Journal of Colloid and Interface Science 253, 171–179 (2002)
doi:10.1006/jcis.2002.8508
Donnan Equilibrium of Ionic Drugs in pH-Dependent Fixed Charge
Membranes: Theoretical Modeling
Patricio Ramı́rez,∗ Antonio Alcaraz,† Salvador Mafé, ‡,1 and Julio Pellicer‡
∗ Departament de Fı́sica Aplicada, Universitat Politècnica de València, Camino de Vera s/n, E-46022 València, Spain; †Departament de Ciències Experimentals,
Universitat Jaume I, Apdo. 224, E-12080 Castelló, Spain; and ‡Departament de Termodinàmica, Facultat de Fı́sica,
Universitat de València, E-46100 Burjassot, Spain
Received December 7, 2001; accepted May 22, 2002
We have studied theoretically the partition equilibrium of a
cationic drug between an electrolyte solution and a membrane with
pH-dependent fixed charges using an extended Donnan formalism.
The aqueous solution within the fixed charge membrane is assumed
to be in equilibrium with an external aqueous solution containing
six ionic species: the cationic drug (DH+ ), the salt cations (Na+ and
Ca2+ ), the salt anion (Cl− ), and the hydrogen and hydroxide ions.
In addition to these mobile species, the membrane solution may
also contain four fixed species attached to the membrane chains:
strongly acid sulfonic groups (–SO−
3 ), weakly acid carboxylic groups
in dissociated (–COO− ) and neutral (–COOH) forms, and positively
charged groups (–COO · · · Ca+ ) resulting from Ca2+ binding to dissociated weakly acid groups. The ionization state of the weak electrolyte groups attached to the membrane chains is analyzed as a
function of the local pH, salt concentration, and drug concentration in the membrane solution, and particular attention is paid to
the effects of the Ca2+ binding to the negatively charged membrane fixed groups. The lipophilicity of the drug is simulated by the
chemical partition coefficient between the membrane and external
solutions giving the tendency of the drug to enter the membrane
solution due to hydrophobic interactions. Comparison of the theoretical results with available experimental data allows us to explain
qualitatively the effects that the pH, salt concentration, drug concentration, membrane fixed charge concentration, and Ca2+ binding exert on the ionic drug equilibrium. The role of the interfacial
(Donnan) electric potential difference between the membrane and
the external solutions on this ionic drug equilibrium is emphasized
throughout the paper. C 2002 Elsevier Science (USA)
Key Words: ionic drug partition equilibrium; fixed charge membranes; pH effects; Donnan potential; theoretical modeling.
INTRODUCTION
Determining the partition equilibrium of an ionic drug between a membrane solution and an external electrolyte solution
(1–15) constitutes a difficult problem not only because of the
multi-ionic character of the system (the ionic drug, the salt ions,
and the hydrogen and hydroxide ions) but also due to the great
1
To whom correspondence should be addressed. E-mail: smafe@uv.es.
171
number of effects (the lipophilicity of the drug, the pH, salt,
and drug concentrations of the external solution, the nature and
concentration of the membrane fixed charge, the ion binding of
certain divalent ions to the membrane fixed charges, etc.) that
are present simultaneously. However, an understanding of the
above effects is needed in most practical applications.
The partition equilibrium of a cationic drug in a membrane
with pH-dependent fixed charges will be studied theoretically
using the Donnan equilibrium formalism (16, 17). This formalism considers the aqueous solution in the fixed charge membrane to be in equilibrium with the external aqueous solution containing six ionic species: the cationic drug, the salt
cations, the salt anion, and the hydrogen and hydroxide ions.
The Donnan equilibrium leads to the constancy of the electrochemical potential through the membrane/solution interface for
each ionic species (16, 17). In addition to the above mobile
species, the membrane solution may also contain four fixed
species attached to the membrane chains: strongly acid sulfonic groups (–SO−
3 ), weakly acid carboxylic groups in dissociated (–COO− ) and neutral (–COOH) forms, and positively
charged groups (–COO · · · Ca+ ) resulting from Ca2+ binding
to dissociated weakly acid groups. The ionization state of the
weak electrolyte groups attached to the membrane chains is analyzed as a function of the local pH, salt concentration, and
drug concentration in the membrane solution. The lipophilicity
of the drug is simulated by the chemical partition coefficient
between the membrane and external solutions. This coefficient
measures the tendency of the drug to enter the membrane solution due to specific (e.g., hydrophobic) interactions. Comparison of the theoretical results with available experimental
data (1–3, 5–9) allows us to explain qualitatively the effects
that the pH, salt, drug, and membrane fixed charge concentrations exert on the ionic drug equilibrium. In particular, we show
that the Donnan potential difference between the membrane
and the external solutions is crucial for the understanding of
this equilibrium.
The results presented should be especially relevant to those
experimental situations where the partition equilibrium of the
ionic drug must be controlled by the external pH and salt concentration (e.g., drug delivery systems and separation processes
0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
All rights reserved.
172
RAMÍREZ ET AL.
in biotechnology (1–4, 7–13, 15, 18, 19)). Indeed, although control release and separation processes usually involve both equilibrium and transport phenomena (8, 9), the partition equilibria
between the membrane and the external solutions often constitutes a crucial step that is not completely understood (2, 3, 7,
9–13, 20). In this context, it should be emphasized that while
experiments are often conducted using a buffer solution and/or
a supporting electrolyte, the measured multi-ionic system properties are sometimes analyzed in terms of simplified binary salt
system equations, and the pH value in the membrane solution
is often assumed to be close to the pH value in the external solution. These assumptions may introduce severe errors and will
not thus be invoked in the theoretical model.
The pH of the external solution is controlled by adding either
HCl or NaOH. Therefore, in the most general case, the ionic
mobile species present in the system are the cationic form of the
drug (DH+ ), the salt ions (Na+ , Ca2+ , Cl− ), and the hydrogen
(H+ ) and hydroxide (OH− ) ions. In the following, ci stands for
the concentrations of species i (i = DH+ , Na+ , Ca2+ , H+ , Cl− ,
and OH− ) and NaCl, CaCl2 , and DHCl in the external solution
phase. Overbars denote membrane solution concentrations. The
external solution is assumed to be perfectly stirred, and the whole
system is considered isothermal.
For a given value of pH, the concentrations of the H+ , OH− ,
DH+ , and Ca2+ ions in the external solution are
THEORETICAL APPROACH
cH+ = 10−pH ,
[2]
cOH− = K W /cH+ ,
[3]
cDH+ = cDHCl ,
[4]
cCa2+ = cCaCl2 ,
[5]
The model system considered is shown schematically in
Fig. 1. The aqueous solution within the fixed charge membrane
is assumed to be in equilibrium with an aqueous external solution containing a mixture of NaCl, CaCl2 , and a drug of the form
DHCl. We assume that within the pH range studied, the drug is
fully dissociated according to the scheme
where K W = 10−14 M2 .
The electroneutrality condition in the external solutions leads to
DHCl → DH+ + Cl− .
cNa+ + 2cCa2+ + cH+ + cDH+ = cCl− + cOH− .
[1]
[6]
In the case pH ≤ 7, HCl is added. The Na+ concentration remains
constant,
cNa+ = cNaCl ,
[7]
and then Eqs. [4]–[6] give
cCl− = cNaCl + cDHCl + 2cCaCl2 + cH+ − cOH− .
[8]
In the case pH > 7, NaOH is added in order to fix the pH value
of the external solution. The Cl− concentration is then given by
cCl− = cNaCl + 2cCaCl2 + cDHCl ,
[9]
and Eqs. [4]–[6] yield
FIG. 1. Schematic representation of the model system studied. The aqueous
solution within the fixed charge membrane is assumed to be in equilibrium with
an external aqueous solution containing six mobile ionic species: the cationic
drug (DH+ ), the salt cations (Na+ and Ca2+ ), the salt anion (Cl− ), and the
hydrogen and hydroxide ions. c̄i and ci are the concentrations of these species
in the membrane and the external solutions, respectively. In addition to these
mobile species, the membrane solution may contain also four fixed species
attached to the membrane chains: strongly acid sulfonic groups (–SO−
3 ), weakly
acid carboxylic groups in dissociated (–COO− ) and neutral (–COOH) forms,
and positively charged groups (–COO · · · Ca+ ) resulting from Ca2+ binding to
dissociated weakly acid groups. XS is the concentration of sulfonic groups and
XCT is the total concentration of carboxylic groups, which can be in negative
(X CN ), neutral (X C0 ), or positive (XPC ) forms. φ = φ̄ − φ is the Donnan potential
difference between the membrane and external solutions resulting from the ionic
partition equilibrium.
cNa+ = cNaCl − cH+ + cOH− .
[10]
In addition to the six mobile species, the membrane may also
contain four fixed species attached to the membrane matrix.
These species are strongly sulfonic acid groups (–SO−
3 ), weakly
carboxylic acid groups in dissociated (–COO− ) and neutral
(–COOH) forms, and positively charged groups (–COO · · · Ca+ )
resulting from Ca2+ binding to dissociated weakly acid groups
(see Fig. 1). X S and X CT are, respectively, the concentration of
sulfonic groups and the total concentration of carboxylic groups
within the membrane. In the range of pH values studied, we assume that the sulfonic groups are fully dissociated. The neutral
and dissociated forms of the carboxylic groups are assumed to
173
DONNAN EQUILIBRIUM OF IONIC DRUGS
be in equilibrium with the H+ ions in the membrane solution
according to
k
N
–COOH ←−
−→ –COO− + H+ ,
k
kB =
X CN c̄H+
,
X C0
X CP
,
X CN c̄Ca2+
[13]
[14]
where X CN , X C0 , and X CP are, respectively, the concentration of negative (–COO− ), neutral (–COOH), and positive
(–COO · · · Ca+ ) forms of the carboxylic groups. From Eqs [13]
and [14] together with
X CT = X CP + X CN + X C0 ,
[15]
the membrane fixed charge concentration (with its sign) yields
−X S + X CP − X CN = − X S −
1 − kB c̄Ca2+
X CT . [16]
1 + c̄H+ /kN + kB c̄Ca2+
Note that for pH pka the membrane fixed charge concentration is −X S − X CT for kB c̄Ca2+ 1 and −X S + X CT for
kB c̄Ca2+ 1. Binding constants for Ca2+ adsorption to fixed
charge groups appear to be in the range 10–103 M−1 (9, 21),
though significantly higher values up to 105 M−1 have been reported for trivalent (La3+ ) cations (21).
The electroneutrality condition within the membrane solution
leads to
c̄Na+ + c̄H+ + 2c̄Ca2+ + c̄DH+
1 − kB c̄Ca2+
= c̄Cl− + c̄OH− + X S +
X CT .
1 + c̄H+ /kN + kB c̄Ca2+
i = Na+ , Ca2+ , H+ , DH+ , Cl− , OH− ,
1 0
0
ki ≡ exp −
µ̄ − µi ,
RT i
i = Na+ , Ca2+ , H+ , DH+ , Cl− , OH−
The concentrations of the ionic species in the membrane (c̄i )
and the external (ci ) solutions are connected by the Donnan
[19]
is the chemical part of the partition coefficient of species i (note
that ki does not include the effect of the electric potential difference across the interface, as is clearly shown by Eq. [18]). In
Eq. [19], µ̄i0 − µi0 is the standard chemical potential difference
between the membrane and the external solutions of species i.
It is reasonable to assume that ki = 1 for all ionic species except for the cationic drug DH+ , for which kDH+ is a (chemical)
partition coefficient that measures the tendency of the drug to
enter the membrane solution due to specific (e.g., hydrophobic)
interactions (9). Thus, the lipophilicity (µ̄0DH+ < µ0DH+ ) of the
drug is simulated by a chemical partition coefficient kDH+ > 1
between membrane and external solutions. The more hydrophobic the drug, the larger the difference µ0DH+ − µ̄0DH+ and the larger
the value of kDH+ = exp[(µ0DH+ − µ̄0DH+ )/RT ]. The coefficient
kDH+ is considered here a phenomenological parameter and no
attempt will be made to evaluate it in terms of a microscopic
model for the drug–membrane-specific interactions.
From Eqs. [18] and [19] we obtain
c̄DH+
c̄Na+ c̄H+
c̄ 2+ 1/2 cCl− cOH−
=
=
= Ca
=
=
.
kDH+ cDH+ cNa+ cH+
cCa2+
c̄Cl− c̄OH−
[20]
Substitution of Eq. [20] in Eq. [17] yields
2cCa2+ u 2 + (kDH+ cDH+ + cNa+ + cH+ )u − (cCl− + cOH− )
−X S −
1 − kB cCa2+ u 2
X CT = 0,
1 + cH+ u/kN + kB cCa2+ u 2
1
u
[21]
where
u≡
[17]
[18]
where φ̄ − φ is the Donnan potential difference between the
membrane and the external solutions resulting from the ionic
partition equilibrium, z i is the charge number of species i, constants F, R, and T have their usual meaning, and
[12]
where kB is the equilibrium constant for the binding reaction. The
resulting –COO · · · Ca+ group acts as an anion-exchange group.
Calcium binding to fixed charge groups is relevant not only in
membranes (16) and ion-exchange fibers (9) but also in phospholipid monolayers (21) and biomacromolecules (22).
From Eqs. [11] and [12] we obtain
10−pka = kN =
zi F
(φ̄ − φ) ,
c̄i = ki ci exp −
RT
[11]
where kN is the equilibrium constant for the dissociation
reaction.
The divalent ion Ca2+ is known to bind strongly to carboxylic
groups, but not to sulfonic groups (3, 9). We assume that Ca2+
binds to –COO− groups according to the scheme
B
–COO− + Ca2+←−
−→ –COO · · · Ca+ ,
equilibrium equations
c̄DH+
.
kDH+ cDH+
[22]
Equation [21] can be solved for u = 0 using, e.g., a NewtonRaphson procedure. This procedure is very convenient even for
low values of u provided that a reasonable initial guess for the
174
RAMÍREZ ET AL.
lower and higher solution bounds is made. Once u has been calculated, the Donnan potential difference between the membrane
and external solutions is
RT
RT
cDH+
kDH+ cDH+
φ̄ − φ =
.
=
ln
ln kDH+ + ln
c̄DH+
c̄DH+
F
F
[23]
Equation [21] yields a very simple solution when cCaCl2 = 0
and X CT = 0 (no carboxylic acid groups)
cDH+
=
c̄DH+
X S2 + 4(cNa+ + cH+ + cDH+ )(cNa+ + cH+ + kDH+ cDH+ ) − X S
.
2kDH+ (cNa+ + cH+ + cDH+ )
[24]
RESULTS AND DISCUSSION
Figure 2 shows the ratio between the cationic drug concentration in the external and membrane solutions, cDH+ /c̄DH+ , vs
the external pH for X S = 10−1 M (sulfonic groups, –SO−
3 ) and
cDHCl = 10−2 M, with X CT = 0 (carboxylic groups, –COO− )
and kDH+ = 1 (µ0DH+ = µ̄0DH+ , no hydrophobic effects in the
cationic drug equilibrium). The curves are parametric in the
NaCl (continuous line) and CaCl2 (discontinuous line) salt concentrations. pH values higher than 10 are not included because the drug could be in a neutral form in this case. On the
other hand, the lower pH values in the figure are seldom employed experimentally. They are included only to show the limiting behavior of cDH+ /c̄DH+ when the hydrogen concentration
cH+ = 10−pH > cDHCl = 10−2 M. In this limit, the hydrogen substitutes for the cationic drug in the membrane solution, and then
FIG. 2. The ratio between the cationic drug concentration in the external
and membrane solutions, cDH+ /c̄DH+ , vs the external pH for X S = 10−1 M and
cDHCl = 10−2 M, with X CT = 0 and kDH+ = 1. The curves are parametric in the
NaCl (continuous lines) and CaCl2 (dashed lines) salt concentrations.
FIG. 3. The ratio cDH+ /c̄DH+ vs the external solution concentration cNaCl
(continuous lines) and cCaCl2 (dashed lines) for X S = 10−1 M and cDHCl =
10−2 M, with X CT = 0 and kDH+ = 1. The curves are parametric in the external pH.
the negative fixed charge groups –SO−
3 are no longer compensated by DH+ but by H+ . This ion-exchange process makes the
drug concentration in the membrane and external solutions similar, and cDH+ /c̄DH+ tends to unity. Comparison of the theoretical
curves for the salt cations Na+ and Ca2+ shows that the incorporation in the external solution of the divalent cation Ca2+ instead
of Na+ significantly decreases the drug content in the membrane (for a fixed salt concentration, the ratio cDH+ /c̄DH+ < 1 is
closer to one for CaCl2 than for NaCl). This is in qualitative
agreement with experimental results in ion-exchange fibers that
show an increase in the drug release from the fiber as the salt
concentration increases (9). Note finally that the above effect
disappears when the salt concentration is made much higher
than the fixed charge concentration X S = 10−1 M since in this
limit the effect of the membrane on the ionic equilibrium is
negligible.
Figure 3 shows the ratio cDH+ /c̄DH+ vs the external solution concentrations cNaCl (continuous line) and cCaCl2 (discontinuous line) for X S = 10−1 M (sulfonic groups, –SO−
3 ) and
cDHCl = 10−2 M, with X CT = 0 and kDH+ = 1. The curves are
parametric in the external pH. Figure 4 gives the Donnan potential difference between the membrane and external solutions,
φ = φ̄ − φ, vs the external solution concentrations cNaCl (continuous line) and cCaCl2 (discontinuous line) for the same conditions as those of Fig. 3. For high pH values, increasing the
NaCl concentration causes the drug concentration in the membrane to decrease (the ratio cDH+ /c̄DH+ increases) because of the
gradual substitution of DH+ in the membrane by Na+ from the
solution. As could be expected, this trend is even more apparent for the case of CaCl2 because of the higher charge number
of the salt cation. For low pH values, however, the above effects are not so marked because it is the hydrogen and not the
DONNAN EQUILIBRIUM OF IONIC DRUGS
FIG. 4. The Donnan potential difference between the membrane and external solutions, φ = φ̄ − φ, vs the external solution concentration cNaCl (continuous lines) and cCaCl2 (dashed lines) for X S = 10−1 M and cDHCl = 10−2 M,
with X CT = 0 and kDH+ = 1. The curves are parametric in the external pH.
salt cation that conpensates for the negative fixed charge in the
membrane. Figure 4 shows that φ closely follows the behavior
of the ratio cDH+ /c̄DH+ in Fig. 3, as could be anticipated from
Eq. [23]. Note finally that the results of Fig. 4 show clearly
the effect of salt concentration on the cationic drug content in
the membrane (see Fig. 3). Since the membrane fixed charge is
negative, the Donnan potential is also negative (see Fig. 1) and
tends to keep the cationic drug in the membrane solution in the
case of low salt concentrations. However, the absolute value of
the (negative) Donnan potential decreases with the external salt
concentration and this lowers the cationic drug concentration
in the membrane. The theoretical predictions of Figs. 3 and 4
are in good agreement with previously reported experimental
results (2, 5–7, 9) showing the influence of salt concentration
on ionic drug release, though other effects not included in our
model (e.g., conformational changes in the membrane chains
and membrane swelling) may also be important in some cases
(2, 18). Therefore, the Donnan potential is a useful concept for
the understanding of the ionic drug partition equilibrium, as it
is observed with ion-exchange membranes (16, 23) and fixed
charge conducting polymers in aqueous solution (24, 25).
Figure 5 shows the ratio cDH+ /c̄DH+ vs the external drug concentration cDHCl for pH = 7 and cNaCl = 0 = cCaCl2 , with X CT = 0
and kDH+ = 1. The curves are parametric in the fixed charge
concentration X S . In each case, the drug concentration in the
membrane solution is essentially equal that of the external solution concentration (cDH+ /c̄DH+ = 1) for cDHCl X S . On the
other hand, the drug concentration in the membrane solution
attains values much higher than those in the external solution
(cDH+ /c̄DH+ 1) in the opposite limit cDHCl X S . Therefore,
the higher the (negative) fixed charge concentration, the more
difficult should be the (cationic) drug release. Figure 6 shows
175
FIG. 5. The ratio cDH+ /c̄DH+ vs the external drug concentration cDHCl
for pH = 7 and cNaCl = 0 = cCaCl2 , with X CT = 0 and kDH+ = 1. The curves are
parametric in the fixed charge concentration X S .
the ratio cDH+ /c̄DH+ vs the external solution concentration cCaCl2
for pH = 7 and cDHCl = 10−2 M, with X CT = 0, cNaCl = 0, and
kDH+ = 1. The curves are parametric in the membrane fixed
charge concentration X S . For a fixed value of X S , the incorporation of the divalent cation Ca2+ to the membrane solution
significantly decreases the drug content in the membrane. This is
essentially an ion-exchange process where the counterion DH+
in the membrane solution is exchanged by the counterion Ca2+
from the external solution. As could be expected, this effect is
more marked the higher the fixed charge concentration (2, 9).
FIG. 6. The ratio cDH+ /c̄DH+ vs the external solution concentration cCaCl2
for pH = 7 and cDHCl = 10−2 M, with X CT = 0, cNaCl = 0, and kDH+ = 1. The
curves are parametric in the membrane fixed charge concentration X S .
176
RAMÍREZ ET AL.
FIG. 7. The ratio cDH+ /c̄DH+ vs the external pH for X CT = 10−1 M
(pka = 4), cCaCl2 = 10−2 M, and cDHCl = 10−2 M, with X S = 0, cNaCl = 0, and
kDH+ = 1. The curves are parametric in the Ca2+ ion binding constant kB .
Figure 7 shows the ratio cDH+ /c̄DH+ vs the external pH for
X CT = 10−1 M (carboxylic groups, –COO− , pka = 4), cCaCl2 =
10−2 M, and cDHCl = 10−2 M, with X S = 0, cNaCl = 0, and
kDH+ = 1. (As in Fig. 2, the results for the lower pH values are
included only to show the limiting behavior of the drug concentration ratio in this limit.) The curves are parametric in the Ca+2
binding constant kB . Ca+2 is known to adsorb strongly to carboxylic groups, but not to sulfonic groups (3, 9), with binding
constants in the range 10–103 M−1 (9, 21), though significantly
higher values (105 M−1 ) have been reported for trivalent (La3+ )
cations (21). The most important consequence of Ca2+ binding is that the membrane fixed charge concentration can change
from the negative value −X CT for kB c̄Ca2+ 1 to the positive
value +X CT for kB c̄Ca2+ 1. For low kB values, the membrane
is negatively charged and then cDH+ /c̄DH+ < 1. The behavior of
cDH+ /c̄DH+ with the external pH is similar to that of Fig. 2. For
high enough binding constants, however, the fixed charge in the
membrane can become positive due to the –COO · · · Ca+ groups,
and this causes the release of the cationic drug in the membrane: the partition equilibrium shifts now to cDH+ /c̄DH+ > 1 (see
Fig. 7). Again, the theoretical results are in agreement with literature experimental data concerning Ca2+ binding to carboxylic
groups in ion-exchange fibers (9). Interestingly enough, Fig. 7
shows that the Ca2+ binding is pH dependent, with important
changes in the ratio cDH+ /c̄DH+ for pH < pka = 4. This theoretical prediction appears to be in agreement with previous experiments (3) where the interaction of Ca2+ with poly(acrylic acid)
resins was studied as a function of solution pH (see in particular
Fig. 2 in this reference).
Figure 8 shows the ratio cDH+ /c̄DH+ vs the external solution concentration cCaCl2 for X CT = 10−1 M (pka = 4), pH = 7,
and cDHCl = 10−2 M, with X S = 0, cNaCl = 0, and kDH+ = 1. The
curves are parametric in the Ca2+ ion binding constant kB . For
low CaCl2 concentrations, increasing cCaCl2 causes the cationic
drug concentration in the membrane to decrease (and thus the
ratio cDH+ /c̄DH+ increases) because of the gradual substitution
of DH+ in the membrane by Ca2+ (see Fig. 2). This can also be
explained in terms of the Donnan potential: the absolute value of
the (negative) Donnan potential decreases with the external salt
concentration and this decreases the cationic drug concentration
in the membrane. For high CaCl2 concentrations and not too
low binding constants, however, Ca2+ binding to the carboxylic
groups causes the fixed charge in the membrane to become positive, and thus the Donnan potential (see Fig. 1) is also positive,
acting to increase cationic drug release. Since increasing the salt
(CaCl2 ) concentration decreases the (now positive) value of the
Donnan potential, the partition equilibrium shifts to higher membrane cationic drug concentrations (cDH+ /c̄DH+ decreases) in this
case. This is also the behavior observed for high enough binding constants leading to positive Donnan potentials, especially
in the limiting case kB = ∞ (see Fig. 8). Note also the existence
of a maximum in the curves of Fig. 8 for intermediate values of
kB . According to the above interpretation, the maximum appears
when the trend of decreasing c̄DH+ (increasing cDH+ /c̄DH+ ) with
cCaCl2 is reversed for high enough CaCl2 concentrations causing
the Ca2+ binding.
Figure 9 shows the ratio cDH+ /c̄DH+ vs the external solution concentration ratio % CaCl2 = 100 cCaCl2 /(cCaCl2 + cNaCl )
for (cCaCl2 + cNaCl ) = 10−2 M, X CT = 10−1 M (pka = 4), pH = 7,
and cDHCl = 10−2 M, with X S = 0 and kDH+ = 1. The curves are
parametric in the Ca2+ ion binding constant kB . Note that for
high values of the binding constant, a relatively small amount
of CaCl2 can increase the ratio cDH+ /c̄DH+ (increase the cation
drug release) significantly. This result has been confirmed experimentally (9). The changes of cDH+ /c̄DH+ with % CaCl2 can now
FIG. 8. The ratio cDH+ /c̄DH+ vs the external solution concentration
cCaCl2 for X CT = 10−1 M (pka = 4), pH = 7, and cDHCl = 10−2 M, with X S = 0,
cNaCl = 0, and kDH+ = 1. The curves are parametric in the Ca2+ ion binding
constant kB .
DONNAN EQUILIBRIUM OF IONIC DRUGS
FIG. 9. The ratio cDH+ /c̄DH+ vs the external solution concentration
ratio % CaCl2 = 100 cCaCl2 /(cCaCl2 + cNaCl ) for (cCaCl2 + cNaCl ) = 10−2 M,
X CT = 10−1 M (pka = 4), pH = 7, and cDHCl = 10−2 M, with X S = 0 and
kDH+ = 1. The curves are parametric in the Ca2+ ion binding constant kB .
be readily explained from Figs. 3 (see also Fig. 4) and 8. Figure 3
shows that when Ca2+ binding can be ignored, substitution of
the divalent cation Ca2+ for Na+ in the external solution significantly decreases the drug content in the membrane (the ratio
cDH+ /c̄DH+ increases then with % CaCl2 in Fig. 9), and Fig. 8
shows the effects of Ca2+ binding on the cDH+ /c̄DH+ vs cCaCl2
curves. We emphasize finally that recent experimental data with
ion-exchange fibers (9) have shown that the trend of increasing drug release (increasing cDH+ /c̄DH+ here) with increases in
the fraction of CaCl2 can be reversed for high enough values of
% CaCl2 , confirming thus the theoretical predictions of Figs. 8
and 9.
Figure 10 shows the ratio cDH+ /c̄DH+ vs the external solution concentration cNaCl for X S = 10−1 M, pH = 7, and cDHCl =
10−2 M, with X CT = 0 and cCaCl2 = 0. The curves are parametric in the drug partition coefficient kDH+ . The drug partition
between the membrane and external solutions is affected by
specific drug–membrane interactions, and kDH+ is a chemical
partition coefficient that measures the tendency of the drug to
enter the membrane solution due to hydrophobic interactions
(9). The more hydrophobic the drug, the larger the value of kDH+
(2, 7, 9), for instance, kDH+ = 300 in the simulation of experiments with tacrine in ion-exchange fibers having carboxylic acid
groups (9). As could be expected, Fig. 10 shows that increasing
kDH+ produces a significant increase in the drug concentration
in the membrane. Indeed, the decrease in the drug concentration
in the membrane solution (increase of cDH+ /c̄DH+ ) with cNaCl
shown in Fig. 10 for kDH+ = 1 becomes less pronounced when
kDH+ > 1. When the drug has a high lipophilicity (kDH+ 1),
Fig. 10 shows that its concentration in the membrane solution
remains high (cDH+ /c̄DH+ 1) even for high salt concentrations
cNaCl , contrary to the case kDH+ = 1. This behavior can be read-
177
FIG. 10. The ratio cDH+ /c̄DH+ vs the external solution concentration cNaCl
for X S = 10−1 M, pH = 7, and cDHCl = 10−2 M, with X CT = 0 and cCaCl2 = 0.
The curves are parametric in the drug partition coefficient kDH+ .
ily explained by means of Fig. 11, where the Donnan potential
φ = φ̄ − φ is plotted vs the external solution concentration
cNaCl for the conditions of Fig. 10; φ changes its sign, switching from negative to positive values, when kDH+ > 10. Note that a
positive Donnan potential acts to decrease the drug content in the
membrane solution (i.e., to increase the cationic drug release).
Figure 11 shows that since increasing the salt (NaCl) concentration decreases the positive value of the Donnan potential, the partition equilibrium shifts to higher membrane cationic drug concentrations (cDH+ /c̄DH+ decreases) with increasing cNaCl when
kDH+ > 10. (This salt concentration behavior is similar to that
FIG. 11. The Donnan potential φ vs the external solution concentration cNaCl for X S = 10−1 M, pH = 7, and cDHCl = 10−2 M, with X CT = 0 and
cCaCl2 = 0. The curves are parametric in the drug partition coefficient kDH+ .
178
RAMÍREZ ET AL.
kB
ki
KW
µi0
µ̄i0
pka
R
T
X C0
X CN
X CP
FIG. 12. The ratio cDH+ /c̄DH+ vs the external solution concentration cCaCl2
for X CT = 10−1 M (pka = 4), pH = 7, and cDHCl = 10−2 M, with X S = 0 and
cNaCl = 0. The curves are parametric in the Ca2+ ion binding constant kB and
the drug partition coefficient kDH+ .
observed for high enough binding constants leading to positive
Donnan potentials in Fig. 8.) It has been reported (9) that high
values of drug lipophilicity made a cation-exchange fiber behave
as an anion exchanger, in agreement with the above theoretical
results for the Donnan potential. As was the case in Figs. 3 and
4, the comparison of Figs. 10 and 11 shows that the Donnan
potential is a useful magnitude in understanding the ionic drug
partition equilibrium in fixed charge membranes (see also (9)).
Figure 12 shows finally the ratio cDH+ /c̄DH+ vs the external solution concentration cCaCl2 for X CT = 10−1 M (pka = 4), pH = 7,
and cDHCl = 10−2 M, with X S = 0 and cNaCl = 0. The curves are
parametric in the Ca2+ binding constant kB and the drug partition
coefficient kDH+ . Figure 12 points out that the drug partition equilibria in fixed charge membranes can show very rich behavior
because many effects are present simultaneously. We emphasize
finally that this behavior could be qualitatively explained on the
basis of the results obtained in Figs. 3, 8, and 10 making use of
relatively simple, fundamental concepts.
APPENDIX: LIST OF SYMBOLS
ci
c̄i
φ
F
φ
φ̄
kN
concentration of species i in the external solution
phase (M)
concentration of species i in the membrane solution
phase (M)
Donnan electric potential difference between the
membrane and external solution (mV)
Faraday constant (C/mol)
electric potential in the external solution phase (mV)
electric potential in the membrane solution phase (mV)
equilibrium constant for the dissociation reaction
of the carboxylic groups (M)
X CT
XS
zi
equilibrium constant for the binding reaction
of the carboxylic groups (M−1 )
chemical partition coefficient of species i
ionic product of water (M2 )
standard chemical potential of species i in the external
solution phase (J/mol)
standard chemical potential of species i in the membrane
solution phase (J/mol)
pK a value for the dissociation reaction of the carboxylic
groups
gas constant (J/mol K)
absolute temperature (K)
concentration of the carboxylic groups (neutral forms)
within the membrane (M)
concentration of the carboxylic groups (negative forms)
within the membrane (M)
concentration of the carboxylic groups (positive forms
because of Ca2+ binding) within the membrane (M)
total concentration of carboxylic groups
within the membrane (M)
concentration of sulfonic groups within the
membrane (M)
charge number of species i
ACKNOWLEDGMENTS
Financial support from the DGICYT, Ministry of Education and Science
of Spain (Project No. PB98-0419), and from Fundació UJI-Bancaixa (Project
P1B98-12) are gratefully acknowledged.
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