Synthetic Membranes: Volume II - American Chemical Society

23 Polymer Solute Rejection by Ultrafiltration Membranes L E O S Z E M A N and M I C H A E L W A L E S Downloaded by UCSF LIB CKM RSCS MGMT on November 25, 2014 | http://pubs.acs.org Publication Date: May 27, 1981 | doi: 10.1021/bk-1981-0154.ch023

Abcor, Inc., 850 Main Street, Wilmington, MA 01887

Ultrafiltration membranes are used, both on industrial s c a l e as w e l l as in l a b o r a t o r i e s , f o r fractionation, purification, s e p a r a t i o n and c o n c e n t r a t i o n of water s o l u t e or water dispersible m a t e r i a l s . R e j e c t i o n o f the s o l u t e (or dispersed c o l l o i d ) is, together w i t h permeate flux, one of the two key performance parameters of any ultrafiltration membrane. The values of rejection coefficients are o f crucial importance in many a p p l i c a t i o n s of ultrafiltration. The o b j e c t i v e o f this c o n t r i b u t i o n is t o consider and analyze the i n d i v i d u a l f a c t o r s a f f e c t i n g rejection o f polymer s o l u t e s by ultrafiltration membranes. The f a c t o r s that will be considered i n c l u d e steric rejection ( s i e v i n g ) , s o l u t e velocity l a g and solute-membrane interaction. Our a n a l y s i s e x p l o i t s h e a v i l y a model concept of a s p h e r i c a l s o l u t e in a cylindrical capillary and we do not want t o dispute the simplicity o r inadequacy inherent i n this model. However, we want t o demonstrate that even w i t h i n the framework of this idealized model, u s e f u l p r e d i c t i o n s about the membrane rejection behavior can be made. We are not going t o d i s c u s s here the e f f e c t s of f o u l i n g (adsorption) o r of electrostatic charge, even if we have t o bear in mind that these may be overwhelmingly important in many situations. The d i s c u s s i o n will a l s o consider only a case of a preponderantly convective s o l u t e transport w i t h a negligible contribution due t o diffusion. Steric

Rejection

In h i s w e l l known d e r i v a t i o n of a formula f o r s t e r i c r e j e c t i o n , J . D. Ferry (1) a r r i v e d i n 1936 a t a simple r e l a t i o n between the s o l u t e r e j e c t i o n c o e f f i c i e n t , R 2 , and the s o l u t e t o pore diameter r a t i o , A, where R

2

3 0097-6156/81/0154-041l$06.00/0 © 1981 American Chemical Society

In Synthetic Membranes: Volume II; Turbak, A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

412

SYNTHETIC

R

2

= 1-4

/ o

MEMBRANES:

3

HF

AND

2

($-3 )dB = ( X ( 2 - X ) ) ; f o r X1

(lc)

These r e l a t i o n s can be used as rough estimates of s t e r i c r e j e c t i o n , i f the s o l u t e and membrane pore dimensions are known. The d e r i v a t i o n i s based on a s t r i c t l y model s i t u a t i o n (see Figure 1) and a long l i s t of necessary assumptions can be w r i t t e n . Apart from the s i m p l i f i e d geometry (hard sphere i n a c y l i n d r i c a l pore), i t was a l s o assumed that the s o l u t e t r a v e l s a t the same v e l o c i t y as the surrounding l i q u i d , that the s o l u t e c o n c e n t r a t i o n i n the a c c e s s i b l e p a r t s of the pore i s uniform and equal t o the c o n c e n t r a t i o n i n the feed, that the f l o w p a t t e r n i s laminar, the l i q u i d i s Newtonian, d i f f u s i o n a l c o n t r i b u t i o n to s o l u t e t r a n s port i s n e g l i g i b l e (pore P e c l e t number i s s u f f i c i e n t l y h i g h ) , c o n c e n t r a t i o n p o l a r i z a t i o n and membrane-solute i n t e r a c t i o n s are absent, e t c . Solute V e l o c i t y Lag In g e n e r a l , a sphere moving through a c y l i n d r i c a l pore does not move w i t h the same v e l o c i t y as the surrounding f l u i d . Consequently, the formula ( l b ) has t o be c o r r e c t e d f o r t h i s e f f e c t . I n 1975, Paine and Scherr (2) c a l c u l a t e d drag coeff i c i e n t s k , k i which weight the c o n t r i b u t i o n s of the sphere and f l u i d v e l o c i t i e s t o the drag f o r c e F: 2

F = -6TTTVL

(k v 2

2

- kivx)

(2)

The drag c o e f f i c i e n t s k]^ and k are both a f u n c t i o n of X and 3 (dimensionless d i s t a n c e of sphere's center from the pore's a x i s ) . The dependence of k]^ and k on 3 can be neglected without too much e r r o r (2) and the r a t i o of k i / k can be considered t o depend only on X. In a steady-state s i t u a t i o n , the drag f o r c e has to be zero and a constant s o l u t e v e l o c i t y l a g can be described by an equation: 2

2

2

vo k-. 2 — = i~ = exp (-0.7146X ) l 2 V

(3)

k

The right-hand s i d e of Equation (3) was obtained by a l e a s t square f i t of a f u n c t i o n (exp (-°cX) on the values of k]^ and k reported by Paine and Scherr (2) f o r d i f f e r e n t values of X. Paine and S c h e r r s values represent a r i g o r o u s t h e o r e t i c a l s o l u t i o n f o r a c e n t e r - l i n e motion of a r i g i d sphere i n s i d e a c y l i n d r i c a l tube. Applying t h i s c o r r e c t i o n , we can then w r i t e 2

2

f


23.

ZEMAN

Polymer Solute Rejection

AND WALES

413

for solute rejection: R

2

2

= 1 - ( 1 - ( A ( 2 - A ) ) ) exp (-0.7146A )

2

(4)

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The magnitude of the s o l u t e v e l o c i t y l a g c o r r e c t i o n i s shown i n F i g u r e 2. As seen, f o r a r i g i d sphere i n a c y l i n d e r , t h i s c o r r e c t i o n i s not too l a r g e . N e v e r t h e l e s s , we w i l l keep c o n s i d e r i n g i t i n f u r t h e r d i s c u s s i o n s . The c o r r e c t i o n c a l c u l a t e d from Equation (16b) of Anderson and Quinn ( 7 ) , a p p l i c a b l e f o r A

3

3

f

atom-3

M 3

6 x

0

t

where

3

and 2

= (s sin

2

2

0 + 3 -23s sin0 s i n ( ) ) )

1/2

(28)

The c l u s t e r of constants i n Equation (27) can be s i m p l i f i e d by u s i n g the Hamaker constant, H 3 3 : P

N

17

3 a

2 •

H33 = ( - V - ) 2

2

-

(29)

and t h e r e f o r e * (B,A) =

H

3 3

' I (B X)

(30)

f

TT

The f a c t that the s o l u t e (component 2) and the membrane (component 3) a r e made from two d i f f e r e n t m a t e r i a l s separated by the solvent (component 1) w i l l be r e f l e c t e d by the use of a composite Hamaker constant. 1/2 H

213

=

( H

1/4 2 (H

11

- 22

V

>

(31)

The energy of i n t e r a c t i o n f o r a s p h e r i c a l s o l u t e of r a d i u s X i n a s o l v e n t - f i l l e d i n f i n i t e c y l i n d r i c a l pore w i t h a r a d i u s 3=1 and a t a r a d i a l p o s i t i o n 3 i s t h e r e f o r e

* (3'X) = ^

H

2 1 3

I (3'X)

(32)

TT

The s o l u t e c o n c e n t r a t i o n p r o f i l e w i t h i n the a c c e s s i b l e p a r t of the pore w i l l be determined by the Boltzmann d i s t r i b u t i o n law.


422

MEMBRANES:

HF

AND

UF


SYNTHETIC

Figure 6b.

Schematic of a spherical solute within the pore


USES

23.

Polymer

Z E M A N AND WALES

C

Solute

Rejection

423

= C exp (-$/kT)

2

(33)

2

o -14 The value o f kT considered (25 C) i s 4.12 x 10 e r g . R a d i a l d i s t r i b u t i o n o f s o l u t e molecules i n the pore does not change the average s o l u t e c o n c e n t r a t i o n i n the pore and t h e r e f o r e ,2 Tr(l-A)

C

I"*

_

= C -2ir / exp

2 j F

2

-

$

(

dg,

(34)


wherefrom: _ c

(1-A)

C 2

=

"

2

(35)

* i=r

2

e x p ( " T j ) 6 dg

/

Equations (32), (33) and (35) a l l o w us t o c a l c u l a t e C / C p as a f u n c t i o n o f B and A f o r d i f f e r e n t values of H ^3 (Hamaker cons t a n t ) . Due t o the complexity o f c a l c u l a t i o n ( e v a l u a t i o n o f two t r i p l e i n t e g r a l s ) , t h i s i s best done on a computer. T y p i c a l r e s u l t s of computer c a l c u l a t i o n s are shown i n F i g u r e 7. The e f f e c t of Van der Waals a t t r a c t i v e f o r c e s w i l l lead t o an accumulation of s o l u t e molecules near the w a l l s f o r s m a l l values of A but a t l a r g e values of A , the p o s i t i o n s c l o s e t o the pore a x i s s t a r t being preferred. The value o f H ^3 chosen i n our c a l c u l a t i o n i s r a t h e r l a r g e . The expected magnitude of Hamaker constants would be between 2 x 1 0 " - 5 x l O " ^ erg depending on the " h y d r o p h i l i c i t y " o f both the s o l u t e and the membrane. To c a l c u l a t e r e j e c t i o n c o e f f i c i e n t s , we use the formula 2

2

2

2

13

1

/

R

9 1

= 1 -£ C

1-A J

/ C (B-B ) dB

2,F o

9

(36)

1

that accounts f o r the presence of a c o n c e n t r a t i o n gradient w i t h i n the a c c e s s i b l e part o f the pore. R e j e c t i o n curves c a l c u l a t e d from Equations (32), (33), (35) and (36) are shown i n F i g u r e 8. I t i s seen t h a t the e f f e c t i s most pronounced below A = 0.7 and i t s magnitude depends on the value o f the r e s p e c t i v e Hamaker constant. The r e j e c t i o n c o e f f i c i e n t i s increased by the a t t r a c t i v e f o r c e s between the s o l u t e and the membrane. I t i s t o be expected that f o r l a r g e values o f H, a d s o r p t i o n a l s o occurs and the pore dimensions are changed correspondingly. The e f f e c t s of adsorbed s o l u t e l a y e r s may be very important, but these were not considered i n our a n a l y s i s o f the solute-membrane interaction effects.


424

MEMBRANES

HF

AND

UF

USES


SYNTHETIC

Figure 7.

Example of calculated concentration profiles for the value of H i X 10~ erg and different values of A and f$

2 3

13


= 1.5

23.

ZEMAN


AND WALES

Concentration

425

Polarization

T y p i c a l r e s u l t s of an u l t r a f i l t r a t i o n experiment a l s o r e f l e c t the presence of c o n c e n t r a t i o n p o l a r i z a t i o n . This phenomenon, i . e . accumulation of s o l u t e i n f r o n t of the membrane, was described i n great d e t a i l by others (Refs. 3, 4 ) . A consequence of c o n c e n t r a t i o n p o l a r i z a t i o n i s a strong dependence of measured r e j e c t i o n c o e f f i c i e n t s on transmembrane f l u x e s . An i l l u s t r a t i o n of the e f f e c t i s presented i n Figure 9, which shows the measured "apparent" r e j e c t i o n c o e f f i c i e n t s (R ) as a f u n c t i o n of transmembrane f l u x f o r two water-soluble polymers (Tetronic 707 and Carbowax 4000). I t i s c l e a r from F i g u r e 9 that i f we want t o minimize the e f f e c t s of c o n c e n t r a t i o n p o l a r i z a t i o n , we have to conduct experiments at very low values of transmembrane f l u x .


a

Theory of S t e r i c R e j e c t i o n and Experimental

Results

The experimental r e s u l t s are going to be presented i n d e t a i l elsewhere and only a b r i e f p r e s e n t a t i o n w i l l be given below. The p r e d i c t i v e power of Equation (4) was t e s t e d w i t h defined polymeric s o l u t e s and track-etched Nuclepore f i l t e r s . The polymers used were: l i n e a r polyethylene oxides, (Carbowaxes s u p p l i e d by Union Carbide C o r p o r a t i o n ) , and Dextran T f r a c t i o n s , (Pharmacia Fine Chemicals). For Carbowaxes, the s o l u t e r a d i i used i n c a l c u l a t i o n s were mean r a d i i of g y r a t i o n c a l c u l a t e d from molecular weights v i a the Flory-Fox equation and u s i n g the Mark-Houwink constants given i n reference (5). For n o n - l i n e a r dextrans, we used the Stokes r a d i i c a l c u l a t e d from molecular weights using the c o r r e l a t i o n of data reported by Granath and K v i s t (6). Both types of polymer have very narrow d i s t r i b u t i o n s of molecular weight. The s o l u t e r a d i i are summar i z e d i n Table I I I . Table I I I Solute R a d i i

Solute

M

w

(Daltons)

R a d i u s

(

Carbowax 4000

4,010

25.4

Carbowax 6000

7,000

34.6

Dextran T10

10,500

23.8

Dextran T40

39,500

44.4

Dextran T70

68,500

57.5

&

}

Mean r a d i u s of g y r a t i o n Stokes Radius

Nuclepore f i l t e r s used have s t r a i g h t - t h r o u g h pores w i t h diameters s p e c i f i e d by Nuclepore Corporation as 150, 300 and 500A. These are the s o - c a l l e d " r a t e d " pore diameters and they represent



426

SYNTHETIC MEMBRANES: HF AND UF USES

Figure 8. Effect of solute-membrane interaction on rejection. Rejection curves calculated for H = 0, H = 0.8 X IO' erg, and H = 2.0 X iO' erg. 13

13

213

213

1.0

213

THEORY

- = 1-R

K

— e (Brian's Model) 1(dalM tWons)Q(GPM)(cm/s) iR OTetronic 707 12000 3.5 6.4xl0' 0.97 ACarbowax 4000 3500 1.7 3.7xl0" 0.83 V Membrane: ABC0R HFM 100 a

Rl

k

R

O

3

3

0.5

-

0

100 200

300 400 500 600 700 800 FLUX (J), GFD

900 1000

Figure 9. Effect of concentration polarization. Theoretical curves calculated for the values of R i and k specified in the figure. Experimental data for Tetronic 707 (O) and Carbowax 4000 (A) ultraflltered through the ABCOR HFM 100 membrane.



23.

ZEMAN

AND

WALES


All

the maximum v a l u e . According to Nuclepore l i t e r a t u r e , the a c t u a l pore s i z e s should not vary more than +0% to -20% from the r a t e d values. For each f i l t e r - s o l u t e combination, A was c a l c u l a t e d as a r a t i o of the s o l u t e r a d i u s to the " r a t e d " pore r a d i u s . The p r e d i c t e d value of the r e j e c t i o n c o e f f i c i e n t was then c a l c u l a t e d from Equation (4). The comparison between the p r e d i c t e d values and those a c t u a l l y measured w i t h Carbowaxes 4000 and 600 i s shown i n F i g u r e 10a and b, r e s p e c t i v e l y . The measurements were c a r r i e d out at s e v e r a l values of AP i n order to assess the importance of c o n t r i b u t i o n from c o n c e n t r a t i o n p o l a r i z a t i o n . Using the GPC a n a l y s i s of feed and permeate s o l u t i o n s (Figure 11 a,b), we a l s o c a l c u l a t e d a p a r t of the s o l u t e r e j e c t i o n curve for the given f i l t e r s . The example of such a curve i s a Carbowax r e j e c t i o n curve f o r Nuclepore 150% (diameter) f i l t e r (Figure 11c). The agreement between experimental data and the p r e d i c t i o n of Equation (4) ( s o l i d l i n e ) i s very good. A s i m i l a r t e s t was performed w i t h Dextran T f r a c t i o n s and Nuclepore 150&, 300&, 500X (diameter) f i l t e r s . The r e s u l t s are summarized i n F i g u r e 12. The experimental p o i n t s were obtained both from s i n g l e s o l u t e measurements (+ - r e j e c t i o n of T10 and T70 by the 150& f i l t e r ) and from a n a l y s i s of the GPC t r a c e s ( F 1 ~ 500& f i l t e r , A-300& f i l t e r , o-1508 f i l t e r ) . The agreement between the experimental data and the p r e d i c t i o n of Equation (4) ( s o l i d l i n e ) i s again s u r p r i s i n g l y good, c o n s i d e r i n g the crudeness of assumptions i n v o l v e d i n i t s d e r i v a t i o n . Simultaneous R e j e c t i o n Measurements The GPC a n a l y s i s of feed and permeate s o l u t i o n i s i d e a l l y s u i t e d f o r r a p i d simultaneous r e j e c t i o n measurements. S i m u l t a eous r e j e c t i o n i s of great importance i n u l t r a f i l t r a t i o n p r a c t i c e . As an example, we show here a simultaneous measurement of r e j e c t i o n of p r o t e i n s and of l a c t o s e i n whey u l t r a f i l t r a t i o n (Figure 13). The membrane used was the ABCOR HFK membrane and the feed s o l u t i o n had a t y p i c a l composition of a p a r t i a l l y concentrated whey stream. The feed ( ) and the permeate (- - -) s o l u t i o n s were analyzed by GPC (Waters 1-125 columns) w i t h simultaneous monitoring of uv absorbance ( A 2 8 0 ) * °^ r e f r a c t i v e index d i f f e r e n c e (ARI). The a n a l y s i s shows a q u a n t i t a t i v e r e j e c t i o n (R=100%) of a l l whey p r o t e i n s and a very low r e j e c t i o n (R=8%) of l a c t o s e . anc

Conclusions According to our a n a l y s i s , the predominant e f f e c t c o n t r o l l i n g r e j e c t i o n of polymeric s o l u t e s by uncharged u l t r a f i l t r a t i o n membranes i s the s t e r i c f a c t o r determined by the v a l u e of parameter A. The c o n t r i b u t i o n s to r e j e c t i o n from hydrodynamic l a g , Van der Waals a t t r a c t i o n between the s o l u t e and the membrane can be r e garded as of r e l a t i v e l y minor importance.


SYNTHETIC

428

MEMBRANES:

H F A N D U F USES


N150A

N300A ^ 10

20

30

AP(psi)

N500A

40

N150A

N300A

A N500A

10

"S

TT

4,

1

20

30

i

AP(psi)

40

L

Figure 10. R(%) as a function of A P for Nuclepore 150 A (Q), Nuclepore 300 A (A), and Nuclepore 500 A ([7]): (a) 0.1 Carbowax 4000; (b) 0.1 Carbowax 6000. Solid lines show theoretical predictions according to Equation 4.



23.

ZEMAN

AND


WALES

429

R(%)

o o /

40

/

20

1

10

1 20

30

40

50

RADIUS OF G Y R A T I O N (A)

Figure 11. Measured apparent rejection of polyethylene oxide (Carbowax) at AP = 50 psi: (a) GPC trace of a blend solution containing 0.02% of each Carbowax 1000, 1400, 1540, 4000, and 6000; (b) GPC trace of a permeate obtained by UF at AP == 50 psi through Nuclepore 150 A membrane; (c) apparent rejection calculated from GPC traces shown in a and b, (Q), as a function of solute radius of gyration. The solid line shows theoretical prediction according to Equation 4.



430

SYNTHETIC

MEMBRANES:

HF

AND

UF

USES

Figure 12. Measured apparent rejection of dextrans by Nuclepore filters calculated from GPC traces as a function of A. Points calculated from Nuclepore 150 A (O), 300 A (A), and 500 A ([J) traces. The solid line shows theoretical prediction according to Equation 4. Rejection coefficients measured for single dextran fractions (T10 and T70) and the Nuclepore 150 A filter are shown also (+).



23.

ZEMAN

AND

WALES


431

AR.I.

20 15 —ELUTION VOLUME, ML Figure 13. Simultaneous rejection measurement by GPC. The GPC profiles: ARI trace (upper,) and A o trace (lower) for whey feed and permeate obtained by UF through the ABCOR HFK membrane. The peak labelled IgG corresponds to whey immunoglobulines; peaks labelled oc and /3 correspond to oc -lactalbumin and /3-lactoglobulin, respectively. 28


432

SYNTHETIC

MEMBRANES:

HF AND U F

USES

The t y p i c a l u l t r a f i l t r a t i o n r e s u l t s w i l l r e f l e c t e f f e c t s of the r e s p e c t i v e s i z e d i s t r i b u t i o n s of both the s o l u t e and the membrane pores, as w e l l as o f c o n c e n t r a t i o n p o l a r i z a t i o n . A l l of these e f f e c t s should be expected to lower the membrane r e j e c t i o n coefficient. Acknowledgement


We thank Mr. Michael Morin of ABCOR, INC. f o r i n v a l u a b l e help i n performing the computer c a l c u l a t i o n s . L i s t of Symbols 2 A

Membrane area, cm

A^

C o e f f i c i e n t s i n Equation (19), dimensionless

a

o -b Constant i n Equation ( 7 ) , A d a l t o n

b

Constant i n Equation ( 7 ) , dimensionless -3

C^

Solute c o n c e n t r a t i o n i n the feed, g cm

F

C^ p f(M«)

-3 Solute c o n c e n t r a t i o n i n the permeate, g cm Solute d i f f e r e n t i a l molecular weight dalton'

distribution,

1

F

Viscous drag f o r c e , dyne

H

Hamaker constant, e r g

I

Integrals Solvent f l u x , cm s ^ -2 -1

J

Solute f l u x , g cm s

2

k

Mass t r a n s f e r c o e f f i c i e n t , cm s ^ (Figure 9)

k

Boltzmann constant, e r g °K

k^,k^

Drag c o e f f i c i e n t s , dimensionless o

1^

Pore l e n g t h , A

M^

Solute molecular weight, d a l t o n

M^

Number average s o l u t e molecular weight, d a l t o n


23.

ZEMAN

M

AND WALES

0

z,w


433

Weight average s o l u t e molecular weight, d a l t o n Membrane polymer molecular weight, d a l t o n -2 Number of pores per u n i t membrane area, cm -2

N


» N

Number of holes per u n i t membrane area, cm

N^

Arogadro number, mole ^

r

D i s t a n c e , cm

r^ R or R^

o Solvent molecule r a d i u s , A o Solute r a d i u s , A o Membrane pore r a d i u s , A Solute r e j e c t i o n c o e f f i c i e n t , dimensionless

R

Apparent r e j e c t i o n c o e f f i c i e n t , dimensionless

r^ r^

a

s

D i s t a n c e , cm o

T

Absolute temperature,

K

vv^

Solvent v e l o c i t y , cm s ^ Solute v e l o c i t y , cm s ^

V

Volume, cm

W^

Solute hindrance c o e f f i c i e n t , dimensionless

x


y


3

Angle, r a d i a n

$

Van der Waals i n t e r a c t i o n energy, e r g

0)

Angle, r a d i a n

Subscripts

1

Solvent

2

Solute

3

Membrane

Literature Cited 1.

F e r r y , J. D., Chem. Rev., 1936, 18, 373

2.

Paine, P. L.; Scherr, P., B i o p h y s i c a l J., 1975, 15, 1087

3.

B r i a n , L. T., " D e s a l i n a t i o n by Reverse Osmosis", Merten, U., Ed., The MIT P r e s s , Cambridge, Massachusetts, 1966, 101

4.

B l a t t , W. F.; David, A.; M i c h a e l s , A. S.; Nelsen, L., "Membrane Science and Technology", F l i n n , J . E., Ed., Plenum P u b l i s h i n g C o r p o r a t i o n , New York, 1970, 47

5.

Ring, W., Cantow, H. J., H o l t r u p , H., European Polymer J., 1966, 2, 151

6.

Granath, K. A.; K v i s t , B. A., J. Chromatogr., 1967, 28, 69-81

7.

Anderson, J. L.; Quinn, J. A., Biophys. J., 1974, 14, 130-150

RECEIVED

December

4,

1980.


Synthetic Membranes: Volume II - American Chemical Society

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