Rejection of polyelectrolytes from microporous membranes

Journal of Membrane Science, 5 (1979) 77-102 o Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

REJECTION OF POLYELECTROLYTES MEMBRANES

WILLIAM Eastman

D. MUNCH Kodak

LAWRENCE Chevron

FROM MICROPOROUS

Co., Rochester,

New York 14650

(U.S.A.)

P. ZESTAR

Research

Co., Richmond,

CA 94802

(U.S.A.)

and JOHN L. ANDERSON* Carnegie-Mellon

University,

Pittsburgh,

Pennsylvania

15213

(Received July 31, 1978; accepted in revised form January

(U.S.A.) 23, 1979)

summary The rejection coefficient for rigid (bovine serum albumin) and flexible (hydrolyzed polyacrylamide) polyelectrolytes during ultrafiltration was experimentally determined by a transient material balance technique. Parameters such as supporting electrolyte (NaCl) concentration, pore radius, and pore flow velocity were varied to study their effects on the degree of rejection. The membranes were fabricated by the track-etch process and offered the advantage of a well-defined pore structure in which constrictions and tortuosity effects were minimal. At a given pore radius, the NaCl concentration strongly affected the rejection coefficient by altering the long-range electrostatic forces between pore wall and macromolecule for the rigid polyelectrolyte, and by substantially altering the molecular conformation of the flexible polyelectrolyte. In all cases, the rejection coefficient increased as the ionic strength decreased. With the flexible macromolecules, the shear rate in the pore also appeared to be a significant factor. Our most surprising result is that the flexible polyelectrolyte was able to pass through pores of radius less than one-half the hydrodynamic (Einstein) radius of the macromolecule, indicating that flexible macromolecules cannot be modeled as rigid spheres in predicting the rejection coefficient.

Introduction Studies of membrane ultrafiltration have tended to focus on the convective-diffusion phenomena associated with concentration polarization on the high pressure surface of the membrane (for example, refs. [1,2] ), and membranes which are totally impermeable to the macrosolute have generally been used in such ultrafiltration experiments. Relatively little effort has been expended to determine what physical properties are most important in determining the degree of rejection of macromolecules from *Person to whom correspondence

should be addressed.

77

78

porous membranes. Aside from being of interest from a purely fundamental viewpoint, an understanding of the rejection process is of potential use in an engineering sense, for example, as applied to rapid fractionation and concentration of a solution of various macrosolutes, and to estimating polymer losses in microporous substrates during tertiary oil recovery operations. Our objective in this paper is to study the role of solution ionic strength and, to a lesser extent, pore dimension in determining the rejection of polyelectrolytes from membranes with well-defined pore structure. To gain physical insight into rejection phenomena, it is instructive to model a porous membrane as a series of uniform parallel capillaries of length I and radius ro, such that Z/r,,S 1. As solvent is driven through the pores by an applied pressure difference across the membrane, macromolecules are rejected because of their interactions with the membrane. The macroscopically observable result of these interactions is a decrease in the flux (IV) of macromolecules, and the rejection coefficient (or, reflection coefficient [3] ), IS, is a convenient parameter to quantitatively describe this decrease : N=(l-a)U,C,

(1)

where U, is the superficial solution velocity based on membrane area, and C, is the solute concentration in the bulk solution on the high pressure side. u is generally considered to be independent of U, and C, for rigid solutes and only a function of membrane structure and solute properties, as long as concentration polarization is avoided (see Appendix B). There are two basic mechanisms by which rejection may occur, as discussed in more detail in Appendix A. The first of these is an equilibrium partitioning of the macromolecules between solution just inside and just outside the pore entrance. The equilibrium spatial distribution inside the pore couples with the non-uniform velocity field to produce a flux of macromolecules which is smaller than would exist without this spatial distribution [4, 51 resulting in u < 1. In the second mechanism, a sieving effect at the pore entrance occurs because the macromolecules are trapped in the flow streamlines and the value of u is determined primarily by the hydrodynamics at the pore entrance. Which of these two mechanisms is controlling is determined by the entrance Peclet number which is defined as

where is the average flow velocity in each pore, and D is the diffusion coefficient of the macromolecule. PeE can be thought of as a ratio of diffusion time (ro2 /D) to convection time (r,/(Up)) for a macromolecule to traverse a distance of order r,,. If PeE < 1 then the flow is slow enough such that the macromolecule is able to freely diffuse in the vicinity of the pore entrance and hence can establish its equilibrium spatial distribution.

79

At the opposite extreme (PeE S l), Brownian motion is so slow relative to convection that the macromolecule is rejected by a sieving action at the pore entrance. Theoretical relationships between rejection coefficient and macromolecule size for both partitioning control and sieving control are plotted in Fig. 1. The theoretical development leading to these curves assumes the macromolecules are rigid spheres, the pores are circular cylinders, and there are no long-range forces between macromolecule and pore wall.

0.8 -

0.6 o-

0

0.2

0.6

0.8

1.0

Fig. 1. Rejection coefficient (a) versus size for rigid spheres in circular pores. See eqns. (Ad), (A5), and (A7) from which curves A, B and C were computed. Although there exists at least a basic understanding of the mechanism behind the rejection of rigid macromolecules, very little is known of the mechanism for flexible macromolecules. Dimarzio and Guttman [6] have analyzed the interaction between a random flight polymer and a parabolic velocity field in a long pore. By conserving the linear and angular momentum of the polymer, they found that its mean translational velocity is the same as for a rigid sphere of radius R,, the root-mean-square radius of gyration of the polymer. Unfortunately, the rigor of this analysis is diluted by the following three assumptions: (1) there is no hydrodynamic interaction among segments of the polymer chain; (2) no part of the macromolecule is near the pore wall; and, most important, (3) the configuration of the macromolecule is unperturbed by the flow field and hence is spherically isotropic. Furthermore, Dimarzio and Guttman failed to consider the partitioning of macromolecules between pore and bulk solutions

80

because their objective was to mdoel exclusion chromatography and not ultrafiltration. The experimental results shown later in this paper provide strong evidence that linear polyelectrolytes cannot be modeled as rigid spheres of radius R,, as determined by intrinsic viscosity measurements, when correlating rejection coefficient with pore dimension. We report here our results of studying the dependence of rejection coefficient on solution ionic strength and pore size for two polyelectrolytes of quite different structural properties: bovine serum albumin (BSA), a rigid, approximately spherical particle of hydrated radius 36 A; and Separan @ AP273, a linear polyacrylamide which is about one-third hydrolyzed with an R, of about 2000 a at .1 M NaCl. The ultrafiltration membranes were produced by the track-etch process and possessed a welldefined pore geometry. We operated at low membrane velocities (U, < 10e4 cm/s) in an attempt to reduce the influence of boundary layer polarization on the observed rejection. In the case of the rigid BSA we expected that variations in the electrolyte (NaCl) concentration would affect u by altering the length scale of repulsive interactions between the macromolecule and pore wall. On the other hand, with Separan the effect of added electrolyte should be a change in molecular configuration; that is, a reduction in NaCl concentration should cause an expansion of the flexible polyelectrolyte and hence an increase in the rejection coefficient. These expectations are confirmed, at least qualitatively, by the experimental results. Experiments A transient ultrafiltration scheme was devised because the time required to reach steady state conditions at the very low membrane velocities studied here would be of the order of one day or longer. Also, the initial stages of the rejection more truly reflect the interactions between the membrane and macromolecules, because during an ultrafiltration experiment the membrane continues to foul by adsorption of contaminants and the macromolecules themselves. The apparatus used for the Separan @ experiments is shown in Fig. 2, and a detailed drawing of the ultrafiltration cell is given in Fig. 3. The apparatus used for the BSA experiments is very similar to what is shown in Figs. 2 and 3, except that the flow across the membrane was generated by a constant pressure rather than by a syringe pump which produced a constant flow rate. Complete experimental details are given in the theses upon which this paper is based (BSA: Munch, 1977; Separan @: Zestar, 1977. Copies of these theses can be obtained from J.L. Anderson.) Table 1 outlines the operating conditions for both sets of experiments. In the Separan @ experiments a steady flow of polymer-free solution was initially forced through the membrane, with polymer-free solution contained in both reservoirs I and II of the cell in Fig. 3. At time zero the flow was switched to the solution containing Separan @, and the polymer con-

81

FILTRATION CELL,

PRESSURE

TRANSDUCER

8 RECORDER

I’ SYRINGE

PUMP MAGNETIC STIRRER

Fig. 2. Ultrafiltration

TEFLON STIRRER

apparatus.

BEARING

8

SUPPORT

MAGNETIC

STIRRING

NEOPRENE

GASKET

GLASS

SAMPL

SHAFT

I NG

PORT

S

Fig. 3. Ultrafiltration

cell.

centration began to rise in reservoir I. After a lag time corresponding to the dead volume in the system (due to connecting tubing), the polymer began to appear in the outlet mixing chamber from which solution was periodically sampled using a microliter syringe. These samples were assayed for the polymer concentration using a total carbon analyzer (Beckmann, Model 915A). The pressure drop across the membrane was measured using a differential transducer and demodulator (Validyne Engineering Corp.,

82 TABLE 1 Summary of experimental

conditions

Property

Bovine serum albumin

Separan @ AP 273

Molecular conformation Membrane Pore radius Pore area fraction Membrane velocity( U, ) NaCl concentration PH Temperature Stirring rate (see Fig. 3)

Compact, MW = 68,000 Track-etched mica 7 pm thick 220 + 30 R 0.64.0 X 1O-3 3.6-6.1 X lo-‘cm/s lo-3,10-2,10-l M 6.8 25% 150 rpm

Linear, MW - 5 X lo6 Nucleopore @ 5 or 10 pm thick 400,500,1000 a 0.02-0.10 1-3X lo4 cm/s 0.0-0.1 M 6.3 25% 100 rpm

Models CP15 and CD15). The dynamics of this system are modeled using stirred tank analyses as described in Appendix C, with the only unknown being the rejection coefficient defined by eqn. (1). Substituting total volume throughput (V) for time, plots of output concentration of polymer versus V were constructed with cr as a free parameter. By plotting the data on the same graph, a value of L-J was obtained by inspection. These material balance curves are shown in Fig. 4, and data for BSA are plotted to illustrate the correctness of the transient analysis since u = 0 for the large pores (1000 a radius) of these membranes. Nuclepore @ membranes (Nuclepore Corporation) with nominal pore radii 400, 500, and 1000 a were used for the Separan JQexperiments. These membranes are made by irradiating thin (5-10 pm) sheets of polycarbonate with heavy fission fragments and then etching the resulting tracks into well-defined pores which are slightly tapered toward the center of the membrane [ 71. During production the collimation of the radiation beam is not good, so that the pores may intersect to some degree inside the membrane, although on the surfaces of the membrane the pores appear very uniform. Flow rate-pressure drop experiments with polymer-free solutions were used to check the pore radius using Poiseuille’s equation (the pore cross-section is circular), and in most cases our value for r. agreed within 20% of the nominal value. In some cases the discrepancy was larger, but we accepted the nominal value of pore radius as being correct because of uncertainties in pore number density, membrane thickness, and degree of tortuosity. Because all polymer-free solutions were pre-filtered with a Nuclepore 0 membrane identical to the one used in the ultrafiltration experiment, we were able to achieve very good stability in the hydraulic permeability (Um /A&‘) over periods of hours when there was no Separan @ in the solution. Porous mica membranes [ 81 were used for the BSA experiments. They offer the advantage of a well-defined pore structure: all the pores of any

83

one membrane have the same dimensions and are parallel to one another, the pore cross-section is uniform along the axis with no taper, and the pore length equals the membrane thickness. The number of pores is controlled by the irradiation time, while the radius is controlled by the etching time and conditions. The pore cross-section is a 60” rhombus, and we define the radius r. to be the radius of a circle of equivalent area. For each membrane, r. was determined from the hydraulic permeability for BSA-free solution. If the rhombic cross-section is assumed to be an ellipse of axis ratio 2.55, so that the ellipse has the same perimeter-to-area ratio as the 60” rhombus, then the area-equivalent pore radius is computed from r, =l.lO[

yAT]

(3)

where the number of pores per unit area (n) and pore length (= membrane thickness, I) are known beforehand. The pore radius was determined for each mica membrane by using eqn. (3) with the measured value of U, /AP, and the results agreed with sizes determined by both electron microscopy and electrolyte conduction [ 81 . As with the Nuclepore @ membranes, careful prefiltration of all aqueous solutions through small-pore (- 400 A radius) membranes allowed us to achieve good stability in the hydraulic permeability for BSA-free solutions (e.g., less than a 4% reduction in permeability after one cm3 of solution was forced through the mica membrane at a rate of 2 cm3 /h). The water used for all experiments with both BSA and Separan had been distilled and then purified by passing it through a Milli-Q2 @ adsorption/ ion-exchange system (Millipore Corp.) with a 0.22 pm pore size Millipore @ filter at the end. The BSA was purchased as a crystallized and lyophilized fraction from the Sigma Chemical Co. Aqueous solutions of 400 ppm were prepared with NaCl concentrations of 0.001, 0.01 and 0.1 M; these solutions were dialyzed for at least 24 h at 4°C. No buffer was ever added, but the pH of the solution was always 6.8 + 0.3. Before each ultrafiltration experiment the cell with membrane clamped in it was filled with the 400 ppm BSA solutions and allowed to sit for 1% h to allow adsorption to approach equilibrium. The low pressure side (reservoir II) was then flushed with BSAfree solvent just before flow through the membrane was initiated. The ultrafiltrate samples were assayed for BSA concentration using the Lowry test [9] which is accurate to below 20 ppm. The Separan @ AP 273 was supplied to us as a dry powder courtesy of the Dow Chemical Co. This polymer is a polyacrylamide approximately 33% hydrolyzed with an average molecular weight of about 3 X lo6 with less than 0.3% monomer content [lo] . Nearly all of the ultrafiltration experiments were conducted with a 20 ppm polymer concentration in pre-filtered water containing NaCl. A stock solution was made by slowly dissolving 200 mg of dry polymer into one liter of the salt solution with intermittent stirring and then shaking gently for three days at 5” C in an

84

0

4

8

12

16

20

24

V-V,(cm’) Fig. 4. Ultrafiltrate concentration (C,) of BSA versus net volume flow (V- V,) for Nucleopore @ membrane with pore radius 1000 A. The mathematical analysis from which the curves were constructed is given in Appendix C.

incubator shaker (New Brunswick Scientific Co.). Strong agitation had to be avoided to prevent degradation of the dissolved polymer. The stock solution was then filtered through No. 54 Whatman paper to remove small amounts of undissolved gelatinous masses of polymer. The stock solution was stored at 5” C and diluted as necessary to reach the appropriate polymer concentration for the ultrafiltration and viscosity experiments. In an effort to partially characterize the dissolved Separan, viscosity measurements of O-100 ppm solutions at 25” C were made using an Ubbelohde capillary viscometer with a mean shear rate for pure water of about 900 s-’ The linearity of plots of viscosity versus polymer concentration was quite good. Intrinsic viscosities were determined from the slopes for various NaCl concentrations. Simple stirring experiments were conducted to test the degradation level of the macromolecules. The stirring configuration is shown in Fig. 3. With BSA, the solutions were stable for periods of 18 h at stirring rates below 500 rpm. There was negligible degradation of Separan at 20 ppm over a period of four hours at 100 rpm, but at 100 ppm and 200 rpm the solution viscosity dropped by 40% over 12 h. All Separan ultrafiltration measurements were performed at a stirring rate of 100 rpm to prevent degradation. In all the ultrafiltration experiments a clean membrane was used, and

85

a membrane was never used more than once. The reason for this was that adsorption of the macromolecules during ultrafiltration added uncertainty to the pore geometry so that reuse of the membrane would produce data which would be difficult to interpret. Control experiments were performed in the ultrafiltration cells to check for effects on material balances of macromolecule adsorption to the cell internal surfaces; no such effects were ever found. Results - BSA An example of the time course of one experiment is shown in Fig. 5. The equivalent pore radius was determined using eqn. (3) and the measured values of membrane velocity and pressure drop. After the 1% h period of contact between the 400 ppm BSA solution and the mica membrane (without flow), the pore radius dropped from 336 A (before contact with BSA) to 276 A. As filtration of the BSA solution occurred, the equivalent radius continued to decrease toward an asymptotic value of 177 A. After the cell was flushed with 500 cm3 of BSA-free solution, a measurement of hydraulic permeability yielded an equivalent radius of 228 A. A summary of all adsorption data for BSA is presented in Table 2. The change in pore radius (Ar, (l )) due to primary adsorption of BSA, that is, after static contact without flow, appears to be independent of ionic strength. This result is somewhat surprising considering that both BSA and the mica pore walls carry a net negative charge. Apparently there are specific adsorption sites on the pore wall which saturate with BSA irrespective of the electrostatics. The mean change in equivalent pore radius for all three NaCl concentrations studied is 48 A. If the adsorption is modeled as a layer of uniform thickness at the wall, this thickness equals about 40 A when the rhombic geometry of the pore cross-section is accounted for (thickness = 0.82 Ar,), in reasonable agreement with the non-hydrated diameter of BSA equal to 54 A [ll] . Other experiments utilizing electrolyte conduction (no bulk flow) to determine pore radius indicate a thickness of about 50 A for BSA adsorbed onto the pore wall in mica membranes [ 121. The fact that there was a gradual decrease in the membrane hydraulic permeability as ultrafiltration proceeded, as shown in Fig. 5, is puzzling to us because such continuous fouling of membrane is thought to be caused by constrictions and tortuosity in the pore structure. However, with mica membranes the pores are parallel to one another and do not taper, SO that such structural problems do not exist. It may be that the decrease in permeability was due to a secondary adsorption of BSA onto the pore wall, induced by a shear flow mechanism thought to be important to the

coagulationrate of microcolloids [ 131. High relative velocities (- 0.1 cm/s)

bOveen dissolved BSA and adsorbedBSA thepores couldhav within

resulted in collisionswhich would overcomethe stabilizingrepulsiveforces

86

0.2 -

0

Y 3

A

Pore Radius

0

Filtration

Dato Data

(5)

G 2

(C,/C,)

0.1 -

V (cm31

Fig. 5. Time course of a typical BSA ultrafiltration experiment with a mica membrane. The dead volume in this particular cell was 0.67 cm3. TABLE 2 BSA adsorption onto pore walls of mica membranesa. pH = 6.8, 25%. NaCl cont. (M)

Initial pore radius (A) (Before contact with BSA)

*r

0.001

266 271 302

48 46 41

55 87 48

46 71 38

45

63

52

43 40 60 60

51 64 135 159

49

55 123 108

51

102

84

52 35 55

93 115 140

91 108

47

116

100

Ave. 0.01

210 250 268 336 Ave.

0.1

222 271 301 Ave.

(I? O

A&

Ar 0 (3)

d

‘All pore radii determined from flow rate measurements using eqn. (3). Each membrane membrane was never used for more than one experimental initial radius minus radius at the beginning of the ultrafiltration experiment (membrane had been contacted with 400 ppm BSA solution for I?4 h without flow). ‘Ar (*) = initial radius minus radius at the end of the ultrafiltrati n a Aroc3) = experiment (1.5 cz3 of BSA solution had been forced through the membrane). initial radius minus radius after 500 cm3 of BSA-free solution was flushed through ultrafiltration cell upon termination of the ultrafiltration experiment.

87 TABLE 3 Summary of rejection data for BSA Mean pore radius (A)

NaCl cont. (M)

Mean pore velocity (cm/s)

u

258 214 212

0.001 0.001 0.001

0.05 0.12 0.27

0.95 0.8 0.9

200 235

0.01 0.01

0.08 0.18

0.5 0.7

208 212

0.1 0.1

0.06 0.24

0.2 0.5

between BSA molecules that tend to stabilize a static dispersion. The effect of ionic strength on Ar, (* ), shown in Table 2, is consistent with this hypothesis: at a lower ionic strength the electrostatic repulsive forces between BSA molecules are stronger because of the increase in Debye length parameter, and hence the same relative velocity between two BSA molecules is not as effective in causing their coagulation. The third column (AroC3)) in Table 2 shows that this secondary adsorption can reverse itself when BSA is removed from solution; conceivably this reversal could have gone to completion had we waited long enough (i.e., > one day). It is also possible, however, that a shear-induced polymerization (and degradation) of the adsorbed BSA occurred, since typical shear rates within the pores were - 105 s-l. The concentration of BSA in the ultrafiltrate versus volume throughput is also shown in Fig. 5. The apparent rejection coefficient seems to decrease with throughput as one would expect because of the decrease in equivalent pore radius. It is important to remember that the ultrafiltrate concentration lagged the pore radius determination by the dead volume of the apparatus which equalled 0.67 cm3 (tubing plus membrane support), and hence the concentration data should really be plotted against A V = (V - 0.67) to correspond to the pore radius determination. To interpret our data we have averaged the pore radius over V = 0 to 0.8 cm3 and the rejection coefficient over A V in the same range, and a summary is found in Table 3. If the BSA molecule behaved as a hard sphere of 36 A radius and did not experience a long-range interaction with the pore wall, then for a pore radius of 220 A we would predict 0 = 0.1 (see Fig. 1) assuming partitioning control. Our experimental determinations are significantly higher than this. The variation of cr with ionic strength implies that electrostatic interactions between BSA molecules and the pore wall (or, more properly, the BSA coated wall) were important to the rejection process. The results of Table 3 are plotted in Fig. 6 to emphasize the ionic strength

88

effect. The pore radius for all membranes from which these data were taken was about 220 ? 30 a. If the rejection was controlled by partitioning of BSA between pore and bulk solutions at the high pressure surfaces of the membranes, then eqn. (A2) could in principle be used to compute u versus NaCl concentration, given the energy profile E(r) for a BSA molecule whose center is at radial position r in the pore. Such a calculation is not feasible at the present time because E(r) is not known even for DebyeHiickel-type electrostatic interactions between a sphere and the pore wall. A simple but promising way to account for the electrostatic interactions is to alter the pertinent rigid dimensions by adding the Debye length parameter K-I to the BSA radius and subtracting it from the pore radius. The rationale behind this approach is that surface charges are only screened effectively beyond distances of order K -‘, and hence the repulsive energy of two surfaces whose charges are the same sign is very large if the separation is less than about 2~ -l. The Debye length parameter is related to the supporting electrolyte (NaCl) concentration C, by K

-1 =

C

EkT

8n Z2 e2

NA&

I

142

(4)

The modified ratio of hard sphere macromolecule-to-pore A*

=

a r.

radius becomes

+K-1 -

-1

K

The curve drawn in Fig. 6 was taken from Fig. 1 (curve A) with h* substituted for h. Changes in C, cause variations in K -I, so h* is a function

1.0

/

0 0 0.8 -

CORRECTED

HARD

SPHERE

0

0.4-

HARD b=36i ---‘__-_---OI

SPHERE

r,, =22Oi)

______ I

I

-3

I

J

- I qo-;NocI]

Fig. 6. Rejection coefficient of BSA versus NaCl concentration.

89

of Nail concentration. The agreement between the data and theory in Fig. 6 is perhaps better than expected considering the crudeness of the analysis, but it does indicate that the physical basis of our hypothesis regarding the importance of electrostatic effects on rejection is sound. Our results are consistent with ionic strength effects observed for the partitioning [ 141 and diffusion [ 151 of polyelectrolytes in porous media. The membrane velocities were low enough in our ultrafiltration experiments that concentration polarization was not a problem (see Appendix B). The value of the concentration polarization parameter Q for most of the BSA experiments was - 0.1, so that according to Fig. 11 the measured rejection coefficient equalled the true membrane coefficient. Consistent with this conclusion is the fact that we could not detect any systematic effect on the rejection by varying the membrane velocity from 3-5 X 10e5 cm/s_ Fig. 6 does give the impression that ITdepends on pore velocity, which was varied by using membranes of nearly the same pore size but with different numbers of pores. The pore entrance Peclet number, defined in eqn. (2), increased from about 0.5 to 2 in going from the low to the high pore velocities in our experiments. In this range we might expect a transition from partitioning control (curve A or B of Fig. 1) to sieving control (curve C) and hence an increase in CJwith
(6)

where M is the molecular weight, and the parameters K and c depend on the degree of ionization and supporting electrolyte concentration. No significant dependence on shear rate was observed in the range 100-600 s-l as long as the salt concentration was 0.025 M or greater. Using their values of K and c for a degree of ionization equal to 0.4 (which probably corresponds roughly to our 33% hydrolyzed polyacrylamide), and taking our measurements of [n] at 0.07, and 0.1 M NaCl, we compute the molecular weight of our Separan to be 2.5 X lo6 and 1.0 X lo’, respectively, in agreement with the supplier’s statement that the average molecular weight

90 I

I

I

I 20

2.c E s E 0 P 0 x

I.C

T

0

IO c;k

(&Z

)

Fig. 7. Intrinsic viscosity of Separan @ AP273 versus NaCl concentration.

is greater than 2 X lo6 . The larger spread in the estimate of the molecular weight of our sample is probably due in part to the variation in c with ionic strength, which tends to magnify small errors, and to a molecular weight distribution. From the above calculations we estimatelour average molecular weight to be 5 X 106. Table 4 lists the equivalent dimensions of the polyelectrolyte as calculated from the intrinsic viscosity data. The hydrodynamic radius (I?,) was computed using Einstein’s value of 2.5 for the intrinsic viscosity in terms of volume fraction of rigid spheres. The root-mean-square radius of gyration TABLE 4 Characterization

of Separan @ AI’273 by intrinsic viscosity’.

pH = 6.3. 20°C

NaCl cont. (M)

[771 (cm3 1s)

R, (A) b

R, (&C

0.0000 0.0020 0.0114 0.0930 0.1000

21.6 16.0 8.3 4.6 3.1

2580 2330 1870 1540 1350

3220 2920 2340 1920 1690

x X x x x’

lo3 lo3 lo3 lo3 lo3

‘Determined with Ubbelohde viscometer at a shear rate of 900 S’ ; polymer concentration aried from 0 to 100 ppm. B Based on hard sphere equivalence in Einstein’s model assuming M = 5 X 1 O6 :

‘Assumed R, = 1.25 R, .

91

(Rg) should equal r;-’ R, according to the theory of Kirkwood and Riseman [ 201, who originally predicted .$ = 0.94, while Berry and Casassa [ 211 quote values of 0.86 for neutral random flight macromolecules and 0.74 for linear polyelectrolytes. As a compromise we have chosen 5 = 0.80 with the realization that uncertainty in this parameter is perhaps trivial compared to those uncertainties introduced by the polydispersity of the Separan. The data from one set of experiments are plotted in Fig. 8 and show without question that ionic strength strongly influences the rejection properties of this type of linear polyelectrolyte. Reproducibility was excellent, that is, two different experiments at the same NaCl concentration using different membranes with the same nominal pore size typically yielded the same profile of ultrafiltrate concentration versus volume throughput. However, nearly all experiments were characterized by small but significant increases in apparent rejection coefficient at the ultrafiltration proceeded, as in the BSA experiments. We are not certain why these increases occurred, but one can speculate that some pores were being partially blocked by contaminants and aggregated polymer molecules. At a constant membrane velocity the measured pressure drop did increase with volume throughput, also indicating partial blockage, but some of this increase was probably due to the intrinsic viscosity of the polymer itself since its concentration also increased on the high pressure side as time

/

I

O-80-

0

o.

v

0.0 114 Molar 0.002 Molar 0.000 Molar

l

Cl

IOO

I

I

I

Molar NaCI No Cl NaCI No Cl

/

OIA

0

4

8

Fig. 8. Ultrafiltration data for Separan @ AP273. analysis from which the curves were constructed.

16

20

See Appendix C for the mathematical

92

progressed. We chose the value of e whose curve best represented the ultrafiltrate concentration data at a volume throughput of about 10 cm3 . Our results for rejection coefficient versus NaCl concentration, pore radius, and membrane velocity are summarized in Table 5. Section A of the table was taken from Fig. 8 and shows that u increased as the NaCl concentration decreased. There are two explanations for this behavior. From Table 4 one sees that the effective size of the polyelectrolyte is greater at lower ionic strengths, and the expanded macromolecule finds it more difficult to penetrate the pores. The second explanation is found in the results for BSA: at lower ionic strengths the Debye length parameter is larger and hence the macromolecule experiences a stronger repulsion from the pore wall. However, in these experiments the first explanation (conformational changes) is probably most responsible for the rejection data because the mean radius of the macromolecule is so much greater than the Debye length even at IO-’ M NaCl. The size effect is also illustrated in Section B of Table 5, since u increases as pore size decreases at fixed NaCl concentration.

TABLE

5

Summary of rejection data for Separan @ AP273 Pore radius (A) s

NaCl cont. (M)

Membrane velocity (cm/s)

Pore Velocity (cm/s 1

.b

1000 1000 1000 1000

0.000 0.002 0.011 0.100

1.9 1.9 1.9 1.9

2.0 2.0 2.0 2.0

1o-3 1o-3 10” 1o-3

0.85 0.7 0.4 0.0

B

400 500 1000

0.093 0.093 0.093

2.9 x lo4 2.9 x lo4 2.9 x lo4

9.5 x 1o-3 1.2 x 1o-2 3.0 x 1o-3

0.6 0.55 0.2

C

400 400 400

0.100 0.100 0.100

1.1 x lo4 1.9 x lo4 2.9 x lo4

3.7 x 1o-3 6.3 x 1O-3 9.5 x 1o-3

0.8 0.6 0.5

D

500 500 500

0.093 0.093 0,093

1.1 x lo4 1.9 x lo4 2.9 x lo4

4.6 x 10” 7.9 x lo9 1.2 x 1o-2

0.35 0.30 0.55

E

1000 1000 1000

0.100 0.100 0.100

1.1 x lo4 1.9 x lo4 2.9 x lo4

1.2 x 1o-3 2.0 x 1o-3 3.0 x 1o-3

0.2 0.0 0.0

A

‘Nominal pore radius, ‘Reproduced membrane for each experiment.

x x x x

lo4 lo4 lo4 lo4

x x x x

in at least two different experiments,

with a new

93

The dependence of rejection on membrane and pore velocities is demonstrated by Sections C-E of Table 5. There are two possible interpretations of these data. The first of these involves boundary layer effects at the membrane surface. The analysis found in Appendix B can be used to at least estimate the importance of concentration polarization to our observations. Jamieson and Presley [22] measured the diffusion coefficient of Pusher 700, a 20% hydrolyzed polyacrylamide of molecular weight greater than lo6 manufactured by the Dow Chemical Co., using a quasielastic light scattering technique. They were able to discern two coefficients which were interpreted as modes of diffusion parallel and perpendicular to an axis of rotation for the macromolecule. The statistically weighted average of these COefficients is 1.6 X lo-’ cm’/s at 25°C. The values of a for the three membrane velocities studied are 1 .l,1.8, and 2.7, so that concentration polarization may have been an important factor in the flow rate dependence of u (see Figs. 10 and 11). The second interpretation of the velocity effect is that the conformation of the polyelectrolyte is shear dependent. The shear rate variation is the pores varied from 300 to 6000 s-’ in our experiments. At high pore shear rates the polyelectrolyte may align itself more with the pore axis and hence would have a better probability of passing through the pore. Shear alignment must have played an important role in the ultrafiltration of Separan because, as seen in Table 4, the effective hydrodynamic dimensions at all ionic strengths are considerably larger than the pore dimensions. Shear degradation was probably not a factor in passage of the polyelectrolyte through the pores, however. Abdel-Amin and Hamielic [ 231 measured the shear stability of polyacrylamide at concentrations of 0.2% and 0.7%. The critical molecular weight was defined as the value below which no

shear degradation occurred, and the experimental results produced the followingcorrelation: M, =

3.59 x 1oa 7o.41

where the shear stress 7 is expressed in dynes/cm’ . At the highest pore shear rate in our experiments (6000 s-l) and assuming a relative pore viscosity of two times that of pure water, M, = 5 X 10’ and hence our polymer should have been stable even inside the pores. Furthermore, eqn. (7) is perhaps an underestimate of M, for our system since our concentrations were two orders of magnitude below 0.2%. Because the number of pores per unit area is relatively constant for the commercially available Nuclepore membranes, it was not possible for us to uncouple membrane velocity from pore velocity, as was done with the mica membranes in the BSA experiments. Furthermore we could not increase the boundary layer mass transfer rate (i.e., SI - see Appendix B) by simply increasing the stirring rate (o ) in the ultrafiltration cell because

HARD

SPHERE Eq.

1.0

1.5 X =

THEORY

0

(A41

2.0

2.5

3.0

R,/r,

Fig. 9. Rejection coefficient of Separan @ AP273 versus ratio of hydrodynamic radius (R,) to pore radius (rO), where R, was calculated from the intrinsic viscosity determinations using Einstein’s theory for rigid spheres.

degradation of the Separan would have occurred. Therefore, it is not possible for us to quantitatively discuss the exact importance of the shear alignment effect since the role played by concentration polarization was uncertain. The rejection coefficients in Table 5 for the 1000 a pore radius membranes are plotted against the ratio of molecular hydrodynamic radius to pore radius in Fig. 9. Although u seems to increase with the ratio Re/ro, it is clear that neither R, nor R, can be interpreted to be a hard sphere radius for the purpose of modeling rejection from porous membranes. At the pore shear rates of our experiments, the hard sphere “equivalent” radius, appropriate to rejection phenomena, appears to be less than onehalf of the hydrodynamic radius. The rejection process for linear macromolecules is quite complex, and much more experimentation is required before generalizations can be made regarding the relationship among rejection coefficient, molecular conformation, and shear rate. Conclusions By transient material balance techniques we have determined the rejection coefficient for both rigid and flexible polyelectrolytes as a function of supporting electrolyte concentration, pore size and shear rate. Because of adsorption of the macromolecules to the membrane, there is some uncertainty in our results and the reported u values are probably accurate only

95

to * 0.1. However, several conclusions are justified by the data. Electrostatic phenomena affect the rejection of rigid macromolecules by controlling the length scale of repulsive interactions between molecules and pore wall. As the solution ionic strength is lowered this length scale, the Debye length, increases and the rejection coefficient also increases. These effects can be approximately correlated by adjusting the dimensions of macromolecule and pore using the Debye screening length, as demonstrated in eqn. (5), and then applying the theory described by curve A in Fig. 1. Several interesting adsorption phenomena were observed with the BSA ultrafiltration, most notably, a primary adsorption thickness which was independent of ionic strength, and also the formation of a secondary adsorbed layer probably caused by flow-induced collisions between dissolved and adsorbed BSA molecules inside the pores. The rejection of flexible polyelectrolytes appears to be a very complex phenomenon involving conformational changes with solution properties (e.g., ionic strength) and shear rate effects in the pores. Our measured rejection coefficients are much less than unity for these polyelectrolytes even when the equivalent hydrodynamic radius, as determined from intrinsic viscosity measurements, is more than twice the pore radius. It can be concluded from Fig. 9 that this hydrodynamic radius, or the root-mean-square radius of gyration, cannot be interpreted as an equivalent hard sphere radius for ultrafiltration. However, there is a direct relationship between rejection coefficient and intrinsic viscosity in that rejection is greatest when the polymer is in an expanded state, as is the case at low ionic strengths. Our data also indicate that shear rate is an important factor in determining the rejection of linear polyelectrolytes, but unfortunately some uncertainty as to the extent of concentration polarization in our experiments precludes any quantitative analysis of the shear rate effect. Obviously, there is a need for more study into the mechanism of rejection for flexible macromolecules. Appendix A - Rejection of hard sphere macromolecules Whether or not the rejection of a macromolecule is controlled by what happens inside the pore depends on how quickly the equilibrium spatial distribution of the macromolecules can be established in the vicinity of the pore entrance. Consider a circular cylindrical pore of radius r0 and length 1. The characteristic length scale which represents how far macromolecules must be transported to achieve the equilibrium distribution is r, . The residence time of a typical macromolecule in the vicinity of the pore entrance is of order rO/(UP). The driving force to achieve the equilibrium spatial distribution arises from Brownian motion. If the diffusion coefficient of the macromolecule is denoted by D, then the time required to establish the spatial distribution is governed by the rate of Brownian displacement and hence is about ro2 I-0. An entrance Peclet number can be defined as

96

the ratio of diffusion-toconvection

time constants:

ro2 /D r,(Up> P@E = = r,/(U,> D

(Al)

If PeE < 1 then the spatial distribution of macromolecules is unaffected by the flow field, and the equilibrium partitioning of macromolecules couples with the velocity field in the pore to determine the rejection coefficient. On the other hand, if PeE 3 1 then the macromolecules have insufficient time to relax to the equilibrium spatial distribution but rather are swept into the pore; in this case the rejection of macromolecules occurs by a sieving action at the pore entrance. The arguments presented by Bresler et al. [24] to the effect that the sieving mechanism is always controlling are erroneous because in defining membrane subelements in series, they neglected diffusional contributions to the membrane flux when the subelements become thin enough such that axial Peclet numbers are of order one or less. For partitioning control (PeE < 1) the flux of spherical particles through a pore is computed by integrating the local unperturbed fluid velocity U,(r) over the relative distribution of particles, which is given by the Boltmann expression with E(r) representing the potential energy of interaction between one particle whose center is at r and the pore wall [4,5] . From the definition of the rejection coefficient found in eqn. (1) and assuming there is no concentration polarization (see Appendix B), this flux expression leads to the following: a

=l-

r, - a

1

2r Up(r) G(r)e-E(r)‘kT

ro2
dr

G42)

0

where G is a hydrodynamic coefficient which accounts for the fact that the particle will not move precisely with the undisturbed velocity U,(r). The upper limit of the integral in eqn. (A2) means that the particle center cannot get closer than its radius to the pore wall. If the particle is small compared to the pore radius and is far from the pore wall, then G is given by Faxen ‘s law: ifa/r,
(a/r, )’

G=I-2

3

[I - (rkJ2

1

(A3)

Calculations for G(0) are available for all values of a/r, [25] . If one approximates G everywhere by G(0) and assumes only hard sphere interactions (E = 0) then eqn. (A2) can be integrated to obtain: u = 1 - 2G(O) (1-X)’

+ G(0) (1-X)4

(A4)

where X equals a/r,-,. This equation is plotted in Fig. 1 as curve A. If the lag coefficient G(0) is set equal to one then (3 = [l-

(l--h)212

= (l-@)2

(A5)

97

where the pore/bulk partition coefficient (a) equals (1-h)’ for hard sphere interactions. Eqn. (A5) is plotted as curve B in Fig. 1. At values of PeE S 1 a sieving action at the pore entrance determines how many particles enter the pore with the solvent. To estimate the rejection coefficient, assume that the velocity profile at the entrance is essentially the same as for creeping flow through a very thin circular orifice of radius r, (see p. 153 of Happel and Brenner [25] ). Since the spherical particle cannot get any closer to the rim of the orifice than its radius (a), only that fraction of the solvent which flows through a circle of radius r,-a, concentrically located in the plane of the orifice, is available to the particles. Let QO be the total solvent flow rate through the orifice and Q the flow rate through the inner region available to the particles. The rejection coefficient is given by

(T

=

I---

Q

(A6)

Q*

Integration of the velocity area results in u = [l-

(l-X)2]

field over the total orifice area and the inner

3’2

(A7)

This result should be compared with eqn. (A5) for partitioning control since no hydrodynamic effect by the membrane on the particle was considered in the derivation of eqn. (A7). Eqn. (A7) is plotted as curve C in Fig. 1. It appears that the rejection coefficient for sieving conditions is slightly higher than that which occurs when conditions of equilibrium partitioning exist for a given ratio of particle-to-pore size. Hence, an increase in mean pore velocity might cause u to increase when conditions are such thatPeE N 1. Appendix

B - Boundary

layer effects

Because macromolecules are rejected at the membrane surface, the concentration there (C,) is higher than in the bulk well-stirred regions far from the membrane (C,). At higher membrane velocities the polarization factor y , defined as Cm/Cm, decreases toward zero, implying that the concentration of macromolecules at the membrane surface bears little relationship to the bulk concentration. If C, becomes too large, a gel layer may even form on the high pressure side of the membrane. Convectivediffusion transport of macromolecules tends to reduce C, by sweeping away the excess solute accumulating at the membrane surface. Although there exist many analyses of boundary layer phenomena for simple flow configurations, the hydrodynamics of a stirred diaphragm cell are far too complicated to permit an exact theoretical development from which the degree of polarization can be computed. The ultrafiltration scheme most

98

similar to our diaphragm cell that has been examined is the rotating disk membrane [2] . If constant physical properties of the solution can be assumed, and if the macromolecule concentration is sufficiently low so that mass concentration ratios can be substituted for mole fraction ratios with little error, then the polarization can be computed from uoNu

=

Re”2

q

SC”~,

031)

Y

where u, is the true rejection coefficient based on the macromolecule concentration at the membrane surface: N = (1 - a,)UmCm

(B2)

Comparing eqn. (B2) with eqn. (1) yields a relationship between the apparent (measured) coefficient (u) and the true membrane coefficient: 1 - u, l_o =y

(B3)

The dimensionless groups in eqn. (Bl) are defined by: 2U,b Nu = ~, D

Re=WbZ,

SC=+

WI

V

where b is the membrane radius, w the angular velocity of the impeller (four-bar configuration in our experiments), v the solution kinematic

0

0.2

0.4

0.6

0.6

1.0

o-

Fig. 10. Polarization factor (7) versus apparent rejection coefficient (a).

99

0.6 a, 0.4

0.;

0

Fig. 11. True membrane rejection coefficient is baaed on reservoir concentrations.

(u,) versus the apparent value (u) which

viscosity, and D the diffusion coefficient of the macromolecule. Combining eqns. (Bl), (B3), and (B4) one obtains an expression from which the polarization factor can be computed: +;+

= [l-

(l-

U)T] S-J,

(B5)

where Nu Rel /z SC’ /3 Fig. 10 shows plots of y versus the measured rejection at different values of a, as calculated from eqn. (B5). From these results and eqn. (B3), the true rejection coefficient can be calculated, and plots of u, versus u are shown in Fig. 11. If the experimental value of L’Zis below 0.25, the measured rejection coefficient appears to be a good estimate of the true membrane coefficient. However, any interpretation of the experimental results when R > 1 contains a considerable amount of uncertainty because the observed rejection under these conditions may reflect the boundary layer phenomena rather than the fundamental interactions between membrane and macromolecule.

Appendix C - Transient material balances

TLe system

for the Separan ultrafiltration experiments is modeled as a

100

flow at concentration C, into a reservoir (side I, see Fig. 3) which is wellmixed and connected by the membrane to a second well-mixed reservoir (side II). The “ultrafiltrate ” is the solution which leaves side II through the sampling valve. The flow rate through the membrane is denoted by q . The reservoir balances are VI

dC1 -=

qc,

dCII

= (I-

dt

VII----

c,(t)

- (I-

dt

=

(J) 4 CI

t*=

CII(t*),

(Cl)

0) q CI

-

cc21

q GI

(C3)

t--D

where C, is the concentration of Separan in the ultrafiltrate sample, and in is the dead time given by V,/q with V, = 0.44 cm3 (inlet and outlet tubing). The initial condition on the reservoir concentrations is: Ci = Cn = 0 for t < 0. The solution to eqns. (Cl)-(C3) in terms of the volume throughput V is:

+[71~1,,]exp-j(V~I~‘~, forV> t V = s

q dt’,

7I =

VI (1 - o)q ’

vn

711 =-

(C4)

VII 4

The curves in Figs. 4 and 8 were computed from eqn. (C4) with ff as a free parameter. An analysis very similar to the above was performed for the BSA ultrafiltration experiments. The major difference was in the initial condition for the high pressure side: C,(O) = C,. Furthermore, the dead volume in this case included the porous support for the membrane as well as the connecting tubing . Acknowledgement We wish to thank the Dow Chemical Co. for providing us with a sample of Separan @ AP273, and especially Dr. G.D. Rose who advised us as to the properties of this polymer. This research was financially supported by NSF Grants ENG75-13440 and ENG76-21921. The experimental work was performed at Cornell University, and we thank the University for its support.

101

Nomenclature radius of a rigid macromolecule, cm radius of membrane, cm Mark-Houwink exponent c Cr, C_ macromolecule concentration in well-stirred

a

b

region of side I (see

Fig. 3), ppm macromolecule concentration in feed solution to side I, ppm macromolecule concentration in solution at the membrane surface on side I, ppm macromolecule concentration in ultrafiltrate sample, ppm NaCl concentration, g moles/l macromolecule diffusion coefficient, cm2 /S fundamental charge of a proton, 4.8 X 10-l ’ esu potential energy of a macromolecule whose center is at radial position r inside a pore, erg hydrodynamic slip coefficient in eqn. (A2) Boltzmann constant, 1.38 X 10-l 6 erg/OK Mark-Houwink prefactor, cm3 /g pore length, cm some type of average polymer molecular weight number of pores per unit membrane area, cmW2 flux of macromolecules through membrane, g/cm2 /s Avogadro’s number, 6 X 1O23 pressure drop across membrane, dyne/cm’ entrance Peclet number defined by eqn. (Al) flow rate through membrane, cm3 /s pore radius, cm equivalent hydrodynamic radius of the flexible macromolecule computed from Einstein’s coefficient (2.5) for intrinsic viscosity, cm root-mean-square radius of gyration for the flexible macromolecule, time, s temperature, *K membrane velocity (q/area), cm/s local fluid velocity inside a pore, cm/s mean fluid velocity inside a pore, cm/s total volume flowed through membrane after time t, cm3 dead volume of apparatus, cm3 C_ ICnl solution dielectric constant shear viscosity of solution, g/cm/s intrinsic viscosity, cm3 /g Debye parameter (see eqn. (4)), cm-’ 0, or R,lr,

cm

102

A* : a 0, T LI

parameter defined by eqn. (5) kinematic viscosity, cm2 /s &JR, rejection coefficient defined by eqn. (1) intrinsic membrane rejection coefficient defined by eqn. (B2) shear stress, g/cm/s2 concentration polarization parameter defined below eqn. (B5)

References

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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