<" . < • r +,

k+,. , +~ , 'i-

ELSEVIER Journal of Membrane Science 135 (1997) 135-144

~M rnal°f BRANE

,~IENC£

Model calculations of ion transport against its concentration gradient when the driving force is a pH difference

across a charged membrane

P a t r i c i o R a m f r e z a, A n t o n i o A l c a r a z a, S a l v a d o r M a f ~ b'*

aDepartament de Cikncies Experimentals, Universitat "Jaume I" de Castell6, 12080 Castell6, Spain bDepartament de Termodin~rnica, Facultat de Ffsica, Universitat de Valkncia, 46100 Burjassot, Spain

Received 13 March 1997; received in revised form 23 May 1997; accepted 23 May 1997

Abstract

Model calculations of the steady-state ion transport against its external concentration gradient when the driving force of this transport is a pH difference across a charged membrane are presented. We have solved numerically the exact Nernst-Planck equations without any additional simplifying approximation, such as the Goldman constant field assumption within the membrane. The validity of this assumption for a broad range of pH values, and salt and membrane fixed charge concentrations was analyzed critically. The membrane characteristics studied are the ionic fluxes and the membrane potential. Special attention is paid to the physical mechanism which leads to the ion transport against the concentration gradient, and the experimental conditions for which this transport can occur. The case of a system with ions of different charge numbers is also considered.

Keywords: Charged membranes; Modeling; Ion transport; Concentration gradient; Nernst-Planck equations

1. Introduction

The study of ion transport against its external con- centration gradient is a classical topic in the mem- brane field (see e.g. the literature review presented in the Introduction sections of Refs. [1-5] as well as [6-10]). Many studies of this phenomenon using synthetic membranes have been focused on liquid membranes [6-8], though more recently much atten- tion has been paid to the case of polymer ion-exchange membranes [ 1-5]. The electrodiffusion of ions against

*Corresponding author• Fax: + 34 6 364 23 45.

0376-7388/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI I S 0 3 7 6 - 7 3 8 8 ( 9 7 ) 0 0 1 3 6 - 1

an external concentration gradient is also a multi-ionic transport problem of great interest in membrane biophysics, especially in the case of active transport across biological membranes [8-10].

The ion transport against its concentration gradient is certainly a complicated phenomenon that involves not only passive electrostatic barriers (due to mem- brane fixed charges), but also active elements like carriers and pumps [8-10] in the case of biological membranes. We propose here to study theoretically an ideal case which considers only the steady-state trans- port of an ion against its external concentration gra- dient when the driving force of this transport is a pH

136 P. Ram[rez et al./Journal of Membrane Science 135 (1997) 135-144

difference across the charged membrane. This case serves as a simplified physical model that could be of relevance for the pH controlled ion transport and drug delivery through synthetic and biological membranes [11-13]. Though many theoretical and experimental work has recently been carried out in similar problems [1-5], we believe that model calculations can still be useful for the analysis of future experiments. Our treatment differs from previous ones in the following questions:

(i) we have solved numerically the Nernst-Planck equations (in the approximated dilute solution form and with no convection term) without invoking the Goldman constant field assumption within the mem- brane [1-5,14], and analyzed critically the validity of this assumption for a broad range ofpH values and salt and membrane fixed charge concentrations;

(ii) we have clarified the physical mechanism which leads to the ion transport against its concentration gradient and given the experimental conditions for which this transport can occur; and

(iii) we have considered in our study the case of a system with ions of different charge numbers.

The theoretical model employed for studying the pH dependence of the steady-state ion transport against a concentration gradient is based on the Nernst-Planck equations. The physical basis and lim- itations of these equations as applied to membrane systems can be found elsewhere [ 15,16]. The complete system of electrical charges formed by: (i) the mem- brane fixed charge, and (ii) the four mobile charges (the two salt ions and the hydrogen and hydroxide ions) have been fully taken into account in the model. The membrane characteristics studied are the ionic fluxes and the membrane potential.

2. Theoretical model

The system considered is shown schematically in Fig. 1. The cation exchange membrane extends from x=0 to x=d, and separates two solutions of the same electrolyte (KC1 in our case). The membrane contains fixed charge groups of molar concentration XM homo- geneously distributed throughout its volume, ci(x) stands for the concentration of the ith species at a point of coordinate x within the membrane (i=1 for salt cations, i=2 for salt anions, i=3 for hydrogen ions,

PHL

HC1

KCI

CiL

ClL------ C L

j

J1

/

J

m ~ / / ( ~

Ci(0) ci(d)

PH R

CiR

ClR---- C R

KC1

HC1

0 d x

Fig. 1. Schematic representation of the physical mechanism which leads to the phenomenon of ion transport against its external concentration gradient in charged membranes. The K + (1) ion is transported across the membrane against its external concentration gradient due to the counter-transport of the H + (3) ion. Note that the concentration gradient (see the continuous curve for c~) and the electric potential gradient (see the discontinuous curve for ~) within the membrane act together to give potassium transport from the right bulk solution to the left bulk solution.

and i=4 for hydroxide ions), cij denotes the concen- tration of the ith species in the bulk of the jth solution (j=L for the left solution and j : R for the right solution). PHi (j=L,R) refers to the pH value of solution j. We assume that the solutions are perfectly stirred, and the whole system is assumed to be iso- thermal and free from convective movements. For the sake of simplicity we will present the theoretical equations corresponding to the case of a uni-univalent electrolyte. The case of a system with ions of different charge numbers can be generalized in a straightfor- ward manner, and some model calculations for this case will also be given later.

The basic equations describing the transport of the four mobile ions through the charged membrane are the Nernst-Planck equations [15]

Ji = - D i [~-~ q- (-1)i+lci ~ - ~ ] , i = 1 , . . . ,4

(1)

the equation for the (zero) electric current density

J1 - J2 Aw ,/3 - J4 = 0 (2)

the equilibrium condition for the H + and O H - ions

C3jC4j = C3C4 = K w , j = L, R (3)

P. Ram[rez et al./Journal of Membrane Science 135 (1997) 135-144 137

with Kw=10-14M 2, and the local electroneutrality assumption

Clj + csj = czj + c4j, j -- L, R (4a)

Cl + c3 = c2 + ca + XM (4b)

in the bulk of the two bathing solutions and in the membrane phase, respectively. Here Di and Ji are the diffusion coefficient in the membrane and the flux of the ith species, respectively, ~b is the local electric potential, and the constants F, R and T have their usual meaning. The membrane is assumed to have negative fixed charge groups in Eq. (4b).

Substituting Eq. (3) into Eq. (1) for i=3 and 4, and taking into account Eq. (2), we obtain

J2 -- J1 J3 = 1 + KwD4/Dac 2 (5a)

D4Kw , J4 -- ~ a3 (5b)

O3c 3

Eq. (5) show that the fluxes of H + and O H - ions are not constant through the membrane in the general case where c3 changes with x.

According to the well-known Donnan equilibrium [ 17,18], both ionic concentration and electric potential are assumed to be discontinuous at the membrane- solution interfaces. The concentrations at the inner boundaries of the membrane are related to those at the outer boundaries through the equations

CiL ci(O) -- 2(ClL + CSL)

× ((--1)i+IXM-I-¢X2+4(ClLq-C3L)2],

i = 1 , . . . , 4 (6a)

fiR ci(d) = 2(cm + C3R)

× [(--1)i+IXM+¢X2+4(ClR+C3R)2],

i---- 1 , . . . , 4 (6b)

while the electric potential drops (Donnan potentials) at the interfaces are

A~bL = R T l n CIL F cl (0) (7a)

A4~R = R T ln cl (d) (7b) F ClR

Note that, as a first approximation, ion activity coeffi- cients and single ion-partition coefficients [15] have not been included in the Donnan relationships [ 19,20].

Eqs. (1)-(5) must be solved using Eq. (6) as bound- ary conditions in order to obtain the ion concentration and electric potential profiles, ci(x) and q~(x), and the ion fluxes Ji. We have employed the following itera- tive procedure: first we assume some initial values for the ion fluxes and integrate Eq. (1) using a fourth- order Runge-Kutta method with the boundary condi- tions at the interface x=0. Then, we check if the solutions satisfy the boundary conditions at x = d or not. If not, the initial estimation is changed until the boundary conditions at x = d are satisfied. Once ci(x), g~(x), and Ji have been obtained, the membrane poten- tial is computed as

where AO~D ---- q~(d) - q~(0) is the diffusion potential in the membrane.

In the general case, the procedure mentioned above requires considerable numerical effort. A useful ana- lytical approximation for the calculation of the mem- brane potential and ion fluxes can, however, be obtained using the so called Goldman constant field approximation [14,20,21], which assumes that

dc~ /k~bo - - (9)

dx d

in Eq. (1). With this assumption, integration of Eq. (1) for i=1 and 2 is straightforward, and gives

D1A~o Cl (0) exp(--Ak~D) -- Cl (d) J1 = - - (10a)

d 1 - e x p ( - A ~ o )

D2,/kkOO c2(d) e x p ( - - A ~ D ) -- c2(0) J2 = - - - (lOb)

d 1 - exp(-Ak~o)

where k~-F~/RT. Integration of Eq. (1) for i=3, leads to

ca(d) d _ f 1 +Ac~ dc3 ~ 2

D3 ABc~ + G(1 + Ac~)c3 ca(O)

x/~B

G v/ AB 2 - 4G 2

1. c3(d) Glnc3(0 )

a + a[Oc~(d) + 8]~(d) in-6 + A [Oc3 (o) + 8]~ (o)

(11)

138 P. Ramfrez et al./Journal of Membrane Science 135 (1997) 135-144

where

D3 A ~ (12a)

D4Kw

B ~ J2 - Jt (12b)

G =- D3A~D (12c) d

Substitution of Eq. (10) in Eq. (11) leads to a trans- cendental equation in A ~ D that can be solved using a standard numerical procedure. In the case D3c2/DaKw >> 1 (which corresponds to pH values within the membrane lower than 7 i fD3~D4), Eq. (11) can be written in the form

d d lnJ2 - Jt + (D3Agdo/d)c3(d) D3 D3A~D J2 Jl + (D3Aq~D/d)c3(O)

(13)

Since in this situation J4~0 and -/3 ~ ./2 - Jl (see Eq. (5)), Eq. (13) gives approximately

D t A ~ o c3(0) exp(--Affgo) -- c3(d) J3 ~ - - (14)

d 1 - e x p ( - - A ~ D )

and the dimensionless diffusion potential can be com- puted in this l imiting case as

lnD2C2(d) + DlCt(0) + D3c3(0) A ¢ , D ( 1 5 )

D2c2(0) q- OlCl (d) q- O3c3(d)

3. R e s u l t s a n d d i s c u s s i o n

In this section we present a set of model calculations (obtained by numerical solution of the exact transport equations) for the ionic fluxes and membrane potential across the charged membrane, and compare them with the approximated analytical equations. Fig. 1 shows schematically the charged membrane and the bathing solutions containing KC1 as electrolyte. We have assumed d = 1 0 -2 cm for the membrane thickness in all the calculations. The physical mechanism which leads to the phenomenon of ion transport against the external concentration gradient across the charged membrane is shown in Fig. 1. The potassium ion is transported through the membrane against its external concentration gradient due to the counter-transport of the hydrogen ion. The profile of the potassium con- centration (Cl) within the membrane and the resulting

1.0

• 0.0

-1.0

-2.0

-3.0

-4.0

i I i i i i

500~

L .... ' I:

2 3 4 7 PH L

Fig. 2. The K + flux Jl across the charged membrane vs pilL for pHR=7. The membrane fixed charge concentration is XM=I M. The numbers in the curves give the values of the KC1 concentration ratio CLICR, where CR=10 -3 M. The continuous curves correspond to the values of J1 calculated from the exact numerical solution of the transport equations. The discontinuous curves correspond to the values of J1 calculated from the approximated solution based on the Goldman constant field assumption. Negative values of the flux give potassium transport against its external concentration gradient (see Fig. 1).

• 2 . 0 / , ! , , ! , ! , ,

i l 0 0 -

0.0 i 50 i - - - - -

"" -2.0 "~ i i [ [

-4.0

-6.0

1 2 3 4 5 6 7 PH L

Fig. 3. The K + flux Jx across the charged membrane vs pilL for pHR=7. The membrane fixed charge concentration is now XM=0.1 M. Other conditions and parameters as in Fig. 2.

electric potential (q~) gradient are also included in Fig. 1. Figs. 2 -4 show the potassium flux J1 across the charged membrane vs pi lL for pHR=7. We have assumed the infinite dilution aqueous solution values [22] D t = l . 9 5 × 10 -5, D2=2.03 × 10 -5, D3=9× 10 -5, and D4=5 × 10 -5 cm 2 s -1 for the diffusion coefficients

P. Ram[rez et al./Journal of Membrane Science 135 (1997) 135-144 139

~ - 2.0 / ' , I , I , , ,

[ o.o , l o i . . i !

~, i /, i i i ~' i" 51 i i

- 2 . o ...... ; - - 7 - . - ~ - ~ . . . . . . . . . . . i . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . i ................. T ..................

-4.0 i

-6.0

-8.0 ,

1 2 3 4 7 pl i L

Fig. 4. The K + flux Jl across the charged membranes vs p i l l for pHR=7. The membrane fixed charge concentration is now XM=0.01 M. Other conditions and parameters as in Fig. 2.

of the ionic species within the membrane. The num- bers in the curves give the values of the KC1 concen- tration ratio ClL/ClR -- CL/CR, where CR=10 -3 M. The membrane fixed charge concentration is XM----1 M. In all curves, there is a critical value pHi, at which potassium transport against its concentration gradient appears (when this happens, Jt changes its sign from positive to negative; see Fig. l). We see that the higher the ratio CL/CR the lower the value of pHi. This result could be anticipated, since according to Fig. 1 the driving force for the potassium transport against its concentration gradient is the pH difference across the charged membrane. The continuous curves of Figs. 2- 4 give the values of Jt calculated from the exact numerical solution of the transport equations, and the discontinuous curves correspond to the values of Jl calculated from the approximated solution based on the Goldman constant field assumption. The contin- uous and discontinuous curves of Fig. 2 coincide each other, which shows that this assumption is very good for the membranes with high fixed charge concentra- tion, i.e. when XM >> CL,R and XM > 10 -pH. This in agreement with previous studies on the validity of the Goldman constant field assumption in charged mem- branes [20].

Figs. 3 and 4 show the potassium flux J1 across the charged membrane vs pilL for the same conditions and parameters as in Fig. 2, except for the membrane fixed charge concentrations, which are now XM----0.1 M (Fig. 3) and XM=0.01 M (Fig. 4). When decreasing

the membrane fixed charge concentration, three effects become apparent: (i) the flux of potassium transported against the concentration gradient decreases significantly, (ii) there is a shift of the critical value pH i towards lower pHt. values, and (iii) the Goldman constant field assumption becomes poorer, especially in the range of concentrations where XM ~ £L,R and XM ~ 10 -pH. Indeed, significant devia- tions between the flux derived from this assumption and that obtained from the numerical integration of the exact transport equations appear in Fig. 4.

Fig. 5(a) and (b) show the concentration profiles within the membrane in the case CL/CR=5,

1.8

1.5

1.2

0.9

0.6

0.3

0.0 (a)

~- 1.6

1.2

0.8

0.4

0.0

-0.4

, I , I , I , I ,

C2L

C2L

ClL

Cl

C3L X ~

ClL

0.0 0.2 0.4 0.6 0.8 1.0 x (10 .2 cm)

, I , I , I , I

X M

C 1RC2R

3R

ClRC2R C3L C 3 C3R

' I ' I ' I ' I ' 0.0 0.2 0.4 0.6 0.8 1.0

(b) X (10 .2 cm)

Fig. 5. (a) The concentration profiles across the membrane system (see Fig. 1) of the mobile ions (K+=I, C1-=2, H+=3) for the case CL/CR=5, CR=10 -3 M with XM=0.01 M, pHR=7, and pilL=2, (b) the concentration profiles across the membrane system for the same conditions as in (a) except that pilL=4. Note the important differences in the concentration profiles of potassium (1) and hydrogen (3) ions of Figs. 5(a) and 5(b).

140 P. Ram(rez et al./Journal of Membrane Science 135 (1997) 135-144

~ 10o

~- 80

6o

40

20

0

-20

-40

I ~ I ~ I L I ~

pH L= 2 ~ I

pHL = 4

0.0 0.2 0.4 I i I '

0.6 0.8 1.0 x (I0 -2 cm)

Fig. 6. The electric potential profiles across the membrane system (see Fig. 1) for the same cases as in Fig. 5(a) (pilL=2, dashed curve) and 5(b) (pile=4, continuous curve). From Figs. 1, 6 and 7 (see below), the role played by the electric potential gradient through the membrane on the potassium transport becomes clear.

1.0

o.o

~:~ -1.0

-2.0

-3.0

-4.0

i i i ! L I i ~

...... ,, i . . . . . . . . . . . i . . . . . . . . . . . . . i . . . . . . . . . . i

'~ ...................................../i ix~,--l~ . . . . . .

. . . . . i"/ i . . . . . i . . . . . . . . . . . . . i \ i Z

\ i

I I I I

2 3 4 5 6 7 PH L

150

~r 130 ~

110

90

70

50

Fig. 7. The K + flux J1 and the membrane potential AOM across the charged membranes vs pile for pHR=7. The membrane fixed charge concentration is XM=I M, and CL/CR=10 with CR=10 3 M. Negative values of the flux correspond to potassium transport against the external concentration gradient. The influence of the membrane potential A ~ on the direction of the potassium flux J1 across the membrane is shown clearly (see also Figs. 1 and 6).

cR=10 -3 M, XM=0.01 M, pHR=7, pilL=2 (Fig. 5(a)), and pilL=4 (Fig. 5(b)), (see also Fig. 4 for CL/CR=5). Concentration profiles have been presented previously by Higa et al. [2]. Fig. 6 shows the profiles of electric potential across the membrane for the same cases than in Fig. 5(a) (pilL=2, dashed line) and 5(b) (pilL----4, continuous line). Fig. 5(a) and Fig. 6 (dashed line) show clearly the conditions under which ion transport against its con- centration gradient can occur. Figs. 5(b) and 6 (con- tinuous line) correspond to a typical situation where this transport can not occur. Note also that the Gold- man constant field assumption is not valid when pilL=2 in Fig. 6.

Fig. 7 gives the potassium flux Jl and the membrane potential A0M across the charged membrane vs pilL for pHR=7. The membrane fixed charge concentration is XM=I M, and CLICR=IO with CR=10 -3 M. The influence of the membrane potential A0M on the direction of the potassium flux Ja across the membrane is shown clearly: the potassium transport against the external concentration gradient appears just at the pHL value at which the membrane potential begins to increase steadily from its constant (Nernstian) value (RTIF) ln(CL/CR=lO)~6OmV. Note that the latter value corresponds to an ideally selective membrane in an experimental situation where the effects of the water ions are negligible (pHL=4-7_<7=pHR). The

role played by the characteristic electric field E=Acko/d created by the diffusion potential across the membrane on the potassium transport is thus crucial, as we anticipated in Fig. 1 (see also Fig. 6 where the electric potential profile through the membrane is shown). Fig. 7 shows clearly the physical mechanism which leads to the potassium flux observed experimentlly [1,3,4].

It is in order now to give the sufficient conditions for which ion transport against its concentration gradient appears in a given experimental situation. As stated above, this phenomenon begins at a critical value pH i at which Jl changes its sign. Since J1 is a continuous function, we can calculate pH~ by imposing J r=0 with AffJo¢0 in Eq. (10a), which yields

c3(0) + v/X~ + 4(CtL + C3L) 2

0 2 CILC2L - - CIRC2R (16)

o3 ct (d)(1 - (¢tL/C3L)(C3R/Cl~))

(Note that according to Fig. 7, A~bM deviates signifi- cantly from (RT/F) ln(cLICR) and thus A~D#0 at the onset of the potassium transport against its concentra- tion gradient.) For fixed values of ctj, c2j (j=L, R) and c3R, Eq. (16) leads to a polynomial equation in C3L that must be solved using a numerical procedure in the most general case. Once Eq. (16) has been solved,

P Ramfrez et al./Journal of Membrane Science 135 (1997) 135-144 141

PHI is obtained as --IOgl0(C3L ). In the limiting case (C1L,R -~- C3L,R) 2 ( ( XM (membrane with high f i xed

charge concentration) and pHR=7,

C3 (0) ~ C 3 ~ L X M ( 1 7 a ) C1L --[- C3L

cl(d) .~ CI~RXM ~ XM (17b) C1R -{- C3R

Eq. (16) transforms into the second order equation

[ C2L -~- 2ClL ClRC2R D3 " 'M] C3 L _ ClRC2R _{_ C~ L CIL D2 ClLJ

D3c3RY2 0 ( 1 8 ) 0 2 ClR '~M =

that can be solved easily for C3L. Fig. 8(a) and (b) show the values of pHI for different ratios CL/CR with CR=10-3 M (Fig. 8(a)) and Cg=10-2M (Fig. 8(b)).

These pH values correspond to the condition Jr=0 in Figs. 2-4, and can then be obtained in each case by solving Eq. (16) (continuous curves) or Eq. (18) (dashed curves). Note that although these equations were derived under the assumption of constant electric field through the membrane, they constitute excellent approximations in the range of pH values for which J l ~ 0 (i.e., in the vicinity of pHi), according to the results of Figs. 2-4. From Fig. 8(a) and (b), we see that increasing the external concentration gradient of the potassium ion requires an increase also of the pH gradient through the membrane for potassium transport against its concentration gradient to occur. The higher the membrane fixed charge concentration, the lower the pH gradient which is needed for this potassium transport. Fig. 8(a) and (b) give the characteristic values of the external pH which lead to potassium transport against its concentration gradient in each experimental situation, and could

7.o then be of great practical applicability in the design : of membrane processes involving this transport phe-

%-~'~ 6.0 nomenon. 5.0 ........... i ........ ~ ......... i .......... i .......... Finally, Figs. 9-11 show the calcium flux Jl across 4.0 ............... i .............. i . . . . . . . . . . . i ................... ~. .................... the charged membrane separating two solutions of

3.0

~ l i i ! i 0.0 1.0 2,0 3.0 ' 9 0 .0 ~ f f 7 : i i

(a) Log I o(CL/CR) 0 = i i i i

- -0 .5 . . . . . . . . . . . . . . . . . . . . i ..................... i ..................... i .................... i . . . . . . . . . . . . . . . . . . .

N i ~i c R --10-2M

6.0 - 1.0

5.0 M i

~ - ~ X [ [ ~ :[ -1.5 i i ' ' ' 3.0 ~i ..................... t~, .................. ~ ....................... ~ . . . . . . . . . . . . . . .

2.0 ................. ~ ............. x.~ -! .................. i ...............

1.0 0.0 1.0 2.0

(b) LOglo(C&R)

Fig. 8. The critical values of p i l l giving transport against its concentration gradient vs loglo (CL/CR) for (a) CR=10 3M and (b) CR=10 -2 M. The numbers in the curves are the membrane fixed charge concentrations XM. Other conditions and parameters as in Fig. 2. The continuous curves correspond to the solutions of Eq. (16), while the dashed curves give the solutions of Eq. (18).

1 2 3 4 5 6 7 PH L

Fig. 9. The Ca 2+ flux Jl across the charged membranes vs pilL for pHR=7. The membrane fixed charge concentration is XM= 1 M. The numbers in the curves give the values of the CaC12 concentration ratio CL/CR, where Ca=10-3M. The continuous curves correspond to the values of Jl calculated from the exact numerical solution of the transport equations. The discontinuous curves correspond to the value of Jl calculated from the approximated solution based on the Goldman constant field assumption. Negative values of the flux correspond to calcium transport against its external concentration gradient (see Fig. 1)

142 P. Ramfrez et aL /Journal o f Membrane Science 135 (1997) 135-144

1.0 where

j 0.5 C1(0 ) f C2L x~2 C3(0 ) f C4L x 2 . . . .

,~ 2 2 _ c I (d) ( C2R "~ c3(d) f C4R x . . . . .

- - 0 5 c--g-=-- t -7[i7) = c - i - . = t, tL°o ,

-1.0 Eq. (19) can be solved either analytically [23] or -1 .5 numerically, and their solutions replace the boundary

conditions given in Eq. (6). The ion fluxes and mem- -2.o brane potential can then be calculated following the

1 2 3 4 5 6 7 PH L

Fig. 10. The Ca 2+ flux Jl across the charged membranes vs p i l l

for pHR=7. The membrane fixed charge concentration is XM=0.1 M. Other conditions and parameters as in Fig. 9.

i 2.0

~ 1 . o ............... i ............ ol ............... i ............ i ................ i ..............

g o . o ............. ; . . . . . . . . . ~i ............. t i i

" -1 .0 ..... ~,,.-~- ~ ~ ............ i ..................... i ..................... ~. .................... ~ ....................

-2.0

-3.0

-4.0 1 2 3 4 5 6

PH L

Fig. 11. The Ca 2+ flux J1 across the charged membrane vs pilL for pHR=7. The membrane fixed charge concentration is now XM=0.01 M. Other conditions and parameters as in Fig. 9

CaC12 vs pilL with pHR=7 for the membrane fixed charge concentrations XM=I M (Fig. 9), XM=0.1 M (Fig. 10) and XM--0.01 M (Fig. 11). The generaliza- tion of Eqs. (1)-(5) to the case of a 2:1 electrolyte is straightforward. In this case, the Donnan equilibrium conditions at the membrane/solution interfaces and the electroneutrality assumption throughout the entire system lead to the equation

2 u j + ( c 3 J ~ u ! / 2 = ( 2 + c 3 J ~ u ! - l / 2 ) +XM, j = L , R \ c t d i \ c~d ~ Cl;

(19)

same numerical procedure than that used in the case of a 1:1 electrolyte. In all the calculations, we have assumed the infinite dilution aqueous solution values D l = l x l 0 -5, D2=2.03x10 -5, D3=9x10 -5, and D4=5x 10 -5 cm 2 s -1 for the diffusion coefficients of the ionic species within the membrane. Again, the numbers in the curves give the values of the ratio CL/CR (with CR=I0 3 M) and the continuous and dis- continuous lines have the same meaning than in Figs. 2-4. The effects of decreasing the membrane fixed charge concentration are similar to those observed in Figs. 2-4. Using a procedure qualitatively similar to that giving Eqs. (16)-(18), we might also obtain the curves analogous to those of Fig. 8(a) and (b) in this case, except for the fact that now the concentrations of mobile ions at the inner membrane solution interfaces must be solved using a numerical procedure.

We have not attempted a direct comparison of our results with experimental data from the literature. In order to analyze ion transport data quantitatively, complete experimental data on the membrane fixed charge concentration and the ion diffusion coefficients in the membrane are needed [4]. In addition, many experimental studies include transient experimental data [2,4,5] obtained without keeping constant the initial concentration gradients, and we have focused here only on the steady-state behavior under constant concentration gradients. Finally, concentration polar- ization effects may be important here (multi-ionic systems have a marked tendency to control the partial diffusion boundary layer; see, e.g., Helfferich [17]), and we assumed in the calculations that the solutions were perfectly stirred because we wished to concen- trate only in the questions stated in Section 1. The introduction of concentration polarization effects will produce fluxes lower than those predicted theoreti-

P Ramfrez et al./Journal of Membrane Science 135 (1997) 135-144 143

cally in Figs. 2--4 and Figs. 9-11, though the physical mechanism responsible for the ion transport against the concentration gradient should be essentially the same. Qualitative agreement with theoretical and experimental studies [1-5] has been found in the following points: (i) the K + ion transport against its concentration gradient is an electrochemical phenom- enon that can be realized here because of the H + counter-transport and the electrodiffusion potential gradient established through the membrane; (ii) high enough membrane fixed charge concentra- tions and pH gradients are necessary for this transport to occur, and (iii) the mechanism is also operative for Ca 2+ provided that the pH gradient be high enough to ensure zero electric current through the membrane.

In closing, although we are aware that excellent work has already been carried out in this field [1-5], we believe that the model calculations presented here can contribute to a better understanding of the physical mechanism underlying the phenomenon of steady- state ion transport against its concentration gradient when the driving force for this transport is pH differ- ence across a charged membrane (see in particular Figs. 1, 5(a), 6 and 7). Also, the sufficient conditions given in Figs. 8(a) and (b) for this transport to occur could provide new ideas for the design and analysis of future steady-state experiments.

p~ pH~

R T uj

x

XM

pH value of the jth solution Critical value of pH in the left bulk solution at which ion transport against its concentra- tion gradient appears Gas constant (J mo1-1 K -1) Absolute temperature (K) Auxiliary dimensionless variable at the membrane-solution interface Coordinate within the membrane (cm) Membrane fixed charge concentration (M-- mol 1-1=10 -3 mol cm -3 )

4.1. Greek symbols

ACD AM

a~M A~R

A ~ D

¢(x)

~V(x)

Diffusion potential in the membrane (mV) Donnan electric potential drop at the left membrane-solution interface (mV) Membrane potential (mV) Donnan electric potential drop at the right membrane-solution interface (mV) Dimensionless diffusion potential in the membrane Local electric potential (mV) at a point of coordinate x Dimensionless local electric potential at a potential of coordinate x

4.2. Subscripts

4. List of Symbols

A Auxiliary parameter (12 mol-2= 10 6 cm 3 mo1-2) B Auxiliary parameter (mol cm -2 s -1) ci(x) Local concentration of species i at a point of

coordinate x within the membrane (M-- mol 1-1=10 -3 mol cm -3)

c 0 Concentration of the ith species in the bulk of the jth solution (M--mol 1-1=10 -3 mol cm -3)

d Membrane thickness (cm) Di Diffusion coefficient of the ith species

(cm 2 s -1) E Electric field (mV cm -1) F Faraday constant (C mol- 1) G Auxiliary parameter (cm s-1) Ji Flux of the ith species (mol cm -2 s -1) Kw Equilibrium dissociation constant of water

(mol 21-2=10 -6 mol cm -6)

1 Salt cation 2 Salt anion 3 Hydrogen ion 4 Hydroxide ion i ith species j jth phase L Left bulk solution M Membrane phase R Right bulk solution

Acknowledgements

Financial support from the DGICYT, Ministry of Education and Science of Spain under Project No PB95-0018 and from the Generalitat Valenciana under Project No. 3242/95 are gratefully acknowl- edged.

144 P. Ramfrez et al./Journal of Membrane Science 135 (1997) 135-144

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