ELSEVIER Journal of Electroanalytical Chemistry 436 (1997) 119-125

Effects of pH on ion transport in weak amphoteric membranes

Patricio Ramirez a, Antonio Alcaraz a, Salvador Maf6 h.° '~ Del~nTamen! de Ci~m'ies E.~perimentals, Unit'ersitat 'Jaume !' de Castelld, Apdo. 224, 12080 Castelh'~, Spain

h Delmrtcmu'nt de Tcnnodit:'zmh'a, Ftwtdtm de F~sica, Unicer,~'itat de Val~'ncia, 46100 Bt.ja,~.~ot, Spain

Received 7 March 19t~7: received in i~vised fiu'm IO June 1997

Abstract

We have sludied Iheorelically Ihe efl¢cls o1' pH on Ihe ion Irausport lhrougll amphoteric polymer membranes compo~ed of we~,k polyeleclmlytes whel~ the charged groups are randomly distributed along the axial direction of the membrane. This system serve~ a~ a simpliiied model fi~r the pH controlled ion Iraltspoll alld drug delivery through membranes of biological inlerest. The |heoretical approach employed is based on the Nernst-Planck equations. The complete system of electrical charges formed by: (i) the pH de~nden|. amphoteric membrane fixed charge, and (it) the four mobil~ charges (the two salt ions and the hydrogen and hydroxide ions) have been taken into account without any additional assumption. The model predictions show that the ionic fluxes and the membrane potential are very sensitive to the external pH, and the po~, : ~tial utility of these predictions for the analysis of experiments involving pH dependent passive transport through membranes is emphasi~:ed. © 1997 Elsevier Science S.A.

I(evword.~: Weak ampholeric membranes: Ion transport; El'feels of pH: Theore|ical modelling

1. Introduction

Amphoteric membranes contain both positively and negatively fixed charge groups chemically bound to the membrane constituents, Early studies on amphoteric memo hranes were carried out by Sollner [I ], Neihof and Sollner [2], Soliner [3], Weinstein and Caplan [4], and Weinstein et al. [5.6]. who considered the case of the charge-mosaic membranes on the basis of the Non-Equilibrium Thermo- dynamics formalism developed by Kedem and Katchalsky [7]. !t is also possible to prepare amphotefic polymer membranes composed of weak polyelectrolytes where the charged groups are randomly distributed along the axial direction of the membrane, as has been done by Yamauchi et al. [8], Hirata et al. [9], Saito and Tanioka [ 10], and Saito et al. [I I]. in the latter studies, it was shown that the membrane properties were very sensitive to the external pH, and then amphoteric membranes could be of potential utility for biochemical sensors and pH-controlled drug delivery systems.

Polymer membranes composed of silk fibroin have also been reported to have amphoteric properties [I 2,13] due to the existence of weakly acidic carboxyl groups and weakly

" Corresponding author. E-mail: smafeC,~uv.es.

0022-0728/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. Pil S0022-0728(97)00310-0

basic amino groups. Since ionizable protein amino acid residues (i.e. carboxylic acid groups) and protonated amine groups are very common in biological systems, the study of ion transport in weak amphoteric membranes can be of relevance also for the biophysical chemistry ot' biological membranes. This is the case of the cornea, fi~r instance. where the ion permselectivity appears to ~ controlled by the degree of protonation of ionizable sites within the tissue [14]. In the above study, the permselectivity was analyzed in terms of diffusion and streaming potential measurements, and the effect of the external pH on these potentials suggested that the cornea contains acidic and basic charge groups that could be responsible for the observed differences in the permeabilities hetwet:n cationic and anionic species [14],

Although the permselectivity of biological membranes is certainly a complicated phenomenon that involve,~ not only passive electrostatic barriers (due probably to memo brane fixed charges) but also active elements like carrier~ and pumps [15,!6], we propose here to study theoretically an ideal case which considers only one major effect: the pH dependence of passive transport thr~ugh a weak amo photeric membrane carrying both positive and negative fixed charge groups. This case serve~ a~ a ~implified physical model for many real systems, and can thu~ be of relevance for the biophysical chemistry of pH controlled

120 P. Ram~re: et aL / Journal ~f Eh,cmmnalyticai Chemistr3. " 436 (19971 ! 19- t25

i ~ transport and drug delivery through thick membranes (~mbrane thickness of the order of or higher than tO- ~ era) of biological interest. For these membranes, bulk rather than sud'ace effects are rate limiting for ion trans- port.

The theoretical model employed for studying the pH dependence of ion trans~rt in weak amphoteric mem- branes is ba~d on the Nemst-Planck equations. The physical basis and limitations of the.~ equations as applied to n~mbrane systems can be found elsewhere [17.18]. ~veral u~f id theoretical m~xlels ba~d on a number of a~sumptions are already available [1ool3]. The pres~ent modelling differs from most of the~ models in that we con~ider the tour nonlinear (difl~tsion + migration) Nemst=Planck equations which are strongly coupled by the thp:~" local chetnical reactions occur|°ing in the menlo brahe (the two membrane fixed charge dissociation equao tion:~ and the wat¢= dissociation equation). The full system of nonlinear differential equations is solved numerically without any additional simplifying appt~ximation (such as the Goldman constant field assumption [I 5.16]) in order to obtain the h~¢a/profi&s across the membrane of all the physical variables (the Ibur ion concentrations and tbe el~tric potential) involved. The membrane characteristics studied a~ the ionic fluxes and the membrane potential.

We believe that the model predictions can be useful for the analysis of future experiments because of the tbllowing reasons: (i) ex~riments involving electrolyte transport through biomembranes, hi.heroical sensors and pH~cow trolled drng delivery systems are at'ten conducted using buffer solutions anti the measured ternary ion system properties ate analyzed in many cases in terms of hinat~ system equations [14, I~]; (ii) simplifying assumptions such as equal pH values inside and outside the membrane arm a cortstant el~trical field wiflfin the membrane [15,16] are s o ~ f i ~ s U ~ withotlt a clear verification of the validity of these assum~ions; and ( i i i ) most of the ex~riments an ~,'~ducted with ~h¢ ~uu¢ oH value in the two external solutions I~tthi~ the membxane [10,11,14,19~21L and thus the effects of imposing a pH gradient through the meat° beane are ~ well known,

~ ~ ! mead

The system eonsider~xi is shown schematically in Fig. I. The amphcaeric a~mb~ne extends i~m r ~ 0 to a ~ d, and ~parates l~o solutions of tile same uni~univalent e~tro!yte. ~ solutions are assuaged ~o be ~rfectly sl i~d, q (x ) stands for the ¢oneenteation of the ith-species at a poim of coordinate x within the t ~ m b r a ~ (i .~ I for ~ h c~ior~ i ~ 2 for mlt anions, i == 3 i't~¢ hydrogen ions, and / - 4 for hydroxide ions), % denotes the concentra- tion of the ith-spe¢ies in the bulk of t ~ jth-solution {j = L for the left hand ~lution and j = R for the fight

solution), pH~ ( j = L, R) refers to the pH value of

OL / Membrane i *r

.................. - 1 - . . . . . " . . . . . I

PHt ] "| pt{ r c I t , ,0 , . , c,,d,[ . . . . . ___~

0 d x

Fig. I. Schematic reprc~ntation of the system studied, The atuphotedc membrane extends from x = O to .t = d and mparates two solutions of the same uni-univalent electrolyte. X(.~,) is the fixed c h a ~ e concentra- tion at a pain! of ¢~x~rdinate x within the tuembrane, and c,~ is the concentration o( II~ ith,.sp,2cies t i ~ I lbr sal| cathms, i ~ 2 for salt anions, i ~ 3 ftw hydrogen ions, and i ~ 4 for hyda~',ide ions) in the bulk of the jlh~o!tl|ioJ~ () '~ 1, fi~f the left hand st~lution and j ~ R for the right hand ~olutio~O. pH ~ I ,j ~* L R} rel~ar.~ t~ the pit value of solu|ion j.

solution .j, The whole system is assumed to be isothermal and flee t't~tm convective nIoviHllell|s,

The amphoteric membrane conlains acidic iN) and ba- sic (P) groups homogeneously distribuled throughout its ~olume. These gtxmps are assumed to be monovalent, and can ~ charged or not del~nding oil the local value of the pH wifllin file membone. We denote by N c and N n the concent~tions of the charged (N ") and neutral (N")acidic gixmps, and by Pc and Pn the concentrations of the charged (P*) and neutral (P") basic groups, res~ctiveiy. The equations describing the h~al equilibria ~tween neu- tral and charged glx~ups are:

N" x,, N + ~i ' , ( la)

P ' ~ + H . ( I b )

where K s and Kp al~ the equilihrimn constants. From Eqs, (la) arm {IbL the local concentrations of charged gtx~ups within the membrane are:

I NT, (2a)

¢ ~ / K p P,. = PT . (2b)

I + c~/Kp

and {he Ioca| concentration X of the net fixed charge in the membrane is defined as:

X = e ¢ - ,V,,, (3)

whe~ we have included the sign (positive or negative) of this charge in tile value of X, and N-r ~ Nn + Nc and Pr ~ Pn + P~" a~ the total concentrations of acidic and basic gnmps, Tlle concentrations of H * and OH- ions verify tile equilibrium condition:

¢.w~ = Kw. (4)

with Kx~,~ = I0 ~ a~ molt l~z through tile whole system. The ion concentrations obey the electroneutrality condi-

tions:

clj + c3i = c ~ + %j. j = L,R, (5a)

g + c3 + X = c 2 + c.t, (5b)

P. Rambrz el al. ! lounud of Electro~malyticai Chemistn, 436 ~ 1997J ! 19-125 12 !

in the bulk of the two bathing solutions and in the mem- brane phase, respectively.

Bulk solution and inner (membrane phase) concentra- tions are connected through the tbllowing Donnan equilib- num conditions [15.16] at the interfaces x = 0 and x = d:

Cl(O) CZL C3(0) C4L

C,L ca(O ) C,~L c4(0 ) (6a)

"l(d) ¢,r ¢,(d) ",,r ¢,,(d) (6b)

From Eqs. (2a), (2b). (3), (4), (5a), (5b), (6a) and (6b). it tbllows that:

Ki, KN g i, g n g i, g n I + ~',~,,ft'in j f

4' ] I {':!~ i':~ ~ +

l K i, Kn ! (Pl .... j,,7

I + c,,~, i c, j

+ +

g i, KN

N+/,',, ] It j °°°

I + c~,/cl, I°~°O.j~°~l . . . . R,

(7) where u L ~ c,(O)/gt, and ur = C l ( d ) / C | R " Eq. (7) can be solved numerically for UL and UR using a standard proce- (lure. Then, the inner membrane phase concentrations g(O) and g(d), i = 2, 3. 4, can be obtained lh~m Eqs. (6a) and (6b) in terms of the electrolyte concentrations and the pH values in the external bathing solutions. Once all the concentrations c,(O) and c,(d) have been determined, the Donnan potential differences at tlae left (x = O) and right (x ~ d) interfaces, A S t and A 4~a, can be computed as:

RT cl i, a,t,L 7 log ,.,( . (8a)

RT c , (d) a *R = - ~ log . (Sb)

t'l R

The electric potential and the ion fluxes in the bulk of the amphoteric membrane can be calculated solving the Nernst=Planck equations [ 17,18]:

[de, F dtk ] J, =D, Tx+l l)' J = c, RT d x

i = ! , . . . , 4 , (9)

with the condition of zero total current:

J, - J: + J~ - J4 = 0. (10)

In Eq. (9), D i and .i i stand Ik, u the .... ' " ' "~' OIlII, I~|UII ¢OetliCl~flt in the membrane and the flux of the ith-species, respectively,

is the local electric potential, and the constants F. R and 1" have their usual meaning.

Fluxes Jt and .I 2 correspond to the salt ions (the sodium and chloride ions coming from the NaCI dissocia- tion in water, for instance) and are constant through the membrane under steady state conditions because of the

continuity (mass conservation) equations: there are neither local sources nor local sinks for these ions due to the complete dissociation of the strong electrolyte in pure water. Fluxes J~ and ,/4 correspond to the water (hydrogen and hydroxide) ions and are not constant through the membrane because these ions participate in the local disso- ciation reactions at the membrane fixed charge groups. The dissociation reactions constitup, local sources and sinks for these ions (see Eqs. (1 a) and ( I b)).

Substituting Eq. (4) into Eq. (9) for i = 3 and 4, and taking into account Eq. (!0), we obtain:

J2 - ,1 I

J~ = I + Kw l ~/D~c; 9 ~" ( l l a )

D.~ Kw '/°' ~ Dj c ~! ,/~. ( I I b )

Eqs, (I la) and ( l ib) show that the fluxes of t t ' and Oil ..... ions are not constant through lhe membrane in th ~:eneral case where ¢~ changes with x.

We have used the fi)liowing ilerative procedure to integrate the system of differential equations given by Eq, (9). First, we assume some initial values for the ion fluxes and integrate Eq. (9) using a fourth-order Runge~Kutta method with the boundary conditions at the interface x = O. Then, we check whether the solutions satisfy the hmndary conditions at x = d or not. if not, the initial estimation is changed until the boundary conditions at x = d are satiso fled. This allows us to obtain the ion concentration and electric potential profiles, c;(x) and 4~(x), and the ion fluxes J,. Then, the profile of the fixed charge concentrao tion X(x) through the membrane can be determined from Eqs. (2a), (2b) and (3). Finally, the membrane potential i.~ computed a:,:

A~m ~ Adh~ + A~I, + Advt. (12)

where A St) ~ d~(d) = d~(()) is the ditTusion potential in the membrane.

3. Results and discussion

in this section, we present a set of model calculations concerning the fixed charge c o n c e n t r a t i o n profile~0 the membrane potential and the ionic fluxes in the amphoteri¢ membrane, in all the calculations, we have considered the symmetrical case N T = P+ = 5 × 10 ~ tool I '~. and a,~° sumed the values d = I 0 '~ cm for the membrane thick° ness and pKn = 4 for the pK value of tl~e acidic groups, The particular concentration and pH values employed here are those typical of the cornea ion trlln~poll and pH°com trolled drug delivery experiment~ [14,19=21], The pK,, values used (see Fig. 2) are characteristic of io,izab!e protein amino acid residues ~ie carboxylic ~cid group~) and protonated amine groups [10,11,14,19=21]. We have also used the tbllowing values for the diflhsion c~ffi o cients of the ionic species within the membrane: D~ ~ 10 7

12'2 P. Ramire: et al. / Journal of Eie,'troanal.vtical Chemistry 436 ( 199D 119-125

L 7.. 5,0-~

- 4 x 25 %

<

ll,o

°2~5

0 2 4 6 8 1o 12 14

ra,~

pHi= ~ p14~ ~ pH, We have eron~idered the ,:a~e N< r ~ Pt ~ ~ × I 0 llt~)J l : ~, g ~ ~'ll ~ ~ 7 II) ~ ill~tl I I ,rid pK~ ~ 4, The cl;il'We~ lll'¢ p~ff~lill~'lfi¢ tn p K i,,

>

2 0 -

4 0 ~ ' ! . . . . t . . . . i . . . . . . . . . . .~ . . . . . . . . . . . . . 1 . . . . . . . . . .

-20

=40

0 ~ \

J

0 2 4 6 8 10 12 #

Fig. 4 OMoJlalcd itl¢tlihl'at!~, l~lentiM ~ff lh¢ amphotedc ll~vah~ane vs. pH ti~' I1~¢ ~a|n¢ ,:ondiiion~ an given in Fig. 3. The ¢onlinuims lille ,:~ve~pond~ to the eXa¢l ilUliit~fieal I~..'.~tllls~ ~ i¢ dashed and dolled:dashed Iine~ ¢ot~'~l~md Io the app.tximal ion~ giveil by 17;II~ t 131i} and Eq~. 112l aild (1411), i~,q~,¢livtfly,

cm ~ ~ '. l)~ ~ 21)1, l)~ ~gD~ lind l)t ~ 5Di, The~e ion difl~UiiiOll ¢i~ft]cient~ are one/two orders of magnitude lower than thilse co~e~ponding to a ti'ee aqueous solution. ~c~co~mg to the typical experimental results obtained lbr ehar~,~ membrai~cs [ 15.16].

steady stale values t~'or the membrane potential and the i,m fluxes are zero when t ~ pH values and the el~trolyte concentrations in the left and right hand bathing solutions an equal (pHi, ¢ pH r and cl~ ~ c r ). In Ihis case. the concentrations of charged acidic and basic gnu, ups an constant thrt'mgh the membrane, and the net fixed charge concentration X within the membrane is homogeneous, Fig, 2 ~how~ thi~ fixed cl't;wge concentration w, ph i , o~ pH ~ ~ pH in the el!so q:. ~ c r ~ 5 >~ 10 ~ too! I ::~ . The nttm~rs in the curves corresl~-~ to the pKp vMues ¢onside~d, The concenlrations t~, and N¢ have ~ l l ealculatcM solving Eq, (7) t:tw u t ~ % and s u b s l i l u i h l g the i~sult in ~ , (~) , (~ ) , (2at and l{2b), As eXl~'Ct~, X ,~ Px I X ~ = N a, ) t~or pit < pg n (pH > pKi,) a l l X ~= P~.12 ( X ~ = NT/2) forpH ~ pK n (pH ~ pgt,), It is also ~ls¢i~'c'M that the memb~-~ne h~ornes unehttrged in the v~inity of t ~ ist~le¢lri¢ point, pl ~ (PRn + pKi,}12, and

o n 4 5

. . . . . . . . . . . . . . 6 !

g

i ,

~.O 02 0,4 0 6 O.S l,t~ x&t

F~. 3. ~ |'i~,.d ~ ¢o~"¢~tral.i,ox, profit¢.~ within the ampholeric

IO-: m~l i ~ = 5 ~ . p E n = 4 and pg~, = 6 in the model calculations. ~ are pararm~tri¢ in pH~ = pH~ ~ pH.

that the pli value region when the net fixed charge ,n~:centration i~ pt~ctically zero I~'xomes enlarged as the diffennce ~tween pK I, ~md pK N is increased,

Figs. 3=5 illustrate the case pH u ~ pH r ~ pH and c u 5c r. The m,~el calculalions have ~en perl'onned using *'t~ ~ 5 × I 0 : tool I -~ ~ and pK t, = 6. Fig, 3 shows the net fixed charge concentration profiles wilhin the membrane, The curves are parametric in the pH value, which is indicated in each curve. Again, we see that X ~ Pr (X ~ N t) fi~r pH < pK n (pH > pKl,) and that X = 0 near the ist~leclric poinl pl ~ (Pgn + pKI,)/2 ~ 5, However. in the ngion~ pH ~ pK n and pH ~ pKp, the net fixed charge concentration pn~files ~come inhomogeneous due to local variations in the pH values within the membrane.

Fig, 4 shows the calculated membrane pot~mtial vs. pH tbr the same conditions as in Fig. 3. The continuous line ¢ornsl~mds to the results obtained using the nul~rical pn~ :~un dc~rihed in the parlous ~clion, The dashed line has been calculated assuming that the pH value within the membrane is equal to that of the bathing solutions and tilat the fluxes of th{ H ~ aud OH- ions can ~ neglected. Under ihe,~ assumptions, the net fixed charge concentra- tion X within the membrane is homogeneous and can be

s

0 2 4 6 8 !0 12

Fig. 5. Ion fluxes (exact numerical values) vs. pH for the same conditions as given in Fig. 3 and Fig. 4.

P. Ramirez et aL / Journal of Electroanalytical Chemisto' 436 { 1997) ! 19= 125 123

calculated from Eqs. (2a), (2b) and (3) for each pH value. The membrane potential and the fluxes of salt ions can be obtained using the equations [15]:

e r { CL (X~" + 4c~ - X A thin = -'7" / log ~ ~ ,

c r V X ' + 4 c ~ . - X

+ U log ,.-7~., , ' ( 1 3 a ) V'X" + 4c[. + UX

o,o, a, = a, = " + 4c~ - ~X 2 + 4c~

" D ~ + D = d "

f ~ ~ X ~/x ~ + 4c~ + UX

+ UX log ~ , , ~ . . . . l (13b) ~:X "~ + 4c~. + VX

whelv U ~ (D~ ~-" D~)/(D~ + D~), This approximation gives very good results in the range 8 < pH < 10, where Ihe membrane has a negative value of X (X = r ~ N r. see Fig. 3) and the membrane pmential is nearly constant (Ad~l ~ 22 mV). Also, introducing X ~ 0 in Eqs. (13a) and (13b) gives approximately the same value Adh, ~ = - 14 mV obtained with the numerical procedure near the iso- electric point. However, the results given by this approxb marion deviate from the exact numerical ones in the low pH region because in this region the flux of the it + ions is no longer negligible. The deviations in the pH range 4 < pH < 8 must be ascribed to the fact that the net fixed charge concentration within the membrane is not homoge- neous.

A more accurate analytical approximation for the calcu- lation of the membrane potential can be achieved using the so called constant field approximation [15,16] which as- sumes that d ~ / d x ~ A CkD/d ~. constant. In this case, Eq. (9) can be readily integrated for i = I, 2 and 3 in the low pH region (note that in this limit ./4 ~ 0 and ,l~ = J~ = J~ = constant, see Eqs. (10), (I la) and (i lb)). Under these assumptions, we obtain [ 15,16]:

RT D~c;(O) + D~t'~( d) + D,c,(O) &~6 o = - ~ - log D.~c~(d) + D~c2(O) + D~co(d) ° (14a)

F D,AtbD c , ( O ) e x p [ - F A ' b D / r T ] - c , ( d ) Ji =

J2 ~

J3 "~

RT d I - exp[-FAdpD/RT] (14b)

F D : A ~ D

RT d c2( d)exp[-FAt~D/RT] - ca(O)

X . ( 1 4 c ) I - e x p [ - F A 4 ~ D / R T ]

F D~A&D c3(O)exp[-FAcbD/RT] - c ~ ( d )

R 7 d I - e x p [ - FA#~D/RT] (14d)

Substituting Eq. (14a) in Eq. (12), the membrane potential can be calculated. A similar procedure can be followed ~o obtain the constant field approximation lbr the membrane potential in the limit pH :~ 7. The dotted-dashed line in Fig. 4 cor:esponds to the results given by this approxima- tion.

Fig. 5 shows the dependence of the ion fluxes on the pH for the same conditions as in Fig. 4. The results have been obtained using the numerical procedure described in the previous section. As expected, we see that J~ = J: and J3 = J4 = 0 in the intermediate range 4 < pH < 10. it can also be observed that the ion fluxes J~ and J~ attain a local maximum at the pH value where Atbm ~ 0 in Fig. 4 (this pH value does not coincide exactly with pl ~ 5 because the pH within the membrane is not equal to the external pH). Again, the constant value J, ~ J: ~ 4.9 × 10 '~7 real cm ~: s '~ in the region 8 < p H < l O a n d the value ,I I = ,1~ = 5.3 X 10 -7 real cm ~ s ~j for pH = p l ~

5 can be anticipated froln Eq. (13b) taking X = .= N t and X ~ 0. respectively. As discussed above, we see that the fluxes J, and Ja can not ~ neglected for the low and high pH values (compare Figs. 4 and 5). In these limits, the constant fiel0 approximation given by Eqs. (14a). (14b). (14c) and (14d) can provide good analytical approxima- tions for the ion fluxes.

The case c L --5c a and pH L ~ pH a is considered in Figs. 6-8. We have fixed c L ~ 5 × IO -~ real I -~, pH k ~ 6 and pKp ~ 9 in the calculations. Fig. 6 shows the profiles of the net fixed charge concentration within the membrane. The numbers close to each curve correspond to the pHL value considered. We see that the profiles are highly inhomogeneous except for the case pHL ~ pH r ~ 6. where the membrane is practically uncharged (X = 0). As could be anticipated, the inhomogeneity of the membrane ino creases as PHu decreases w i t h respect to the value in the right hand solution, pH r = 6.

Fig. 7 shows the membrane potential of the amphoteric membrane tbr the same case a~ in Fig. 6. The exact

~o .... J = = i = ~ r ¸

0,0 ~ ~ = T - ~ = ~ ~=~ '~ ~ '

0,0 0,2 0,4 0,6 0 8 I 0

Fig. 6. Net fixed charge co|lceIltrafio|i profile~ wilhi~ llle ~mlpll,w~ri~ membrane in the ca~¢ phi. ~ phi, W~ have ur+cd N+ ~ P+ + 5× I0 : real I' J, % ~ 5× I0 =2 11101 1 I ~ 5ci~, pg~ ~ 4 ~md pK v ~ 9 in tlI~

model calculalion.~. The curve~ ore paramelri¢ in pHi.° wilh pH u ~ 6.

The case pHL = pHr = 6 giver X = O.

124 P. Rmn[rez ez aL / Journal t~ Eleczroanalytical Chemisto' 436 (1997~ !19-125

> 40

~,~o

0 - - ~ l 2 ~ 4 ~ 6 7 s

~L

FIIII0 7, C~!ol~t~d ~,m!~.~ i'~te~tial of l:he ampholeri¢ n~mbfane w.

l i~). ~1 re~tttt~ t~ F~l.~, (12) ~.d (14~) (d~ted=d~d It~),

nu~fical ~ul is have ~ n plotted in a continuous I;ne, The dashed and the dottedo~da~hed lines correspond to the approximations given by ~+ (13a) and Eqs. (12) and (14a). rrespeclively. ~ exact numerical results for the ion fluxes vs, pH~ are shown in Fig, 8, As in the previous ¢ ~ ( ~ Figs, 4 ~ d ~). the behavior of the membrane potential is cio~ly r e l a ~ to the behavior of the ionic fluxes through the membrane, We can distinguish three different regions in Fig. 7: first, the membrane potential decreases with pHi. then attains a plateau region characo t e d ~ by a construct value A~u ~ - ~ mV. and. finally, decrea~s again with pHi, The~ regions correspond to J~ > J,. J~ ~ J~ ~nd J~ ~ Jt ~ J~, in Fig, 8, respectively.

The apl~xima~e solution for ~ u given by ~ , (13a) deviates significantly from the exact ~lution except for the ~H region $ < ~H < 7, where X ~ O, J, ~ J~ ~ $,3 × 10 ° reel em ~ ~ , ~and Y ~ 0 ( ~ Figs, 6~8), Tbe ~sutts fo¢ ~ ~b u o b t a i ~ using ~ s , (12) and ( I ~a) do not

with ~ exact ~medeal results over the entire ran~ of pH~ vMues ten,clerk, and thus the constant field a ~ i o n ~ P m t e s a poor ~ , x i m a t i o n when a pH ~ m exists ~ough the membrane,

We ~ve ee~ied out also other ¢Mculations for different vMums of ~ ~ ~ s involved, but we obtained qualita+

l ~ 3 4 ~ 6 ? 8

Hg. ~. ~ ~ e s (~xact ~ values) vs. pH I~r the same conditioas

tively similar results for reasonable choices of these pa- rameters (for instance, an increase in the membrane thick- ness d does not change the membrane potential values but decreases the ion fluxes as I/d, as could be anticipated for relatively thick membranes).

Two additional effect~ mat could be present in weak amphoteric membranes have not ~ n taken into account in our study. ~ of ~ s e is the variation of the water content of the membrane as~ia ted wid~ the change of the ionization d e g ~ of the fixed charge groups. The other is the possibility of having cross electrical neutralization between ionized fixed groups of opposite charge, Let us discuss then questions qu~flitafively now. The ionization of the wnembrane fixed charge groups produces an uptake of additional water from the solution [22]. Since the mem- brane swells, ~ effeelive fixed charge volume concentmo tion X decreases, The d ~ l ; ~ in the effective value of X with the ~ g ~ of hydration of charged membranes is well d~un~nted [23] and would make less dramatic some of the pH effects disused in our study, We must recognize that the omission of the swelling effect is a m~jor limita- tion of our result, though this effect should not be so important here as il is in ~ ca~ of ampholytic hydrogds of low crosslinker content [24]. Nolo that there is a limit in the charged membrane swelling ~aus¢ of the restoring elastic forces exerted by the erosslinker chains inserted I~..tween ~ fixed cha~e chains [22], The ~cond effect mentioned above, the cross elec|rical neutralization be- tween ionized membrane groups of different charge, can p ~ b l y be d i ~ d e d on the basis of the following reason- ing [25.26], Since the fixed charge groups are randomly distributed in the membrane, an avenge distance between g~ups is ? ( I /XN^ )'/~' where N, is Avogadro's con- stant, If X 10 ~ = reel I = , then r ~ 50 A. Now. a typical Deby¢ length for electrical ~ ¢ n i n 8 within the membrane is L a ' ( v R T / 2 F c m) ~-" 20 ~ for ~ 4 0 ~ o [27] and c m ~ I0 ~ reel I =~ where ~o is the vacuum electrical permiuivily and ¢.~ is a typical mobile ion concentration within the membrane. The electrical ~reening factor [25,26] is then exl~-r/L D) ~ 0.08. This factor is small enough to neglect the electrostatic attraction between near- est neighbor membrane fixed groups of opposite charge [25,26|, Therefore. the condition of local electroneutrality in the membrane must be assured by the mobile electrolyte ions taken from the solution (,~e Eq. (5b)) and the electri- cal state of the fixed charge groups must be dictated by the I~al concentrations of the mobile :ons in the vicinity of the fixed charge groups (see Eqs. (Ix) and (Ib)).

In summary, we have pre~nted a ~ t of model calcula- tions concerning the membrane potential and the ionic fluxes in weak amphoteric membranes. The whole system of equations including the four Nernst-Planck equations and the equations for the local dissociation equilibria within the membrane have been solved numerically wifl~ out any additional simplifying assumption. The compari- son of the exact results with those obtained by using

P. Ramirez et al. / Journal of Eleclroanalytical C~misto" 436 (1~7) ! 19-12.'; 125

several approximations has shown the potential utility of the numerical solution, especially when a pH gradient is imposed through the membrane. Since the pH dependence of passive transport in weak amphoteric membranes is a subject of great impoaance in thick biological membranes [14,15] and pH-controlled drug delivery systems [19-21], the model calculations presented here could be of interest for the analysis of future experiments.

Acknowledgements

Financial support from the DGICYT, Ministry of Edu- cation and Science of Spain under Project No, PB95-0018 and from the Oeneralitat V~denciana under Project No. 3242/95 l~re gratefully acknowledged,

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