Ind. Eng. Chem. Res. 199433,2473-2482
2473
Electrostatic and Electrokinetic Interactions during Protein Transport through Narrow Pore Membranes Narahari S. Pujar and Andrew L. Zydney‘ Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Experimental data are obtained for bovine serum albumin transport through an asymmetric poly(ether sulfone) ultrafiltration membrane over a wide range of salt concentrations. The actual membrane sieving coefficients are determined from the measured filtrate concentrations using a stagnant film model that accounts for bulk mass transfer effects in the stirred device. The relative contributions of the diffusive, convective, and “electrophoretic” transport t o the overall protein flux were then evaluated using a theoretical model for transport in charged cylindrical pores. The electrophoretic transport arises from the streaming potential that is generated by the convective solvent flow through the membrane, with this electrophoretic effect being comparable t o convection at very low salt concentrations. The intrinsic membrane transport parameters are then compared with available hydrodynamic predictions. The dominant effect of the electrical interactions on protein transport is through the alteration of the partition coefficient, although the hydrodynamic drag factors also appear to be a function of salt concentration.
Introduction Membrane filtration is used commercially for the purification of many biological molecules, but the widespread use of these membrane devices has generally been limited to applications in which the species of interest are widely different in size. The relatively poor separation obtained in most previous studies of membrane filtration for the fractionation of complex protein mixtures has been attributed to the inadequate transport characteristics of available membranes (Sieberth et al., 1983;Oda and Inoue, 1983)as well as the presence of solute-solute interactions and bulk mass transfer limitations. Most fundamental studies of protein transport through these semipermeable membranes have focused solely on size selectivity, but there is also considerable theoretical and experimental evidence that long range electrostatic interactions can play a critical role in the filtration of these protein solutions. The presence of a net charge on the protein and/or the pore walls can have a significant effect on both solute (protein) and solvent transport even in the absence of any externally applied electric fields. For example, the fixed charge on the pore wall causes an excess of counterion (i.e., the Na+ ion for a negatively charged pore) in the electric double layer adjacent to the pore wall. More importantly, the electrostatic interactions between the protein and the pore wall (mediated by their respective electrical double layers) can significantly alter the chemical potential of the protein in the pore, which is manifested as a change in the equilibrium partition coefficient between the bulk solution and the membrane (Smith and Deen, 1980, 1983). The situation in the presence of a net fluid flow through the pore (i.e., in the presence of an applied pressure gradient) is significantly more complicated due to the development of a variety of electrokinetic phenomena under these conditions. In particular, the fluid flow generates an electrical (streaming) potential across the membrane due to the unequal partitioning of the positive and negative ions in the pore. This streaming potential generates a flux of counterions opposed to the convective
* Author to whom correspondence should be addressed Dr. Andrew L. Zydney, Department of Chemical Engineering, University of Delaware, Newark, DE 19716. Phone: 302-8312399. FAX. 302-831-1048. E-mail:
[email protected]. OSS8-5885/94/2633-2473$04.50/0
flow, with this counterion flux reducing the overall solvent flow from that which would be expected based simply on the applied pressure gradient (this effect is often referred to as counter electroosmosis). This streaming potential can also give rise to an “electrophoretic” transport of the charged protein (even in the absence of any externally applied electric field) that is in addition to the normal diffusive and convective protein transport, although this effect has generally been neglected in previous analyses of membrane transport. These electrical phenomena can also alter the hydrodynamic interactions between the solute and pore walls through the development of Maxwell (electrical) stresses in the fluid and the modification of the viscous stresses. These effects are discussed in more detail subsequently. The objectives of this study were (1)to obtain quantitative experimental data for the effects of these electrostatic and electrokinetic phenomena on the transport of bovine serum albumin (BSA) through asymmetric ultrafiltration membranes, (2) to evaluate the diffusive, convective (sieving), and electrophoretic contributions to solute transport using a model that properly accounts for the effects of the induced streaming potential on protein transport through these membranes, and (3) to compare these experimental results for the intrinsic membrane properties with appropriate theoretical descriptions.
Previous Work Most fundamental studies of electrostatic interactions in membrane transport have focused only on solute diffusion, thereby eliminating most of the electrokinetic phenomena discussed above. These experiments have generally been performed using track-etched polycarbonate membranes with well-defined cylindrical pores or tracketched mica membranes with rhomboidal pores. For example, Malone and Anderson (1978) measured the hindered diffusion coefficient of latex particles through several track-etched mica membranes for ratios of the solute to pore radii (A = rs/rp) less than 0.3. The latex diffusivity through the membrane a t low electrolyte concentrations was significantly smaller than that at high salt concentrations, which was attributed to the reduction in the solute partition coefficient due to the decrease in electrostatic shielding. The magnitude of this effect was 1994 American Chemical Society
2474 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
in good qualitative agreement with theoretical calculations of the partition coefficient determined using the potential between a charged sphere and an infinite flat plate along with an appropriate definition for the effective “radius” of the rhomboidal pores in these mica membranes. Deen and Smith (1982)made similar measurements with molecular-size fractions of polyelectrolytes (ficoll sulfate and dextran sulfate) using polycarbonate track-etched membranes with X = 0.08-0.29. The data were in good qualitative agreement with theoretical predictions for the partitioning of a porous sphere in a cylindrical pore with the pore size and membrane charge evaluated from independent experimental measurements. Johnson et al. (1989) found similar results for the hindered diffusion of ionic micelles in track-etched polycarbonate membranes with X < 0.1. Experimental data at very low salt concentrations were above the predictions given by Smith and Deen’s analysis, which was attributed to the neglected effects of the diffusion potential arising from the unequal diffusivities of the ionic micelles and the salt ions (Johnson et al., 1989). Lin and Deen (1991) examined the effects of electrostatic interactions on the diffusion of linear flexible polymers (potassium polystyrene sulfonate) through track-etched polycarbonate membranes with X < 0.1. The data were in qualitative agreement with theory (Lin and Deen, 19901, with the reduction in the diffusion coefficient at low salt concentrations determined primarily by the partitioning effect. Although there have been no fundamental studies of the effects of electrostatic interactions on convectivesolute transport (i.e., solute sieving or rejection), several investigators have shown that the observed rejection in a variety of filtration systems is dependent on the electrolyte concentration. For example, Fane et al. (1983) found that the rejection of BSA was minimum at the protein isoelectric point and was greater at low salt concentrations under conditions when the BSA was charged. These effects were attributed primarily to differences in protein adsorption and deposition on the membrane, although they are at least qualitatively consistent with the expected effects of protein charge and electrolyte concentration on the protein partition coefficient and in turn the sieving coefficient. Likewise, Bil’dyukevich et al. (1989) found large effects of ionic strength on the rejection of a number of different proteins and enzymes through asymmetric ultrafiltration membranes. The data appeared to be correlated with the protein solubility, with the very large rejection a t low ionic strengths attributed to electrostatic repulsion from the membrane pores. More recently, Kim et al. (1993)showed that both flux and solute rejection during the ultrafiltration of silver colloidal particles were strongly affected by the salt concentration. The increase in rejection at low salt concentrations was attributed to electrostatic exclusion of the charged colloids from the membrane pores, while the increase in flux under these conditions was attributed to the increased permeability of the particle cake on the membrane caused by the electrostatic repulsion between adjacent colloids. No attempt has been made to compare any of these results with available theoretical models. Although these studies clearly demonstrate that electrostatic interactions can strongly affect the rate of diffusive and convectivetransport through semipermeable membranes, the data have generally been limited to relatively large pore membranes (i.e., X < 0.3), thus providing no verification of the existing theories under more sterically hindered conditions. In addition, all of the quantitative diffusion studies have been performed with model solutes (e.g. dextrans, ficolls, or micelles) in
model (track-etched) membranes, with no results available for proteins in the type of asymmetric membranes used in almost all commercial applications. Finally, there are really no quantitative data of any kind available for the effects of electrostatics on solute sieving during pressuredriven membrane filtration despite the importance of these phenomena in applications of membrane filtration for protein separations.
Theoretical Development Most theoretical analyses of solute transport in liquid filled pores are developed by equating the gradient in the chemical potential of the solute (-VpJ to the hydrodynamic drag force acting on the solute in the pore (Anderson and Quinn, 1974; Deen, 1987): -Vps = f m K ( U -GV)
(1)
where f m is the friction coefficient of the solute in the bulk solution, and U and V are the solute and fluid velocity, respectively,both evaluated relative to a coordinate system fixed with respect to the pore walls. The lag coefficient (G) and the enhanced drag coefficient ( K )account for the hydrodynamic interactions between the solute and pore walls and are complex functions of the solute and pore size, the solute position in the pore, and the presence of any long range (e.g., electrostatic) interactions. The gradient of the chemical potential of the solute in the presence of a potential field can be written as (Evans, 1979) Vp, = RT V In C,
+ V\k
(2)
where \k is the potential energy of interaction and C, is the solute concentration in the pore at radial position r and axial position z. In writing eq 2 we have implicitly assumed that solute-solute interactions are negligible, i.e., that the solute concentration in the pore is infinitely dilute. Equation 2 is substituted into eq 1 which can then be solved explicitly for the solute flux (N,) yielding
N , = UC,= -K“Dm[VC,
+m C, V\k]+ GVC,
(3)
where Dm= RT/fmis the free solution diffusion coefficient at infinite dilution. The area averaged solute flux through the membrane for a spherical solute (radius r,) in a cylindrical pore (radius rp) can thus be evaluated as
where @ = r / r p . The upper limits for the integrals in eq 4 are fixed at 1- X due to the steric exclusion of the solute from the region within one solute radius of the pore wall. The solute concentration is a function of radial position in the pore due to the radial dependence of the potential energy of interaction between the solute and pore wall. This radial concentration profile can be described by a Boltzmann (equilibrium) distribution as
where \k(r=O) and C,(r=O) are the potential energy of interaction and solute concentration at r = 0. Equation 5 can also be developed by integration of eq 3 with
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2475
V = 0 and N, = 0 in the radial direction. Deen (1987) provides an excellent discussion of the validity of the radial equilibrium approximation for membrane systems. Equation 4 can be rewritten in terms of the radially averaged solute concentration in the membrane pore
yielding
where ( V) and ( d \ k / d z ) are the radially averaged velocity and potential energy gradient. The Coefficients K,, Kd, and Ke are thus given as
d\k, - u$T --E, d.2
D,
u$T V) D,
= CUP(
where E, is the streaming potential and u, is the bulk electrophoretic mobility of the protein. The streaming potential in this system is proportional to the average solution velocity with the proportionality constant ( a )a complex function of the pore radius, the Debye length, and the solution conductivity as discussed in the Appendix. Note that d\kz/dz as given by eq 12 is a constant independent of axial position, so that a t steady state (N,= constant) eq 7 becomes a linear first-order ordinary differential equation for ( C,) with constant coefficients. In addition, since \ k 2 is independent of r, K , must equal Kd from eqs 9 and 10. In order to relate the solute flux to the solute concentrations at the upstream (Cw)and downstream (Cf)surfaces of the membrane, eq 7 has to be integrated across the membrane. The boundary conditions at the upper ( z = 0) and lower (z = 6,) surfaces were developed by assuming that the solute concentrations across the interface are in equilibrium yielding (Deen, 1987):
The results are conveniently expressed in terms of the actual membrane sieving coefficient (Sa),which is defined as the ratio of the solute concentration in the filtrate (Cf) to that a t the upper surface of the membrane (C,) (Opong and Zydney, 1991):
S,(1- w ) exp[Pe,(l - w)]
The solute flux (eq 7) thus has contributions from convection, diffusion, and the axial derivative of the potential energy of interaction (which is associated with the induced electric field in the axial direction), with the magnitude of these contributions reduced from their bulk solution values by the coefficients K,, Kd, and Ke. A detailed expression for the potential energy of interaction Q for a protein in a cylindrical pore in the presence of an axial electric field is currently unavailable. In order to proceed further, we assume that the actual potential is given by the pairwise summation of the potential energies arising from \kl(r), the interaction between the protein and pore in the absence of any flow and thus in the absence of a streaming potential, and \kz(z), the interaction between the protein and the streaming potential in an unbounded system with the streaming potential assumed to be unaffected by the presence of the protein. This approximation thus neglects the possible effects of the streaming potential on the structure of the double layer surrounding the protein, as well as any effects of the protein on the streaming potential. These phenomena are discussed in more detail subsequently. Under these conditions we can write
W , z ) = Ql(r) + \k2W
(11)
where 91 is only a function of r and 9 2 is only a function of z. Smith and Deen (1980, 1983) have developed expressions for \kl(r), the electrostatic potential energy of interaction, under conditions of both constant surface charge and constant surface potential. The derivative of the potential energy of interaction associated with the streaming potential is assumed to be that of an equivalent electric field acting on an isolated protein in an unbounded system:
Cf sa=-= C, S,(I - w ) + exp[Pe,(l-
w)] - 1
(14)
where
Thus a complete description of solute transport (sieving) during membrane filtration requires the knowledge of three parameters: the asymptotic sieving coefficient (S,), which describes the convective contribution to the solute flux; the membrane Peclet number (Pe,), which describes the relative importance of solute convection to diffusion inside the membrane; and the electrophoretic ratio ( w ) , which describes the relative importance of electrophoretic transport to convectioninside the membrane. At very low Peclet numbers (corresponding to very small filtration velocities), eq 14 predicts that the actual sieving coefficient is unity for any values of S, and w , as long as the membrane is at least partially permeable to the solute of interest. Under these conditions solute transport is governed by diffusion, which tends to equalize the solute concentrations on the two sides of the membrane causing the sieving coefficient to approach a value of one. At high Peclet numbers, transport is dominated by convectionand "electrophoresis" with the relative importance of electrophoresis determined by the ionic strength of the solution, the surface charge densities on the solute and the pore, and the solute and pore size (all of which affect the magnitude of both the protein partition coefficient and the parameter 0). The
2476 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
dominant electrical effect is on the partition coefficient, with 4 decreasing dramatically when the charges on the solute and the pore are of like sign. The actual sieving coefficient when w is negative (Le., when the solute carries a charge of the same sign as the pore walls) is higher than that predicted in the absence of any electrophoretic transport, with the sieving coefficient at infinite Peclet numbers attaining a value of S,(1 - w). S, is thus equal to the asymptotic value of S a at high Pem only when o = 0. For positive values of w , i.e., for oppositely charged solutes and pores, there are two cases of interest: w < 1 and w > 1. When w < 1the actual sieving coefficient a t high Peclet number again approaches a value of S,(1 - w ) , but in this case the high Pe, limit is smaller than S,. In contrast, when w > 1the exponentials in eq 14 go to zero at high Pem causing S a to approach zero in this limit due to the strong electrophoretic transport that acts in the opposite direction of the convective flow.
Materials and Methods All ultrafiltration experiments were performed with solutions of bovine serum albumin prepared by dissolving BSA powder (Sigma A-7906, Cohn fraction V, Sigma Chemicals) in NaCl solutions ranging from 0.0015 to 0.15 M. The solution pH was adjusted to approximately neutral pH using small amounts of 0.1 M NaOH or HC1as required. The pH was measured to within 0.1pH units using a FisherScientific Accumet 915 pH meter (Fisher Scientific, Pittsburgh, PA). All protein solutions were prefiltered through 100 000 (100K) molecular weight cut-off membranes (Filtron Technology Corporation, Marlborough, MA) at a constant pressure of 13.5 kPa to remove any BSA aggregates and small oligomers prior to use. These prefiltered BSA solutions were stored at 4 "C and used within 2 days of preparation. Protein concentrations were determined spectrophotometrically by reaction of BSA with bromcresol green (Sigma Chemicals). The absorbance of the resulting bluegreen complex was measured at 628 nm using a PerkinElmer Lambda 4B spectrophotometer (Perkin-Elmer, Norwalk, CT) and compared to that of BSA standards of known concentration. Protein concentrations could be accurately measured to within 0.1 g/L. Protein filtration experiments were performed using a 100 000 molecular weight cut-off OMEGA poly(ether sulfone) ultrafiltration membrane provided by Filtron Technology Corporation (Marlborough, MA). These membranes are anisotropic consisting of: (1)an ultrathin poly(ether sulfone) skin (approximately 0.5 pm thick) which determines the sieving properties of the membrane, (2) a porous poly(ether sulfone) substructure (approximately 50 pm thick), and (3) a porous support matrix (approximately 230 pm thick) made of Tyvek (DuPont Co., Wilmington, DE). The membrane was flushed with filtered deionized distilled water to remove any wetting agents using at least 100 L of water per m2 of membrane area. Previous data for BSA adsorption to the poly(ether sulfone)membranes indicated that equilibrium adsorption was not attained until after 10-12 h due to diffusional limitations within the porous membrane (Robertson and Zydney, 1990). The membrane was thus soaked in a BSA solution of the same concentration as that to be used in the sieving experiments for approximately 24 h at 4 "C prior to use. This protein adsorption was done using a 0.15 M ionic strength BSA solution to obtain maximum protein adsorption. The membrane was then gently rinsed
in a 0.15 M NaCl solution to remove any labile protein, and the hydraulic permeability was evaluated as described below. Protein Filtration. All filtration experiments were conducted using a 25 mm diameter Amicon UF cell (Model 8010, Amicon Corporation, Danvers, MA) connected to a 1 L reservoir containing either the saline or protein solution. The transmembrane pressure was set by adjusting the height of the solution reservoir (for pressures less than 14 kPa) or by air pressurization. The pressure on the filtrate side was approximately atmospheric under all conditions. All experiments were conducted at room temperature (22 f 3 "C). The membrane was mounted in the stirred cell, and the entire apparatus was carefully filled with saline taking care to eliminate any entrapped air bubbles in the cell and associated tubing. The hydraulic permeability of the membrane was evaluated from the slope of data for the saline flow rate measured as a function of applied pressure for AP = 3-7 kPa. The filtrate port was then clamped, the stirred cell and reservoir were emptied and refilled with the protein solution, and the desired pressure was established by adjusting the reservoir height or the air pressure regulator. The stirrer was then set at the desired rotational speed (previouslycalibrated using a strobe light), after which the feed and filtrate ports were unclamped. Data collection was begun after the system attained stable operation, i.e., after filtration for a minimum of 2 min and after collection of a minimum of 500 pL of filtrate, with the latter required to wash-out the dead volume downstream of the membrane in the stirred cell (approximately 200 pL). Filtrate flow rates were evaluated using timed collection with the filtrate mass determined using a Sartorius digital balance with accuracy of f l mg (Model 1518 Sartorius, Westbury, NY). The BSA concentration in the filtrate was determined spectrophotometrically as described previously. The filtrate port was then clamped and a small sample (approximately 100pL) taken directly from the stirred cell to evaluate the bulk protein concentration. The stirred cell was then emptied and refilled with saline. The saline flux was re-evaluated, with the experiments continued only if the flux differed by less than 5% from the value determined at the start of the experimental run. The protein solution was then returned to the stirred cell and the entire procedure repeated at a new value of the applied pressure. In order to minimize protein deposition effects, data were obtained by monotonically increasing the flux (i.e., by increasing the applied pressure), with repeat measurements then performed at selected (lower) pressures. These experiments were then repeated using albumin solutions of different NaCl concentrations.
Results and Analysis Experimental data for the observed sieving coefficients (defined as the ratio of the filtrate to bulk protein concentrations) for filtration of BSA through a single OMEGA lOOK membrane at a stirring speed of 600 rpm are shown as a function of filtrate flux (J,) in Figure 1. Data are shown at five different ionic strengths, with the pH and bulk protein concentrations given in Table 1.The higher bulk protein concentration used for the run at the lowest ionic strength provided greater accuracy in measuring the small filtrate concentrations (Le., small sieving coefficients) under these conditions. The membrane was preadsorbed overnight with a 5 g/L BSA solution at pH 7.0 and an ionic strength of 0.15 M NaCl prior to these experiments, with the membrane stored in 0.15 M NaCl
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2477 100
IO.'
10-2
Filtrate Flux. J, (inis)
Figure 1. Flux dependenceof the observed BSA sieving coefficient (So = Cf/Cb) at NaCl concentrations from 0.0015 to 0.15 M. Experimental conditions summarized in Table 1.
Figure 2. Flux dependence of the actual BSA sieving coefficient (Sa = Cf/Cw)at NaCl concentrationsfrom 0.0015to 0.15 M.Experimental conditions and transport parameters summarized in Table 1.
Table 1. Summary of Experimental Conditione and Mass Transfer Parameters for BSA Filtration
0
0
e A
ionic strength (M) 0.15 0.05 0.015 0.005 0.0015
cb
pH 6.8 6.7 6.9 7.4 7.8
(g/L) 8.4 9.7 7.2 7.2 13.0
diffusivity (m2/s X 1011) (Dohertyand Benedek, 1974) 6.0 8.5 13 22 28
at pH 7 and 4 "C in between the runs at the different ionic strengths. The data were obtained beginning at the lowest salt concentration and proceeding up to the 0.15 M NaC1. The saline flux at 5 kPa, evaluated using 0.15 M NaCl at pH 7 in between the different ionic strength experiments, all agreed to within 10?6 indicating that protein adsorption, desorption, or any other type of irreversible membrane fouling was negligible under these experimental conditions. The observed sieving coefficients were clearly a very strong function of the NaCl concentration, increasing monotonically as the NaCl concentration increased. For example, the observed sieving coefficient at J, = 6.0 X 10" m/s increased by more than 2 orders of magnitude (from So = 0.0035 to So = 0.38) as the ionic strength was increased from 0.0015 to 0.15 M NaC1. This dramatic change in So is due to the electrostatic exclusion of the negatively charged BSA from the membrane pores, with the repulsion decreasing with increasing ionic strength due to the electrostatic shielding provided by the electrolyte. The observed sieving coefficients were also a function of the filtrate flux, appearing to decrease with increasing J, at low flux with the opposite trend seen at high flux. The solid curves in Figure 1 are model calculations for this flux dependence which are described subsequently. The observedsieving coefficients in the stirred cell reflect not only the rate of membrane transport but also the effects of bulk concentration polarization, the accumulation of retained protein in the concentration boundary layer above the membrane. The complexhydrodynamics in the stirred cell does not permit the development of an analytical solution for the protein concentration profiles in this device, thus astagnant film model (Blatt et al., 1970;Opong and Zydney, 1991)was used to evaluate the actual sieving coefficient ( S a = Cf/C,) from the data for the observed sieving coefficient (So = Cf/Cb):
sa=
S O
(1- So) exp(J,/k)
+ So
(18)
The average mass transfer coefficient in the stirred cell (k)was evaluated using the correlation developed by Smith et al. (1968):
mass transfer coefficient (m/s X 10s) 4.8 6.1 7.9 11.3 13.5
S d -w )
t&KaDJL(m/s)
0.16 0.02 0.15 f 0.02 0.037 f 0.002 (4.1 0.9)X 109 (1.1 0.5)X 109
(7.6f 0.9)X 10-7 (3.8f x 10-7 (4.3 0.4) X l e (9.7f 1.4) X 10-a (9.5f 0.5) X 10-a
*
(2)
*
*
0.4
= X~e0.667~C0.33
where Re = Orc2/vis the Reynolds number, Sc = v / D is the Schmidt number, rc is the radius of the stirred cell, O is the stirring speed, and Y is the kinematic viscosity. The parameter x was evaluated for the Amicon stirred cell as x = 0.27 using data obtained with BSA in a 0.15 M ionic strength solution (Opong and Zydney, 1991). The bulk BSA diffusivity at each ionic strength (values in Table 1) were evaluated from the quasi-elastic light scattering data of Doherty and Benedek (1974). These data were all obtained at a BSA concentration of 50 g/L, which is characteristic of the high protein concentrations that exist in the immediate vicinity of the membrane at the higher filtration velocities. The mass transfer coefficients at the other ionic strengths (listed in Table 1)were evaluated from eq 19 using these values for the diffusion coefficient, with the results in good qualitative agreement with independent data for the mass transfer coefficients determined experimentally in the stirred cell using the procedure outlined by Opong and Zydney (1991). The mass transfer coefficient increased by almost a factor of 3 as the salt concentration was reduced from 0.15 to 0.0015 M due to the increase in the diffusion coefficient arising from the increased electrostatic repulsion between the proteins at low ionic strength. The actual BSA sieving coefficients were determined from the observed sieving coefficient data in Figure 1using eqs 18 and 19 with the mass transfer coefficients shown in Table 1. The results are plotted as a function of the filtrate flux in Figure 2. The actual sieving coefficients at each salt concentration decreased monotonically with increasing flux approaching a constant asymptotic value at high flux, which is consistent with the behavior predicted by eq 14. The solid curves in Figure 2 are the calculated values of Saevaluated using eq 14 with the best fit values of the parameters Sm(l - w ) and (w#&Dm)/6, determined by minimizing the sum of the squared residuals between the calculated and experimental values for S a using the method of steepest descent. The membrane porosity (E) appears in the results since the filtrate flux, J, = e( V), is used instead of the average fluid velocity in the pore ( V ) . The best fit values and standard deviations in the fitted parameters are summarized in Table 1,with the standard
2478 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 Table 2. Calculated Values of the Membrane Transport Parameters at Different Ionic Strengths electrophoretic diffusion ionic charge strength (M) (esu) coefficient (m2/s) mobility (m2/(V 8 ) ) a ((V s)/m2) w S, 4Kd -1.1 X 10-8 (5.6 f 0.2) X 106 (-5.0 f 1.0) X ' o l 0.16 f 0.02 (1.3 f 0.1)X 5.9 x lo-" 0.15 -16.6 (2.4i 0.1)X lo6 (-1.2 f 0.2)X 103 0.15f 0.19 -1.2 X 10-8 (6.3f 0.7)X 5.9 x lo-" 0.05 -12.3 -1.5 X 10-8 (1.1 f 0.04) X lo7 (-3.2f 0.4) X 103 (3.7f 0.2)X 1k2 (7.3f 0.6)X 5.7 x lo-" 0.015 -10.8 -2.5 X 10-8 (3.8i 0.2) X lo7 (-4.4f 1.1) X 1k2 (3.9f 0.9)X 10-9 (1.8f 0.3)X 5.3 x lo-" 0.005 -14.0 -3.4 X 1o-S (1.2f 0.1)X lo8 -2.0 f 2.9 (3.7f 5.2)X lo-' (1.9f 0.1)X 4.9 x 10-11 0.0015 -15.7
deviations determined from the covariance matrix using the approach described by Himmelblau (1970). The good agreement between the experimental data and the calculated values for both So and Saover the entire range of filtrate flux in Figures 1 and 2 indicates that eq 14 accurately describes the flux dependence of the actual membrane sieving coefficient and that eqs 18 and 19 accurately describe the concentration polarization phenomena in the stirred cell. The scatter in the data at high J, is probably due to small errors in the mass transfer coefficients; however, the best fit values of the parameters were found to be relatively insensitive to small changes in k. The observed sieving coefficients are approximately equal to the actual sieving coefficients at low flux but then increase at high flux due to the accumulation of protein in the concentration polarization boundary layer. This results in the minima in the observed sieving coefficients seen in Figure 1,with the position of the minima shifting to higher flux at lower salt concentrations due to the increase in the bulk mass transfer coefficient with decreasing ionic strength. In order to evaluate the intrinsic membrane properties (#&, S,, and w ) from eqs 15-17 it is first necessary to determine the value of a (eq A7). This requires knowledge of the membrane surface charge density, which was determined experimentally from the variation in the hydraulic permeability of the membrane (after preadsorption with BSA) with NaCl concentration. These permeability data are shown in Figure 3 with the solid curve representing the theoretical prediction given by eq A6for flow through a cylindrical pore with constant surface charge. This expression was developed using the DebyeHiickel approximation which should be valid at the low surface potentials (
