSA, MW = 69,000) adsorption on polyethersulfone ultrafiltration membranes with molecular weight cut-offs from 30K to 300K [ 121 show alterations in the transport coefficients that are in good qualitative agreement with the type of pore constriction model discussed in this study, with no evidence of pore blockage even for the small pore 30K membrane.

Acknowledgements

This work was supported in part by grants from the National Science Foundation and the National Institutes of Health.

List of symbols

a,$, expansion coefficients in hydrodynamic model (values in Table 1)


CLl

C,

c,

bulk solute concentration (g/l) filtrate solute concentration (g/l) radially-averaged solute concentration in pore (g/l) solute concentration at membrane surface (g/l) free solution diffusion coefficient (m2/sec )

asymptotic sieving coefficient average asymptotic sieving coefficient radially-averaged solution velocity in pore (J,/c) (m/set) filtration velocity (m/set) pore volume (A” ) axial position in pore (A)

F

GSD*

Greek letters

n(r) 4

thickness of membrane (A) membrane porosity solute equilibrium partition coefficient ratio of solute radius to pore radius (R,/r) effective solute to pore size ratio (defined in eqn. 1) solvent viscosity (g/m/set ) geometric standard deviation of log-normal distribution standard deviation of Gaussian distribution

N*

geometric standard deviation determined from sieving data volumetric filtrate flux (m/set) hindrance factor for convection hindrance factor for diffusion hydrodynamic functions in Bun- gay and Brenner’s analysis membrane hydraulic permeability (m) average membrane hydraulic permeability (m ) number of pores of radius r number of pores at the maximum in the distribution function local solute flux across membrane

(g/m2/=) average solute flux (g/m”/sec ) transmembrane pressure drop (kpa) cylindrical pore radius (A) adsorbed layer thickness (A) radius below which all pores are blocked (A) pore radius at the maximum in the distribution function (A) solute radius (A ) solute radius at S,=O.5 (A)

mean projected solute radius (A) ratio of pore volume to pore surface area (VP/S,) (A) average actual sieving coefficient (WC,)

References

1.

2.

3.

4.

5.

pore surface area (A” )

K.J. Kim,A.G.Fane, C.J.D.Fell,T. SusukiandM.R. Dickson, Quantitative microscopic study of surface characteristics of ultrafiltration membranes, J. Mem- brane Sci., 54 (1990) 89. P. Diets, P.K. Hansma, 0. Inacker, H.-D. Lehmann and K.-H. Herrmann, Surface pore structure of micro- and ultrafiltration membranes imaged with the atomic force microscope, J. Membrane Sci., 65 (1992) 101. A.R. Cooper and D.S. van Derveer, Characterization of ultrafiltration membranes by polymer transport measurements, Sep. Sci. Technol., 14 (1979) 551. A.S. Michaels, Analysis and prediction of sieving curves for ultrafiltration membranes: A universal correlation?, Sep. Sci. Technol., 15 (1980) 1305. T. Horiuchi, P.S. Malchesky, M. Emura, M. Usami and Y. Nose, Solute/pore changes in plasma filtration, in: Y. Nose et al. (Ed.), Progress in Artificial Organs, ISA0 Press, Cleveland, OH, 1986, pp. 877- 882.

Seiichi Mochizuki and Andrew L. Zydney / J. Membrane Sci. 82 (1993) 211-227 227

6. K.H. Youm and W.S. Kim, Prediction of intrinsic pore properties of ultrafiltration membrane by solute rejection curves: Effects of operating conditions on pore properties, J. Chem. Engr. Jpn., 24 (1991) 1.

7. J.K. Leypoldt, Determining pore size distribution of ultrafiltration membranes by solute sieving: Mathe- matical limitations, J. Membrane Sci., 31 (1987) 289.

8. R.P. Wendt and E. Klein, Membrane heteroporosity and the probability function correlation, J. Mem- brane Sci., 17 (1984) 161.

9. J. Kassotis, J. Schmidt, L.T. Hodgins and H.P. Gre- gor, Modeling of the pore size distribution of ultrafiltration membranes, J. Membrane Sci., 22 (1985) 61.

10. P.M. Aimar, M. Meireles and V. Sanchez, A contribution to the translation of retention curves into pore size distributions for sieving membranes, J. Mem- brane Sci., 54 (1990) 321.

11. M. Meireles, P. Aimar and V. Sanchez, Effects of protein fouling on the apparent pore size distribution of sieving membranes, J. Membrane Sci., 56 (1991) 13.

12. S. Mochizuki and A.L. Zydney, Dextran transport through asymmetric ultrafiltration membranes: Com- parison with hydrodynamic models, J. Membrane Sci., 68 (1992) 21.

13. B.C. Robertson and A.L. Zydney, Hindered protein. diffusion in asymmetric ultrafiltration membranes

with highly constricted pores, J. Membrane Sci., 49 (1990) 287.

14. W.S. Gpong and A.L. Zydney, Diffusive and convective protein transport through asymmetric membranes, AIChE J., 37 (1991) 1497.

15. J.C. Giddings, E. Kucera, C.P. Russell and M.N. Myers, Statistical theory for the equilibrium distribution of rigid molecules in inert porous networks. Exclusion chromatography, J. Phys. Chem., 72 (1968) 4397.

16. S. Mochizuki and A.L. Zydney, Effect of protein adsorption on the transport characteristics of asymmetric ultrafiltration membranes, Biotechnol. Prog., 8 (1992) 553.

17. W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J., 33 (1987) 1409.

18. P.M. Bungay and H. Brenner, The motion of a closely fitting sphere in a fluid filled tube, Int. J. Multiph. Flow, 1 (1973) 25.

19. K.D. Knierim, M. Waldman, and E.A. Mason, Bounds on solute flux and pore-size distributions for non-sieving membranes, J. Membrane Sci., 42 (1989) 87.

20. E.A. Mason, L.F. de1 Castillo, and R.F. Rodriguez, Coupling-constant description of coupled flow and diffusion, J. Membrane Sci., 74 (1992) 253.

theoretical analysis of pore size distribution effects on membrane transport

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