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j o u r n a l of MEMBRANE SCIENCE ELSEVIER

Journal of Membrane Science 105 (1995) 203-215

Pore size distribution effects on electrokinetic phenomena in semipermeable membranes Skand Saksena, Andrew L. Zydney * Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA Received 14 November 1994; accepted 27 February 1995

Abstract Electrokinetic phenomena occurring in charged ultrafiltration membranes can significantly influence the transport characteristics of these membranes. Theoretical calculations have been performed to evaluate the effects of different log-normal pore size distributions on the solvent flow rate, the induced streaming potential, and the membrane zeta potential. The solvent flow rate increased with an increase in the breadth of the pore size distribution, with this effect being much more pronounced for pressuredriven flow than for electrically-driven flow. The streaming potential induced by the convective fluid flow significantly alters the flow profiles through the membrane due to the very different dependence of the pressure-driven and counter-electroosmotic flow on the pore radius. This causes a relatively large negative (reverse) flow to develop through the smallest pores in the distribution. The membrane zeta potential was also a function of the pore size distribution, even for membranes having the same hydraulic permeability and surface charge density. These results provide important insights into the effects of different pore size distributions on membrane transport and on the proper interpretation of these electrokinetic phenomena. Keywords: Pore size distribution; Electroosmosis; Liquid permeability and separations; Theory; Ultrafiltration

1. Introduction Electrostatic and electrokinetic phenomena can significantly affect the performance of many membrane processes by altering the rate o f both solvent and solute transport through the membrane. Most polymeric and ceramic ultrafiltration membranes carry a net negative charge in aqueous solutions at neutral pH. Thus, electrostatic interactions will tend to increase the concentration o f the positively charged (counter-) ions while excluding the negatively charged (co-) ions from the membrane pores. Electrokinetic phenomena are directly associated with the motion o f the fluid; e.g. the ~"Corresponding author. Phone: 302-831-2399, FAX: 302-8311048, E-mail: [email protected] 0376-7388/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

SSDI0376-7388(95)00060-7

solvent flow that is generated by an externally applied electric field (electroosmosis) and the induced electric field caused b y the pressure-driven flow of an electrolyte solution through a charged membrane (the streaming potential). This induced electric field develops because the convective flux of the counter-ions is greater than that of the co-ions due to the preferential partitioning of the counter-ions into the membrane pores; the induced field generates an electrophoretic flux o f the counter-ions that is opposed to the convective flux thereby satisfying the constraint that there is no net current flow through the membrane. These effects are discussed in more detail subsequently. Recent studies of protein transport through semipermeable ultrafiltration membranes have demonstrated that electrostatic interactions can significantly

204

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

alter the protein partition coefficient between the membrane and bulk solution, leading to large changes in the overall rate of protein transport through the membrane [ 1 ]. Pujar and Zydney [ 1 ] have also shown that the streaming potential generated during ultrafiltration can cause a significant electrophoretic protein flux in addition to the normal convective flux. These electrostatic interactions can be used to dramatically increase the selectivity for the membrane separation of protein mixtures by exploiting differences in the net charge of the individual proteins [ 2]. Applied electric fields have been used to control the rate of both solute and solvent transport in a process known as electrofiltration [ 3 ]. Electrokinetic phenomena have also been used quite extensively to evaluate the surface charge characteristics of different membranes [4,5]. For example, Nystrom et al. [5] correlated changes in adsorption and fouling behavior for surface-modified membranes in terms of the membrane zeta potential, with the zeta potential evaluated experimentally from measurements of the streaming potential generated by the pressure-driven flow of an appropriate electrolyte solution through the membrane pores. Theoretical analyses of these electrokinetic phenomena were initially performed for charged circular cylinders with small surface potentials, i.e. under conditions where the classical Debye-Htickel approximation is valid [6]. More recent work has extended these results to higher potentials and various non-circular cross-sections by numerically solving the nonlinear Poisson-Boltzmann equation [7]. One of the major difficulties in applying these theoretical results to solute and solvent transport in real ultrafiltration membranes is the presence of a fairly broad pore size distribution. Mochizuki and Zydney [8] have examined the effects of such pore size distributions on solute transport through semipermeable ultrafiltration membranes by numerically integrating available expressions for the solute flux over an assumed pore size distribution (both a truncated Gaussian and a log-normal distribution). More recently, Causserand et al. [9] performed a series of theoretical calculations for the effects of electrostatic interactions on solute transport through membranes consisting of a log-normal distribution of slit-shaped pores. However, both of these studies neglected the effects of the electrokinetic phenomena on solute and solvent transport,

thereby limiting the range of applicability of these results. The overall objective of this study was to examine the effects of different pore size distributions on both solvent and ion transport through membranes consisting of a parallel array of cylindrical pores. In particular, calculations were performed to determine the effects of the pore size distribution on: ( 1 ) the pressure-driven and electrically-driven solvent flow across the membrane, (2) the development of a streaming potential and its effect on the overall flux of the electrolyte solution, and (3) the evaluation of the membrane surface charge from experimental measurements of these electrokinetic phenomena.

2. Theoretical development The membrane is treated as an array of parallel cylindrical pores, each with a constant circular cross-section. The governing equations for the ion distribution and solvent flux in the individual pores at small values of the surface potential have been presented by Newman [ 10] with the key results summarized here. The electrical potential (qb) is given by Poisson's equation: ~O=

Pc

(1)

where E is the permittivity of the solvent, Pe = FEzici is the local charge density, F is Faraday's constant, and zi and ci are the ion valence and local concentration, respectively. The second derivative of • with respect to axial position (z) is negligible for a pore with a large aspect ratio since the axial electric field [Ez = ( 0 ~ / Oz) ] is assumed to be a constant. The radial concentration profile for the ions in the capillary is described by a Boltzmann distribution: ci = c°exp(

- zi 1/~)

(2)

where c o is the ion concentration at the pore axis (r = 0) and F ~F= ~-~( ~ - ~r=o)

(3)

is the dimensionless potential with R the ideal gas constant, T the absolute temperature, and ~r= o the potential evaluated at the pore axis. Eqs. ( 1 ) - ( 3 ) can be

205

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

solved analytically in the limit of small potentials, i.e. under conditions where the exponentials in Eq. (2) can be approximated by the first term in their Taylor series expansions (this is the classical Debye-Htickel approximation). The solution for a cylindrical pore with a constant surface charge density (%) is given as [ 10]:

~=

qeF [1 --lo(Kr) ] eKRTI1( Krp)"

(4)

where rp is the capillary radius and Io and Ij are modified Bessel functions of the first kind of order zero and one, respectively. K is the reciprocal of the Debye length and is given as:

The local fluid velocity in the axial direction (v~) is evaluated by solving the Navier-Stokes equation including the electrical force term:

dP~-tzO-~-(r-~)+E~p~=O dz r or\ or/

(6)

where /z is the solution viscosity, (dP/dz) is the applied pressure gradient, and Ez is the electric field. Eq. (6) is integrated twice with respect to r using Eq. (2) for the ion concentrations (actually the first term in the Taylor series expansions for c,). The average solvent velocity in the pore (V) is then evaluated by integrating the resulting expression for Vzover the pore cross-section yielding:

V=

[~_e][

Iz(KrP)

( Krp)/1(Krp)

]Ez_ [~]

dP

~

(7)

where/2 is the modified Bessel function of the first kind of order two. The electric field and the solvent flow both give rise to an axial flux of ions through the pore: N i = uiciE z -f- UzC i

(8)

where ui is the ionic mobility. For simplicity, we have assumed that the ions have equal mobilities (as would be the case for KCI solutions); the extension of this analysis to electrolytes with different ionic mobilities is relatively straightforward although algebraically messy. The average current density (equal to the dif-

ference between the flux of the positive and negative ions multiplied by Faraday's constant) is evaluated by integrating Eq. (8) over the cylindrical cross-section using the previously developed expressions for the solvent velocity and ion concentration profiles yielding:

(i~)=Ez

[

A°+

~{_'o(Krp)I2(Krp)ll 1

I~(Krp)

)J

~- q-z I2('%)

(9)

dz K/z Ii (Krp) where A ° = FZ]Ez~uic° is the conductivity of the electrolyte solution evaluated at r = 0 . Eqs. (7) and (9) completely describe the solvent flow rate and ion flux (current density) for a single capillary pore in terms of the electrical and pressure driving forces.

3. Log-normal pore size distribution Although the detailed pore size distribution of available ultrafiltration membranes is unknown, a number of previous investigations of pore size distribution effects have employed the log-normal density function to characterize the pore size distribution and/or evaluate its effect on membrane transport [8,9,11 ]. The log-normal probability density function is a particularly convenient distribution since it is defined only for positive values of the pore radius (as opposed to the standard normal which includes pore radii from positive to negative infinity). A number of different functional forms for the log-normal distribution have appeared in the literature. Although these forms can be shown to be mathematically equivalent [ 11 ], it is most convenient to express the results explicitly in terms of the mean (r-) and standard deviation (tr) of the distribution:

fR(rp) = n(rp) no 1 ~exp

rpl/ 2~'b

2b

(10)

where the parameter b is given as (11)

206

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

r

0.16

o

a

The average solvent flux through a membrane with a log-normal pore size distribution is evaluated by integrating the expression for V [Eq. (7) ] over the distribution:

0.12 0.05 0.2 0.5

0.08

Vn ( re) m~ppdrv n o Q.

- o

0.04

(12)

f n (rp) 7rr~pdrp 0

0 1.0 c/~ = 0.05 o

0.8

~

Note that (V) is the volumetric flow rate per unit pore area and is thus independent of the membrane porosity. The filtrate flux (Jv), which is typically defined as the volumetric flow rate per total membrane area, is equal to E(V) where Eis the membrane porosity. In a typical ultrafiltration system, the filtrate solution is electrically neutral, i.e. there is no net current flow across the membrane. For a membrane with a pore size distribution, this implies that:

1.0

"r_ 0

o (3.

0.6

0.4 0.2 ~ ~ ,

0

i

0 0

03

1,0

1.5

2.0

rpl~ Fig. 1. Representative plots of the log-normal probability density function (top panel) and the cumulative probability function (bottom panel) for several values of the reduced standard deviation. Normalized Pore Radius,

The pore size distributions described by the lognormal density function are shown in Fig. 1 for several values of the reduced standard deviation, or/F. The top panel shows the actual probability density function fR(rp), while the bottom panel shows the cumulative probability function FR(rp), which is defined as the integral offR(rp) and is thus equal to the fraction of pores in the distribution with radius R < rp. As shown in the upper panel, the most likely pore radius (the maximum in the probability density function) occurs at a value of rp/F< 1, with the difference between the most likely radius and the mean pore size increasing with an increase in the breadth of the distribution. This behavior arises from the asymmetry in the log-normal distribution, which has a relatively long tail at large rp corresponding to a significant number of very large pores in the distribution. This can be seen very clearly in the bottom panel for the curve with o-/F= 1 in which case more than 10% of the pores have radii rp > 2F.

f(i~)n(rp)

7~pdrp = 0

(13)

0

with the current density (iz) given by Eq, (9). Since the positive and negative ions partition differently into the charged pore, the pressure-driven solvent flow causes the convective flux of the counter-ions to be greater than that of the co-ions. This gives rise to an induced streaming potential, which causes an electrophoretic ion flux that balances the excess convective flux of the counter-ions. The net result is that the system as a whole satisfies the constraint that there is no net current flow across the ultrafiltration membrane (in the absence of any externally applied electric field). The magnitude of this induced electric field can be evaluated from Eq. (13) using the previously developed expression for the current density [Eq. (9) ] yielding:

_

where

(qoi \KIM

(14)

S. Saksena,A. Zydney/ Journalof MembraneScience105 (1995)203-215 1.0

Q-

i

oIr

.

207 .

,

.

I

~

i

•

=q

"~ o.8

n,/z

r

x"',

J t ] t Krp)

~=~

p/

""p~'p

0

.~

0.6

f[A°+~{1-l°(i~)I~;rr)}]n(rp)Trr~pdr p 0.4

0

(15)

~

All integrals in Eqs. ( 1 1 ) - ( 1 5 ) were evaluated numerically on an IBM RISC 6000 computer system using available I M S L routines.

it=

0.2

0

"

~

1

,

2

3

I

4

S

Dimensionless Pore Radius, rpJr

Fig. 3. The fractional cumulative flow rate as a function of the pore radius for pressure-driven flow through membranes with different log-normal pore size distributions.

4. Results and analysis The effects o f the membrane pore size distribution on the average solvent flux for purely pressure-driven flow (Ez = 0 ) are shown in Fig. 2. The solvent flux has been non-dimensionalized using the mean pore radius ~, the average value o f the square o f the pore radius ( ~ ) [as suggested by the pressure-driven flow term in Eq. ( 7 ) ] , and the effective pore size s, where s is defined as twice the ratio o f the pore volume to surface area:

f n(rp)~drp f n(rp)rpdrp 0

S ~

(16)

0

40

i

i

i

i

8p. o

30

-~ < r:>

/

.

<

This type of effective pore size is used in the Koze n y - C a r m a n equation to describe the pressure-driven solvent flow through random porous media. Note that for a membrane with a single pore size (rp = r-), s is simply equal to E In each case, the dimensionless solvent flux increases with an increase in the breadth of the pore size distribution due to the large contribution to the flow associated with the largest pores in the distribution. This effect is not related to any changes in membrane porosity since the solvent flux (V) has been defined using the pore area and not the membrane area. The increase in (I,1) with increasing o ' / F i s much less dramatic when the flux is non-dimensionalized by (~p) or s due to the corresponding increase in these parameters with increasing o ' / E There is, however, still a factor of 8 increase in the flux as o ' / F g o e s from 0 to 1 even when (V) is made dimensionless using the parameter s. Thus, the use o f the pore volume to surface area ratio significantly underestimates the effects of the largest pores in the log-normal distribution on the solvent flow. The effect of the pore size distribution on the solvent flux can also be seen by examining the behavior of the fractional cumulative flow rate: rp

E lo

fVn(rp) 7rr~vdrp

0 Z i

0 0

0.2

I

0.4

I

0.6

Scaled Standard Deviation,

fQ=O

I

0.8

(17)

1.0

0/~

Fig. 2. Effect of the log-normalpore size distribution on the pressuredriven solvent flux made dimensionless using the mean pore radius, the average value of the square of the pore radius, and the effective pore size, s [defined by Eq. (16) ].

f Vn( rp) zrr~pdrp 0

which is equal to the fraction o f the total flow accounted for by all pores with radii r_< rp. Fig. 3 shows the results

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

208 1,00

i

p. qpEza 0.75

0.50 .--

~:r = 0 . 2 . .--

.... _~

0.25

m

o

,~

1.00

g 8,

.

K~ = 1.0

.......................

~

= 5.0

I

I

I

I

i

i

i

i

i

/ qpE z r o

0.75

z

0.50

0.25

K~ = 5.0

°o

o12 Scaled

o'., Standard

o18 Deviation,

,.o

motic velocity (V) under these conditions is determined entirely by the thickness of the double layer and is thus independent of the pore radius (the ratio of the Bessel functions in Eq. (7) approaches a value of 1 as KF--, ~ with V being proportional to K- 1 under these conditions). The average solvent flux at large KFis thus completely unaffected by the details of the pore size distribution and is therefore independent of or/F (Fig. 4). The variation of the fractional cumulative flow rate with pore radius at KF----5 (bottom panel in Fig. 5) is thus due to the distribution of the pore area, which is more strongly weighted by the larger pores in the distribution than is FR(rp) due to the ~ dependence of the pore area in the integrals in Eq. (12). The electroosmotic solvent velocity increases as the Debye length increases due to the greater fraction of the pore area occupied by the double layer. Thus for a membrane with a single pore size (i.e. or/F=0), the solvent flux increases by about a factor of 2 as KFgoes from 5 to 0.2. In the limit of very small KF, the electroosmotic velocity becomes linearly proportional to

~/F

Fig. 4. Effectof the log-normalpore size distribution on the electrically-drivenflowfor severalvaluesof the Debyelength. Results are shown for the flux made dimensionless using both the mean pore radius (bottompanel) and the effectivepore size (top panel). for the purely pressure-driven flow (Ez = 0) for several values of the scaled standard deviation. Note that a membrane with a uniform pore size would have a fractional cumulative flow rate given by the Heaviside function with the step increase occurring at rp/F= 1. When or~f= 0.2 almost all of the flow occurs through pores with radii less than 2E However, when or/F= 1 more than 80% of the flow occurs through pores with radii r>_5?. Thus under these conditions, the large majority of the flow occurs through less than 1% of the total number of pores in the distribution (see Fig. 1 ). The corresponding calculations for electricallydriven flow ( d p / d z ) are shown in Fig. 4 and Fig. 5 for several values of the dimensionless Debye length (Kr-). The solvent flux has again been made dimensionless based on the expression in Eq. (7) using both the mean pore radius (bottom panel of Fig. 4) and the effective pore size (top panel in Fig. 4). At high ionic strengths (large Kr-), the electrical double layer extends only a short distance from the pore wall, with most of the pore remaining electrically neutral. The electroos-

0.8

o.42 0.6

g:

0.2

o ,'r

I

0

0

1

2

Dimensionless

i

I

3

4

Pore Radius,

rp/r"

5

Fig. 5. The fractional cumulative flow rate as a function of pore radius for electrically-driven flow through membranes with different log-normal pore size distributions at K f = 5 (bottom panel) and KF=0.2 (top panel).

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

the pore radius (the ratio of Bessel functions in Eq. (7) is equal to (Krp)/4 under these conditions). This leads to a much stronger dependence of the average velocity on the pore size distribution at small rF; thus for K~= 1 the dimensionless solvent flux increases by about a factor of 2.2 as tr/Fgoes from 0 to 1 while for KF= 0.2 the average flux increases by more than a factor of 3.5 over this same range. Note that if the solvent flux is made dimensionless by s (instead of F), the dimensionless flux is relatively independent of o'/Fat KF-----1 (there is actually a slight increase in the flux at small tT/~with a maximum occurring at o'/F= 0.6 due to the competing effects of the pore size distribution on the electroosmotic flow and on the ratio of the pore volume to surface area). In contrast, the flux made dimensionless by the effective pore size decreases with increasing t r / f a t KF= 5 due to the increase in s ( Vis independent of rp at high Krp), but the flux at KF= 0.2 increases with increasing t r / f d u e to the linear dependence of V on rp at small Krp. These different'effects of the pore size distribution on the electroosmotic flow in different ionic strength electrolytes (i.e. at different K) can also be seen in the fractional cumulative flow rate (Fig. 5). The fractional cumulative flow rate is a much stronger function of the pore radius at KF= 0.2 than at KF= 5 due to the much stronger dependence of V on rp at small KE However even at KF= 0.2, the electrically-driven flow has a much weaker dependence on the pore radius than was seen previously for the pressure-driven flow (Fig. 2 and Fig. 3), reflecting the linear dependence of the velocity on rp for the electrically-driven flow (at small Xrp) compared to the ~ dependence for the pressure-driven flow. As discussed previously, the pressure-driven flow of an electrolyte solution through a charged membrane generates an induced electric field (the streaming potential) in order to satisfy the constraint that there is no net current flow through the membrane. The streaming potential can be evaluated from Eqs. (14) and (15), with the results shown in Fig. 6 as a function of K? for several values of o-/E The results have been presented in terms of a dimensionless streaming potential,

~_ ~=

10 0

,

,

209

, ....

i

g ~o e=

,

, ( , / ~_ , , ,

0.5

10"1

E

10-2 c o "a

g ._E a

10"3

,

10 "1

.

,

.

, ,,,I

. . . . . .

10 0

10

Dimensionless Debye Length, ~ F F i g . 6. D i m e n s i o n l e s s s t r e a m i n g p o t e n t i a l a s a f u n c t i o n o f t h e d i m e n sionless Debye

length for several different log-normal

pore size

distributions.

with this particular non-dimensionalization chosen to best highlight the dependence on the solution ionic strength ( K- 1). All calculations were performed using a dimensionless membrane charge density of I

2~-2

7

q r- / _ 2 etzuiRTd

with qualitatively similar results obtained at both higher and lower charge densities. The dimensionless streaming potential displays a relatively weak dependence on the solution ionic strength for KF< 0.3, but it decreases sharply at larger KFdue to the reduction in the thickness of the electrical double layer and the corresponding increase in the centerline conductivity (A°). The streaming potential at very large KF is independent of the pore size distribution since Ez is determined entirely by the thickness of the double layer under these conditions. The effect of the pore size distribution on the streaming potential is much more pronounced at small KF, with the dimensionless streaming potential at KF=0.1 increasing by about a factor of four as t~/F increases from 0 to 1. This increase in the streaming potential with increasing tr/F can be understood as follows. At a constant value of K, the excess convective flux of the counter-ions associated with the pressuredriven flow is most significant in the largest pores in the distribution since they have the largest convective velocity. The increase in tr/F thus causes an overall increase in the excess convective counter-ion flux

S. Saksena, A. Zydney l Journal of Membrane Science 105 (1995) 203-215

210 1.0 A

.

.

.

.

.

.

.

.

,

.

.

.

.

,

>">~

08

1.0

0.6 ~ <

0.4

z

0

.

10 "1

10 0

101

Dimensionless Debye Length, 1

0.50

///

'10

b

so 4o

o o.31

c d

30 10

0.48 0.95

a

III /1 / 0

z

0.25 10

10

0

10

Dimensionless Debye Length, ~ r elf

Fig. 9. Dimensionless streaming potential (top panel) and the normalized solvent velocity (bottom panel) as a function of the inverse Debye length for membranes with different pore size distributions but having the same permeability when qp = 0. Solid symbols are experimental data from [ 1].

defined using the calculated hydraulic permeability, i.e.

re~= ~

I--g-]

(18)

with the results in Fig. 9 shown for reff= 50 ,&. In contrast to the behavior seen in Fig. 6, the dimensionless streaming potential decreases with an increase in the breadth of the pore size distribution, although this effect is fairly small even when rreff

Lihat lebih banyak...
Journal of Membrane Science 105 (1995) 203-215

Pore size distribution effects on electrokinetic phenomena in semipermeable membranes Skand Saksena, Andrew L. Zydney * Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA Received 14 November 1994; accepted 27 February 1995

Abstract Electrokinetic phenomena occurring in charged ultrafiltration membranes can significantly influence the transport characteristics of these membranes. Theoretical calculations have been performed to evaluate the effects of different log-normal pore size distributions on the solvent flow rate, the induced streaming potential, and the membrane zeta potential. The solvent flow rate increased with an increase in the breadth of the pore size distribution, with this effect being much more pronounced for pressuredriven flow than for electrically-driven flow. The streaming potential induced by the convective fluid flow significantly alters the flow profiles through the membrane due to the very different dependence of the pressure-driven and counter-electroosmotic flow on the pore radius. This causes a relatively large negative (reverse) flow to develop through the smallest pores in the distribution. The membrane zeta potential was also a function of the pore size distribution, even for membranes having the same hydraulic permeability and surface charge density. These results provide important insights into the effects of different pore size distributions on membrane transport and on the proper interpretation of these electrokinetic phenomena. Keywords: Pore size distribution; Electroosmosis; Liquid permeability and separations; Theory; Ultrafiltration

1. Introduction Electrostatic and electrokinetic phenomena can significantly affect the performance of many membrane processes by altering the rate o f both solvent and solute transport through the membrane. Most polymeric and ceramic ultrafiltration membranes carry a net negative charge in aqueous solutions at neutral pH. Thus, electrostatic interactions will tend to increase the concentration o f the positively charged (counter-) ions while excluding the negatively charged (co-) ions from the membrane pores. Electrokinetic phenomena are directly associated with the motion o f the fluid; e.g. the ~"Corresponding author. Phone: 302-831-2399, FAX: 302-8311048, E-mail: [email protected] 0376-7388/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

SSDI0376-7388(95)00060-7

solvent flow that is generated by an externally applied electric field (electroosmosis) and the induced electric field caused b y the pressure-driven flow of an electrolyte solution through a charged membrane (the streaming potential). This induced electric field develops because the convective flux of the counter-ions is greater than that of the co-ions due to the preferential partitioning of the counter-ions into the membrane pores; the induced field generates an electrophoretic flux o f the counter-ions that is opposed to the convective flux thereby satisfying the constraint that there is no net current flow through the membrane. These effects are discussed in more detail subsequently. Recent studies of protein transport through semipermeable ultrafiltration membranes have demonstrated that electrostatic interactions can significantly

204

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

alter the protein partition coefficient between the membrane and bulk solution, leading to large changes in the overall rate of protein transport through the membrane [ 1 ]. Pujar and Zydney [ 1 ] have also shown that the streaming potential generated during ultrafiltration can cause a significant electrophoretic protein flux in addition to the normal convective flux. These electrostatic interactions can be used to dramatically increase the selectivity for the membrane separation of protein mixtures by exploiting differences in the net charge of the individual proteins [ 2]. Applied electric fields have been used to control the rate of both solute and solvent transport in a process known as electrofiltration [ 3 ]. Electrokinetic phenomena have also been used quite extensively to evaluate the surface charge characteristics of different membranes [4,5]. For example, Nystrom et al. [5] correlated changes in adsorption and fouling behavior for surface-modified membranes in terms of the membrane zeta potential, with the zeta potential evaluated experimentally from measurements of the streaming potential generated by the pressure-driven flow of an appropriate electrolyte solution through the membrane pores. Theoretical analyses of these electrokinetic phenomena were initially performed for charged circular cylinders with small surface potentials, i.e. under conditions where the classical Debye-Htickel approximation is valid [6]. More recent work has extended these results to higher potentials and various non-circular cross-sections by numerically solving the nonlinear Poisson-Boltzmann equation [7]. One of the major difficulties in applying these theoretical results to solute and solvent transport in real ultrafiltration membranes is the presence of a fairly broad pore size distribution. Mochizuki and Zydney [8] have examined the effects of such pore size distributions on solute transport through semipermeable ultrafiltration membranes by numerically integrating available expressions for the solute flux over an assumed pore size distribution (both a truncated Gaussian and a log-normal distribution). More recently, Causserand et al. [9] performed a series of theoretical calculations for the effects of electrostatic interactions on solute transport through membranes consisting of a log-normal distribution of slit-shaped pores. However, both of these studies neglected the effects of the electrokinetic phenomena on solute and solvent transport,

thereby limiting the range of applicability of these results. The overall objective of this study was to examine the effects of different pore size distributions on both solvent and ion transport through membranes consisting of a parallel array of cylindrical pores. In particular, calculations were performed to determine the effects of the pore size distribution on: ( 1 ) the pressure-driven and electrically-driven solvent flow across the membrane, (2) the development of a streaming potential and its effect on the overall flux of the electrolyte solution, and (3) the evaluation of the membrane surface charge from experimental measurements of these electrokinetic phenomena.

2. Theoretical development The membrane is treated as an array of parallel cylindrical pores, each with a constant circular cross-section. The governing equations for the ion distribution and solvent flux in the individual pores at small values of the surface potential have been presented by Newman [ 10] with the key results summarized here. The electrical potential (qb) is given by Poisson's equation: ~O=

Pc

(1)

where E is the permittivity of the solvent, Pe = FEzici is the local charge density, F is Faraday's constant, and zi and ci are the ion valence and local concentration, respectively. The second derivative of • with respect to axial position (z) is negligible for a pore with a large aspect ratio since the axial electric field [Ez = ( 0 ~ / Oz) ] is assumed to be a constant. The radial concentration profile for the ions in the capillary is described by a Boltzmann distribution: ci = c°exp(

- zi 1/~)

(2)

where c o is the ion concentration at the pore axis (r = 0) and F ~F= ~-~( ~ - ~r=o)

(3)

is the dimensionless potential with R the ideal gas constant, T the absolute temperature, and ~r= o the potential evaluated at the pore axis. Eqs. ( 1 ) - ( 3 ) can be

205

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

solved analytically in the limit of small potentials, i.e. under conditions where the exponentials in Eq. (2) can be approximated by the first term in their Taylor series expansions (this is the classical Debye-Htickel approximation). The solution for a cylindrical pore with a constant surface charge density (%) is given as [ 10]:

~=

qeF [1 --lo(Kr) ] eKRTI1( Krp)"

(4)

where rp is the capillary radius and Io and Ij are modified Bessel functions of the first kind of order zero and one, respectively. K is the reciprocal of the Debye length and is given as:

The local fluid velocity in the axial direction (v~) is evaluated by solving the Navier-Stokes equation including the electrical force term:

dP~-tzO-~-(r-~)+E~p~=O dz r or\ or/

(6)

where /z is the solution viscosity, (dP/dz) is the applied pressure gradient, and Ez is the electric field. Eq. (6) is integrated twice with respect to r using Eq. (2) for the ion concentrations (actually the first term in the Taylor series expansions for c,). The average solvent velocity in the pore (V) is then evaluated by integrating the resulting expression for Vzover the pore cross-section yielding:

V=

[~_e][

Iz(KrP)

( Krp)/1(Krp)

]Ez_ [~]

dP

~

(7)

where/2 is the modified Bessel function of the first kind of order two. The electric field and the solvent flow both give rise to an axial flux of ions through the pore: N i = uiciE z -f- UzC i

(8)

where ui is the ionic mobility. For simplicity, we have assumed that the ions have equal mobilities (as would be the case for KCI solutions); the extension of this analysis to electrolytes with different ionic mobilities is relatively straightforward although algebraically messy. The average current density (equal to the dif-

ference between the flux of the positive and negative ions multiplied by Faraday's constant) is evaluated by integrating Eq. (8) over the cylindrical cross-section using the previously developed expressions for the solvent velocity and ion concentration profiles yielding:

(i~)=Ez

[

A°+

~{_'o(Krp)I2(Krp)ll 1

I~(Krp)

)J

~- q-z I2('%)

(9)

dz K/z Ii (Krp) where A ° = FZ]Ez~uic° is the conductivity of the electrolyte solution evaluated at r = 0 . Eqs. (7) and (9) completely describe the solvent flow rate and ion flux (current density) for a single capillary pore in terms of the electrical and pressure driving forces.

3. Log-normal pore size distribution Although the detailed pore size distribution of available ultrafiltration membranes is unknown, a number of previous investigations of pore size distribution effects have employed the log-normal density function to characterize the pore size distribution and/or evaluate its effect on membrane transport [8,9,11 ]. The log-normal probability density function is a particularly convenient distribution since it is defined only for positive values of the pore radius (as opposed to the standard normal which includes pore radii from positive to negative infinity). A number of different functional forms for the log-normal distribution have appeared in the literature. Although these forms can be shown to be mathematically equivalent [ 11 ], it is most convenient to express the results explicitly in terms of the mean (r-) and standard deviation (tr) of the distribution:

fR(rp) = n(rp) no 1 ~exp

rpl/ 2~'b

2b

(10)

where the parameter b is given as (11)

206

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

r

0.16

o

a

The average solvent flux through a membrane with a log-normal pore size distribution is evaluated by integrating the expression for V [Eq. (7) ] over the distribution:

0.12 0.05 0.2 0.5

0.08

Vn ( re) m~ppdrv n o Q.

- o

0.04

(12)

f n (rp) 7rr~pdrp 0

0 1.0 c/~ = 0.05 o

0.8

~

Note that (V) is the volumetric flow rate per unit pore area and is thus independent of the membrane porosity. The filtrate flux (Jv), which is typically defined as the volumetric flow rate per total membrane area, is equal to E(V) where Eis the membrane porosity. In a typical ultrafiltration system, the filtrate solution is electrically neutral, i.e. there is no net current flow across the membrane. For a membrane with a pore size distribution, this implies that:

1.0

"r_ 0

o (3.

0.6

0.4 0.2 ~ ~ ,

0

i

0 0

03

1,0

1.5

2.0

rpl~ Fig. 1. Representative plots of the log-normal probability density function (top panel) and the cumulative probability function (bottom panel) for several values of the reduced standard deviation. Normalized Pore Radius,

The pore size distributions described by the lognormal density function are shown in Fig. 1 for several values of the reduced standard deviation, or/F. The top panel shows the actual probability density function fR(rp), while the bottom panel shows the cumulative probability function FR(rp), which is defined as the integral offR(rp) and is thus equal to the fraction of pores in the distribution with radius R < rp. As shown in the upper panel, the most likely pore radius (the maximum in the probability density function) occurs at a value of rp/F< 1, with the difference between the most likely radius and the mean pore size increasing with an increase in the breadth of the distribution. This behavior arises from the asymmetry in the log-normal distribution, which has a relatively long tail at large rp corresponding to a significant number of very large pores in the distribution. This can be seen very clearly in the bottom panel for the curve with o-/F= 1 in which case more than 10% of the pores have radii rp > 2F.

f(i~)n(rp)

7~pdrp = 0

(13)

0

with the current density (iz) given by Eq, (9). Since the positive and negative ions partition differently into the charged pore, the pressure-driven solvent flow causes the convective flux of the counter-ions to be greater than that of the co-ions. This gives rise to an induced streaming potential, which causes an electrophoretic ion flux that balances the excess convective flux of the counter-ions. The net result is that the system as a whole satisfies the constraint that there is no net current flow across the ultrafiltration membrane (in the absence of any externally applied electric field). The magnitude of this induced electric field can be evaluated from Eq. (13) using the previously developed expression for the current density [Eq. (9) ] yielding:

_

where

(qoi \KIM

(14)

S. Saksena,A. Zydney/ Journalof MembraneScience105 (1995)203-215 1.0

Q-

i

oIr

.

207 .

,

.

I

~

i

•

=q

"~ o.8

n,/z

r

x"',

J t ] t Krp)

~=~

p/

""p~'p

0

.~

0.6

f[A°+~{1-l°(i~)I~;rr)}]n(rp)Trr~pdr p 0.4

0

(15)

~

All integrals in Eqs. ( 1 1 ) - ( 1 5 ) were evaluated numerically on an IBM RISC 6000 computer system using available I M S L routines.

it=

0.2

0

"

~

1

,

2

3

I

4

S

Dimensionless Pore Radius, rpJr

Fig. 3. The fractional cumulative flow rate as a function of the pore radius for pressure-driven flow through membranes with different log-normal pore size distributions.

4. Results and analysis The effects o f the membrane pore size distribution on the average solvent flux for purely pressure-driven flow (Ez = 0 ) are shown in Fig. 2. The solvent flux has been non-dimensionalized using the mean pore radius ~, the average value o f the square o f the pore radius ( ~ ) [as suggested by the pressure-driven flow term in Eq. ( 7 ) ] , and the effective pore size s, where s is defined as twice the ratio o f the pore volume to surface area:

f n(rp)~drp f n(rp)rpdrp 0

S ~

(16)

0

40

i

i

i

i

8p. o

30

-~ < r:>

/

.

<

This type of effective pore size is used in the Koze n y - C a r m a n equation to describe the pressure-driven solvent flow through random porous media. Note that for a membrane with a single pore size (rp = r-), s is simply equal to E In each case, the dimensionless solvent flux increases with an increase in the breadth of the pore size distribution due to the large contribution to the flow associated with the largest pores in the distribution. This effect is not related to any changes in membrane porosity since the solvent flux (V) has been defined using the pore area and not the membrane area. The increase in (I,1) with increasing o ' / F i s much less dramatic when the flux is non-dimensionalized by (~p) or s due to the corresponding increase in these parameters with increasing o ' / E There is, however, still a factor of 8 increase in the flux as o ' / F g o e s from 0 to 1 even when (V) is made dimensionless using the parameter s. Thus, the use o f the pore volume to surface area ratio significantly underestimates the effects of the largest pores in the log-normal distribution on the solvent flow. The effect of the pore size distribution on the solvent flux can also be seen by examining the behavior of the fractional cumulative flow rate: rp

E lo

fVn(rp) 7rr~vdrp

0 Z i

0 0

0.2

I

0.4

I

0.6

Scaled Standard Deviation,

fQ=O

I

0.8

(17)

1.0

0/~

Fig. 2. Effect of the log-normalpore size distribution on the pressuredriven solvent flux made dimensionless using the mean pore radius, the average value of the square of the pore radius, and the effective pore size, s [defined by Eq. (16) ].

f Vn( rp) zrr~pdrp 0

which is equal to the fraction o f the total flow accounted for by all pores with radii r_< rp. Fig. 3 shows the results

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

208 1,00

i

p. qpEza 0.75

0.50 .--

~:r = 0 . 2 . .--

.... _~

0.25

m

o

,~

1.00

g 8,

.

K~ = 1.0

.......................

~

= 5.0

I

I

I

I

i

i

i

i

i

/ qpE z r o

0.75

z

0.50

0.25

K~ = 5.0

°o

o12 Scaled

o'., Standard

o18 Deviation,

,.o

motic velocity (V) under these conditions is determined entirely by the thickness of the double layer and is thus independent of the pore radius (the ratio of the Bessel functions in Eq. (7) approaches a value of 1 as KF--, ~ with V being proportional to K- 1 under these conditions). The average solvent flux at large KFis thus completely unaffected by the details of the pore size distribution and is therefore independent of or/F (Fig. 4). The variation of the fractional cumulative flow rate with pore radius at KF----5 (bottom panel in Fig. 5) is thus due to the distribution of the pore area, which is more strongly weighted by the larger pores in the distribution than is FR(rp) due to the ~ dependence of the pore area in the integrals in Eq. (12). The electroosmotic solvent velocity increases as the Debye length increases due to the greater fraction of the pore area occupied by the double layer. Thus for a membrane with a single pore size (i.e. or/F=0), the solvent flux increases by about a factor of 2 as KFgoes from 5 to 0.2. In the limit of very small KF, the electroosmotic velocity becomes linearly proportional to

~/F

Fig. 4. Effectof the log-normalpore size distribution on the electrically-drivenflowfor severalvaluesof the Debyelength. Results are shown for the flux made dimensionless using both the mean pore radius (bottompanel) and the effectivepore size (top panel). for the purely pressure-driven flow (Ez = 0) for several values of the scaled standard deviation. Note that a membrane with a uniform pore size would have a fractional cumulative flow rate given by the Heaviside function with the step increase occurring at rp/F= 1. When or~f= 0.2 almost all of the flow occurs through pores with radii less than 2E However, when or/F= 1 more than 80% of the flow occurs through pores with radii r>_5?. Thus under these conditions, the large majority of the flow occurs through less than 1% of the total number of pores in the distribution (see Fig. 1 ). The corresponding calculations for electricallydriven flow ( d p / d z ) are shown in Fig. 4 and Fig. 5 for several values of the dimensionless Debye length (Kr-). The solvent flux has again been made dimensionless based on the expression in Eq. (7) using both the mean pore radius (bottom panel of Fig. 4) and the effective pore size (top panel in Fig. 4). At high ionic strengths (large Kr-), the electrical double layer extends only a short distance from the pore wall, with most of the pore remaining electrically neutral. The electroos-

0.8

o.42 0.6

g:

0.2

o ,'r

I

0

0

1

2

Dimensionless

i

I

3

4

Pore Radius,

rp/r"

5

Fig. 5. The fractional cumulative flow rate as a function of pore radius for electrically-driven flow through membranes with different log-normal pore size distributions at K f = 5 (bottom panel) and KF=0.2 (top panel).

S. Saksena, A. Zydney / Journal of Membrane Science 105 (1995) 203-215

the pore radius (the ratio of Bessel functions in Eq. (7) is equal to (Krp)/4 under these conditions). This leads to a much stronger dependence of the average velocity on the pore size distribution at small rF; thus for K~= 1 the dimensionless solvent flux increases by about a factor of 2.2 as tr/Fgoes from 0 to 1 while for KF= 0.2 the average flux increases by more than a factor of 3.5 over this same range. Note that if the solvent flux is made dimensionless by s (instead of F), the dimensionless flux is relatively independent of o'/Fat KF-----1 (there is actually a slight increase in the flux at small tT/~with a maximum occurring at o'/F= 0.6 due to the competing effects of the pore size distribution on the electroosmotic flow and on the ratio of the pore volume to surface area). In contrast, the flux made dimensionless by the effective pore size decreases with increasing t r / f a t KF= 5 due to the increase in s ( Vis independent of rp at high Krp), but the flux at KF= 0.2 increases with increasing t r / f d u e to the linear dependence of V on rp at small Krp. These different'effects of the pore size distribution on the electroosmotic flow in different ionic strength electrolytes (i.e. at different K) can also be seen in the fractional cumulative flow rate (Fig. 5). The fractional cumulative flow rate is a much stronger function of the pore radius at KF= 0.2 than at KF= 5 due to the much stronger dependence of V on rp at small KE However even at KF= 0.2, the electrically-driven flow has a much weaker dependence on the pore radius than was seen previously for the pressure-driven flow (Fig. 2 and Fig. 3), reflecting the linear dependence of the velocity on rp for the electrically-driven flow (at small Xrp) compared to the ~ dependence for the pressure-driven flow. As discussed previously, the pressure-driven flow of an electrolyte solution through a charged membrane generates an induced electric field (the streaming potential) in order to satisfy the constraint that there is no net current flow through the membrane. The streaming potential can be evaluated from Eqs. (14) and (15), with the results shown in Fig. 6 as a function of K? for several values of o-/E The results have been presented in terms of a dimensionless streaming potential,

~_ ~=

10 0

,

,

209

, ....

i

g ~o e=

,

, ( , / ~_ , , ,

0.5

10"1

E

10-2 c o "a

g ._E a

10"3

,

10 "1

.

,

.

, ,,,I

. . . . . .

10 0

10

Dimensionless Debye Length, ~ F F i g . 6. D i m e n s i o n l e s s s t r e a m i n g p o t e n t i a l a s a f u n c t i o n o f t h e d i m e n sionless Debye

length for several different log-normal

pore size

distributions.

with this particular non-dimensionalization chosen to best highlight the dependence on the solution ionic strength ( K- 1). All calculations were performed using a dimensionless membrane charge density of I

2~-2

7

q r- / _ 2 etzuiRTd

with qualitatively similar results obtained at both higher and lower charge densities. The dimensionless streaming potential displays a relatively weak dependence on the solution ionic strength for KF< 0.3, but it decreases sharply at larger KFdue to the reduction in the thickness of the electrical double layer and the corresponding increase in the centerline conductivity (A°). The streaming potential at very large KF is independent of the pore size distribution since Ez is determined entirely by the thickness of the double layer under these conditions. The effect of the pore size distribution on the streaming potential is much more pronounced at small KF, with the dimensionless streaming potential at KF=0.1 increasing by about a factor of four as t~/F increases from 0 to 1. This increase in the streaming potential with increasing tr/F can be understood as follows. At a constant value of K, the excess convective flux of the counter-ions associated with the pressuredriven flow is most significant in the largest pores in the distribution since they have the largest convective velocity. The increase in tr/F thus causes an overall increase in the excess convective counter-ion flux

S. Saksena, A. Zydney l Journal of Membrane Science 105 (1995) 203-215

210 1.0 A

.

.

.

.

.

.

.

.

,

.

.

.

.

,

>">~

08

1.0

0.6 ~ <

0.4

z

0

.

10 "1

10 0

101

Dimensionless Debye Length, 1

0.50

///

'10

b

so 4o

o o.31

c d

30 10

0.48 0.95

a

III /1 / 0

z

0.25 10

10

0

10

Dimensionless Debye Length, ~ r elf

Fig. 9. Dimensionless streaming potential (top panel) and the normalized solvent velocity (bottom panel) as a function of the inverse Debye length for membranes with different pore size distributions but having the same permeability when qp = 0. Solid symbols are experimental data from [ 1].

defined using the calculated hydraulic permeability, i.e.

re~= ~

I--g-]

(18)

with the results in Fig. 9 shown for reff= 50 ,&. In contrast to the behavior seen in Fig. 6, the dimensionless streaming potential decreases with an increase in the breadth of the pore size distribution, although this effect is fairly small even when rreff

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