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At study of ion permeation across a charged membrane in multicomponent ion systems as a function of membrane charge density
Science, 49 (1990) 145-169 Elsevier Science Publishers B.V.. Amsterdam - Printed in The Netherlands
145
Journal of Membrane
A STUDY OF ION PERMEATION ACROSS A CHARGED MEMBRANE IN MULTICOMPONENT ION SYSTEMS AS A FUNCTION OF MEMBRANE CHARGE DENSITY
MITSURU HIGA, AKIHIKO TANIOKA and KEIZO MIYASAKA Department
of Organic and Polymeric
Meguro-ku,
Tokyo 152 (Japan)
Materials,
Tokyo Institute
of Technology,
Ookayama,
(Received September 7,1988; accepted in revised form July 25,1989)
Summary The authors have described an ion flux across a charged membrane in multicomponent systems by the theories (based on the Donnan equilibrium and the Nernst-Planck equation) in the previouspaper. In the present study, the permeation coefficients of ions in various kinds of electrolyte systems are calculated as functions of membrane charge density by this theory. The theory predicts that permeability coefficients of multi-valent ions will be lower than those of univalent ions according to the low charge density, and that the permeability coefficients of multi-valent ions will become negative if the membrane has high charge density. That is, they are transported against their concentration gradient. This phenomenon in l-l- and 2-l-type electrolyte systems was examined for negatively charged membranes, which were composed of poly (vinyl alcohol) (PVA) and poly (styrenesulfuric acid) (PSSA) mixtures, and a positively charged one, which was composed of a PVA and poly(allylamine) (PAAm) mixture. The theoretical prediction agreed well with the experimental results. The uphill transport of a multivalent ion occurred under the appropriate membrane charge density. This result shows that the direction of multivalent ion transport changes with charge density. This phenomenon is applicable in a new mechanism to control ion transport direction.
1. Introduction Recently it has become necessary to analyze precisely the ion transport phenomena in multicomponent ion systems across a charged membrane from the standpoint of industrial and medical applications. Naturally, almost all membranes in a living body have electrical charges. A cell membrane consisting of a phosphorus lipid and protein, which can be regarded as a charged membrane, has a high selectivity for special solutes, and can transport a solute against its concentration gradient under certain conditions. One reason for this high selectivity is considered to be the charges on the membrane. A neutral membrane separates solutes according to the difference in their size and/or in their interaction with the membrane. In addition to this, a charged membrane can sep-
0376-7388/90/$03.50
0 1990 Elsevier Science Publishers B.V.
146
arate charged solutes according to the differences in their sign and valence, and should thus show a higher selectivity than a neutral membrane. From this point of view, charged membranes have been used for reverse osmosis and artificial dialysis [ 11, replacing neutral membranes. The theoretical analysis of the ion transport phenomena in the system is very important, therefore, in producing a highly selective membrane [2] and explaining the metabolic mechanism in a cell membrane. Ion transport against its concentration gradient is also a very interesting phenomenon. This uphill transport phenomenon has been studied by many authors [ 3-101. In these experimental systems, however, a highly concentrated electrolyte solution is necessary as the driving force. In the present paper, the permeability coefficients of ions in various kinds of electrolyte systems are calculated precisely as functions of membrane charge density by a theory based on the Donnan equilibrium and the Nernst-Planck equation considering practical applications. The theory predicts that membrane charge density will greatly affect the transport phenomena of ions, especially multivalent ions. The theory also predicts that the uphill transport of multivalent ions will occur in the experimental system where a charged membrane is placed between two cells, and one cell has an electrolyte concentration ten times higher than the other. The uphill transports in the systems occur without a highly concentrated electrolyte solution as the driving force. One of these phenomena was examined for negatively charged membranes composed of poly(viny1 alcohol) (PVA) and poly( styrenesulfuric acid) (PSSA) mixtures, and for a positively charged one composed of a PVA and poly (allylamine ) (PAAm) mixture. The theoretical prediction agreed well with the experimental results and the uphill transport of a multivalent ion occurred under the appropriate membrane charge density. This result shows that the direction of multivalent ion transport changes with charge density. This phenomenon is applicable in a new mechanism to control ion transport direction. 2. Theory 2.1 Assumptions
The basic assumptions of our theory are as follows: (a) It is difficult to measure experimentally the ion mobility in a membrane. Mackie and Meares therefore approximately calculated the mobility as a function of the tortuosity of a membrane and the mobility in solution [ 111: uy = uy/e2
(1)
where Uf and UT are the ion mobility in a membrane and an external solution, respectively, and 0 is a tortuosity factor, which is related to the degree of hydration H by:
147
8
=-
2-H H
The membranes used in this study have a higher degree of hydration and a lower membrane charge density than current ion exchange membranes. This assumption is therefore considered to be suitable for these membranes. (b) From assumption (a), membranes with the same degree of hydration have the same amount of ion mobility, even if these membranes have a different ion charge group, a different membrane charge density and a different polymer network configuration. (c ) Osmotic or hydrostatic flow of water is negligible. (d) The membrane surfaces are always at a state of Donnan equilibrium with the same distribution ratio (K) for all ions. (e) The boundary layer effect on membrane surfaces is negligible. (f) For simplicity here, all electrolytes dissolve perfectly and ion activity equals unity in both the external solutions and the membrane. (g ) Also for simplicity, the standard chemical potential in external solutions is equal to that in a membrane. 2.2 Determination of concentration of the ith ion in a membrane The Donnan equilibrium for the ith ion between a membrane and external solutions can be realized as [ 121:
where Ci and Ci are the ith concentrations in the membrane and the solutions, respectively, Kj ( = exp - FA&,,JRT) and Aq&,,j are the Donnan equilibrium constant and the Donnan potential at side j of the membrane surface, respectively, F, R and T are Faraday’s constant, the gas constant and the absolute temperature, respectively, and zi is the valence of the ith ion. An electroneutrality condition in the membrane requires that:
where z, is the valence of fixed charge density and C, is the effective charge density. Equation (5) is obtained by substituting eqn. (3 ) into eqn. (4): CziKf’Ci +z,C~ =O
(5)
Since K is obtained from only Ci and C, by eqn. (5)) the concentration of the ith ion in a membrane, pi, is obtained from K and Ci using eqn. (3 ) .
148
2.3 Calculation of the ith ion permeability coefficient ratio (P.C.R.) in a membrane In an experimental system, a charged membrane is placed between two cells (side I and side II ) and the cells are filled with electrolyte solutions. Thus, each ion flux can be obtained analytically under the assumption of a constant electric field or a constant ion concentration gradient. For constant electric field gradient, each ion flux is given by the following equation [ 131:
(6) where p= exp - FA@,iff/RT and A$diffis the diffusion potential difference from side I to side II, mi is the ion mobility in the membrane, and f?j and Cf’ are the concentrations at side I and side II of the membrane surface, respectively. p is unequal to one and greater than zero when A~diffis unequal to zero. As the systems treated in this paper have no electric current:
where S and d are the membrane area and thickness, respectively. If the valence of ions in the system is three and less, eqn. (7) can be solved for j? analytically [lo] and j? is the solution of the following equation (see appendix) : (B, +4& +9&)P4+ - (2A, +4A, -B,
(2B, +4B, +9& -A,)p3+ +9A3)P-
(2B1 +4&-2A,
-4A,)P2
(A, +4Az +9A3) =O
(8)
where: + x(-q-
pk-’
B, = .&x;+ e;+’ + cm;-
pk-”
A, = Cog+
e;+”
z= 1,2,3
and WE+ and wk- are the mobility of z-valent cations and anions, respectively; C-c+’ and &I are the concentrations of z-valent cations and anions at side I of the membrane surface, and Pk’ I1and Cj?-I1 are those at side II, respectively. For a univalent ion system, eqn. (8) is rewritten as: (P+I)(P”+P+I)(B,P-A,)=0
(9)
Since p is unequal to one, the solution of p is: p=
$L = yJJ--;~~~~: 1
k
k
k
Similarly, in uni- and bi-valent ion systems, eqn. (8) is rewritten as:
(10)
149
(p”+p+l) [ @I +4&)fi2+ (B, -A,)P-
(A, +4&j]
=o
(11)
and the solution of /3 is:
P=
Al -&
+@,
-M2+4(B1
+4B2)641+4A2) (12)
2(&
+4B2)
The ith ion flux (Ji) can be obtained by substituting the concentrations on both surfaces of the membrane into eqn. (6). The ith ion permeability coefficient (Pi) is obtained from the ion flux and external ion concentrations by the following equation: (13) where Cf and Cf’ are the ith ion concentrations in side I and side II cells, respectively. Ion transport phenomena across a charged membrane are a function of both the ion mobility and its charge density. The membranes which have the same ion mobility but different charge densities are necessary when discussing the relation between the phenomena and its charge density. It is difficult, however, to make them by experimentation. The phenomena are therefore discussed in terms of a permeability coefficient ratio (P.C.R.). The P.C.R. is obtained to normalize the permeability coefficient in a charged membrane by the permeability coefficient of potassium chloride in a neutral membrane with the same degree of hydration as the charged membrane. From eqns. (1) and (2), membranes with the same degree of hydration have the same tortuosity, and hence the effect of tortuosity on ion transport can be cancelled out to obtain the P.C.R. The permeability coefficient calculated from eqn. (13) can be normalized by the following permeability coefficient: P,=
2w,oc,RT (WC +mcl)d
(14)
where OK and ocI are the K+ and Cl- ion mobilities in the neutral membrane, respectively, and the mobilities are calculated from eqn. (1) . 2.4 Donnanpotential and diffusionpotentiul The Donnan potential differences from an external solution to the inside of a membrane at both membrane surfaces are calculated from the following equation using the Donnan equilibrium constant (K): A@don,j= -
RT 7 1nKj j=I, II
(15)
150
The total Donnan potential difference (d$ddon)is obtained from the following equation:
The diffusion potential difference from side I to side II (A~diff) is obtained from the following equation: (17) The total membrane potential difference from side I to side II (A@) is obtained from eqns. (16) and (17): A@= A&don + A@diff
(16)
2.5 Permeability coefficient ratio and membrane potential in electrolyte systems containing polycations and polyanions Many polycations and polyanions which are considered not to permeate the glomerular wall can be found in blood plasma. In electrolyte systems containing these polyions, the diffusion and the Donnan potential due to them affect the permeation of mobile ions. In this paper, the ion permeation in the systems is considered, assuming that these ions are not taken into a membrane. In these systems, the electroneutrality condition is required in the external solution:
(19) where C,, and C,, are the total charge densities of polyanions and polycations, respectively, and zpaand zpc are the mean valences of polyanions and polycations, respectively. For simplicity, we set zpa= - 1 and z,, = 1, respectively. In this paper, K+ and Cl- ions are used as the univalent cation and anion, and Ca2+ and SOi- as the bivalent cation and anion, respectively. However, assumptions (d) and (g) suggest that there are no distinctions, except ion mobility, between ions of the same valence. The calculation cited above can thus be performed for different ionic species if their ion mobilities are given. 3. Experimental
3.1 Samples Membranes were cast from 7% PVA [poly (vinyl alcohol); Wako Pure Chemical Industries Ltd., D.P. = 20001 and PSSA [poly (styrenesulfuric acid), Asahi Chemical Industry Co. Ltd.] or PVA and PAAm [poly (allylamine), Nittobo Industries Inc.] aqueous solutions at atmospheric pressure and 20” C, and were annealed at 160 oC for 20 min under vacuum to obtain membranes with a low degree of hydration. Parts of the membranes were crosslinked in an
151
aqueous solution of 20% Na,SO,, 1% H&SO, and 0.1% glutaraldehyde at 20°C to obtain a degree of hydration lower than that of the uncrosslinked samples. The lower the degree of hydration of a membrane, the higher its charge density. 3.2 Membrane potential measurement The membrane potential of the membranes used in this paper was measured to obtain the charge density. Figure 1 shows the apparatus for measurement of the membrane potential. The membrane area was 0.78 cm2 and the volume of each cell was about 300 cm3. The concentration of KC1 electrolyte solution on one side was five times higher than that on the other. The solutions in both cells were stirred by magnetic stirrers to minimize the effect of the boundary layers on the membrane potential. The potential was measured at 19.0 -I-0.1 “C using glass electrodes (TOA HS-205C) with salt bridges (3 N KCl) and a voltmeter (TOA HM-20E). The charge density was given by fitting the experimental results to the following equation [ 14,151:
J
1+
$
F+w x 1 [1+ 2
2
+1
A@=-
-sWln
2
1+ J
9
X
X
J(,
1+
+1
Jc,
3 2+W CX -
(20)
where w= (&& - wcl) / (UK + ocl), and C, and C, are the KC1 concentrations on both sides.
Glass Electrode
Charged Membrane
Fig. 1. Apparatus for membrane potential measurement. The membrane area is 0.78 cm2 and the volume of each cell is ca. 300 cm3. The concentration of the KC1 electrolyte solution on one side is five times higher than that on the other (r = 5).
152
3.3 Measurement of degree of hydration In this paper, the ion mobility in a membrane was not measured experimentally, but calculated from the degree of hydration (D.H.) and the mobility in free solution using eqns. (1) and (2 ). The D.H., defined as the volume fraction of water in a wet membrane, was estimated from the following equation: D.H. =
w, - WJ1.0 water volume WJ1.3 water volume + polymer volume = ( w, - w,)/l.o+
(21)
where W, and W, are the weights of a membrane at the equilibrium swollen and dry states, respectively, and 1.0 and 1.3 are the densities of water and polymer, respectively. 3.4 Permeability coefficient measurement Figure 2 shows the apparatus for the permeability coefficient measurement. A membrane was placed between the two cells and fixed by a clamp. Electrolyte solutions were circulated by tubing pumps (Core-Parmer Instrument Co. PA2113) to minimize the effect of the boundary layers. The side II solution was introduced through a 500 ,~l microsyringe and diluted lOO-fold with distilled water for measurement of the ion concentrations by an atomic absorption spectrophotometer (Shimadzu AA-640-12). The permeability coefficient (Pi) was found from the ion concentration change at side II using the following equation:
v,,
pi=-
ACi (cf’_cy)S,At
(22)
where V, is the volume of the side II cell, ACi is the ith ion concentration change at time interval At, S’ (=SH) is the effective membrane area, S is the crosssectional membrane area and H is the degree of hydration. Membranes used for this experiment have various degrees of hydration.
Reservoir I
Tubing Pump Clamp
Fig. 2. Apparatus for permeability coefficient measurement. The membrane the volumes of side I and side II are 900 cm3 and 50 cm3, respectively.
area is 7.01 cm* and
153
Therefore the permeability coefficient calculated from eqn. (22 ) is normalized by eqn. (14) and a permeability coefficient ratio is obtained. 4. Results and discussion
4.1 Calculated results for permeability coefficient ratio (P. C.R.) and membrane potential 4.1.1 Electrolyte system type l-l The l-l-type electrolyte system shown in Fig. 3(a) is considered. In this system, a charged membrane is placed between two cells and the concentrations of KC1 electrolyte at side I and side II are C, and C,, respectively. The ratio of C, to C, is 10. Figure 4 (a) shows the calculated results of electric potential difference from the high-concentration side to the low-concentration side when C, = 1 x 10e3 mol/l. Solid, broken and dash-dotted lines are the total membrane, the diffusion and the Donnan potential, respectively. The Donnan potential difference is symmetrical. The value of the diffusion potential is small on account of the small mobility difference between the cation and the anion. The value of the total membrane potential is thus almost the same as that of the Donnan potential. The calculated results for the P.C.R. are shown in Fig. 4(b) as a function of the membrane charge density. In this figure, the higher the charge density, the smaller the P.C.R. symmetry in both negatively and positively charged membranes. In the case of a positively charged membrane and a l-l-type electrolyte system, a cation is excluded from the membrane due to a high Donnan potential difference, and an anion is scarcely transported across the membrane as a consequence of an electroneutral condition. Eventually, the permeation of both the cation and the anion is reduced. The same explanation can be used to describe a similar phenomenon in a negatively charged membrane. This shows that a neutral membrane has a higher P.C.R. than a charged membrane. This phenomenon explains qualitatively why a charged reverse osmosis membrane has a higher salt rejection than a neutral one. 4.1.2 Electrolyte system type 2-1 The P.C.R. and the membrane potential of the 2-l-type electrolyte system shown in Fig. 3 (b) are considered. Figure 5 (a) shows the calculated results of the electric potential difference when C, = 1 x 10e3 mol/l. Solid, broken and dash-dotted lines are the total membrane, the diffusion and the Donnan potentials, respectively. The contribution of the Donnan potential to the total membrane potential is large when the charge density is higher than the external concentration, but that of the diffusion potential to the total potential is large when the membrane charge approaches zero. The diffusion potential has
154
Side I
Side I
Side II
Charged
CaCI,(Co)
CaCI,(Cd)
KCl(Co)
KCl(Cd)
Charged
(a) I-l-type
Side 11
Membrane
Membrane
electrolyte
(b) 2-l-type
system
electrolyte
system
KCl(Co)
KCI(Cd)
KCl(Cd)
KCI(Co)
CaCl,(Cd)
CaCI,(Co)
WI&Cd)
CaCI,(Co)
Side II
Side I
Side I
Charged
Charged
(c) 2-l- and I-l-type
Side II
Membrane
Membrane
electrolyte
(d) 2-l- and l-l-type
system
electrolyte
system
I KCl(Cd)
KCI(Co)
K,SO,(W
K,sO,(W
K,SO,(Cd)
K,SO,(Co)
CaCI,(Cd)
CaCI*(Co)
Side II
Side I
Side 1
Charged
&de II
Charged Membrane
Membrane
(e) I-I- and 1.2.type electrolyte
(f) 2-l. and l-Ztypc
system
electrolyte
system
Polycations(Cpc> PoIyanions(Cpa) K,SO,(Cd)
K,SO,(Co)
CaCI,(Cd)
CaC12(Co)
Side I
Charged
K&)&Cd) CaCl,(Cd)
Side I
Side II
Membrane
) 2-I.
Charged
(h) 2-l- and I-2-type electrolyte and l-?-type
electrolyte
system contamlng
Side II
Membrane
polyamons
Fig. 3. The various electrolyte systems considered
in this paper.
system containing
polycatia
155 100
y50
I
Co=le-3 [molill
-
.c ‘;i 0.0
Membr.
P.
---Diff. P. -‘-‘-’ Donnan P. c________________________-___
- ____________________---
-50 -
-1od-.5
I
I
I
-.2.5
0.0 Charge
0.25
0.5
Density [mol/l]
Fig. 4 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (a). 1.5
I
I
Co=le-3
[moVII
1.2 -
0.9 d d a 0.6
-
-
K’ and Cl- IO”
Charge
Density [mol/l]
Fig. 4(b). Permeability coefficient ratio (P.C.R.) in l-l-type 3 (a) as a function of membrane charge density.
electrolyte system shown in Fig.
a negative value because the mobility of a Ca2+ ion is smaller than that of a Cl- ion. In contrast to the l-l-type electrolyte system, the Donnan potential difference here has an asymmetric value. In other words, the absolute value of the Donnan potential in a negatively charged membrane is smaller than that in a positively charged one. The reason is that the counter-ion of a negatively charged membrane has a higher valence than that of a positively charged membrane in the system. This phenomenon affects the P.C.R. Figure 5 (b) shows the calculated results of the P.C.R. as a function of membrane charge density. In both negatively and positively charged membranes, the P.C.R. decreases with the charge density. However, the decrease of the P.C.R. in the latter is larger than in the former. In other words, 2-l-type electrolytes are transported across a negatively charged membrane more than across a positively charged membrane if both membranes have the same absolute
156 1OC
I
I Co=le-3
50
-
s
-1OG -.5
I -.25
-
Membr.
----
Diff.
-‘-‘-’
Donnan
P.
P. P.
I 0.25
I 0.0 Charge Density
[moUll
0.5
[mol/ll
Fig. 5 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (b). 1.0
I
Co=le-3
0.8
-
0.6 d d a 0.4
0.2
0.0
4 I
-. 5
- 25
rmol/l]
Ca2’ and Cl- ion
I_
0.0 Charge Density
0.25 [molll]
Fig. 5(b). Permeability coefficient ratio (P.C.R.) in 2-l-type electrolyte system shown in Fig. 3 (b) as a function of membrane charge density.
charge density and an identical membrane structure, because the absolute value of the Donnan potential in the former is larger than that in the latter. 4.1.3 Electrolyte system type 2-l and l-l The 2-l- and l-l-type electrolyte system shown in Fig. 3(c) is considered. In this system, a charged membrane is placed between two cells, and the concentrations of KC1 and CaCl, electrolytes at side I are C,, and Cd, respectively, and C, and COat side II, respectively. The concentration gradient of K+ ion is in the opposite direction to that of Ca2+ ion, and a counter-flow of ions occurs in the system. Figure 6 (a) shows the electric potential difference as a function of the charge density in this system. The calculated results of the P.C.R. are shown in Fig. 6 (b) as a function of membrane charge density. As the density of the negative charges in the mem-
157
50
I
I Co=le-3
[moH]
F 25 .E
Donnan P.
-‘-‘-’
-50 ’ -.5
I -.25
I 0.0 Charge Density
I 0.25
0.5
[molil]
Fig. 6 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (c).
Charge Density
-
K+
___-
#&z+
_,_._.
CI‘
IO” ion
,o”
[molil]
Fig. 6(b). Permeability coefficient ratio (P.C.R.) in l-l- and P-l-type electrolyte system shown in Fig. 3 (c) as a function of membrane charge density.
brane increases, the number of cations taken into the membrane increases as does the number of anions rejected on account of the Donnan potential at the membrane surfaces. Therefore, the anion permeability is reduced in a negatively charged membrane while cation permeability is enhanced compared with the permeability across a neutral membrane. The permeability coefficient of the univalent cation is about twice as high as that of the bivalent one under an electroneutral condition. 4.1.4 Electrolyte system type 2-l and l-l The 2-l- and l-l-type electrolyte system shown in Fig. 3 (d) is considered. In this system, the concentrations of KC1 and CaCl, electrolytes at both side I and side II are Cd and C,, respectively. Figure 7 (a) shows the electric potential difference in the system. This figure shows that the Donnan potential differ-
158
ence is positive across a negatively charged membrane, which means that the Donnan potential and Donnan distribution constant (K) at the low-concentration side (side II) are higher than those at the high-concentration side (side I). The concentration ratio of an ion in a membrane to that in the external solution is equal to the ion valence power of the Donnan distribution constant from eqn. (3). The ratio of cations at side II is thus greater than at side I. Moreover, the ratio for a bivalent cation is greater than that for a univalent one. For these reasons, the concentration of the bivalent cation at the side II membrane surface is higher than that at side I under an appropriate condition. Hence, the concentration gradient of the bivalent ion in the membrane is in the opposite direction to its external concentration gradient, as shown in Fig. 8. This phenomenon affects the ion permeation in the system.
$
50 -
E
-
z
0
-50
___________--________
-
-1OG -5
-
Membr.
----
Dlff.
-‘- -
Donnan
-.25
P.
P. P.
I 0.0
0.25
Charge Density
Fig. 7 (a). Electric potential differences
0.5
[mol/l] from side I to side II in the system shown in Fig. 3 (d).
Co=le-3 [mol/ll
_--_
K’ &
ion ion
-‘- -
cl-
lo”
1.0 -
Charge Density
Fig. 7(b).
Permeability
coefficient
[mol/l]
ratio (P.C.R.)
in l-l-
in Fig. 3 (d) as a function of membrane charge density.
and 2%l-type electrolyte system shown
159
KCl(Cd)
CaCl,(Cd)
Side I Charged Membrane
Fig. 8. Schematic diagram of the concentration gradient of Ca ‘+, K+ and Cl- ions in a negatively charged membrane when the uphill transport of Ca2+ ion occurs.
The calculated results of ion transport are shown in Fig. 7 (b). The ion permeation phenomenon in the system across a negatively charged membrane is of great interest. The permeability coefficient of a K+ ion across the membrane is higher than that across a neutral one. On the other hand, the permeability coefficient of a Ca2+ ion is smaller than that across a neutral one, and becomes negative as the charge density becomes more negative. That is to say, a Ca2’ ion is transported against its concentration gradient, because the concentration gradient of the bivalent ion in the membrane is in the opposite direction to its external gradient, as shown in Fig. 8. A Ca2+ ion is thus transported against its external concentration gradient as K+ ions diffuse from the high to low concentration side. The absolute value of P.C.R. for K+ is about twice that for Ca2+ because of an electroneutral condition. 4.1.5 Electrolyte system type l-l and l-2 The l-l-and 1-2-type electrolyte system shown in Fig. 3 (e) is considered. The theoretical predictions are shown in Figs. 9 (a) and (b). The opposite phenomenon to the l-l- and 2-l-type electrolyte as shown in Fig. 3 (d) is observed across a positively charged membrane. The permeability coefficient of the univalent anion, Cl- ion, across a negatively charged membrane is higher than that across a neutral one. On the other hand, the permeability coefficient of the bivalent anion, SO:- ion, is smaller than that across a neutral one, and becomes negative as the charge density becomes more positive. 4.1.6 Electrolyte system type 2-l and l-2 The 2-l- and 1-2-type electrolyte system, which contains univalent cation and anion and bivalent cation and anion shown in Fig. 3 (f), is considered. Figure 10 (a) shows the total membrane, the diffusion and the Donnan potential differences as a function of the charge density. The Donnan potential difference is positive for a negatively charged membrane and negative for a positively charged one.
160 100
I
I
I Co=le-3 Imol/lJ
Membr. P. ---- Diff. P. - -‘-’ Donnan P
-50 -
-1od
-.5
-.25
I
I
0.0
0.25
Charge density
I 0.5
[mol/l]
Fig. 9 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (e)
Charge density
[mol/l]
Fig. 9(b). Permeability coefficient ratio (P.C.R.) in l-l and l-a-type electrolyte system shown in Fig. 3 (e) as a function of the membrane charge density.
Figure 10(b) shows the permeability coefficient ratio in this system as a function of the charge density. The permeation phenomenon is a combination of those in Fig. 8(b) (l-l- and 2-l-type electrolyte system) and in Fig. 9(b) (l-l- and 1-2-type electrolyte system). The univalent cation, K+, is transported to a greater extent across a negatively charged membrane and to a less extent across a positively charged one compared with a neutral one, since a negatively charged membrane has a positive Donnan potential difference, as shown in Fig. 10 (a). On the other hand, the opposite phenomena is observed for univalent anion transport. The bivalent cation, Ca2+ ion, is transported against its concentration gradient across a negatively charged membrane and the bivalent anion, SO:- ion, is transported across a positively charged one. 4.1.7 Electrolyte system type 2-l and l-2 containing polyanions The electrolyte system shown in Fig. 3 (g) is considered. This is the electrolyte system where polyanions of 0.1 mol/l total concentration are added to side
161 50
I
-25
I
I
I
Co=le-3
-
Mrmbr
----I-‘-’
Dlff. P. Donnan P.
-50 1 -.5
[mol/I]
P.
t -.25
I 0.0
I 0.25
Charge Density
I 0.5
[molil]
Fig. 10 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (f).
‘...
,/ /’
-1.0 -
-2.0 -.5
.’
I’
,’
/
-
‘... ‘i,
/
___-
K’ Cc?’
ion ion
_.-._. c,- ion ‘..., X. so,2- ion .‘,..._
8’ 8’
,
I
L.,
-.25
0.0
0.25
Charge Density
0.5
[m&l]
Fig. 10(b). Permeability coefficient ratio (P.C.R.) in 2-l- and I-2-type electrolyte system shown in Fig. 3 (f) as a function of the membrane charge density.
I of the Z-l-and 1-2-type electrolyte system shown in Fig. 3 (f). The counterion of the polyanions is a potassium ion in the system. The side I cell has 0.1 mol/l more K+ and polyanions than that in the system shown in Fig. 3 (f). In this system, the polyanions are too large to be taken into a membrane and only their counter-ions can be taken into and transported across it. Figure 11(a) shows the total membrane, the diffusion and the Donnan potential differences as a function of the charge density. In the system, the counterion of the polyanions, K+ ion, is taken into and across a negatively charged membrane. The diffusion and the Donnan potential are therefore about 10 mV higher than those of the 2-l- and 1-2-type electrolyte system. The counterion, however, is hardly transported across a positively charged membrane. The behaviour in this system is thus almost the same as that in Fig. 3(f). This phenomenon affects the ion permeation in the system.
162 I
I
Co=le-3 Cpa=O.l (rnOl/ll
___,__-----. -
Membr. P.
----
Dlff.
- -‘-’
Donna” P.
P.
-20
-40 1 -.5
t 0.0
I 25
Charge Density
I 0.25
I 0.S
[mol/l]
Fig. 11 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (g). I Co=ie-3
I CpsO.1
[mol,!], A,-’ , -’
*.-
-1.0
-
-2.0 -.5
K+ 10” ____ C2’ IOIl -.-.-. c,- ,o” ,......_ SO 2~ion 4
I -.25
: :
‘...
:
‘..,
i
‘... x.
:
:
.-
, 0.0
Charge density
I ‘.._ 0.25
0.5
[mol/l]
Fig. 11 (b). Permeability coefficient ratio (P.C.R.) in 2-l- and 1-2-type electrolyte system containing polyanions shown in Fig. 3 (g) as a function of the membrane charge density.
Figure 11 (b) shows the P.C.R. of ions as a function of the charge density. As shown in this figure, the P.C.R. of ions across a positively charged membrane are almost the same as that shown in Fig. 3 (f). On the other hand, the P.C.R. of ions across a negatively charged membrane differ from those in the 2-l- and 1-2-type electrolyte system because of the difference of the diffusion and Donnan potentials between these systems. That is to say, in the system, the P.C.R. of the Cl- ion is larger, that of the K+ ion is lower, and that of a Ca2+ ion is negatively larger than those in the system shown in Fig. 3 (f). When a membrane has negative charges, the anion transport from side I to side II and the cation transport from side II to side I in the system is easier than that in Fig. 3 (f) because the total membrane potential in the system is higher. 4.1.8 Electrolyte system type 2-l and l-2 containing polycations The electrolyte system shown in Fig. 3 (h) is considered. This is the electrolyte system where polycations of 0.1 mol/l total concentration are added to
163
side I of the 2-l- and 1-2-type electrolyte system shown in Fig. 3 (f). The counter-ion of polyanions in the system is a chloride ion. The side I cell thus has 0.1 mol/l more Cl- and polyanions than that shown in Fig. 3 (f). Figure 12 (a) shows the total membrane, the diffusion and the Donnan potential differences as a function of the charge density. In this system, the counter ion of the polycations, the Cl- ion, is taken into and across a positively charged membrane. The diffusion and the Donnan potentials are about 10 mV lower than those of the system shown in Fig. 3 (f). This counter-ion, however, is hardly transported across a negatively charged membrane. The diffusion and Donnan potentials of the system, therefore, are almost the same as those shown in Fig. 3 (f). Figure 12 (b) shows the P.C.R. of ions as a function of the charge density.
T
I
Co=le-3
-40
-80
-
-.5
-
Membr.
----
Diff.
-‘- -
Donnan
Cpc-0.1
[moVII
P.
P. P.
I -.25
I 0.0 Charge
Density
I 0.25
0.5
[molill
Fig. 12(a). Electric potential differences from side I to side II in the systemshownin Fig.3(h ).
____
K’ lo” Ca2+ ion
___ _. CI- ion
Charge Density
[molil]
Fig. 12(b). Permeability coefficient ratio (P.C.R.) in 2-l- and 1-2-type electrolyte system containing polycations shown in Fig. 3 (h) as a function of the membrane charge density.
164
As shown in this figure, the P.C.R. of ions across a negatively charged membrane are almost the same as those of the system shown in Fig. 3 (f). On the other hand, the P.C.R. of ions across a positively charged membrane in this system differ from those shown in Fig. 3 ( f) because of the difference of the diffusion and the Donnan potentials between these systems. That is to say, in this system, the P.C.R. of the K+ ion is larger, that of the Cl- ion is lower, and that of the SOi- ion is negatively larger than those in the 2-l- and 1-2-type electrolyte system. When a membrane has positive charges, the transport of cations from side I to side II and that of anions from side II to side I in this system is easier than that in the 2-l- and l-2-type electrolyte system because the total membrane potential in the system is higher. 4.2 Experimental results The results of the membrane potential measurements are shown in Fig. 13 (a) for negatively charged membranes and in Fig. 13 (b) for a positively charged membrane. The solid and open circles and triangles indicate the experimental results, while the solid lines show the theoretical results, which are made to fit the experimental results obtained by the Teorell-Meyer-Sievers (T.M.S. ) theory. The experimental and theoretical results correspond well with each other. From these fitting parameters, the charge density is obtained. Table 1 shows the charge density as well as the membrane thickness and the degree of hydration. Various kinds of negatively charged membranes and a positively charged membrane were obtained. The ion permeation experiments of the electrolyte systems shown in Fig. 3 were performed using these membranes. Figure 14 shows the P.C.R. of K+ and Ca2+ ions in the electrolyte system shown in Fig. 3 (c ) as a function of the charge density. The solid lines represent the theoretical predictions, while the solid and open circles show the experimental results. The theoretical predictions agree quantitatively well with the experimental results. The P.C.R. of cations across a positively charged membrane is smaller, while that across the negative membranes is higher, than that across a neutral one. The P.C.R. of the univalent ion, K+, is about twice as high as that of the bivalent ion. Figure 15 shows the P.C.R. of K+ and Ca2+ ions in the electrolyte system shown in Fig. 3 (d) as a function of the charge density. The theoretical results agree well with the experimental ones. The P.C.R. of cations across a positively charged membrane are smaller than those across a neutral membrane. On the other hand, the P.C.R. of the univalent ion across a negative membrane is greater than that across a neutral one. The P.C.R. of the bivalent ion across negatively charged membranes decreases and shows a negative value as the absolute value of the membrane charge density increases. That is to say, the bivalent ion is transported against its concentration gradient for samples C-3 and C-4 shown in Table 1.
165 50
I
’
I -4.0
I -3.0
r=5
I
I
I -2.0
I -1.0
: 20
10
0
-
-
-5.0
l c-4 0 c-3 A C-l A c-2
Log(G)
0.0
[molil]
Fig. 13 (a). Membrane potential for negatively charged membranes, where solid and open circles, and triangles represent the experimental results for samples C-4, C-3, C-l and C-2 shown in Table 1,respectively, and solid are the theoretical results.
-5 F E
-
-15 -
W a
l
A-l
-25 -
-45-5.0
I -4.0
I -3.0 Log(C0)
I -2.0
I -1.0
0.0
[molll]
Fig. 13 (b). Membrane potential for a positively charged membrane where the solid circles are the experimental results for sample A-l shown in Table 1, and the solid line is the theoretical prediction.
TABLE 1
Sample
Membrane thickness (Pm)
D.H.
Charge density (X 1O-2mol/l)
C-l
150 250 140 150 150 100
0.63 0.67 0.33 0.52 0.39 0.60
- 0.4 - 7.0 -20 -36 -30 11
c-2 c-3 c-4 c-5 A-l
2
0
Charge Density
[molill
Fig. 14. Permeability coefficient ratio (P.C.R.) of K+ and Ca”’ cations in the electrolyte system shown in Fig. 3 (c). Solid and open circles are experimental results, and solid lines are theoretical predictions.
3.0
2.0
0 ca*+ IO”
I.0 d d a 0.0
o
-
-1.O- ,/
-2.0 -. 5
\ L:w
0 0 I -.25
I 0.0 Charge Density
I 0.25
0.5
[molil]
Fig. 15. Permeability coefficient ratio (P.C.R.) of K+ and Ca2+ cations in the electrolyte system shown in Fig. 3 (d). Solid and open circles are experimental results, and solid lines are theoretical predictions.
5. Conclusions
The permeation of ions across charged membranes in the various kinds of electrolyte systems has been investigated as a function of the charge density. The calculated predictions for the two electrolyte systems shown in Figs. 3 (b) and (c) were examined by permeability coefficient measurements across various kinds of charged membranes. The theoretical predictions agree well with experimental results, indicating that the theoretical calculation accurately predicts the ion permeation across a charged membrane in multicomponent
167
ion systems. The theory cited above can predict how many ions are transported across any kind of charged membrane in multicomponent systems as long as the concentration of external solutions, the ion mobility in the membrane and its charge density are given. The charge density greatly affects the transport phenomena of an ion, especially a multivalent ion. Multivalent ions are transported against their external concentration gradient under appropriate conditions. For instance, these phenomena are applicable for the recovery of heavy metals from industrial waste water and sea water. The direction of multivalent ion transport changes with charge density. This phenomenon is applicable in a new mechanism to control the ion transport direction. An ion gel changes its volume by the change of pH, temperature, ion concentration, electric field and hydrostatic pressure [ 16-191. This means that a charged gel membrane can change its charge density according to these conditions. Furthermore, an amino acid and a weak acid membrane can change their charge density; the former can also change the sign of its charge by changing the pH of the solution. Therefore the ion transport across these membranes in multicomponent systems can be controlled by varying these conditions. Moreover, this theory can predict ion transport phenomena in systems containing polycations and polyanions on one side. In these systems, uphill transport of multivalent ions occurs more easily than in the system without these polyions. These predictions are applicable in medical and biochemical fields, for example, in artificial kidney membrane design and for purification of biochemical products, because body fluids and biochemical products consist of many kinds of polycations, polyanions and metal ions.
References H. Miyama, K. Tanaka, Y. Nosaka, N. Fijii, H. Tanzawa and S. Nagaoka, Charged ultrafiltration membrane for permeation of proteins, J. Appl. Polym. Sci., 36 (1988) 925. R. Schlogl, Electrolyte separation of ions by ion-exchange membranes, Ber. Bunsenges. Phys. Chem., 82 (1978) 225. M. Planck, iiber die Erregungvon Elektrizitit und W&me in Electrolyten, Ann. Phys. Chem., 39 (1890) 161. M. Planck, Uber die Potentialdifferenz zwischen zwei verdtinnten Lijsungen biniirer Electrolyten, Ann. Phys. Chem., 40 (1890) 561. T. Teorell, Studies on the “diffusion effect” upon ionic distribution. I. Some theoretical considerations, Proc. Natl. Acad. Sci. U.S.A., 21 (1935) 152. J. Straub, Membrangleichgewichte und Harmonien, Kolloid. Z., 64 (1933) 72. T. Teorell, Transport processes in ionic membranes, Prog. Biophys. Biophys. Chem., 3 (1953) 305. M. Nakagaki and M. Kobayashi, Diffusion and reverse diffusion of ions in mixed solutions of strong electrolytes, Nippon Kagaku Kaishi, 4 (1973) 635. P. Schwahn and D. Woermann, Transport of ions against their concentration gradient across cation exchange membranes, Ber. Bunsenges. Phys. Chem., 90 (1986) 773.
168 10 11 12 13 14 15 16 17 18 19
M. Higa, A. Tanioka and K. Miyasaka, Simulation of the transport of ions against their concentration gradient across charged membranes, J. Membrane Sci., 37 (1988) 251. J.S. Mackie and P. Meares, The diffusion of electrolytes in a cation-exchange resin membrane, Proc. R. Sot. London, Ser. A, 232 (1955) 498A. H.-U. Demisch and W. Pusch, Ion exchange capacity of cellulose acetate membrane, J. Electrochem. Sot., 123 (1976) 370. G.E. Goldman, Potential, impedance and rectification in membrane, J. Gen. Physiol., 26 (1943) 37. T. Teorell, An attempt to formulate a quantitative theory of membrane permeability, Proc. Sot. Exp. Biol. Med., 33 (1935) 282. K.H. Meyer and J.-F. Sievers, La permeabilite des membranes. I. Theorie de la permeabilitk ionique, Helv. Chim. Acta, 19 (1936) 649. T. Tanaka, D. Fillmore, S.-T. Sun, I. Nishio, G. Swislow and A. Shah, Phase transitions in ionic gels, Amer. Phys. Sot., 45 (1980) 1636. T. Tanaka, I. Nishio, S.-T. Sun and S. Ueno, Collapse of gels in an electric field, Science, 218 (1982) 467. I. Ohmine and T. Tanaka, Salt effects on the phase transition of ionic gels, J. Chem. Phys., 77 (1982) 5725. J. Ricka and T. Tanaka, Swelling of ionic gels: quantitative performance of the Donnan theory, Macromolecules, 17 (1984) 2916.
Appendix
If the valence of ions in electrolyte systems is smaller than three, eqn. (9) can be rewritten as the following:
(77p+n4+1p2
+&X,+
p2--1 c;+II
+ CO;+
+ cop-
-_lyp2
/P-l Q-1
_cf+lp”
/P-l
CF-1
+&x-
_~;-“I~”
p3-1
(A-1)
1=o
Since j? is unequal to one, both parts of this equation can be divided by - FSRTb$/d and multiplied by
[O~+(e,!+ll-~+‘p,+c(,Z-(~i’-‘-~Z-ll~)] [w:+,C;+ii
_c;+rp7
+,;-
+g(P+1)[~~+(~P”‘-eq+I/j3)+~~-(~~--I_~-~I~3)] =(P+NP”+P+l)(A,-&P)+4(P2+P+U(A,--&P2) +9(P+1)(A3-B3P3)=0 where
@;-I
__cp-“p”)] (-4-z)
169
A, =
‘@‘+pk+I1 +
B, =pB,+
cm;- pk-’
C;+I+Cci&-
c;-”
z=l,
2,3
Equation ( 10) is obtained be rewriting eqn. (A-2 ) .
145
Journal of Membrane
A STUDY OF ION PERMEATION ACROSS A CHARGED MEMBRANE IN MULTICOMPONENT ION SYSTEMS AS A FUNCTION OF MEMBRANE CHARGE DENSITY
MITSURU HIGA, AKIHIKO TANIOKA and KEIZO MIYASAKA Department
of Organic and Polymeric
Meguro-ku,
Tokyo 152 (Japan)
Materials,
Tokyo Institute
of Technology,
Ookayama,
(Received September 7,1988; accepted in revised form July 25,1989)
Summary The authors have described an ion flux across a charged membrane in multicomponent systems by the theories (based on the Donnan equilibrium and the Nernst-Planck equation) in the previouspaper. In the present study, the permeation coefficients of ions in various kinds of electrolyte systems are calculated as functions of membrane charge density by this theory. The theory predicts that permeability coefficients of multi-valent ions will be lower than those of univalent ions according to the low charge density, and that the permeability coefficients of multi-valent ions will become negative if the membrane has high charge density. That is, they are transported against their concentration gradient. This phenomenon in l-l- and 2-l-type electrolyte systems was examined for negatively charged membranes, which were composed of poly (vinyl alcohol) (PVA) and poly (styrenesulfuric acid) (PSSA) mixtures, and a positively charged one, which was composed of a PVA and poly(allylamine) (PAAm) mixture. The theoretical prediction agreed well with the experimental results. The uphill transport of a multivalent ion occurred under the appropriate membrane charge density. This result shows that the direction of multivalent ion transport changes with charge density. This phenomenon is applicable in a new mechanism to control ion transport direction.
1. Introduction Recently it has become necessary to analyze precisely the ion transport phenomena in multicomponent ion systems across a charged membrane from the standpoint of industrial and medical applications. Naturally, almost all membranes in a living body have electrical charges. A cell membrane consisting of a phosphorus lipid and protein, which can be regarded as a charged membrane, has a high selectivity for special solutes, and can transport a solute against its concentration gradient under certain conditions. One reason for this high selectivity is considered to be the charges on the membrane. A neutral membrane separates solutes according to the difference in their size and/or in their interaction with the membrane. In addition to this, a charged membrane can sep-
0376-7388/90/$03.50
0 1990 Elsevier Science Publishers B.V.
146
arate charged solutes according to the differences in their sign and valence, and should thus show a higher selectivity than a neutral membrane. From this point of view, charged membranes have been used for reverse osmosis and artificial dialysis [ 11, replacing neutral membranes. The theoretical analysis of the ion transport phenomena in the system is very important, therefore, in producing a highly selective membrane [2] and explaining the metabolic mechanism in a cell membrane. Ion transport against its concentration gradient is also a very interesting phenomenon. This uphill transport phenomenon has been studied by many authors [ 3-101. In these experimental systems, however, a highly concentrated electrolyte solution is necessary as the driving force. In the present paper, the permeability coefficients of ions in various kinds of electrolyte systems are calculated precisely as functions of membrane charge density by a theory based on the Donnan equilibrium and the Nernst-Planck equation considering practical applications. The theory predicts that membrane charge density will greatly affect the transport phenomena of ions, especially multivalent ions. The theory also predicts that the uphill transport of multivalent ions will occur in the experimental system where a charged membrane is placed between two cells, and one cell has an electrolyte concentration ten times higher than the other. The uphill transports in the systems occur without a highly concentrated electrolyte solution as the driving force. One of these phenomena was examined for negatively charged membranes composed of poly(viny1 alcohol) (PVA) and poly( styrenesulfuric acid) (PSSA) mixtures, and for a positively charged one composed of a PVA and poly (allylamine ) (PAAm) mixture. The theoretical prediction agreed well with the experimental results and the uphill transport of a multivalent ion occurred under the appropriate membrane charge density. This result shows that the direction of multivalent ion transport changes with charge density. This phenomenon is applicable in a new mechanism to control ion transport direction. 2. Theory 2.1 Assumptions
The basic assumptions of our theory are as follows: (a) It is difficult to measure experimentally the ion mobility in a membrane. Mackie and Meares therefore approximately calculated the mobility as a function of the tortuosity of a membrane and the mobility in solution [ 111: uy = uy/e2
(1)
where Uf and UT are the ion mobility in a membrane and an external solution, respectively, and 0 is a tortuosity factor, which is related to the degree of hydration H by:
147
8
=-
2-H H
The membranes used in this study have a higher degree of hydration and a lower membrane charge density than current ion exchange membranes. This assumption is therefore considered to be suitable for these membranes. (b) From assumption (a), membranes with the same degree of hydration have the same amount of ion mobility, even if these membranes have a different ion charge group, a different membrane charge density and a different polymer network configuration. (c ) Osmotic or hydrostatic flow of water is negligible. (d) The membrane surfaces are always at a state of Donnan equilibrium with the same distribution ratio (K) for all ions. (e) The boundary layer effect on membrane surfaces is negligible. (f) For simplicity here, all electrolytes dissolve perfectly and ion activity equals unity in both the external solutions and the membrane. (g ) Also for simplicity, the standard chemical potential in external solutions is equal to that in a membrane. 2.2 Determination of concentration of the ith ion in a membrane The Donnan equilibrium for the ith ion between a membrane and external solutions can be realized as [ 121:
where Ci and Ci are the ith concentrations in the membrane and the solutions, respectively, Kj ( = exp - FA&,,JRT) and Aq&,,j are the Donnan equilibrium constant and the Donnan potential at side j of the membrane surface, respectively, F, R and T are Faraday’s constant, the gas constant and the absolute temperature, respectively, and zi is the valence of the ith ion. An electroneutrality condition in the membrane requires that:
where z, is the valence of fixed charge density and C, is the effective charge density. Equation (5) is obtained by substituting eqn. (3 ) into eqn. (4): CziKf’Ci +z,C~ =O
(5)
Since K is obtained from only Ci and C, by eqn. (5)) the concentration of the ith ion in a membrane, pi, is obtained from K and Ci using eqn. (3 ) .
148
2.3 Calculation of the ith ion permeability coefficient ratio (P.C.R.) in a membrane In an experimental system, a charged membrane is placed between two cells (side I and side II ) and the cells are filled with electrolyte solutions. Thus, each ion flux can be obtained analytically under the assumption of a constant electric field or a constant ion concentration gradient. For constant electric field gradient, each ion flux is given by the following equation [ 131:
(6) where p= exp - FA@,iff/RT and A$diffis the diffusion potential difference from side I to side II, mi is the ion mobility in the membrane, and f?j and Cf’ are the concentrations at side I and side II of the membrane surface, respectively. p is unequal to one and greater than zero when A~diffis unequal to zero. As the systems treated in this paper have no electric current:
where S and d are the membrane area and thickness, respectively. If the valence of ions in the system is three and less, eqn. (7) can be solved for j? analytically [lo] and j? is the solution of the following equation (see appendix) : (B, +4& +9&)P4+ - (2A, +4A, -B,
(2B, +4B, +9& -A,)p3+ +9A3)P-
(2B1 +4&-2A,
-4A,)P2
(A, +4Az +9A3) =O
(8)
where: + x(-q-
pk-’
B, = .&x;+ e;+’ + cm;-
pk-”
A, = Cog+
e;+”
z= 1,2,3
and WE+ and wk- are the mobility of z-valent cations and anions, respectively; C-c+’ and &I are the concentrations of z-valent cations and anions at side I of the membrane surface, and Pk’ I1and Cj?-I1 are those at side II, respectively. For a univalent ion system, eqn. (8) is rewritten as: (P+I)(P”+P+I)(B,P-A,)=0
(9)
Since p is unequal to one, the solution of p is: p=
$L = yJJ--;~~~~: 1
k
k
k
Similarly, in uni- and bi-valent ion systems, eqn. (8) is rewritten as:
(10)
149
(p”+p+l) [ @I +4&)fi2+ (B, -A,)P-
(A, +4&j]
=o
(11)
and the solution of /3 is:
P=
Al -&
+@,
-M2+4(B1
+4B2)641+4A2) (12)
2(&
+4B2)
The ith ion flux (Ji) can be obtained by substituting the concentrations on both surfaces of the membrane into eqn. (6). The ith ion permeability coefficient (Pi) is obtained from the ion flux and external ion concentrations by the following equation: (13) where Cf and Cf’ are the ith ion concentrations in side I and side II cells, respectively. Ion transport phenomena across a charged membrane are a function of both the ion mobility and its charge density. The membranes which have the same ion mobility but different charge densities are necessary when discussing the relation between the phenomena and its charge density. It is difficult, however, to make them by experimentation. The phenomena are therefore discussed in terms of a permeability coefficient ratio (P.C.R.). The P.C.R. is obtained to normalize the permeability coefficient in a charged membrane by the permeability coefficient of potassium chloride in a neutral membrane with the same degree of hydration as the charged membrane. From eqns. (1) and (2), membranes with the same degree of hydration have the same tortuosity, and hence the effect of tortuosity on ion transport can be cancelled out to obtain the P.C.R. The permeability coefficient calculated from eqn. (13) can be normalized by the following permeability coefficient: P,=
2w,oc,RT (WC +mcl)d
(14)
where OK and ocI are the K+ and Cl- ion mobilities in the neutral membrane, respectively, and the mobilities are calculated from eqn. (1) . 2.4 Donnanpotential and diffusionpotentiul The Donnan potential differences from an external solution to the inside of a membrane at both membrane surfaces are calculated from the following equation using the Donnan equilibrium constant (K): A@don,j= -
RT 7 1nKj j=I, II
(15)
150
The total Donnan potential difference (d$ddon)is obtained from the following equation:
The diffusion potential difference from side I to side II (A~diff) is obtained from the following equation: (17) The total membrane potential difference from side I to side II (A@) is obtained from eqns. (16) and (17): A@= A&don + A@diff
(16)
2.5 Permeability coefficient ratio and membrane potential in electrolyte systems containing polycations and polyanions Many polycations and polyanions which are considered not to permeate the glomerular wall can be found in blood plasma. In electrolyte systems containing these polyions, the diffusion and the Donnan potential due to them affect the permeation of mobile ions. In this paper, the ion permeation in the systems is considered, assuming that these ions are not taken into a membrane. In these systems, the electroneutrality condition is required in the external solution:
(19) where C,, and C,, are the total charge densities of polyanions and polycations, respectively, and zpaand zpc are the mean valences of polyanions and polycations, respectively. For simplicity, we set zpa= - 1 and z,, = 1, respectively. In this paper, K+ and Cl- ions are used as the univalent cation and anion, and Ca2+ and SOi- as the bivalent cation and anion, respectively. However, assumptions (d) and (g) suggest that there are no distinctions, except ion mobility, between ions of the same valence. The calculation cited above can thus be performed for different ionic species if their ion mobilities are given. 3. Experimental
3.1 Samples Membranes were cast from 7% PVA [poly (vinyl alcohol); Wako Pure Chemical Industries Ltd., D.P. = 20001 and PSSA [poly (styrenesulfuric acid), Asahi Chemical Industry Co. Ltd.] or PVA and PAAm [poly (allylamine), Nittobo Industries Inc.] aqueous solutions at atmospheric pressure and 20” C, and were annealed at 160 oC for 20 min under vacuum to obtain membranes with a low degree of hydration. Parts of the membranes were crosslinked in an
151
aqueous solution of 20% Na,SO,, 1% H&SO, and 0.1% glutaraldehyde at 20°C to obtain a degree of hydration lower than that of the uncrosslinked samples. The lower the degree of hydration of a membrane, the higher its charge density. 3.2 Membrane potential measurement The membrane potential of the membranes used in this paper was measured to obtain the charge density. Figure 1 shows the apparatus for measurement of the membrane potential. The membrane area was 0.78 cm2 and the volume of each cell was about 300 cm3. The concentration of KC1 electrolyte solution on one side was five times higher than that on the other. The solutions in both cells were stirred by magnetic stirrers to minimize the effect of the boundary layers on the membrane potential. The potential was measured at 19.0 -I-0.1 “C using glass electrodes (TOA HS-205C) with salt bridges (3 N KCl) and a voltmeter (TOA HM-20E). The charge density was given by fitting the experimental results to the following equation [ 14,151:
J
1+
$
F+w x 1 [1+ 2
2
+1
A@=-
-sWln
2
1+ J
9
X
X
J(,
1+
+1
Jc,
3 2+W CX -
(20)
where w= (&& - wcl) / (UK + ocl), and C, and C, are the KC1 concentrations on both sides.
Glass Electrode
Charged Membrane
Fig. 1. Apparatus for membrane potential measurement. The membrane area is 0.78 cm2 and the volume of each cell is ca. 300 cm3. The concentration of the KC1 electrolyte solution on one side is five times higher than that on the other (r = 5).
152
3.3 Measurement of degree of hydration In this paper, the ion mobility in a membrane was not measured experimentally, but calculated from the degree of hydration (D.H.) and the mobility in free solution using eqns. (1) and (2 ). The D.H., defined as the volume fraction of water in a wet membrane, was estimated from the following equation: D.H. =
w, - WJ1.0 water volume WJ1.3 water volume + polymer volume = ( w, - w,)/l.o+
(21)
where W, and W, are the weights of a membrane at the equilibrium swollen and dry states, respectively, and 1.0 and 1.3 are the densities of water and polymer, respectively. 3.4 Permeability coefficient measurement Figure 2 shows the apparatus for the permeability coefficient measurement. A membrane was placed between the two cells and fixed by a clamp. Electrolyte solutions were circulated by tubing pumps (Core-Parmer Instrument Co. PA2113) to minimize the effect of the boundary layers. The side II solution was introduced through a 500 ,~l microsyringe and diluted lOO-fold with distilled water for measurement of the ion concentrations by an atomic absorption spectrophotometer (Shimadzu AA-640-12). The permeability coefficient (Pi) was found from the ion concentration change at side II using the following equation:
v,,
pi=-
ACi (cf’_cy)S,At
(22)
where V, is the volume of the side II cell, ACi is the ith ion concentration change at time interval At, S’ (=SH) is the effective membrane area, S is the crosssectional membrane area and H is the degree of hydration. Membranes used for this experiment have various degrees of hydration.
Reservoir I
Tubing Pump Clamp
Fig. 2. Apparatus for permeability coefficient measurement. The membrane the volumes of side I and side II are 900 cm3 and 50 cm3, respectively.
area is 7.01 cm* and
153
Therefore the permeability coefficient calculated from eqn. (22 ) is normalized by eqn. (14) and a permeability coefficient ratio is obtained. 4. Results and discussion
4.1 Calculated results for permeability coefficient ratio (P. C.R.) and membrane potential 4.1.1 Electrolyte system type l-l The l-l-type electrolyte system shown in Fig. 3(a) is considered. In this system, a charged membrane is placed between two cells and the concentrations of KC1 electrolyte at side I and side II are C, and C,, respectively. The ratio of C, to C, is 10. Figure 4 (a) shows the calculated results of electric potential difference from the high-concentration side to the low-concentration side when C, = 1 x 10e3 mol/l. Solid, broken and dash-dotted lines are the total membrane, the diffusion and the Donnan potential, respectively. The Donnan potential difference is symmetrical. The value of the diffusion potential is small on account of the small mobility difference between the cation and the anion. The value of the total membrane potential is thus almost the same as that of the Donnan potential. The calculated results for the P.C.R. are shown in Fig. 4(b) as a function of the membrane charge density. In this figure, the higher the charge density, the smaller the P.C.R. symmetry in both negatively and positively charged membranes. In the case of a positively charged membrane and a l-l-type electrolyte system, a cation is excluded from the membrane due to a high Donnan potential difference, and an anion is scarcely transported across the membrane as a consequence of an electroneutral condition. Eventually, the permeation of both the cation and the anion is reduced. The same explanation can be used to describe a similar phenomenon in a negatively charged membrane. This shows that a neutral membrane has a higher P.C.R. than a charged membrane. This phenomenon explains qualitatively why a charged reverse osmosis membrane has a higher salt rejection than a neutral one. 4.1.2 Electrolyte system type 2-1 The P.C.R. and the membrane potential of the 2-l-type electrolyte system shown in Fig. 3 (b) are considered. Figure 5 (a) shows the calculated results of the electric potential difference when C, = 1 x 10e3 mol/l. Solid, broken and dash-dotted lines are the total membrane, the diffusion and the Donnan potentials, respectively. The contribution of the Donnan potential to the total membrane potential is large when the charge density is higher than the external concentration, but that of the diffusion potential to the total potential is large when the membrane charge approaches zero. The diffusion potential has
154
Side I
Side I
Side II
Charged
CaCI,(Co)
CaCI,(Cd)
KCl(Co)
KCl(Cd)
Charged
(a) I-l-type
Side 11
Membrane
Membrane
electrolyte
(b) 2-l-type
system
electrolyte
system
KCl(Co)
KCI(Cd)
KCl(Cd)
KCI(Co)
CaCl,(Cd)
CaCI,(Co)
WI&Cd)
CaCI,(Co)
Side II
Side I
Side I
Charged
Charged
(c) 2-l- and I-l-type
Side II
Membrane
Membrane
electrolyte
(d) 2-l- and l-l-type
system
electrolyte
system
I KCl(Cd)
KCI(Co)
K,SO,(W
K,sO,(W
K,SO,(Cd)
K,SO,(Co)
CaCI,(Cd)
CaCI*(Co)
Side II
Side I
Side 1
Charged
&de II
Charged Membrane
Membrane
(e) I-I- and 1.2.type electrolyte
(f) 2-l. and l-Ztypc
system
electrolyte
system
Polycations(Cpc> PoIyanions(Cpa) K,SO,(Cd)
K,SO,(Co)
CaCI,(Cd)
CaC12(Co)
Side I
Charged
K&)&Cd) CaCl,(Cd)
Side I
Side II
Membrane
) 2-I.
Charged
(h) 2-l- and I-2-type electrolyte and l-?-type
electrolyte
system contamlng
Side II
Membrane
polyamons
Fig. 3. The various electrolyte systems considered
in this paper.
system containing
polycatia
155 100
y50
I
Co=le-3 [molill
-
.c ‘;i 0.0
Membr.
P.
---Diff. P. -‘-‘-’ Donnan P. c________________________-___
- ____________________---
-50 -
-1od-.5
I
I
I
-.2.5
0.0 Charge
0.25
0.5
Density [mol/l]
Fig. 4 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (a). 1.5
I
I
Co=le-3
[moVII
1.2 -
0.9 d d a 0.6
-
-
K’ and Cl- IO”
Charge
Density [mol/l]
Fig. 4(b). Permeability coefficient ratio (P.C.R.) in l-l-type 3 (a) as a function of membrane charge density.
electrolyte system shown in Fig.
a negative value because the mobility of a Ca2+ ion is smaller than that of a Cl- ion. In contrast to the l-l-type electrolyte system, the Donnan potential difference here has an asymmetric value. In other words, the absolute value of the Donnan potential in a negatively charged membrane is smaller than that in a positively charged one. The reason is that the counter-ion of a negatively charged membrane has a higher valence than that of a positively charged membrane in the system. This phenomenon affects the P.C.R. Figure 5 (b) shows the calculated results of the P.C.R. as a function of membrane charge density. In both negatively and positively charged membranes, the P.C.R. decreases with the charge density. However, the decrease of the P.C.R. in the latter is larger than in the former. In other words, 2-l-type electrolytes are transported across a negatively charged membrane more than across a positively charged membrane if both membranes have the same absolute
156 1OC
I
I Co=le-3
50
-
s
-1OG -.5
I -.25
-
Membr.
----
Diff.
-‘-‘-’
Donnan
P.
P. P.
I 0.25
I 0.0 Charge Density
[moUll
0.5
[mol/ll
Fig. 5 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (b). 1.0
I
Co=le-3
0.8
-
0.6 d d a 0.4
0.2
0.0
4 I
-. 5
- 25
rmol/l]
Ca2’ and Cl- ion
I_
0.0 Charge Density
0.25 [molll]
Fig. 5(b). Permeability coefficient ratio (P.C.R.) in 2-l-type electrolyte system shown in Fig. 3 (b) as a function of membrane charge density.
charge density and an identical membrane structure, because the absolute value of the Donnan potential in the former is larger than that in the latter. 4.1.3 Electrolyte system type 2-l and l-l The 2-l- and l-l-type electrolyte system shown in Fig. 3(c) is considered. In this system, a charged membrane is placed between two cells, and the concentrations of KC1 and CaCl, electrolytes at side I are C,, and Cd, respectively, and C, and COat side II, respectively. The concentration gradient of K+ ion is in the opposite direction to that of Ca2+ ion, and a counter-flow of ions occurs in the system. Figure 6 (a) shows the electric potential difference as a function of the charge density in this system. The calculated results of the P.C.R. are shown in Fig. 6 (b) as a function of membrane charge density. As the density of the negative charges in the mem-
157
50
I
I Co=le-3
[moH]
F 25 .E
Donnan P.
-‘-‘-’
-50 ’ -.5
I -.25
I 0.0 Charge Density
I 0.25
0.5
[molil]
Fig. 6 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (c).
Charge Density
-
K+
___-
#&z+
_,_._.
CI‘
IO” ion
,o”
[molil]
Fig. 6(b). Permeability coefficient ratio (P.C.R.) in l-l- and P-l-type electrolyte system shown in Fig. 3 (c) as a function of membrane charge density.
brane increases, the number of cations taken into the membrane increases as does the number of anions rejected on account of the Donnan potential at the membrane surfaces. Therefore, the anion permeability is reduced in a negatively charged membrane while cation permeability is enhanced compared with the permeability across a neutral membrane. The permeability coefficient of the univalent cation is about twice as high as that of the bivalent one under an electroneutral condition. 4.1.4 Electrolyte system type 2-l and l-l The 2-l- and l-l-type electrolyte system shown in Fig. 3 (d) is considered. In this system, the concentrations of KC1 and CaCl, electrolytes at both side I and side II are Cd and C,, respectively. Figure 7 (a) shows the electric potential difference in the system. This figure shows that the Donnan potential differ-
158
ence is positive across a negatively charged membrane, which means that the Donnan potential and Donnan distribution constant (K) at the low-concentration side (side II) are higher than those at the high-concentration side (side I). The concentration ratio of an ion in a membrane to that in the external solution is equal to the ion valence power of the Donnan distribution constant from eqn. (3). The ratio of cations at side II is thus greater than at side I. Moreover, the ratio for a bivalent cation is greater than that for a univalent one. For these reasons, the concentration of the bivalent cation at the side II membrane surface is higher than that at side I under an appropriate condition. Hence, the concentration gradient of the bivalent ion in the membrane is in the opposite direction to its external concentration gradient, as shown in Fig. 8. This phenomenon affects the ion permeation in the system.
$
50 -
E
-
z
0
-50
___________--________
-
-1OG -5
-
Membr.
----
Dlff.
-‘- -
Donnan
-.25
P.
P. P.
I 0.0
0.25
Charge Density
Fig. 7 (a). Electric potential differences
0.5
[mol/l] from side I to side II in the system shown in Fig. 3 (d).
Co=le-3 [mol/ll
_--_
K’ &
ion ion
-‘- -
cl-
lo”
1.0 -
Charge Density
Fig. 7(b).
Permeability
coefficient
[mol/l]
ratio (P.C.R.)
in l-l-
in Fig. 3 (d) as a function of membrane charge density.
and 2%l-type electrolyte system shown
159
KCl(Cd)
CaCl,(Cd)
Side I Charged Membrane
Fig. 8. Schematic diagram of the concentration gradient of Ca ‘+, K+ and Cl- ions in a negatively charged membrane when the uphill transport of Ca2+ ion occurs.
The calculated results of ion transport are shown in Fig. 7 (b). The ion permeation phenomenon in the system across a negatively charged membrane is of great interest. The permeability coefficient of a K+ ion across the membrane is higher than that across a neutral one. On the other hand, the permeability coefficient of a Ca2+ ion is smaller than that across a neutral one, and becomes negative as the charge density becomes more negative. That is to say, a Ca2’ ion is transported against its concentration gradient, because the concentration gradient of the bivalent ion in the membrane is in the opposite direction to its external gradient, as shown in Fig. 8. A Ca2+ ion is thus transported against its external concentration gradient as K+ ions diffuse from the high to low concentration side. The absolute value of P.C.R. for K+ is about twice that for Ca2+ because of an electroneutral condition. 4.1.5 Electrolyte system type l-l and l-2 The l-l-and 1-2-type electrolyte system shown in Fig. 3 (e) is considered. The theoretical predictions are shown in Figs. 9 (a) and (b). The opposite phenomenon to the l-l- and 2-l-type electrolyte as shown in Fig. 3 (d) is observed across a positively charged membrane. The permeability coefficient of the univalent anion, Cl- ion, across a negatively charged membrane is higher than that across a neutral one. On the other hand, the permeability coefficient of the bivalent anion, SO:- ion, is smaller than that across a neutral one, and becomes negative as the charge density becomes more positive. 4.1.6 Electrolyte system type 2-l and l-2 The 2-l- and 1-2-type electrolyte system, which contains univalent cation and anion and bivalent cation and anion shown in Fig. 3 (f), is considered. Figure 10 (a) shows the total membrane, the diffusion and the Donnan potential differences as a function of the charge density. The Donnan potential difference is positive for a negatively charged membrane and negative for a positively charged one.
160 100
I
I
I Co=le-3 Imol/lJ
Membr. P. ---- Diff. P. - -‘-’ Donnan P
-50 -
-1od
-.5
-.25
I
I
0.0
0.25
Charge density
I 0.5
[mol/l]
Fig. 9 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (e)
Charge density
[mol/l]
Fig. 9(b). Permeability coefficient ratio (P.C.R.) in l-l and l-a-type electrolyte system shown in Fig. 3 (e) as a function of the membrane charge density.
Figure 10(b) shows the permeability coefficient ratio in this system as a function of the charge density. The permeation phenomenon is a combination of those in Fig. 8(b) (l-l- and 2-l-type electrolyte system) and in Fig. 9(b) (l-l- and 1-2-type electrolyte system). The univalent cation, K+, is transported to a greater extent across a negatively charged membrane and to a less extent across a positively charged one compared with a neutral one, since a negatively charged membrane has a positive Donnan potential difference, as shown in Fig. 10 (a). On the other hand, the opposite phenomena is observed for univalent anion transport. The bivalent cation, Ca2+ ion, is transported against its concentration gradient across a negatively charged membrane and the bivalent anion, SO:- ion, is transported across a positively charged one. 4.1.7 Electrolyte system type 2-l and l-2 containing polyanions The electrolyte system shown in Fig. 3 (g) is considered. This is the electrolyte system where polyanions of 0.1 mol/l total concentration are added to side
161 50
I
-25
I
I
I
Co=le-3
-
Mrmbr
----I-‘-’
Dlff. P. Donnan P.
-50 1 -.5
[mol/I]
P.
t -.25
I 0.0
I 0.25
Charge Density
I 0.5
[molil]
Fig. 10 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (f).
‘...
,/ /’
-1.0 -
-2.0 -.5
.’
I’
,’
/
-
‘... ‘i,
/
___-
K’ Cc?’
ion ion
_.-._. c,- ion ‘..., X. so,2- ion .‘,..._
8’ 8’
,
I
L.,
-.25
0.0
0.25
Charge Density
0.5
[m&l]
Fig. 10(b). Permeability coefficient ratio (P.C.R.) in 2-l- and I-2-type electrolyte system shown in Fig. 3 (f) as a function of the membrane charge density.
I of the Z-l-and 1-2-type electrolyte system shown in Fig. 3 (f). The counterion of the polyanions is a potassium ion in the system. The side I cell has 0.1 mol/l more K+ and polyanions than that in the system shown in Fig. 3 (f). In this system, the polyanions are too large to be taken into a membrane and only their counter-ions can be taken into and transported across it. Figure 11(a) shows the total membrane, the diffusion and the Donnan potential differences as a function of the charge density. In the system, the counterion of the polyanions, K+ ion, is taken into and across a negatively charged membrane. The diffusion and the Donnan potential are therefore about 10 mV higher than those of the 2-l- and 1-2-type electrolyte system. The counterion, however, is hardly transported across a positively charged membrane. The behaviour in this system is thus almost the same as that in Fig. 3(f). This phenomenon affects the ion permeation in the system.
162 I
I
Co=le-3 Cpa=O.l (rnOl/ll
___,__-----. -
Membr. P.
----
Dlff.
- -‘-’
Donna” P.
P.
-20
-40 1 -.5
t 0.0
I 25
Charge Density
I 0.25
I 0.S
[mol/l]
Fig. 11 (a). Electric potential differences from side I to side II in the system shown in Fig. 3 (g). I Co=ie-3
I CpsO.1
[mol,!], A,-’ , -’
*.-
-1.0
-
-2.0 -.5
K+ 10” ____ C2’ IOIl -.-.-. c,- ,o” ,......_ SO 2~ion 4
I -.25
: :
‘...
:
‘..,
i
‘... x.
:
:
.-
, 0.0
Charge density
I ‘.._ 0.25
0.5
[mol/l]
Fig. 11 (b). Permeability coefficient ratio (P.C.R.) in 2-l- and 1-2-type electrolyte system containing polyanions shown in Fig. 3 (g) as a function of the membrane charge density.
Figure 11 (b) shows the P.C.R. of ions as a function of the charge density. As shown in this figure, the P.C.R. of ions across a positively charged membrane are almost the same as that shown in Fig. 3 (f). On the other hand, the P.C.R. of ions across a negatively charged membrane differ from those in the 2-l- and 1-2-type electrolyte system because of the difference of the diffusion and Donnan potentials between these systems. That is to say, in the system, the P.C.R. of the Cl- ion is larger, that of the K+ ion is lower, and that of a Ca2+ ion is negatively larger than those in the system shown in Fig. 3 (f). When a membrane has negative charges, the anion transport from side I to side II and the cation transport from side II to side I in the system is easier than that in Fig. 3 (f) because the total membrane potential in the system is higher. 4.1.8 Electrolyte system type 2-l and l-2 containing polycations The electrolyte system shown in Fig. 3 (h) is considered. This is the electrolyte system where polycations of 0.1 mol/l total concentration are added to
163
side I of the 2-l- and 1-2-type electrolyte system shown in Fig. 3 (f). The counter-ion of polyanions in the system is a chloride ion. The side I cell thus has 0.1 mol/l more Cl- and polyanions than that shown in Fig. 3 (f). Figure 12 (a) shows the total membrane, the diffusion and the Donnan potential differences as a function of the charge density. In this system, the counter ion of the polycations, the Cl- ion, is taken into and across a positively charged membrane. The diffusion and the Donnan potentials are about 10 mV lower than those of the system shown in Fig. 3 (f). This counter-ion, however, is hardly transported across a negatively charged membrane. The diffusion and Donnan potentials of the system, therefore, are almost the same as those shown in Fig. 3 (f). Figure 12 (b) shows the P.C.R. of ions as a function of the charge density.
T
I
Co=le-3
-40
-80
-
-.5
-
Membr.
----
Diff.
-‘- -
Donnan
Cpc-0.1
[moVII
P.
P. P.
I -.25
I 0.0 Charge
Density
I 0.25
0.5
[molill
Fig. 12(a). Electric potential differences from side I to side II in the systemshownin Fig.3(h ).
____
K’ lo” Ca2+ ion
___ _. CI- ion
Charge Density
[molil]
Fig. 12(b). Permeability coefficient ratio (P.C.R.) in 2-l- and 1-2-type electrolyte system containing polycations shown in Fig. 3 (h) as a function of the membrane charge density.
164
As shown in this figure, the P.C.R. of ions across a negatively charged membrane are almost the same as those of the system shown in Fig. 3 (f). On the other hand, the P.C.R. of ions across a positively charged membrane in this system differ from those shown in Fig. 3 ( f) because of the difference of the diffusion and the Donnan potentials between these systems. That is to say, in this system, the P.C.R. of the K+ ion is larger, that of the Cl- ion is lower, and that of the SOi- ion is negatively larger than those in the 2-l- and 1-2-type electrolyte system. When a membrane has positive charges, the transport of cations from side I to side II and that of anions from side II to side I in this system is easier than that in the 2-l- and l-2-type electrolyte system because the total membrane potential in the system is higher. 4.2 Experimental results The results of the membrane potential measurements are shown in Fig. 13 (a) for negatively charged membranes and in Fig. 13 (b) for a positively charged membrane. The solid and open circles and triangles indicate the experimental results, while the solid lines show the theoretical results, which are made to fit the experimental results obtained by the Teorell-Meyer-Sievers (T.M.S. ) theory. The experimental and theoretical results correspond well with each other. From these fitting parameters, the charge density is obtained. Table 1 shows the charge density as well as the membrane thickness and the degree of hydration. Various kinds of negatively charged membranes and a positively charged membrane were obtained. The ion permeation experiments of the electrolyte systems shown in Fig. 3 were performed using these membranes. Figure 14 shows the P.C.R. of K+ and Ca2+ ions in the electrolyte system shown in Fig. 3 (c ) as a function of the charge density. The solid lines represent the theoretical predictions, while the solid and open circles show the experimental results. The theoretical predictions agree quantitatively well with the experimental results. The P.C.R. of cations across a positively charged membrane is smaller, while that across the negative membranes is higher, than that across a neutral one. The P.C.R. of the univalent ion, K+, is about twice as high as that of the bivalent ion. Figure 15 shows the P.C.R. of K+ and Ca2+ ions in the electrolyte system shown in Fig. 3 (d) as a function of the charge density. The theoretical results agree well with the experimental ones. The P.C.R. of cations across a positively charged membrane are smaller than those across a neutral membrane. On the other hand, the P.C.R. of the univalent ion across a negative membrane is greater than that across a neutral one. The P.C.R. of the bivalent ion across negatively charged membranes decreases and shows a negative value as the absolute value of the membrane charge density increases. That is to say, the bivalent ion is transported against its concentration gradient for samples C-3 and C-4 shown in Table 1.
165 50
I
’
I -4.0
I -3.0
r=5
I
I
I -2.0
I -1.0
: 20
10
0
-
-
-5.0
l c-4 0 c-3 A C-l A c-2
Log(G)
0.0
[molil]
Fig. 13 (a). Membrane potential for negatively charged membranes, where solid and open circles, and triangles represent the experimental results for samples C-4, C-3, C-l and C-2 shown in Table 1,respectively, and solid are the theoretical results.
-5 F E
-
-15 -
W a
l
A-l
-25 -
-45-5.0
I -4.0
I -3.0 Log(C0)
I -2.0
I -1.0
0.0
[molll]
Fig. 13 (b). Membrane potential for a positively charged membrane where the solid circles are the experimental results for sample A-l shown in Table 1, and the solid line is the theoretical prediction.
TABLE 1
Sample
Membrane thickness (Pm)
D.H.
Charge density (X 1O-2mol/l)
C-l
150 250 140 150 150 100
0.63 0.67 0.33 0.52 0.39 0.60
- 0.4 - 7.0 -20 -36 -30 11
c-2 c-3 c-4 c-5 A-l
2
0
Charge Density
[molill
Fig. 14. Permeability coefficient ratio (P.C.R.) of K+ and Ca”’ cations in the electrolyte system shown in Fig. 3 (c). Solid and open circles are experimental results, and solid lines are theoretical predictions.
3.0
2.0
0 ca*+ IO”
I.0 d d a 0.0
o
-
-1.O- ,/
-2.0 -. 5
\ L:w
0 0 I -.25
I 0.0 Charge Density
I 0.25
0.5
[molil]
Fig. 15. Permeability coefficient ratio (P.C.R.) of K+ and Ca2+ cations in the electrolyte system shown in Fig. 3 (d). Solid and open circles are experimental results, and solid lines are theoretical predictions.
5. Conclusions
The permeation of ions across charged membranes in the various kinds of electrolyte systems has been investigated as a function of the charge density. The calculated predictions for the two electrolyte systems shown in Figs. 3 (b) and (c) were examined by permeability coefficient measurements across various kinds of charged membranes. The theoretical predictions agree well with experimental results, indicating that the theoretical calculation accurately predicts the ion permeation across a charged membrane in multicomponent
167
ion systems. The theory cited above can predict how many ions are transported across any kind of charged membrane in multicomponent systems as long as the concentration of external solutions, the ion mobility in the membrane and its charge density are given. The charge density greatly affects the transport phenomena of an ion, especially a multivalent ion. Multivalent ions are transported against their external concentration gradient under appropriate conditions. For instance, these phenomena are applicable for the recovery of heavy metals from industrial waste water and sea water. The direction of multivalent ion transport changes with charge density. This phenomenon is applicable in a new mechanism to control the ion transport direction. An ion gel changes its volume by the change of pH, temperature, ion concentration, electric field and hydrostatic pressure [ 16-191. This means that a charged gel membrane can change its charge density according to these conditions. Furthermore, an amino acid and a weak acid membrane can change their charge density; the former can also change the sign of its charge by changing the pH of the solution. Therefore the ion transport across these membranes in multicomponent systems can be controlled by varying these conditions. Moreover, this theory can predict ion transport phenomena in systems containing polycations and polyanions on one side. In these systems, uphill transport of multivalent ions occurs more easily than in the system without these polyions. These predictions are applicable in medical and biochemical fields, for example, in artificial kidney membrane design and for purification of biochemical products, because body fluids and biochemical products consist of many kinds of polycations, polyanions and metal ions.
References H. Miyama, K. Tanaka, Y. Nosaka, N. Fijii, H. Tanzawa and S. Nagaoka, Charged ultrafiltration membrane for permeation of proteins, J. Appl. Polym. Sci., 36 (1988) 925. R. Schlogl, Electrolyte separation of ions by ion-exchange membranes, Ber. Bunsenges. Phys. Chem., 82 (1978) 225. M. Planck, iiber die Erregungvon Elektrizitit und W&me in Electrolyten, Ann. Phys. Chem., 39 (1890) 161. M. Planck, Uber die Potentialdifferenz zwischen zwei verdtinnten Lijsungen biniirer Electrolyten, Ann. Phys. Chem., 40 (1890) 561. T. Teorell, Studies on the “diffusion effect” upon ionic distribution. I. Some theoretical considerations, Proc. Natl. Acad. Sci. U.S.A., 21 (1935) 152. J. Straub, Membrangleichgewichte und Harmonien, Kolloid. Z., 64 (1933) 72. T. Teorell, Transport processes in ionic membranes, Prog. Biophys. Biophys. Chem., 3 (1953) 305. M. Nakagaki and M. Kobayashi, Diffusion and reverse diffusion of ions in mixed solutions of strong electrolytes, Nippon Kagaku Kaishi, 4 (1973) 635. P. Schwahn and D. Woermann, Transport of ions against their concentration gradient across cation exchange membranes, Ber. Bunsenges. Phys. Chem., 90 (1986) 773.
168 10 11 12 13 14 15 16 17 18 19
M. Higa, A. Tanioka and K. Miyasaka, Simulation of the transport of ions against their concentration gradient across charged membranes, J. Membrane Sci., 37 (1988) 251. J.S. Mackie and P. Meares, The diffusion of electrolytes in a cation-exchange resin membrane, Proc. R. Sot. London, Ser. A, 232 (1955) 498A. H.-U. Demisch and W. Pusch, Ion exchange capacity of cellulose acetate membrane, J. Electrochem. Sot., 123 (1976) 370. G.E. Goldman, Potential, impedance and rectification in membrane, J. Gen. Physiol., 26 (1943) 37. T. Teorell, An attempt to formulate a quantitative theory of membrane permeability, Proc. Sot. Exp. Biol. Med., 33 (1935) 282. K.H. Meyer and J.-F. Sievers, La permeabilite des membranes. I. Theorie de la permeabilitk ionique, Helv. Chim. Acta, 19 (1936) 649. T. Tanaka, D. Fillmore, S.-T. Sun, I. Nishio, G. Swislow and A. Shah, Phase transitions in ionic gels, Amer. Phys. Sot., 45 (1980) 1636. T. Tanaka, I. Nishio, S.-T. Sun and S. Ueno, Collapse of gels in an electric field, Science, 218 (1982) 467. I. Ohmine and T. Tanaka, Salt effects on the phase transition of ionic gels, J. Chem. Phys., 77 (1982) 5725. J. Ricka and T. Tanaka, Swelling of ionic gels: quantitative performance of the Donnan theory, Macromolecules, 17 (1984) 2916.
Appendix
If the valence of ions in electrolyte systems is smaller than three, eqn. (9) can be rewritten as the following:
(77p+n4+1p2
+&X,+
p2--1 c;+II
+ CO;+
+ cop-
-_lyp2
/P-l Q-1
_cf+lp”
/P-l
CF-1
+&x-
_~;-“I~”
p3-1
(A-1)
1=o
Since j? is unequal to one, both parts of this equation can be divided by - FSRTb$/d and multiplied by
[O~+(e,!+ll-~+‘p,+c(,Z-(~i’-‘-~Z-ll~)] [w:+,C;+ii
_c;+rp7
+,;-
+g(P+1)[~~+(~P”‘-eq+I/j3)+~~-(~~--I_~-~I~3)] =(P+NP”+P+l)(A,-&P)+4(P2+P+U(A,--&P2) +9(P+1)(A3-B3P3)=0 where
@;-I
__cp-“p”)] (-4-z)
169
A, =
‘@‘+pk+I1 +
B, =pB,+
cm;- pk-’
C;+I+Cci&-
c;-”
z=l,
2,3
Equation ( 10) is obtained be rewriting eqn. (A-2 ) .