pH and supporting electrolyte concentration effects on the passive transport of cationic and anionic drugs through fixed charge membranes

pH and supporting electrolyte concentration effectson the passive transport of cationic and anionic

drugs through ®xed charge membranes

Patricio RamõÂreza, Antonio Alcaraza, Salvador MafeÂb, Julio Pellicerb,*

aDepartment de CieÁncies Experimentals, Universitat `̀ Jaume I'' de CastelloÂ, Apdo. 224, E-12080 CastelloÂ, SpainbDepartament de TermodinaÁmica, Facultat de FõÂsica, Universitat de ValeÁncia, E-46100 Burjassot, Spain

Received 23 December 1998; accepted 12 March 1999

Abstract

The effects of pH and supporting electrolyte concentration on the passive transport of an ionized (cationic or anionic) drug

through a thick ®xed charge membrane have been theoretically studied. This system constitutes a simpli®ed model for the pH

controlled ion transport and drug delivery through membranes of biological and pharmaceutical interest. Calculations were

carried out for different values of the membrane ®xed charge, supporting electrolyte and drug concentrations covering a broad

range of the conditions usually found in experiments. The theoretical approach employed is based on the Nernst±Planck ¯ux

equations, and all of the species present in the system (the neutral and ionized forms of the drug, the two supporting electrolyte

ions and the hydrogen and hydroxide ions) have been taken into account without any additional assumption. It has been shown

that the Goldman constant ®eld assumption together with the total co-ion exclusion assumption provide good approximated

solutions for high membrane ®xed charge concentrations. The model predictions show that the internal pH within the

membrane, the total drug ¯ux and the membrane potential are very sensitive to the external pH values. Comparison of our

results with available experimental data con®rms the potential utility of the calculations for the analysis and design of

experiments involving the pH dependent passive transport of ionized drugs through a ®xed charge membrane, especially in the

cases of thick biomembranes, biochemical sensors and pH-controlled drug delivery systems. # 1999 Elsevier Science B.V.

All rights reserved.

Keywords: Cationic/anionic drug passive transport through charged membranes; pH and supporting electrolyte effects; Internal pH vs.

external pH; Flux and membrane potential; Theoretical modeling

1. Introduction

Many of the transport phenomena found in pro-

blems of biophysical interest and biotechnological

applications involve the passive transport of amino

acids and proteins [1±3] as well as ionized (cationic or

anionic) drugs [4,5] through ®xed charge membranes.

Experimental studies show that the particular values of

the pH and supporting electrolyte concentration in the

external solutions have a marked in¯uence on the

transport rate. Recently, Chen et al. [4] and Akerman

et al. [5] have shown that the permeability of a ®xed

charge membrane to an ionized drug can change by

Journal of Membrane Science 161 (1999) 143±155

*Corresponding author. Tel.: +34-96-398-3385; fax: +34-96-

398-3385; e-mail: [email protected]

0376-7388/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 6 - 7 3 8 8 ( 9 9 ) 0 0 1 0 1 - 5

several orders of magnitude when the external pH is

varied. Also, pH effects could be very signi®cant in

thick biological membranes like the human skin [6]

and the cornea [7], where differences in the perme-

abilities between cationic and anionic species have

usually been ascribed to the existence of ®xed charges

within the membrane.

We propose here to study theoretically the ¯ux of an

ionic drug through a thick ®xed charge membrane for

a broad range of pH values and membrane ®xed

charge, supporting electrolyte and drug concentra-

tions. The theoretical approach employed is based

on the Nernst±Planck ¯ux equations [8,9] and all

species present in the system (the neutral and ionized

forms of the drug, the two supporting electrolyte ions

and the hydrogen and hydroxide ions) are taken into

account. Although several approximated models

based on a number of assumptions are already avail-

able [5,10,11] the present calculations differ from

previous ones in that we consider the full (diffusion

� migration) Nernst±Planck equations for the ionized

drug and all the ionic species involved. These equa-

tions are coupled by the drug dissociation equilibrium

occurring within the ®xed charge membrane and the

local electroneutrality condition. We solve numeri-

cally the resultant system of non-linear differential

equations without invoking simplifying approxima-

tions such as the Goldman constant ®eld or total

co-ion exclusion assumptions [1,2,11,12] in order

to obtain all the ionic ¯uxes and the membrane

potential. The validity of these assumptions can then

be checked.

We are aware that the permselectivity of many

biological membranes is certainly an intrincated phe-

nomenon that involves not only passive electrostatic

barriers (probably due to membrane ®xed charges) but

also active elements like carriers and pumps [10,11],

but we will focus here on an ideal case which con-

siders only the pH and supporting electrolyte concen-

tration effects on the drug passive transport. This case

serves as a simpli®ed physical model for those systems

involving thick membranes where bulk rather than

interfacial (e.g. hydrophilic/hydrophobic) effects are

rate limiting for ion transport. Within this framework,

interfacial effects other than the Donnan equilibrium

formalism [10,11] will not be considered here,

although they may be incorporated in the model by

introducing the appropriate partition coef®cients in

the equations describing the ionic membrane/solution

interfacial equilibrium in each particular case [5,10].

We expect the theoretical predictions to be of

relevance for the analysis and design of experiments

involving the pH dependent passive transport of an

ionized drug through a ®xed charge membrane, espe-

cially for the case of thick biomembranes containing

aqueous pores, biochemical sensors and pH-controlled

drug delivery systems [1±8,10,11,13]. Experiments

involving ion transport through the above systems

are often conducted using a buffer solution and/or a

supporting electrolyte and the measured multi-ionic

system properties are analyzed routinely in terms of

simpli®ed binary salt system equations. Moreover, the

internal pH within the membrane is assumed to be

approximately equal to the pH of the external solution.

We will show that these procedures introduce severe

errors [14]. We will provide also theoretical results for

the full system of transport equations and show their

potential applicability in two experimental situations

where complete experimental data are available [4,5].

2. Theoretical model

The system considered is shown schematically in

Fig. 1. The ion-exchange membrane extends from

x � 0 to x � d, and separates two solutions containing

the drug in both its neutral and ionized form and the

supporting electrolyte (NaCl in our case). !XM is the

®xed charge concentration of the membrane (! � �1

for an anion exchange membrane and ! � ÿ1 for a

cation exchange membrane), ci(x) is the concentration

of the species i at a point of coordinate x within the

membrane, ci,j gives the concentration of the species i

in the bulk of the solution external j (j � L for the left

solution and j � R for the right solution), and pHj

(j � L, R) refers to the pH value of the solution j. These

solutions are considered to be perfectly stirred, and the

whole system is assumed to be isothermal and free

from convective movements.

We might consider separately the cases of a cationic

drug,

DH� ?KDC

D� H�; (1a)

and an anionic drug,

DH� ?KDC

Dÿ � H�; (1b)

144 P. RamõÂrez et al. / Journal of Membrane Science 161 (1999) 143±155

where KDC and KDA are the equilibrium constants of

the dissociation reactions, but we will analyze here

only the latter case because the case of the cationic

drug can be treated in a similar way.

The pH value of the solutions surrounding the

membrane can be controlled by adding either an acid

or a base (HCl or NaOH in our case) to these solutions.

The addition of the corresponding ionic species

changes the concentration of the mobile ions during

the experiment. Let us assume that the concentration

of supporting electrolyte and the total concentration of

drug in the external solutions take the initial (0)

values:

cS;j � c0Na�;j � c0

Clÿ;j; j � L;R; (2)

and

cDT;j � c0DH;j � c0

Dÿ;j; j � L;R; (3)

respectively. From Eq. (1b), the concentrations of the

neutral and ionized species of the drug in the external

solutions are, respectively,

c0DH;j �

c0H�;j=KDA

1� c0H�;j=KDA

cDT;j; j � L;R; (4)

c0Dÿ;j �

1

1� c0H�;j=KDA

cDT;j; j � L;R; (5)

where c0H�; j is the concentration of H� ions in the

external solutions. The concentrations of H� and

OHÿ ions verify the water dissociation equilibrium:

c0H�;jc

0OHÿ;j � KW; j � L;R; (6)

where KW �10ÿ14 mol2 lÿ2. The electroneutrality

condition in the external solutions leads to

c0Na�;j � c0

H�;j � c0Clÿ;j � c0

OHÿ;j � c0Dÿ;j; j � L;R:

(7)

Substituting Eqs. (2), (5) and (6) in Eq. (7) we obtain

c0H�;j �

KW

c0H�;j� 1

1� c0H�;j=KDA

cDT;j; j � L;R: (8)

Eq. (8) can be solved for c0H�;j using a numerical

procedure (the Newton±Raphson method for instance)

in order to obtain the initial pH values of the external

solutions, pH0j (j � L, R). Once pH0

L and pH0R have

been determined, the pH value of these solutions can

be adjusted to a desired value by adding the required

amount of either HCl or NaOH. For a given pHj value,

the concentrations of the H� and OHÿ ions in the

solutions are

cH�;j � 10ÿpHj ; j � L;R; (9)

cOHÿ;j � KW=cH�;j; j � L;R: (10)

The concentrations of the neutral and ionized forms of

the drug are determined by the new pHj values

imposed as

cDH;j �cH�;j=KDA

1� cH�;j=KDAcDT;j; j � L;R; (11)

cDÿ;j � 1

1� cH�;j=KDAcDT;j; j � L;R: (12)

In the case pHj < pH0j , HCl is added, and therefore:

cNa�; j � c0Na�; j � cS; j; j � L;R; (13)

Fig. 1. Schematic representation of the system studied. The fixed

charge, ion exchange membrane extends from x � 0 to x � d and

separates two solutions containing the drug in its neutral and

ionized forms, the supporting electrolyte (NaCl) and the H� and

OHÿ ions. !XM is the fixed charge concentration of the membrane

(! � �1 for an anion exchange membrane and ! � ÿ1 for a cation

exchange membrane), and ci,j is the concentration of the species i

(i � Na�, Clÿ, H� and OHÿ for ions; DH and Dÿ for the neutral

and ionized species of the anionic drug, and D and DH� for the

neutral and ionized species of the cationic drug) in the bulk of the

solution j (j � L for the left solution and j � R for the right

solution). pHj (j � L, R) refers to the pH value of solution j, and

cDT,j (j � L, R) and cS,j (j � L, R) to the total concentration of drug

and the supporting electrolyte concentration in the external solution

j, respectively. Finally, ci is the concentration of species i in the

membrane solution.

P. RamõÂrez et al. / Journal of Membrane Science 161 (1999) 143±155 145

while the electroneutrality condition leads to:

cClÿ;j � cNa�;j � cH�;j ÿ cOHÿ;j ÿ cDÿ;j; j � L;R:

(14)

In the case pHj > pH0j , NaOH is added, and the

concentrations of the Clÿ and Na� ions are

cClÿ;j � c0Clÿ;j � cS;j; j � L;R; (15)

cNa�;j � cClÿ;j � cOHÿ;j � cDÿ;j ÿ cH�;j; j � L;R:

(16)

Therefore, the concentrations of all species present

in the external solutions are determined by the pH

value, the supporting electrolyte concentration and the

total concentration of drug in these solutions.

External solution and membrane solution concen-

trations are connected through the following Donnan

equilibrium conditions [10,11] at the interfaces x � 0

and x � d:

cNa��0�cNa�;L

� cClÿ;L

cClÿ�0� �cH��0�cH�;L

� cOHÿ;L

cOHÿ�0� �cDÿ;L

cDÿ�0� ;

(17)

cNa��d�cNa�;R

� cClÿ;R

cClÿ�d� �cH��d�cH�;R

� cOHÿ;R

cOHÿ�d� �cDÿ;R

cDÿ�d� :

(18)

Combining Eqs. (17) and (18) with the electroneu-

trality condition within the membrane

cNa� � cH� � !XM � cClÿ � cOHÿ � cDÿ ; (19)

and the following equation is found

u2j �

!XM

cNa�;j � cH�;juj ÿ 1 � 0; j � L;R; (20)

where uL � cH��0�=cH�;L and uR � cH��d�=cH�;R.

Eq. (20) can be solved readily for uL and uR, yielding

uj�ÿ !XM

2�cNa�; j�cH�;j�� 1� !XM

2�cNa�;j � cH�;j�� 2

!1=2

;

j � L;R: (21)

After determining uj, the inner membrane phase con-

centrations ci(0) and ci(d) can be obtained from

Eqs. (17) and (18) in terms of the concentrations of

the different species in the external solutions. Once all

the concentrations ci(0) and ci(d) have been calcu-

lated, the Donnan potential differences through the left

(x � 0) and right (x � d) interfaces, ��L and ��R, can

be computed as

��L � RT

Fln

cH�;L

cH��0�� RT

Fln

1

uL

; (22)

��R � RT

Fln

cH��d�cH�;R

� RT

Fln uR: (23)

The electric potential and the ion ¯uxes through the

membrane can be calculated solving the Nernst±

Planck equations [8,9]:

JNa� � ÿDNa�dcNa�

dxÿ DNa�cNa�

F

RT

d�

dx; (24)

JClÿ � ÿDClÿdcClÿ

dx� DClÿcClÿ

F

RT

d�

dx; (25)

JH� � ÿDH�dcH�

dxÿ DH�cH�

F

RT

d�

dx; (26)

JOHÿ � ÿDOHÿdcOHÿ

dx� DOHÿcOHÿ

F

RT

d�

dx; (27)

JDH � ÿDDHdcDH

dx; (28)

JDÿ � ÿDDÿdcDÿ

dx� DDÿcDÿ

F

RT

d�

dx; (29)

together with the condition of zero total current

JNa� ÿ JClÿ � JH� ÿ JOHÿ ÿ JDÿ � 0; (30)

the dissociation equilibrium equations in the mem-

brane,

KW � cH�cOHÿ ; (31)

KDA � cDÿcH�

cDH

; (32)

the de®nition of the total ¯ux of the drug

JDT � JDH � JDÿ ; (33)

and the electroneutrality condition of Eq. (19). In

Eqs. (24)±(33), Di and Ji are the diffusion coef®cient

in the membrane and the ¯ux of species i, respectively,

� is the local electric potential, and the constants F, R

and T have their usual meaning [10,11].

Fluxes JNa� , JClÿ and JDT 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 the Na� and

Clÿ ions due to the complete dissociation of the strong

electrolyte in pure water, and the total concentration of


drug also remains constant. On the contrary, ¯uxes

JH� , JOHÿ , JDH, JDÿ are not constant through the

membrane because these ions participate in local

dissociation reactions that constitute sources and sinks

for these ions (see Eqs. (31) and (32)).

If the ¯uxes JNa� , JClÿ and JDT are known,

Eqs. (19), and (24)±(33) constitute a set of eleven

coupled ®rst-order differential equations with eleven

unknowns: the local concentrations of the six mobile

species, the ¯uxes JH� , JOHÿ JDH and JDÿ , and the local

electric potential �. But what we actually know are the

concentrations of the mobile species at both mem-

brane/solution interfaces, and not the ¯uxes JNa� , JClÿ

and JDT. We have used the following iterative proce-

dure to solve the problem: ®rst we assume some initial

values for the ion ¯uxes JNa� , JClÿ and JDT, and

integrate Eqs. (19), and (24)±(33) using a fourth-order

Runge±Kutta method [15] 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 satis®ed. This itera-

tive procedure allows to obtain the ion concentration

and electric potential pro®les, ci(x) and �(x), and the

ion ¯uxes Ji. Finally, the membrane potential is com-

puted as

��M � ��Don ��Dif ; (34)

where ��Don � ��L ��R is the total Donnan

potential drop at the membrane±solution interfaces

and ��Dif � ��d� ÿ ��0� is the diffusion potential

drop within the membrane.

The convergence of the iterative procedure

described above depends critically on the initial

values for the ion ¯uxes JNa� , JClÿ and JDT

assumed in the ®rst step. One possible initial

choice is some approximated solution of the system

of differential equations. For instance, if the local

electric ®eld and all ion ¯uxes within the mem-

brane are assumed to be approximately constant,

the system of equations given by Eqs. (24)±(30) leads

to [10,11]:

JNa� �FDNa��Dif

RTd

cNa��0�exp�ÿF��Dif=RT� ÿ cNa��d�1ÿ exp�ÿF��Dif=RT� ;

(35)

JClÿ �

ÿ FDClÿ��Dif

RTd

cClÿ�d�exp�ÿF��Dif=RT� ÿ cClÿ�0�1ÿ exp�ÿF��Dif=RT� ;

(36)

JH� �FDH��Dif

RTd

cH��0�exp�ÿF��Dif=RT� ÿ cH��d�1ÿ exp�ÿF��Dif=RT� ;

(37)

JOHÿ �

ÿ FDOHÿ��Dif

RTd

cOHÿ�d�exp�ÿF��Dif=RT�ÿcOHÿ�0�1ÿ exp�ÿF��Dif=RT� ;

(38)

JDH � DDH

d�cDH�0� ÿ cDH�d��; (39)

JDÿ �

ÿ FDDÿ��Dif

RTd

cDÿ�d�exp�ÿF��Dif=RT� ÿ cDÿ�0�1ÿ exp�ÿF��Dif=RT� ;

(40)

with

3. Results and discussion

3.1. Theoretical predictions

In this section we present a complete set of model

calculations concerning the ionic ¯uxes and the mem-

brane potential across the thick ®xed charge mem-

brane depicted in Fig. 1. In all the calculations of

Figs. 2±10, we have assumed the value d � 10ÿ2 cm

for the membrane thickness and the free aqueous

solution diffusion coef®cients values DNa� � 1:33�10ÿ5 cm2/s, DClÿ � 2:03� 10ÿ5 cm2/s, DH� � 9:30�

F��Dif

RT� ln

DNa�cNa��0� � DH�cH��0� � DClÿcClÿ�d� � DOHÿcOHÿ�d� � DDÿcDÿ�d�DNa�cNa��d� � DH�cH��d� � DClÿcClÿ�0� � DOHÿcOHÿ�0� � DDÿcDÿ�0� : (41)


10ÿ5 cm2/s and DOHÿ � 4:50� 10ÿ5 cm2/s for the

ionic species [10,11]. We have also considered

DDÿ � DDH � 8:50� 10ÿ7 cm2/s as typical values

for the diffusion coef®cients of the ionized and neutral

form of the drug species [5]. All calculations corre-

spond to the case pHL � pHR, cS,L � cS,R and

cDT,R � 0, i.e., the external pH and supporting elec-

trolyte concentration values are the same in the donor

(left) and the receiving (right) external solutions, and

Fig. 2. Total flux of drug across an anion (! � �1) and a cation

(! � ÿ1) exchange membrane vs. the external pH, pHL. The

curves correspond to the case XM � 0.1 M, cS,L � cS,R � 10ÿ5 M,

cDT,L � 10ÿ3 M and cDT,R � 0. The pKa values are pKDC � 6

(cationic drug) and pKDA � 8 (anionic drug). The fluxes show

symmetrical characteristics in the transport of the anionic and

cationic drugs, and can change by several orders of magnitude with

the external pH.

Fig. 3. The pH value at the membrane side of the membrane/

solution interface at x � 0, pH(0), vs. pHL for the system

considered in Fig. 2 in the case of the anionic drug. The straight

line corresponds to the case of a neutral membrane. For ! � �1,

pH(0) > pHL, and the drug is basically in its charged form within

the membrane if pHL > 4.6, because in this case pH(0) > 8 � pKDA.

For ! � ÿ1, pH(0) < pHL, and the drug is basically in its charged

form within the membrane only if pHL > 10, since in this

pHL(0) > 8 � pKDA. Note that the internal pH values pH(0) can

differ significantly from the external pH values pHL.

Fig. 4. Concentrations of the anionic drug species at the membrane

side of the membrane/solution interface at x � 0 vs. pHL for the

system considered in Fig. 2. The correspondence between the total

concentration of drug within the membrane and the fluxes JDT in

Fig. 2 can be clearly seen.

Fig. 5. Ion fluxes vs. pHL for the system considered in Fig. 2 in the

case of the anionic drug and ! � ÿ1. Fluxes JNa� , JClÿ and JDT

have been calculated using the numerical solution of Eqs. (19), and

(24)±(33). Fluxes JH� , JOHÿ , JDÿ and JDH have been determined

using the constant field approximation given in Eqs. (35)±(41). We

can see that ÿJH� � JNa� and the other fluxes are negligible, as

anticipated by the total co-ion exclusion approximation given in

Eqs. (42)±(46).


the drug concentration is zero in the receiving solu-

tion.

Fig. 2 shows the total ¯ux of drug across an anion

(! � �1) and a cation (! � ÿ1) exchange membrane

vs. the pH value (pHL) of the external solutions. All

curves correspond to the case XM � 0.1 M,

cS,L � cS,R � 10ÿ5 M, cDT,L � 10ÿ3 M (a situation

of practical `̀ absence'' of supporting electrolyte, since

cDT;L � cS;L � cS;R). We have considered the pK

values pKDC � 6 for the cationic drug and pKDA � 8

for the anionic drug. The continuous and dashed lines

correspond to the cases of an anionic drug passing

through an anion exchange membrane (! � �1) and

through a cation exchange membrane (! � ÿ1),

Fig. 6. Ion fluxes vs. pHL for the system considered in Fig. 2 in the

case of the anionic drug and ! � �1. All fluxes have been

determined following the same procedure than in Fig. 5. Now, the

total co-ion exclusion predicts JDÿ � JDT � ÿ�JClÿ � JOHÿ � and

the other fluxes are negligible.

Fig. 7. Membrane potential ��M (continuous lines) vs. pHL for the

system considered in Fig. 2 in the case of the anionic drug. The

Donnan potential drops ��Don (dashed lines) and the diffusion

potential drops ��Dif (dotted-dashed lines) have also been plotted.

��Don takes opposite values for ! � ÿ1 and ! � �1, and attains a

maximum in the case ! � ÿ1 (a minimum in the case ! � �1) at

pHL � ÿ1=2 log10�KWKDA=cS;L�. The minimum of ��Dif in the

case ! � ÿ1 corresponds to the external pH value at which

JNa� � JmaxNa� in Fig. 5. The maximum of ��Dif in the case ! � �1

corresponds to JDT � JDÿ � JmaxDT in Fig. 6.

Fig. 8. Total flux of anionic drug across an anion (! � �1) and a

cation (! � ÿ1) exchange membrane vs. pHL. All curves

correspond to the case pKDA � 8, cS,L � cS,R � 10ÿ5 M,

cDT,L �10ÿ3 M and cDT,R � 0. The curves are parametric in XM

(M). The top curves correspond to the case ! � �1, and the bottom

curves to ! � ÿ1. The case of an uncharged membrane, XM � 0,

has also been plotted.

Fig. 9. Total flux of anionic drug across an ion exchange

membrane vs. pHL. All curves correspond to the case pKDA � 8,

XM � 10ÿ1 M, cDT,L � 10ÿ3 M and cDT,R � 0. The curves are

parametric in cS,L � cS,R (M). The top curves correspond to the

case ! � �1 and the bottom curves to ! � ÿ1.


respectively. The dotted and dotted-dashed lines cor-

respond to a cationic drug passing through a cation

exchange membrane and an anion exchange mem-

brane, respectively. Since the curves show symmetri-

cal characteristics in the transport of anionic and

cationic drugs, we will focus on the case of an anionic

drug in the following, bearing in mind that all the

conclusions drawn for this kind of drug can be

extended mutatis mutandi to the case of a cationic

drug.

We see in Fig. 2 that the total ¯ux of the anionic

drug through the membrane depends critically on the

sign of the ®xed charge concentration in the mem-

brane and the pH value of the external solutions.

However, the curves of JDT vs. the external pH

corresponding to the anion and cation exchange mem-

branes attain a common limiting value in the region

pHL < pKDA � 8. For these external pH values, the

drug is in its neutral form (see Eqs. (11) and (12)),

and therefore the sign of the membrane ®xed

charges does not affect at all the total ¯ux of drug,

which is JDT � JDH � DDH�cDH�0� ÿ cDH�d��=d �DDHcDT;L=d � 8:50� 10ÿ11 mol cm2/s. For higher

pH values, however, JDT attains a maximum in the

case of an anion exchange membrane and a minimum

in the case of a cation exchange membrane. We will

show later that the particular characteristics of the JDT

curves are related to the concentrations of drug, cDH(0)

and cDÿ�0�, and the pH value, pH(0), at the membrane

side of the membrane/solution interface at x � 0. This

latter value can be signi®cantly different from the pH

value in the external left solution due to the Donnan

equilibrium (see Eqs. (17)±(21) and Fig. 3 later).

Finally, we emphasize that the drug ¯uxes are very

sensitive to the external pH (note the logarithmic scale

in the ordinate axis), as found experimentally [4].

In order to explain the maximum and minimum of

JDT, Fig. 3 shows the curve pH(0) vs. pHL for the cases

considered in Fig. 2. The case of a neutral membrane

(pH(0) � pHL) has also been represented. For the

anion exchange membrane (! � �1), the internal

(membrane) solution pH value, pH(0), is always

higher than the external solution pH value, pHL. For

instance, pH(0) � pKDA � 8 corresponds to

pHL � 4:6. Thus, if pHL > 4:6, the drug is basically

in its charged form within the membrane and JDT is

determined mainly by JDÿ. Conversely, in the case of

the cation exchange membrane (! � ÿ1), pH(0) is

always lower than pHL, and pH�0� � pKDA � 8 cor-

responds now to pHL > 10. Therefore, if pHL > 10 the

drug is mainly in its charged form within the mem-

brane and again JDT is determined by JDÿ. This can be

seen more clearly in Fig. 4, where the concentrations

of the drug species have been plotted as a function of

the external pH. Fig. 4 explains the positions of the

maximum (for ! � �1) and the minimum (for

! � ÿ1) of JDT shown in Fig. 2. The maximum is

attained at pHL � 8.8 (pH(0) � 10.9 from Fig. 3), and

corresponds to the maximum value of cDÿ�0� in Fig. 4,

cmaxDÿ �0� � XM (note that in our case, cClÿ;L � cS;L � 0,

and according to the electroneutrality condition the

membrane ®xed charge concentration must then be

compensated by the Dÿ ions at this high pH value).

For pHL > 8:8, the OHÿ ions begin to substitute the

Dÿ ions in order to compensate the positive ®xed

charge concentration of the membrane, and, as a

consequence, cDT(0) and JDT begin to decrease with

the external pH value as shown in Figs. 2 and 4. The

minimum of JDT vs. pHL of Fig. 2 corresponds to the

minimum of the total concentration of drug

cDT�0� � cDÿ�0� � cDH�0� of Fig. 4, which is reached

at pHL � 10:5�pH�0� � 8:6 from Fig. 3).

The important conclusions to be drawn from

Figs. 2±4 are: (i) the local pH values within the

membrane dictate the observed ¯ux of the drug,

and (ii) these membrane solution pH values can be

very different from those in the external solution (note

Fig. 10. Total flux of anionic drug across an ion exchange

membrane vs. pHL. All curves correspond to the case pKDA � 8,

XM � 10ÿ1 M, cS,L � cS,R �10ÿ5 M and cDT,R � 0. The curves are

parametric in cDT,L (M). The top curves correspond to the case

! � �1 and the bottom curves to ! � ÿ1.


that one pH unit corresponds to a factor 10 in the

hydrogen concentration). Ignoring this latter fact can

lead to serious errors in the interpretation of experi-

mental data, as pointed out previously in the cases of

variable permeability membranes (chemical valves)

[13], ionic hydrogels [16] and weak-acid membranes

[17].

If the membrane ®xed charge concentrations are

relatively high, the concentrations of the co-ions

within the membrane can be assumed to be negligible

compared to those of the counter-ions (total co-ion

exclusion limit) except for extreme pH values. This

allows to obtain the following approximated solutions

for Eqs. (19), and (24)±(33). In the case ! � ÿ1, we

have

cNa� � cH� � XM; (42)

uj� XM

cNa�;j � cH�;j� XM

cClÿ;j � cOHÿ;j � cDÿ;j; j � L;R

(43)

JClÿ � JOHÿ � JDÿ � 0; (44)

ÿJH� � JNa� : (45)

From Eqs. (24), (26), (42) and (45) we ®nd

JNa�1

DNa�ÿ 1

DH�

� �� ÿXM

F

RT

d�

dx; (46)

which means that the local electric ®eld within the

membrane is approximately constant. Therefore, the

initial values given by Eqs. (35)±(41) constitute here

good approximations for the numerical solution

described in the previous section.

In the case ! � �1, neglecting the contribution of

co-ions leads to

cClÿ � cOHÿ � cDÿ � XM; (47)

uj�cClÿ;j�cOHÿ;j � cDÿ;j

XM� cNa�;j � cH�;j

XM; j � L;R;

(48)

JNa� � JH� � 0; (49)

JClÿ � JOHÿ � JDÿ � 0; (50)

and Eqs. (25), (27), (29) and (47) result in

JClÿ

DClÿ� JOHÿ

DOHÿ� JDÿ

DDÿ� XM

F

RT

d�

dx(51)

which indicates again that the constant ®eld assump-

tion also constitutes a good ®rst approximation to the

problem for high membrane ®xed charge concentra-

tions. Note that, because of the Donnan exclusion,it is

in the case of high ®xed charge concentration where

one would expect the total co-ion exclusion to hold.

Figs. 5 and 6 show the ion ¯uxes vs. pHL in the

cases ! � ÿ1 (Fig. 5) and ! � �1 (Fig. 6). Fluxes

JNa� , JClÿ and JDT have been calculated using the

numerical solution of Eqs. (19), and (24)±(33). Fluxes

JH� , JOHÿ , JDÿ and JDH have been determined

using the constant ®eld approximation given in

Eqs. (35)±(41). We can see that ÿJH� � JNa� and

the other ¯uxes are negligible in Fig. 5 (! � ÿ1)

and JDÿ � JDT � ÿ�JClÿ � JOHÿ� and the other ¯uxes

are negligible in Fig. 6 (! � �1), as anticipated by

the total co-ion exclusion approximation given in

Eqs. (42)±(51). This approximation allows to simplify

further the results provided by the constant ®eld

assumption. In the case ! � ÿ1, Eqs. (42)±(45)

lead to:

��Don � RT

Fln

cNa�;L � cH�;L

cNa�;R � cH�;R; (52)

��Dif�RT

Fln

cNa�;R�cH�;R

cNa�;L�cH�;L

DNa�cNa�;L�DH�cH�;L

DNa�cNa�;R�DH�cH�;R

� �:

(53)

Integrating Eq. (46) from x � 0 to x � d yields

JNa� � ÿJH� � ÿDH� ÿ DNa�

DH�DNa�

XM

d

F��Dif

RT(54)

In our case, Eq. (53) gives F��Dif=RT � 1 in most of

the pH range considered, and therefore

exp�ÿF��Dif=RT� in Eq. (40) can be approximated

by 1ÿ F��Dif=RT. Under this assumption, the ¯ux of

the charged species of the drug is

JDÿ � DDÿ

dcDÿ�0�; (55)

and the total ¯ux of drug gives

JDT � DDH

dcDH�0� � DDÿ

dcDÿ�0� (56)

Eq. (56) is in agreement with results shown in Figs. 2

and 4: if DDH � DDÿ , then JDT is determined by the

total concentration of drug at the membrane side of the

membrane/solution interface at x � 0; cDH�0��cDÿ�0�. Using Eqs. (42) and (43), JDT can be written


approximately in the form

JDT � DDH

dcDH;L � DDÿ

d

cDÿ;LXM

cClÿ;L � cOHÿ;L � cDÿ;L

(57)

In the case ! � �1, Eqs. (47)±(50) lead to

��Don � RT

Fln

cClÿ;R � cOHÿ;R � cDÿ;R

cClÿ;L � cOHÿ;L � cDÿ;L; (58)

and integration of Eq. (51) from x � 0 to x � d

gives

JClÿ

DClÿ� JOHÿ

DOHÿ� JDÿ

DDÿ� XM

d

F��Dif

RT: (60)

Now, the maximum value of JDT vs. pHL in Fig. 2,

JmaxDT , is attained at pHL � 8.8. At this pH value,

cDÿ�0� � XM in Fig. 4 and F��maxDif =RT � 3:1. With

these assumptions, Eq. (40) yields approximately

JmaxDT � Jmax

Dÿ �DDÿXM

d

F��maxDif

RT

� 2:7� 10ÿ8 mol=cm2

s (61)

in agreement with Fig. 2.

Fig. 7 shows the membrane potential ��M (con-

tinuous lines) across the ion exchange membranes vs.

pHL. The contributions of the Donnan potential drops

��Don (dashed lines) and the diffusion potential drops

��Dif (dotted-dashed lines) have also been plotted

separately. As mentioned above, the constant ®eld

approximation together with the total co-ion exclusion

assumption give good approximated solutions for the

membrane potential and the ion ¯uxes for high ®xed

charge concentration. The different contributions to

the total membrane potential can be calculated by

using Eqs. (52) and (53) for ! � ÿ1 and Eqs. (58)

and (59) for ! � �1. Under these assumptions, ��M

gives

��M�RT

Fln

DNa�cNa�;L � DH�cH�;L

DNa�cNa�;R � DH�cH�;R; for !�ÿ1;

(62)

and

��M � RT

Fln

DClÿcClÿ;R � DOHÿcOHÿ;R � DDÿcDÿ;R

DClÿcClÿ;L � DOHÿcOHÿ;L � DDÿcDÿ;L;

for ! � �1; (63)

As expected, ��Don takes opposite values for ! � ÿ1

and ! � �1 (see Eqs. (22) and (23)). In the case

! � ÿ1, the position of the maximum of the curve

��Don vs. pHL (the minimum in the case ! � �1) can

be calculated from Eq. (52), giving

pHmaxL �! � ÿ1� � pHmin

L �! � �1�� ÿ 1

2log10

KWKDA

cS;L� 8:5 (64)

We can also see that the behavior of ��Dif is closely

related to the behavior the ionic ¯uxes through the

membrane (see Figs. 5 and 6). In the case ! � ÿ1,

��Dif attains a minimum at pHL � 6:1. This point

corresponds to the external pH value at which JNa�

reaches a maximum (see Eq. (54) and Fig. 5). In the

case ! � �1, ��Dif attains a maximum at the same

pH value pHL � 8:8 where JDT � JDÿ reaches a max-

imum (see Eq. (61) and Fig. 6).

Fig. 8 shows the effect of the membrane ®xed

charge concentration on the calculated total ¯ux of

anionic drug. The numbers close to each curve denote

the membrane ®xed charge concentration XM (in M)

used in the calculations. The upper curves correspond

to the case ! � �1, and the lower ones to ! � ÿ1.

The case of an uncharged membrane has also been

plotted. The curves show similar characteristics than

those seen in Fig. 2. As expected, the maxima and

minima become less pronounced when XM decreases.

In the case ! � ÿ1, JDT attains a minimum, JminDT ,

whose position is shifted to lower pHL values as XM

decreases. In the case ! � �1, the position of the

maxima of JDT hardly depends on XM, as suggested by

Eqs. (59) and (61) (note that the maximum of the

curve ��Dif vs. pHL does not depend on XM) although

the value of JmaxDT is approximately proportional to XM.

However, it must be emphasized that in the case of

��Dif � RT

Fln

cClÿ;L � cOHÿ;L � cDÿ;L

cClÿ;R � cOHÿ;R � cDÿ;R

DClÿcClÿ;R � DOHÿcOHÿ;R � DDÿcDÿ;R

DClÿcClÿ;R � DOHÿcOHÿ;L � DDÿcDÿ;L

� �; (59)


weakly charged (XM � 10ÿ3 M) and neutral (XM � 0)

membranes the constant ®eld approximated solution

deviates signi®cantly from the exact numerical solu-

tion, and therefore the above discussion is valid only

qualitatively.

Fig. 9 shows the effect of the supporting electrolyte

concentration on the calculated total ¯ux of anionic

drug. The numbers close to each curve denote the

concentration of supporting electrolyte cS;L � cS;R (in

M) used in the calculations. Again, top and bottom

curves correspond to ! � �1 and ! � ÿ1, respec-

tively. We see that increasing cS,L leads to lower

deviations of JDT from the limiting value correspond-

ing to the transport of the drug in its neutral form (the

low pHL limit). This is because increasing the support-

ing electrolyte concentration reduces the Donnan

sorption in the membrane of the ionized form of

the drug. In the case cS;L � XM � cDT;L, the migration

term of Eq. (29) can be neglected, and JDT can be

written approximately in the form

JDT � DDH

dcDH�0� � DDÿ

dcDÿ�0�

for both ! � ÿ1 and ! � �1: (65)

Fig. 10 shows the effect of the concentration of the

drug on the calculated total ¯ux of an anionic drug.

The numbers close to each curve denote the concen-

tration of the drug in the left external solution, cDT,L

(M). We see that increasing cDT,L leads to lower

deviations of JDT from the low pHL limit DDHcDT,L/

d. This is to be expected, since the effect of the

membrane ®xed charges on the ionized drug transport

becomes less noticeable when cDT,L approximates XM.

3.2. Comparison with experiments

It is in order now to check the theoretical predic-

tions in some experimental contexts. Two recent

experimental studies on transport of pharmaceuticals

across ®xed charge membranes are those by Chen

et al. [4] and Akerman et al. [5]. These authors have

found a signi®cant dependence of the transport pro-

perties of different drugs across the membranes on

the pH values of the external solutions, and some

of their experimental results can be discussed on the

basis of the theoretical model described in the

previous section.

Fig. 11 shows the experimental results for the total

¯ux of resorcinol, 5-¯uoracil and vitamin C across a

silk ®broin membrane vs. the pH value of the sur-

rounding solutions [4]. These drugs are anionic, and

their pKDA values are about 9.2 (resorcinol), 8 (5-

¯uoracil) and 4.3 (vitamin C). The symbols represent

the experimental results for JDT obtained from the

reported permeability coef®cients [4]. The experimen-

tal situation was similar to that depicted in Fig. 1, with

cDT,L � 10ÿ3 M and cDT,R � 0. In these experiments,

no supporting electrolyte was added, and then

cS,L � cS,R � 0. The membrane used was found to

present amphoteric properties. However, in the range

represented in Fig. 11 it can be assumed that the

membrane was negatively charged since the isoelec-

tric point of this membrane was close to pHL � 4.5 [4].

The ®xed charge concentration and the membrane

thickness were XM � 1.5�10ÿ2 M and d � 2.7�10ÿ3 cm, respectively [4]. The continuous lines of

Fig. 11 correspond to the theoretical predictions of

the model described in the previous section using the

values for pKDA, cDT,L, cS,L, XM and d listed above [4].

Therefore, the only free parameters used to ®t the

experimental data were the diffusion coef®cients of

Fig. 11. Experimental results for the total flux of resorcinol (open

squares), 5-fluoracil (solid circles) and vitamin C (open circles)

across a silk fibroin membrane vs. pHL [4]. All drugs are anionic,

with pKDA � 9.2 (resorcinol), 8 (5-fluoracil) and 4.3 (vitamin C).

The experimental situation was similar to that depicted in Fig. 1,

with cDT,L � 10ÿ3 M, cDT,R � 0 and cS,L � cS,R � 0. In the pH

range represented, the membrane used was negatively charged with

XM � 1.5�10ÿ2 M and d � 2.7�10ÿ3 cm [4]. The continuous lines

correspond to the theoretical predictions using to DDÿ � DDH �1:6� 10ÿ7 cm2/s (resorcinol), DDÿ � DDH � 10ÿ7 cm2/s (5-fluor-

acil), and DDÿ � DDH � 4:6� 10ÿ8 cm2/s (vitamin C).


the neutral and ionized species of the drugs. The

theoretical results plotted correspond to DDÿ �DDH � 1:6� 10ÿ7 cm2/s (resorcinol), DDÿ � DDH �10ÿ7 cm2/s (5-¯uoracil), and DDÿ � DDH � 4:6�10ÿ8 cm2/s (vitamin C).

Fig. 12 shows the experimental results for the total

¯ux of cimetidine and phthalimide [5] across porous

poly(vinylidene ¯uoride) membranes grafted with

poly(acrylic acid) chains (PVDF±PAA membranes)

[13]. These membranes present cation exchange prop-

erties due to dissociated carboxyl groups in the poly(-

acrylic acid) chains, and their properties can be varied

by exposing the polymer matrix to different degrees of

grafting [13]. Since the pKa value of the poly(acrylic

acid) is about 4, the membranes used in the experi-

ments of Fig. 12 are expected to be negatively charged

in the pHL range studied. Cimetidine is a cationic drug

with pKDC�6.8 while phthalimide is an anionic drug

with pKDA�7.4. The symbols represent the experi-

mental results reported for JDT [5]: solid circles, open

circles and open squares correspond to cimetidine, and

solid triangles to phthalimide. The experimental situa-

tion was again similar to that depicted in Fig. 1, with

cDT,L � 5�10ÿ4 M and cDT,R � 0, but now a support-

ing electrolyte (NaCl) with concentration

cS,L � cS,R � 0.2 M was added to the solutions sepa-

rated by the membrane. In the measurements corre-

sponding to cimetidine, several PVDF±PAA

membranes having different degrees of grafting (in

%) were used: 58% (solid circles), 29% (open squares)

and 15% (open circles) [5]. These degrees of grafting

correspond to the ®xed charge concentrations

XM � 0.35, 0.30 and 0.15 M, respectively [5]. In the

measurements corresponding to phthalimide, only the

membrane having a 58% degree of grafting was used.

The thickness of all membranes was estimated to be

d � 10ÿ2 cm [13].

The lines of Fig. 12 correspond to the theoretical

predictions of the model described in the previous

section using the above values for pKDA, pKDC, cDT,L,

cS,L, XM and d [5]. Again, the only free parameters

used to ®t the experimental data were the diffusion

coef®cients of the neutral and ionized species of the

drug. The theoretical results represented correspond to

DDH� � DD � 5:8� 10ÿ7 cm2/s (cimetidine) for

XM � 0.35 (dotted-dashed line), XM � 0.30 (dashed

line) and XM � 0.15 (continuous line), and

DDÿ � DDH � 2:1� 10ÿ7 cm2/s (phthalimide) for

XM � 0.35 (dotted-dashed line). We can see that

although the theoretical curves follow qualitatively

the experimental points, the experiment shows higher

drug ¯uxes in the case XM � 0.15 (open circles) than

for XM � 0.30 (open squares). This is not consistent

with the theoretical predictions. However, this anom-

alous result can be explained taking into account that

the membrane degree of grafting determines not only

the ®xed charge concentration, but also other impor-

tant characteristics like the effective radius of the

membrane pores [5,13] which could also affect the

transport properties of the membrane [13].

The effective values for the diffusion coef®cients

obtained from the reported drug ¯ux and permeability

data [4,5] are signi®cantly lower than those typical of

small ionic species in charged membranes, which are

usually in the range 10ÿ7±10ÿ6 cm2/s [18]. Aside from

size effects, it should be noted that our diffusion

coef®cients incorporate implicitly the membrane por-

Fig. 12. Experimental results for the total flux of cimetidine

(cationic drug with pKDC � 6.8) and phthalimide (anionic drug

with pKDA � 7.4) across porous PVDF±PAA membranes vs. pHL

[5]. Solid circles, open circles and open squares correspond to

cimetidine data, and solid triangles to phthalimide data. The

experimental situation was similar to that depicted in Fig. 1, with

cDT,L � 5�10ÿ4 M, cDT,R � 0 and supporting electrolyte (NaCl)

concentration cS,L � cS,R � 0.2 M. The membranes used in the

experiments were negatively charged in the pH range studied, with

XM � 0.35 M (solid circles), XM � 0.30 M (open squares) and

XM � 0.15 M (open circles), in the case of cimetidine, and

XM � 0.35 M (solid triangles) in the case of phthalimide [5]. The

theoretical results represented correspond to DDH� � DD � 5:8�10ÿ7 cm2/s (cimetidine) and XM � 0.35 M (dotted-dashed line),

XM � 0.30 M (dashed line) and XM � 0.15 M (continuous line),

and DDÿ � DDH � 2:1� 10ÿ7 cm2/s (phthalimide) and XM �0.35 M (dotted-dashed line).


osity and ion partition coef®cient effects [5]. These

effects were not included explicitly in the model, and

act together to decrease signi®cantly the effective

diffusion coef®cients obtained here. For instance,

some binding between the drug and the membrane

has been reported [5], and this binding increased with

the drug lipophilicity.

In summary, we have presented a complete set of

model calculations concerning the transport of both

cationic and anionic drugs through thick ®xed charge

membranes. At this preliminary stage, we have

restricted our attention to pH and supporting electro-

lyte effects, the ®xed charge concentration being the

only membrane characteristic considered. The whole

system of equations including the six Nernst±Planck

equations and the equations for the local dissociation

equilibria within the membrane have been solved

numerically without any additional simplifying

assumption. The validity of the Goldman constant

®eld and total co-ion exclusion assumptions has been

checked. The comparison of the theoretical predic-

tions with available experimental data has shown the

potential utility of the numerical solution and the

analytic approximations proposed. Since the physical

system considered constitutes a simpli®ed model for

the pH controlled ion transport and drug delivery

through membranes of biological and pharmaceutical

interest, the model calculations presented here could

be of interest for the analysis and the design of future

experiments.

Acknowledgements

Financial support from the DGICYT, Ministry of

Education and Science of Spain, under Project No

PB98 is gratefully acknowledged.

References

[1] M. Minagawa, A. Tanioka, P. RamõÂrez, S. MafeÂ, Amino acid

transport through cation exchange membranes: effects of pH

on interfacial transport, J. Colloid Interface Sci. 188 (1997)

176±182.

[2] M. Minagawa, A. Tanioka, Leucine transport through cation

exchange membranes: effects of HCl concentration on

interfacial transport, J. Colloid Interface Sci. 202 (1998)

149±154.

[3] X. Hu, D.D. Do, Q. Yu, Effects of supporting and buffer

electrolytes (NaCl, CH3COOH and NH4OH) on the diffusion

of BSA in porous media, Chem. Eng. Sci. 47 (1992) 151±164.

[4] J. Chen, N. Minoura, A. Tanioka, Transport of pharmaceu-

ticals through silk fibroin membranes, Polymer 35 (1994)

2853±2856.

[5] S. Akerman, P. Viinikka, B. Svarfvar, K. JaÈrvinen, K.

Kontturi, J. Nasman, A. Urti, P. Paronen, Transport of drugs

across porous ion exchange membranes, J. Control. Release

50 (1998) 153±166.

[6] V. Aguilella, K. Kontturi, L. MurtomaÈki, P. RamõÂrez,

Estimation of the pore size and charge density in human

cadaver skin, J. Control. Release 32 (1994) 249±257.

[7] Y. Rojanasakul, J.R. Robinson, Transport mechanisms of the

cornea: characterization of barrier permselectivity, Int. J.

Pharm. 55 (1989) 237±246.

[8] R.P. Buck, Kinetics of bulk and interfacial ionic motion:

Microscopic basis and limits for the Nernst±Planck equations

applied to membrane systems, J. Membrane Sci. 17 (1984) 1±

62.

[9] R.S. Eisenberg, Computing the field in proteins and channels,

J. Membrane Biol. 150 (1996) 1±25.

[10] N. Laksminarayanaiah, Equations of Membrane Biophysics,

Academic Press, New York, 1984.

[11] S.G. Schultz, Basic Principles of Membrane Biophysics,

Cambridge University Press, Cambridge, 1980.

[12] J. Pellicer, S. MafeÂ, V.M. Aguilella, Ionic transport across

porous charged membranes and the Goldman constant field

assumption, Ber. Bunsenges. Phys. Chem. 90 (1986) 867±

872.

[13] K. Kontturi, S. MafeÂ, J.A. Manzanares, B.L. Svarfvar, P.

Viinikka, Modeling of the salt and pH effects on the

permeability of grafted porous membranes, Macromolecules

29 (1996) 5740±5746.

[14] J.A. Manzanares, G. Vergara, S. MafeÂ, K. Kontturi, P.

Viinikka, Potentiometric determination of transport numbers

of ternary electrolyte systems in charged membranes, J. Phys.

Chem. 102 (1998) 1301±1307.

[15] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling,

Numerical Recipes, Cambridge University Press, Cambridge,

1989.

[16] S. MafeÂ, J.A. Manzanares, A.E. English, T. Tanaka, Multiple

phases in ionic copolymer gels, Phys. Rev. Lett. 79 (1997)

3086±3089.

[17] S. MafeÂ, P. RamõÂrez, J. Pellicer, Activity coefficients and

Donnan coion exclusion in charged membranes with weak-

acid fixed charge groups, J. Membrane Sci. 138 (1998) 269±

277.

[18] F. Helfferich, Ion Exchange, McGraw-Hill, New York, 1962.


Documents

pH and supporting electrolyte concentration effects on the passive transport of cationic and anionic drugs through fixed charge membranes