Membrane potential of separation membranes as affected by ion adsorption

Journal of Membrane Scrence, 71 (1992) 189-200

189

Elsevler Science Publishers B V , Amsterdam

Membrane potential of separation membranes as affected by ion adsorption* Ryosuke Takagl” and Masayuki Nakagaklb “Shukugawa GakuLnJunror College, 6-58 Koshrkuwa-cho, Nzshmomlya 662 (Japan) bHosha Untversrty, Ebara 2-4-41, Shmagawa-ku, Tokyo 142 (Japan)

(Received October 31,1990, accepted m revised form March 2,1992)

Abstract

Recently it has been reported that membrane charge density depends on the bulk concentration In this paper, we mvestigate the effect of ion adsorption on membrane charge and on membrane potential by using the selective ion adsorption model, winch considers that a membrane which has no fixed charge can adsorb anions seleckvely We find that the membrane potentml depends on the bulk concentration as if the membrane Itself had a fixed charge However, the magnitude of the membrane potential m the reDon of low bulk concentration 1s determmed by the adsorptlon coefficient and the saturated ion concentration of adsorbed ions These parameters can be determmed by analysis of the membrane potential using our model After obtaining values of parameters wluch charactenze the membrane, the vmation of the membrane charge density and the mterfaclal potential difference caused by ion adsorption can be predicted theoretically

Keywords

electrochemistry,

ion adsorption,

membrane potent&,

Introduction Smce particles in aqueous mela are usually charged electrically, the filtration efficiency of a separation membrane depends on the electric potential of the membrane surfaces. The interfacial potential difference of the membrane can be determined by measuring the membrane potential, that E., the potential difference between two bulk solutions contacting the faces of the membrane, because the membrane poCorrespondence to Dr R Takag, Shukugawa Gakum Junior College, 6-58 Koshdcnwa-cho, Nlshmomlya 662, Japan *Paper presented at the 5th World Flltratlon Congress, Nice, France, June 5-8,199O

theory, ultrafiltration

tentlal is a function of the interfacial potential difference as well as the &ffuaon potential across the membrane. In order to calculate a correct value of the mterfaclal potential hfference, we have to employ a suitable model for a real membrane, especially for membrane charge, because the interfacial potential difference 1s dependent on the electric charge density on the surface The characterization of charged membranes has been studed by many mvestlgators [l-5]. However, their attention has mostly been focused on the fixed charge of a uniform membrane On the other hand, in stu&es of sorption equilibria for celluloslc membranes, It has been reported that lomc partltlon equlhbrla m highly

0376-7388/92/$05 00 0 1992 Elsevler Science Publishers B V All rights reserved

190

R Takagr and M Nakagakr/J

ddute solutions could not be interpreted satisfactorily in terms of simple Donnan equrhbrium theory [6-g]. The dependence of fixed charge density on bulk concentration for a muscovite mica membrane has also been reported [lo] The present authors reported that the membrane potential of collodion membranes deviates from the Teorell-Meyer-Sievers model in highly dilute bulk solutions [ 111. Many efforts have been made to establish a suitable model for a real membrane. The models that have been proposed include Glueckauf’s model [ 121, the space charge model [ 10,13-151 and the non-equipotential volume membrane model [6,7]. However, we have not yet obtained a quantitatively suitable model. If a membrane, such as.an affimty membrane [16], adsorbs ions selectively, the effective membrane charge will vary with the bulk concentration [ 171 and the partition of ions into the membrane will deviate from the simple Donnan equilibrium. Thus selective ion adsorption could significantly influence the membrane potential. We have already reported the effect of ion adsorption on membrane potential for a membrane which has a homogeneous distribution of fixed charges [ 111. In this paper, we discuss theoretically the effect of selective ion adsorption on the membrane potential of an uncharged membrane in order to make clear the effect of ion adsorption.

0 0

Membrane SCL 71(1992) 189-200

O Oo

0

tree

0

free anlo”

CatIon

0

adsorbed amon

C,

Donnan equlllbrlum

w

Langmuw adsorptlon

Fig 1 Theoretwl model of membrane Membrane IS assumed to have no fixed charge and to adsorb amons selectwely

permeate the membrane The membrane itself is assumed to be symmetrical, with both surfaces havmg the same properties. Among partitioned ions m the membrane pore, either cations or anions are selectively adsorbed by the pore wall accordmg to the Langmuir model of localized adsorption. The adsorbed ions bound at the pore wall give the electric charge to the membrane and affect the Donnan equilibrium. In the stationary state, the adsorbed ions are m equilibrium with free ions in the pore. The Donnan equilibrium is set up between free ions m the bulk solution and at the pore end This Donnan equihbrium corresponds to the contmuous Donnan model [ 181, which is considered convement for the potential distribution across a charged membrane [ 19,201. The electroneutrahty condition is satisfied in all parts of the membrane system. Membrane potential

Theory Membrane model We propose the membrane model shown m Fig. 1. This model is based on a pore model in common use for swollen membranes. This model is available for gel membranes as well: if membrane pores are connected with each other, the pore model is equivalent to a gel model except for the length of the path by which ions

The membrane system is assumed to be in the stationary state in which no electric current flows through the membrane. For simplicity, we treat hereafter a membrane system which contains one kmd of uni-univalent electrolyte (I+, a- ) The concentration of each ion is C1z m bulk solution 1 and C,, m bulk solution 2 The bulk concentration ratio C1n/CPBis r The membrane IS assumed to have no fixed charge and to adsorb anions selectively For this

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Membrane Scl 71(1992) 189-200

membrane system, the model shown in Fig. 1 is expressed as follows: (1)

brane is obtained by integrating eqn. (5 ) over free ion concentrations within the membrane [5,21 I. dE

%4=-

(2)

hb&lM/(l+~[alM)

at~ldbl~

= kaM4/t&)-’

(3)

where the subscripts M and B indicate membrane and bulk solution, respectively. Square brackets indicate the concentrations, with subscripts M and B m~cating the concentrations of free ions in the membrane and m the bulk solution, respectively; 0, is the effective charge density of the membrane and K,,is the adsorption coefficient; [rz,,] is the saturated concentration of adsorbed ion; and l/g, and l/g, are the partition coefficients of the cation 1and the anion a, respectively, in the case where the interfacial potential difference is zero [ 51. In this paper, the values of g, and g, are assumed to be unity, since we are interested in water-swollen membranes [ 71. The free ion concentrations at the pore end are obtained by solving eqns. ( 1) ( 3 ) simultaneously. The membrane potential is given by eqn. (4) : LlEd3~~

-A?&

=A&

+mpJ

(4) Here, subscripts 1 and 2 denote the surface of the membrane at X= 0 and X= L where the membrane surfaces are in contact with bulk solutions 1 and 2, respectively; I&n and E2z are the potentials m bulk solutions 1 and 2, respectively. AE, denotes the dfference of interfacial potential differences, which can be obtained from the Donnan equilibrium condition, dEM denotes the diffusion potential within the membrane, R is the gas constant, T 1s the absolute temperature, and F 1s the Faraday constant The diffusion potential within the mem-

RT&d[ilM M=-F

-B,d[a],

(5)

&[i]M+&[a]M

where EM denotes the potential within the membrane and B, and B, are the electrical mobllities of the cation and the anion, respectively. Using eqns. (1) and (2), eqn. (5) is rewritten as follows:

(6) =--

RT F

xdIa1,

(7)

Equation (7) can be rewritten as eqn. (7’ ): dE

ge

M=-

F

RT -7 where B, [noI 6,

PY+q by2+cy+d

dy

(7’ )

fy(by2+cy+djdy l

b= (B, +B,)K& c=2(B, +&J&) +

d=& +& +B,[nolKo,e= VA&A Ko,f=B1_B,+B,[nolKo,P=Ko,q=2 and y = [a] The indefinite integral of eqn. ( 7’ ) is M

.

given by eqn. (8) for c2-4bd=B,2[h]2Ki

>O.

E +2bq-cp 2b

-2 vd

x arctanh( -$?$$I

(8) Y2

Iby2+cy+dl

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R Takagl and M Nakagakt/J

xarctanh(.$g$=J] Thus, the diffusion potential within the membrane dEM is given by eq. (9): RT

AEMvF

rc%[all~+(2+((1+~)[nolKo/2))

I[ 7

‘xg[a]t

’ P

+ (2+ ((1+7)

[no]~/2))”

ICo[a11M+1+((1+Z)[~l~/2) ‘~ob12M+1+ W+d hJlco/2)

_RT71n[a12M=!i?71n;(11)

M_

F

a ?M 1 a ZM

M=

l-7

l+ ( (1+7)[n,]K0/2)+1+7

Kobh+l XG41~+l

>

I>

(9)

where 7 is (B, -B,)/(B, +B,) Substituting eqn. (9) mto eqn (4), the membrane potential is given by eqn. (10).

RT

r

~klBew(-dW’/RT)

=BkM

(

F

According to the Donnan eqmhbrium condition, the concentratron of free ion at the pore end is given by eqn. (12)) where k denotes all iomc species including cation 1 and amon a:

PI

t

iallM

Adsorptron coeffment

-I[

[

=AE

Equation (11) is the same as that for a uncharged membrane which does not adsorb ions.

+r+ ((l+r) lI%l%/2) 2f (l+r) [&JlKo Xln

In our model, the effective charge distribution 1s non-umform throughout the membrane because the bulk concentration C1e 1s different from C&n,that is, [a] 1Mis different from [a] 2M Asymmetric charge distribution was pointed out by Petropoulos and co-workers [ 71 and by Westermann-Clark and Anderson [lo]. It IS also clear that the effective membrane charge density depends on the bulk concentration. For [no]x,,-+O, that is, the case where selective ion adsorption scarcely occurs, eqn (10) is rewritten as eqn. (11): AE

II

Membrane Scl 71(1992) 189-200

ta12M

AE=-$n(rfallM)+eqn.

(9)

where ED denotes the interfacial potential &fference, PkM the partition coefficrent of the ion k and zk the charge number of the ion k. In our model we neglect the potential distribution on the cross sectional area of the pore Thus, ED m eqn (12) means the difference between the average potential on the cross sectional area at the pore end and the potential m the bulk solution. Substitutmg eqn. (12) into eqn. (2) and rearranging, we obtain eqn. (13) for amon adsorbed by the pore wall at x = 0 and x = L [alMA=-eM

(10)

where r is CIB/C2n; C1n is the bulk concentration in bulk solution 1 and C,, IS the bulk concentration m bulk solution 2

(12)

[kb

=

bolwxpUW/RT)

MB

l+&exp(EDF/RT)

[a]n (13)

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193

Membrane SCL 71(1992) 189-200

where K=K,exp(E,F/RT)=lCOaAM

1 (a)

1

(14)

Equation (13) shows the relationship between the concentration of adsorbed ion at the pore end and the bulk concentration From eqns (13) and (14) it is clear that, in our model, the effect of the interfacial potential hfference on the adsorptlon 1s taken mto account explicitly rc, corresponds to the adsorption coefficient in the case where the interfacial potential difference ED is zero Results and discussion Dependence of membrane potentral on bulk concentratton In eqn. (lo), the parameters needed to calculate the membrane potential are F, z, [no] &/ 2, G[allw and k-,,[alzM From eqns. (l)-(3) It 1s found that the value of K, [a] 1M1s obtained as a function of rco& and the value of K,,[a] 2M as a function of K~C&.Here, C1a is rCzB Thus, when we calculate the membrane potential as a function of KOCPB,the calculation corresponds to all values of h. Figure 2 (a) shows the theoretical curves of membrane potential as a function of log ~~~~~ at 34°C and for r=C1JCZB=2 and r= (B,-B,)/(B,+B,) = -0.14. The value of z= -0.14 corresponds to that of NaCl for cellulose membranes From Fig. 2 (a), It 1s seen that all lines except line a show the bulk concentration dependence as if the membrane itself has a fixed membrane charge Lme a represents the membrane which does not adsorb ions. The absolute magnitude of the membrane potential at the low bulk concentration limit is, however, lower than that expected from the theory for a membrane with homogeneous fixed charge In a homogeneous fixed charged membrane, the absolute magmtude of membrane potential at the low bulk

a b

o-

c

2 d

Y -io-

‘0 b-4

I

Fig 2 (a) Membrane potential at 34°C and for r=2 and 7= (B,-B,)/(B,+B,)= -0 14 Membrane IS assumedto have no fixed charge and to adsorb anions selectively K=z,[no]ko/2=0 (a), -001 (b), -0 1 (c), -1 (d), -10 (e), -100 (f), -1000 (g) (b) Two components of the membrane potential shown by line f m Fig (a) Lines f, fhl and fo show the membrane potential AE (mV), the Qffuslon potential within the membrane AEM and the difference of the mterfaclal potential differencesAE, (m V) , respectively

concentration limit is 18.36 mV for r= 2. Figure 2(b) shows, as an example, the two components, dED anddEM of hne fin Fig. 2 (a). Two components of other lines m Fig. 2 (a) show the same tendency. From Fig. 2 (b ) it 1s seen that the magnitude of the dfference of mterfacial potential differences, LIE,, at the low and high bulk concentration hmlts 1szero. It 1s

194

R Takagl and M Nakagakt/J

also seen that the magmtude of the diffusion potential within the membrane AEMis not zero at the low bulk concentration limit. If the membrane has a homogeneous fixed charge distribution, the membrane potential consists of only AEM at the high bulk concentration limit and only dEn at the low bulk concentration limit. From eqns. (1) and (3)) the partition coefficients of free ions at the pore end are given by eqn. (15) [5,21].

Membrane Scr 71(1992) 189-200

concentration region is given by eqn. (19):

=-

bolKol2J~

(19)

The membrane potential at &a-+0 is given by eqn. (20)) obtained by substituting eqns (15) and (19) into eqn. (10): AE=AEM =-

RT

r+ ((l+d

hh/2)ln[allM

F 1+ ((1+7) [nO1Ko/2)[&?M

(15)

(20)

where 79M

=9,/2&I

(16)

Here, Ca is the bulk concentration. In eqns. (15) and (16), I& and C1z are used for &, and ti,, and C2a are used for j&& In the region of high bulk concentration, the concentration of adsorbed ion is saturated and the effective membrane charge density is equal to - [ rq,]. Therefore the value of AM at C,-+ co is unity because & is zero. The membrane potential is given by eqn. (17) by substituting eqns. (15) and (16) mto eqn. (10): dE=d~!& =r(RT/F)ln(l/r)

(17)

Equation (17) is the same as that for both the homogeneous fixed charged membrane and the uncharged membrane. It is clear from eqn. (12) that the interfacial potential difference En is zero in the high bulk concentration region, since the partition coefficients are umty. In the region of low bulk concentration, the concentration of adsorbed ion is directly proportional to the free ion concentration within the pore. Thus, from eqns. (l)-(3), the effective membrane charge density at the pore end is given by eqn. (18). &=-

bolK,G/~~

(18)

In eqn. (18)) CIB is used for (& and C2afor tiZM From eqns. (16) and (18), 19~ in the low bulk

Equation (20) is the same as that for the nonuniform membrane with respect to the membrane charge density, where elM = r&, [ 231. From eqns. (15) and (19), in the region of low bulk concentration, the partition coefficients of free ions become the constant determined by [no] 6. Since partition coefficient is related to the interfacial potential difference by eqn (12), a constant partition coefficient means a constant interfacial potential difference. Therefore, in the low bulk concentration region, the difference of mterfacial potential differences, AE, ( = ED2 - EDI ), is zero because ED,

=EDP

The free ion concentration at one pore end is different from that at the other pore end, since CIB is not equal to C,,. There ISa concentration gradient of free ions within the membrane, and a &ffusion potential exists even at the low bulk concentration limit. Smce the value of r_& in eqn. (20) is the constant given by eqn (19), the membrane potential in the low bulk concentration region is constant and is determined by M&,randr Dependence ofPkM and eM on bulk concentration Here, we consider the partition coefficient /I,& and the effective membrane charge density

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195

Membrane See 71(1992) 189-200

0, at the pore end m order to elucidate the effect of ion adsorption on the membrane potential Figure 3 shows PkMand log 113~1as a function of log K,,C~~.From Fig 3 it is seen that the effective membrane charge density approaches zero with decrease of the bulk concentration (lines c and c’ ) This phenomenon is due to the desorptlon of the adsorbed ion. The dependence of the membrane charge density has been experimentally reported [ 6,7,10]. However, the value of & 1snot zero but constant m the low bulk concentration region (lines d and d’ ) This is because the value of 0, at the pore end is directly proportional to the bulk concentration in the low bulk concentration region. The value of rYMin the low bulk concentration region is given by eqn. (19) A constant value of tiM means a constant value of PkM,as is shown by eqn. (15). In the region of high bulk concentration, the effect of 1%

ml ~OCZB Fig 3 The dependence of&.,, and log ( [&,,I ) (mol/m3) on bulk concentration for the membrane shown by lme f m Fig 2 (a) In this figure, K,,( m3/mol) 1s assumed to be unity Lines a and a’ show /& and aZM, respectively, where subscript 1 denotes the membrane surface 1 and subscnpt 2 the membrane surface 2 Lines b and b’ show BalMand pazM,respectively Lmes c and c’ show log ( 1f&, 1) (mol/ m3) and log ( I &.,, 1) (mol/m3), respectively Lines d and d’ show - BIMand - tYtM,respectively, where &= 13~~1 2C,a and t&= 0J2C2e

log “OC, Fig 4 The concentration dependence of the fraction of adsorption B&.[n,] at the pore end at 34°C K=.z,[n,]q,/ 2=0 (a), -001 (b), -0 1 (c), -1 (d), -10 (e), -100 (f), -1000 (g) Lmeaisfor [no]+0

the effective membrane charge density is negligibly small and the partition coefficient is unity Therefore the partition coefficient varies from unity to the constant determined by [no] rc, with decrease of the bulk concentration (lines a, a’, b andb’) Figure 4 shows the concentratron dependence of the fraction of adsorption f&/z, [ no] at the pore end as a function of log rc&n In Fig. 4, line a is for [no] +O and corresponds to the normal Langmuu isotherm From Fig. 4 it is seen that the curve shifts to the high bulk concentration as the saturated concentration of adsorption increases. This is because the mcrease of the effective membrane charge density prevents the partition of coion for the effective membrane charge Then, the concentration of the free amon within the pore does not increase as much as the bulk concentration mcreases. This phenomenon is also seen m Fig 2 (a), that is, the curve shifts to high bulk concentration as the value of [no] increases. The positive departure of the product of the partition coefficients of cation and anion from unity has been reported m studies of the sorption equilibria for celluloslc membranes [ 6,7]. In our model, the product of the partition coefficient of cation and amon at the pore end /i,, is defined by eqns (21) and (22).

R Takagl and M Nakagakt/J Membrane SCL 71(1992) 189-200

196

a

4,

(21)

= A%

where 8=

( [i]MA

+

blM)/[ilB9

(22) In the case treated here, [i] MAis zero because the membrane adsorbs anions selectively Substituting eqn. (22) into eqn. (21) and rearranging, we obtain eqn. (23): blMb]MA

A Ia=

+

[i]Bbb

hIb]M

(23)

bh[dB

In eqn. (23 ) , from eqn. (3 ) : (

blM/

b]B)

( blM/

[alB)

=bMfiaM

=I

e4)

Equation (24) is the Donnan equilibrium relationship. It IS clear from eqn. (23) that A,, is equal to or larger than unity. Interfacralpotentml dsfference We consider the mterfacial potential dlfference in the membrane system treated here. The interfacial potential difference results from Donnan exclusion. Therefore the mterfacial potential difference gives important mformation about the ion selectivity of the membrane Figure 5 shows the interfacial potential difference ED as a function of log %$n. From Fig. 5 it is found that the interfacial potential dlfference is not zero except m the high bulk concentration region, even though the membrane itself has no fixed charge. Lme a in Fig 5 shows the membrane which has no fixed charge and adsorbs no ions. It is also found that the magnitude of ED approaches a constant value at low bulk concentration. From eqn. (18), the value of the effective membrane charge density eM at the pore end approaches zero at the low bulk concentration limit. However, the value of tiM (=19,/2C,)

I -3

I

-2

-1

0

1

2

3

4

5

lwl%C,

Fig 5 Dependence of mterfaclal potential difference on the bulk concentration for the membrane shown m Fig 2(a) K=t,[n,]q/B=O (a), -0 01 (b), -0 1 (c), -1 (d), -10 (e), - 100 (f), - 1000 (g)

which affects the partition of ions approaches the constant determined by IC,[n,] . The limiting magnitude of the mterfacial potential &fference at the low bulk concentration limit is, therefore, determined by x0 [no]. If the saturated concentration of adsorbed ion is extremely large, of course, the magnitude of 1ED 1 at the low bulk concentration limit approaches mfimty as well as that for the homogeneous fixed charged membrane Determinatron of the values of the parameters The value of r can be determined from eqn. (17) using the magnitude of the membrane potential at the high bulk concentration limit Figure 6 shows the relation between and log 1IQ, where dEC2B_0denotes A&+o the magnitude of the membrane potential at the low bulk concentration limit and K= - [n,,] KJ 2. Figure 6 is for z= - O.l4andr=2andat34”C If the magnitude of dEc,B+o 1sobtained from the experimental data, the value of K 1s determined from Fig. 6. The determination of the values of t and K makes it possible to calculate the theoretical curve as a function of log &,CZB This calculation corresponds to all values of IC,

R Takagl and M Nakagakl/J

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Membrane SCL 71(1992) 189-200

.______________ --/ -2c3

-2

-1

0

1

2

3

IoglKl

Fig 6 The dependence of A&,,, (mV) on K for the membrane shown m Fig 2 (a) The broken hne shows the magnitude of the membrane potential for the uncharged membrane which does not adsorb ions

as mentioned before. Thus, the value of ~0 is determined from the best fit of the theoretical curve with the experimental results, since CPB is known from experimental data. The determination of the value of IC,and K makes it possible to determine the value of [no], since K= - [n,,] %/2. Consequently, the values of all parameters which characterize a membrane, z, KY,, and [no], are determined. Apphcatwn to the membranepotentsal of a cellulose membrane Here, we apply the theory to the analysis of the membrane potential of the cellulose membrane shown in Fig. 7, which was measured by Nakagaki and Mlyata [ 241. In Fig. 7 the open circles show the experimental results. The measurement was carried out for NaCl and r = 2 and at 34” C In Fig 7, the magnitude of membrane potential at the low bulk concentration hmlt is negative and not - 18.36 mV, which is the theoretical magnitude for the homogeneous and negatively fixed charged membrane This phenomenon indicates that the cellulose mem-

-1

-2o-3

-2

-1

10g~c2&0, 0 1 log aOCZB

1

3 nt-3,

2

3

4

Fig 7 Membrane potential of cellulose membrane for NaCl at 34°C and for r=2 Open circles show the expenmental data The solid line shows the theoretical curve calculated by eqn (10) with 7= -0 14, z.[n,,] = -4 1 mol/m3 and ~,,=2 24 m3/mol The broken line shows the theoretlcal curve calculated by assuming the membrane as a homogeneous and negatively charged membrane where 7= - 0 14 and fixed membrane charge density IS -2 8 mol/m3

brane has no fixed membrane charge and adsorbs anions selectively It is also reported that the membrane charge density depends on the concentration of the Cl- ion for a collodion membrane [ 111 and for a muscovlte mica membrane [lo]. The value of z was determined to be -0.14 using the magnitude of the membrane potential at C,, + co from eqn. ( 17 ). From the experimental data, the value of dEc2B_owas estlmated to be - 14 2 mV The value of K was determined to be -4.6 using Fig. 6 with the value of LU&_~ = - 14.2 mV. The value of IC, was determined to be 2 24 m3/mol from the best tit of the experimental results with the theoretical curve for r= - 0.14 andK= - 4 6 The value of z,[n,] was determined to be -4.1 mol/m3, since K=z, [ no] G/2 In Fig. 7, the solid line shows the theoretical curve calculated from eqn. (10) with z= - 0.14, z, [ no] = -4.1 mol/m3 and &=2.24 m3/mol The broken line shows the theoretical curve

R Takagt and M Nakagakt/J Membrane

198

calculated by assummg the cellulose membrane as a homogeneous membrane with negative fixed charge (its fixed membrane charge density IS - 2.8 mol/m3). Our theoretical curve and the experimental results are in good agreement. The value of the saturated concentration of the adsorbed ion is nearly equal to the fixed membrane charge density determined from the analysis m which the membrane is assumed to be a homogeneous membrane with fixed charge. After obtaming values of the parameters which characterize the membrane, we can predict theoretically the variation of the mterfacial potential difference. Figure 8 shows the variation of the mterfaclal potential difference and of the effective membrane charge density at the pore end of the cellulose membrane shown in Fig. 7. In Fig. 8, in our model (solid line), the absolute value of the effective membrane charge

S’CL

71(1992) 189-200

density is constant m the high bulk concentration region, since the adsorption is saturated, and decreases with the decrease of the bulk concentration. This phenomenon is due to the desorption of the adsorbed ion The value of interfacial potential difference is constant at high bulk concentration and decreases with the bulk concentration. This value approaches the constant determined by the value of the product of the adsorption coefficient and the saturated concentration of adsorbed ion. If we neglect the deviation of the theoretical curve from the experimental results m the low bulk concentration region and treat the membrane as a membrane which has a homogeneous fixed charge and does not adsorb ions, the theoretical prediction changes to the broken line From Fig. 8 it is clear that the theoretical pre&ctlons for the interfacial potential difference and the effective membrane charge density are quite different, even though two theoretical curves of membrane potential are similar except at low bulk concentration region. We ought to consider a selective ion adsorption for which the absolute magnitude of the membrane potential at low bulk concentration is lower, as in Fig. 7, than that expected from the theory for the membrane which has a homogeneous fixed charge and does not adsorb ions Conclusions

-2

-1

0 1 2 log(CB/mot ms3)

3

Fig 8 The vanatlon of the mterfaclal potential difference Eo(mV) and the effective membrane charge density 113,( (mol/m3) at the pore end of the cellulose membrane shown m Fig 7 Lines A and a show the effectwe membrane charge den&y and lines B and b the mterfaclal potentlal difference The solid hne and broken hne have the same meanings as m Fig 7

In order to make clear the effect of ion adsorption on the membrane charge density, we investigated theoretically the effect of ion adsorption on the membrane potential of an uncharged membrane using the selective ion adsorption model, which is based on a pore model in common use for swollen membranes. The positive deviation of the products of the partition coefficients of cation and amon LSwell explained by our theory. It has been pointed out that the membrane potential depends on the bulk concentration as

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Membrane Set 71(1992) 189-200

if the membrane itself has a fixed membrane charge. The absolute magnitude of the membrane potential in the low bulk concentration region is, however, lower than that expected from the theory for the membrane with homogenous fixed charge. In the region of low bulk concentration, the magnitude of the membrane potential depends not only on the concentration ratio of the bulk solutions but also on the adsorption coefficient and the saturated concentration of adsorbed ion. The theory was applied to the membrane potential of a cellulose membrane, and it was shown that the theory was in good agreement with the experimental results by using proper parameter values. After determining values of parameters which characterize the membrane, we can predict the variation of the surface charge density and the interfacial potential difference, which are important m discussing the filtration efficiency. List of symbols

r

ionic species including cation i and anion a bulk concentration ratio, Cln/

zk

charge number of ion k

k

L’2B

B/& [eqn. (21) 1 partition coefficient for total ion [eqn. (22) ] partition coefficient for free ion [eqns. (12) and (15)] effective membrane charge density ( mol/m3 ) &Wn Ieqn. (16) 1 adsorption coefficient (m”/ mol) [ eqn. (14) ]

4, Sk

P kM

0,

VA-&)l(&+&,)

z

Subscrtpts interface 1 interface 2 membrane bulk solution cation anion ionic species including cation i and amon a

1 2 M B 1

[ [

IMP

[

IB

IMA

hl & CB, C1B, &B

AE ED AED

AEM K a i

concentration of free ion (mol/ m3) concentration of adsorbed ion within the pore ( mol/m3 ) saturated concentration of adsorbed ion ( mol/m3 ) electric mobility of ion k within the membrane (m’/sec-V) bulk concentration ( mol/m3) membrane potential (mV) interfacial potential difference (mV) difference of interfacial potential differences, ED2 - EDI (mW diffusion potential within the membrane (mV)

Mdd2 anion cation

;

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