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A study of the transport of ions against their concentration gradient across ion-exchange membranes using the network method
j o u r n a l of MEMBRANE SCIENCE ELSEVIER
Journal of Membrane Science 130 (1997) 183-192
A study of the transport of ions against their concentration gradient across ion-exchange membranes using the network method J. Castilla, M.T. Garc~a-Hermindez, A.A. Moya, A. Hayas, J. Homo* Departamento de Fisica Aplicada, Universidad de Ja~n, Facultad de Ciencias Experimentales, Paraje Las Lagunillas s/n, 23071 Jadn, Spain Received 11 November 1996; received in revised form 2 January 1997; accepted 3 January 1997
Abstract The network method has been used to analyze the conditions that favour the uphill transport across ion-exchange membranes. A model for the Nernst-Planck-Poisson equations describing the ionic transport in such system is proposed, including the Donnan equilibrium relations at the membrane/solution interfaces. With this model and the electric circuit simulation program PSPICE, the transient response of the system under open circuit conditions (1=0) and the response of the system subject to an applied potential difference are simulated. The ionic concentrations and electric potential profiles, as well as the electric current density, the ionic fluxes and the charge density, have been obtained as a function of time.
Keywords: Network method; Uphill transport in membranes; PSPICE modeling
1. Introduction The uphill transport is a special type of transport that occurs when a species moves against its concentration gradient. This type of phenomenon has been studied by many authors [1-5] and includes a wide variety of experimental and theoretical studies. Although the uphill transport can appear in various systems, we will limit its study to charged membranes. When studying the ionic transport processes in charged membranes, the theoretical treatment based both on the Nernst-Planck flux equations and on the electrostatic Poisson's equation together with the Donnan equilibrium relations is one of the most widely used [6-8]. However, the nature of the electrodiffusion equations is such that their analytical *Corresponding author. 0376-7388/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI1 S 0 3 7 6 - 7 3 88 ( 9 7 ) 0 0 0 2 2 - 7
solution is nearly impossible in a great number of interesting physical situations, and it is preferable to use numerical methods [9-12]. In particular, the uphill transport in charged membranes has been numerically treated by Higa et al. [ 13,14] under the assumption of a constant field and current density equal to zero, despite the fact that these approximations are known to be valid only under certain conditions [15,16]. The main purpose of this paper is to propose a method to investigate the uphill transport in charged membranes, taking into account both the Poisson equation and an externally applied electric current or electric potential. This approach (hereafter referred to as the 'network method') has been described elsewhere [17] and consists in modeling a physicochemical process using a graphical representation analogous to circuit diagrams in the framework of the electrical network theory. The equivalent network
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J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
is thus analyzed using the electrical circuit simulation program PSPICE [18]. The network method can be used satisfactorily in the study of problems involving transport of charged particles in membranes [19,20], since it avoids serious difficulties of the mathematical analysis. Moreover, the network method permits to impose any condition on the electric current or the electric potential externally applied. This paper analyzes the transport of ions against their concentration gradient across ion-exchange membranes. To accomplish this, a network model is first proposed for the ionic transport in a charged membrane, including Donnan equilibrium effects. Using this model and the computation program PSPICE, we then study both the influence of charge in the membrane and the response of the system to an externally applied electric potential in a ternary electrolyte system. In particular, the ionic concentrations and the electric potential profiles, as well as the electric current density, the ionic fluxes and the charge density, have been obtained as a function of time. Some of our results confirm the ideas previously advanced by Higa et al. [14], despite the use of different approaches. Other results provide new physical insights into the problem of the influence of an input electrical perturbation on uphill transport.
Bath 1
Bath 2
Membrane
CAB=C*
CAB.~C*
Cc~=r.c*
x=-L,
x=O
x=L
x=L +L2
Fig. 1. A schematic diagram of the system considered.
transported ion and r stands for the initial concentration ratio of driving electrolytes at the bulk phases. On the assumption that volume flux is negligible, the basic dimensionless equations involved in this non-stationary transport problem are the NemstPlanck flux equations, Ji(~, T) : - O i <
O~
@ ZiCi(~, 7)
O~
i=A,B,C
]
(1)
the continuity equations, cgJi(~, T) __
O{
OCi(~, T)
Or
(2)
and the Poisson equation 2. Electrodiffusion processes in charged membranes
OE(~, r) _ ZxCx + ~ zici(x, t) _ p((, r) c9~ e e
(3)
2.1. Mathematical description Let us consider a planar charged membrane and the x-axis normal to the direction of the membrane surface, so that the plane x=0 coincides with the left boundary between the membrane and the solution. Fig. 1 shows a sketch of the membrane system under study. The ion-exchange membrane of thickness L is flanked by two bulk solutions that extend from x=--L1 to 0 and from x=L to x=L+L2. Both baths, which are assumed to be well-stirred (so that the boundary layer effect on membrane su.rface is negligible), include the same concentration, c , of a binary electrolyte ABH20, but in the right bath there is, also a binary electrolyte, CB-H20 of concentration r.c*. B and C are the ions driving uphill transport, A is the uphill
where Ji, Di, ci and zi stand for the ionic flux (in units of Doco/lo), the diffusion coefficient (in units of Do), the molar concentration (in units of Co) and the charge number of ion i, respectively. The electric potential is represented by ¢ (in units of RT/F), the dielectric permitivity by e (in units of F2col2/RT) and the electric field by E=-O¢/O~ (in units of RT/Flo). The letter r denotes the time (in units of 120~Do), denotes the position along the electrodiffusion region (in units of lo), R is the gas constant, T is the thermodynamic temperature, F is the Faraday constant and Z~x (in units of Co) the charge density of the membrane (zx = + 1 or - 1 according to the sign of the charge). Do, Co and lo are scaling factors with the dimensions of the diffusion coefficients, concentration and length,
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J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
respectively. In particular, we have taken lo=L=10 -2 cm, Co=C*=10 -6 mol/cm 3 and Do_10 -7 cm 2 s -1. On the other hand, it is convenient to introduce the overall electric current density, I (in FDoco/lo), as the sum of Faradaic components, If, and the displacement current density, Id
I(T) = Z
ziJi((, T) + e OE(~, T) = If + Ia cOT
GJ~ak(+)
I
GJo~(-)
GJo~(+)
I
J~ ~
rpk
E
c-.
: "v~, •
~
rpk
,~,:
(5)
where ci(0 +, z), ci(0-, r) are, respectively, the ionic concentrations in the phases membrane and solution in the surface (=0, and ci(1-,z), Ci(I+,T) for the surface (----1. KL and KR are the Donnan equilibrium constants, for ~--0 and ( = 1, respectively. Moreover, Donnan potentials (A~L and AxPR), which represent the electric potential difference when the ionic equilibrium at the interfaces (~=0 and ~--1) has been reached, can be connected with the Donnan equilibrium constants by the equations: Ak~L = ~b(0+) - ~b(0-) = --InKL A~R = ~b(1-) - ~b(1+) = --InKR
GJ~Bk(-) JB ~
ci(O +, ~-) ci(O-, 7-) - K~' /(~i
GJo~,(+)
(4)
In this work we have also assumed that the membrane surfaces are always at a state of Donnan equilibrium, in other words, the membrane is assumed to be thick so that the main resistance to the ion fluxes lies in the membrane proper and equilibria on the surfaces have enough time tO be practically established. The Donnan equilibrium for the ith ion can be represented as [6--8]:
Ci(1-,T) Ci(1 + , T)
k
GJ~Ak(-)
(6)
We have also assumed for every interface that all the ions have the same partition coefficients.
2.2. The network model According to network modeling procedures [17,19,20], the network model representative of any transport process can be obtained by dividing the physical region of interest, which we consider to have unit area, into N volume elements or compartments of width 6k (k= 1.... , N), sufficiently small for the spatial
Fig. 2. Network model for the Nernst-Planck and Poisson equations in a single compartment.
variations of the parameters within each compartment to be negligible. The network model for electrodiffusion processes in a volume element is shown in Fig. 2, and a complete explanation of it has been given elsewhere [20]. In this figure, the network elements are as follows: Rdik is the one-port resistor representing the dissipative effect of the diffusion of ion i in compartment k, GJeik(+) is the multivariable voltage-controlled current source modeling the electrical contribution to the ionic flux, Jeik(--) and Jeik(+) are the fluxes entering and leaving compartment k, respectively, Cd~ is the one-port capacitor representing the non-stationary effects of the electrodiffusion process in compartment k, and rpk and GJpk are, respectively, the one-port resistor and the voltage-controlled current source modeling the Poisson equation in compartment k. GJpk now, includes a density of static charge ZxCx. The relations between
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
186
these network elements and the system parameter are given by
being related by ci(O +, 7") = K~i ci(O- , T)
(16)
Rdik = - -
ci(1-, r) = K~ ci(1 +, r)
(17)
(7)
2Di
aJeik(-1- ) = -}-Dizici ( ~k "-}-
(8) (9)
c~k = ~k
6k
rpk = ~-
(10)
£ Any number N of circuit elements like those in Fig. 2 are connected in series to form a network model for the entire physical region undergoing a diffusionmigration process through a charged membrane, which is described by the Nernst-Planck and Poisson equations (Eqs. (1)-(3)). Likewise, to obtain the global network model it is necessary to model the external baths. If these external regions were smaller or if enough time were to elapse, the ionic concentration in each bath would then be like an electrical capacitor of capacitance equal to its volume and in our case, on considering baths of unit area, the capacitances equal the depths of the external baths (11 = L1/lo and 12 = L2/lo) (see Fig. 3). The next step is to include the initial and boundary conditions of system into the model. The initial conditions specified by
ci(~,O)=cio
0<~<
1
(12)
Ci(~, O) = CiL
-- ll < ~ < 0
(13)
Ci(~, O) : CiR
1 < ~ < 1 + 12
(14)
~(~, 0) = (riO
-- l 1 < ~ '~ 1 + 12
(15)
are included by means of the initial voltages at the appropriate nodes of the network (IC in Fig. 2). In order to avoid boundary layer effects, we have assumed that the solutions near the membrane surfaces are well-stirred. Moreover, our system has two membrane-solution interfaces and, consequently, four concentrations are needed, i.e., the external surface concentrations ci(O-, r), ci(1 +, r) and the internal surface concentrations ci(O +, 7-), ci(l-, 7-), all four
as it is given in Eq. (5). These relations between the concentrations at the interfaces (Eqs. (16) and (17)) must be synthesized, for network modeling purpose, by suitable linear voltage-controlled voltage sources of respective outputs: (1 K~i)ci(O -, 7-) and (1 - K ~ ) c i ( 1 +, 7-). Thus, each interface is linked to a controlled source which is the electrical analog of the physical interface. The boundary conditions for the electric potential are -
-
(18)
¢(--/1,7-) = 0
and in the case (a) of an externally applied electric current density, I (Eq. (4)), into account that I is not function of ~, it can be computed at any point in the system. Thus, at ~ = 1 + 12,
OE
e~(1
+ 12,7-) = I(r) -- EZiJi(1 + 12,7-)
= Id(1 + l> 7-)
(19)
Eq. (18) defines the left bath (( --- - I t ) as the origin of the electric potential and is introduced in the model by short-circuiting the node ¢ at ~ = - l l . Implementation of Eq. (19) into the model requires an additional subcircuit in the network: the electrical analogue of Eq. (19) is a capacitor with capacitance of value e. Thus, E(1 +/2, 7-) is the voltage through this capacitor in subcircuit-a of Fig. 3, where the displacement current, Id, has been modeled by a independent-current source of value 1(7-) in parallel with a multivariable current-controlled current source Fly = ~ ziJi(1 + 12, 7-). Now, E( 1 + 12, 7-)can be used as a boundary condition for the Poisson equation by means of the voltage-controlled current source GE----E(1 +12,7-). On the other hand, to consider the case (b) in which the current/is not know a priori, it is necessary to take into account the new boundary condition Eq. (20), instead of Eq. (19): ¢(1 + 12, r) = ¢(7-)
(20)
where ¢(7-) is an externally applied electric potential perturbation. For this purpose, Eq. (20) is modeled by
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J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
Bath 1
Membrane
Interface
Interface
Bath 2
i
l',x~CA~ 12
|1 T CALi(1-KL)CA(0,x) CB(0,~) :
11"~ CBL: (1-KLB)CB(0 "r) T
::
Cc(O,x)i I. >i
.... 1
I(1-K~)Ca(l+,x)iCBR=-~-- 12 ~
--o
11~ CCLi (1-K~c)Cc(Ox)
'
iCc(l÷,~) i ....
,.. X)io'~CR +-L [(1-KR~)Cc(1 T 12
E
.....
iAqJL=-lnKL ~=0-
~=0+
~(~)~ ~=1
E(I+12,x) I('[)+
~d
GE(I+12,x)
~=1÷
I('c) +FIf
®
HE~
TE
+Elf
®
Fig. 3. Network model for the electrodiffusion in charged membranes, assuming a Donnan equilibrium in the interfaces. Details of the structure of boxes 1-N are shown in Fig. 2. Subcircuit-a computes E(1 + 12, r) from the externally controlled electric current. Subcircuit-b computes the total electric current density.
an independent voltage source with a time-dependent value for transient analysis given by ~(r), as shown for -- 1 + 12 in Fig. 3. Given that the total electric current density is not know in this case, it will be necessary to obtain its value from any branch in the network. Eq. (4) at = 1 + 12 allows us to compute the current I through the membrane. Thus, the current I can be modeled by two current elements connected in parallel, a multivariable current-controlled current source modeling the function If(1 + 12, r) -= ~-~ziJi(1 + 12,'r), and the current across a capacitor with capacitance equal to as in subcircuit-b of Fig. 3. It should be noted that modeling the displacement current density, Ia, by the current across a capacitor requires considering E(1 +12, r) as a voltage-type variable which is obtained by a current-controlled voltage source HE = E(1 + 12, r) [20].
Finally, the membrane potential is computed from the diffusion potential in the membrane A ~ d , a s well as the Donnan potentials at the membrane solution interface A~b,~ = A ~ L + A~bd -- A~R
(21)
Donnan potentials in the interfaces (Eq. (6)) are synthesised by two voltage sources of values -lnKc and --lnKR at ~=0 and ~--1, respectively. All these initial and boundary conditions, as well as the transport equations, have been taken into account in the network of Fig. 3, which represents the global network model for the abovementioned electrodiffusion problem in charged membranes, both in the case of electric current (switch-a and subcircuit-a) and the drop of electric potential (switch-b and subcircuit-b) are used as the independent variables.
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
188
3. Results and discussion
from Eqs. (24) and (25):
The network model can be easily simulated with a circuit simulation program. We have found the PSPICE package to be very useful for this purpose. PSPICE is a sophisticated circuit simulation program for non-linear dc, non-linear transient and linear ac analysis, which can be run in personal computers. The format for entering a description of the network into the program is quite simple, and a complete explanation is given in the user's guide [18]. All the simulations presented here have been accomplished with a PC-486/66 by means of the network model of Fig. 3 with the appropriate numerical values for the system parameters. The electrolyte system chosen here (Fig. 1) was A = K +, B = C1- and C = H +. In order to investigate the possible uphill transport in the membrane, a cation-exchange membrane of thickness L = 10 -2 cm is considered. The volume of baths is L a = / ~ = 1 0 cm and c*=10-6mol/cm 3. The parameters ruling the problem are the fixed charge density of the membrane (cx), the concentration of HC1 in bath 2 (r.c*) and the externally applied electric current (/) or electric potential (~ba). The diffusion coefficients considered in the simulation are O i = Diw/(5Do ), where Diw is the diffusion coefficient at infinite dilution (DKw= 1.95×10 -5 cm 2 s -1, DHw--9.18×10 -5 cm 2 s -1 and Dclw=2.03 x 10 -5 cm 2 s 1) and e-- lOeoRT/(F2colo). The initial conditions for concentrations and potential used are CH(~, 0) = 0, CK(~, 0) = Cx, CCl(~, 0) = 0 and ~b(~, 0) = 0. We considered 6k=1/30, which provides excellent accuracy without excessively lengthening the computation time. From the electroneutrality condition in the membrane, that is to say:
E
Zici(O+' 7") -~- ZxCx = 0
(22)
E
ZiCi(1-, 7") + ZxCx = 0
(23)
and taking Eqs. (16) and (17) into account, we have
Z
ziK~ici(O-, 7-)+ ZxCx = 0
(24)
E
zig~i C,( l +, 7") -]- ZxCx = 0
(25)
and therefore, in this situation of three ionic species of charge numbers ZH ----ZK -------ZCl ----1, the Donnan equilibrium constants can be obtained analytically
( ZxCx,2
KL --
ZxCx 2ccl (0-, q-) +
l+t2cci-~,T)]
KR
ZxCx 2CCl(1 +, T) f-
1 + t : 2 c c l ~ .r)
-
-
(z
cx)
(26)
(27)
3.1. Case (a): The current through the system is controlled Let us first study the ionic transport across the charged membrane under open circuit conditions (I-0), a system that Higa et al. have thoroughly studied [3,4,13,14]. The network model in this case will be the one shown in Fig. 3 with switch-a and I=0. The simulated results for the time evolution of the concentration of K + ion in the baths as a function of both the membrane charge density (cx) and the initial concentration ratio of the driving electrolytes (r)are shown in Fig. 4(a) and (b), respectively. It can be observed in these figures that the concentration of K + ion in bath 2 increases to reach a maximum as the result of uphill transport; moreover, this maximum value increases with increasing Cx (Fig. 4(a)). However, the maximum ion concentration does not always increase with increasing r (Fig. 4(b)), as previously pointed out by Higa et al. [14]. Fig. 5, parts (a) and (b), show the temporal evolution of the flux of K + ion at ~= 1 as a function of Cx and r, respectively. As shown in Fig. 5 during the first times, the flux of K + ion is positive (flux entering bath 2) and thus the K + ion is transferred from bath 1 to bath 2 across the membrane against a concentration difference. After this the sign reverts (flux leaving bath 2) and the system evolves to its steady-state [5]. It can also be observed in Fig. 5(a) that the relaxation phenomenon is faster for low charged membranes than for high charged membranes. In Fig. 6 the time evolution of electric potential profiles from the initial value ~b(~,0)=0 are plotted for r=100 and Cx=100. As can be observed in this figure, the electric potential gradient is always negative, so that the migrative flux of the cations will always be positive, that is to say, will always move towards bath 2. Furthermore, the potential drop is greater in the first
189
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
CK 60
3r
2:I
3
(a)
1'5 t
30
(a)
1! . ..............................
0,5
__
0 0
25
50 17
75
0
1O0
I
4
8
12
CK 2
1,5
(b)
.......... ::. . . . . . . . . . . . . . . . . . . . . . . . .
1
, • • •. ,,;'..,,,,,...,
...... iii: ......
. 3
0¸
0'5I2'':':'''''''. ..........1........................... 0
/
0
,
L
I
25
50
_ _ 1
75
100
Fig. 4. Time evolution of the concentration of K + ion in baths ]
(---) and 2 ( ) under open circuit conditions (I=0), in the following two cases: (a) r=100 and Cx=10-z, 10 1, 1 (1), 10 (2), 102 (3) and 103 (4); (b) c~=100 and r=10 (1), 10a (2) and 10z (3).
times, this being the cause of uphill transport of the ion. The time evolution of electric charge density in the baths for r = 1 0 0 and c~=l is shown in Fig. 7. Although does not appear in this figure, we have also obtained that the membrane is locally electroneutral for 0+<(<1 -. Important deviations from electroneutrality can be observed in the baths, which is the origin of the negative value of the electric potential (Fig. 6) and, consequently, of the uphill transport.
0
25
50
Fig. 5. Simulatedtime evolutionof the flux of K+ ion as a function both of charge membrane density (a) and initial concentrationratio of driving electrolytes at the bulk phases (b), for the conditions: (a) r=100 and cx=l (1), 10 (2) and 102 (3); (b) cx=l and r=0.1 (1), 1 (2) and 102 (3).
The above results are in complete agreement with those of Higa et al. [14], despite the use of different approaches. 3.2. Case (b): The electric potential is controlled
We will now study the transient response of the system to an externally applied electric potential. The network in this case is the one shown in Fig. 3 with switch-b.
190
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
CH.10-2 0
11
4
[
-1
-2
-1
k._ M.
0,75
-0.5
0
0,5
a)
J I
-3
,,
.
-0.5
0,25
I
-1
-4
-0,5
i
J
i
0
0,5
1
O0
_ _
i
i
i
25
50
75
100
1,5
CK Fig. 6. Electric potential profiles for the time T=0.1 (1), 1 (2), 10 (3) and 100 (4) in the case I=0, r = 1 0 0 and Cx=100.
1,6
-1
1,3
9 L m ~
~""..
0,2
b)
-05
%
0,7
%
"-.... 0,4
,
i
-1 ,
25
I
,
50
I
,
75
1O0
"C
Ccf 10-2
-0,2 ¸
11 0
25
50
0,75 Fig. 7. Time evolution of the electric charge density in baths 1 (- - -) and 2 ( ) in the case I=0, r = 1 0 0 and cx=l.
The decay of the ionic concentrations in the baths is shown in Fig. 8 for the externally applied electric potential q~a ranging from 0 to - 1 , r--100 and cx=l. The results obtained show that a negative potential favors the uphill transport, the K + ions extraction of the bath 1 increasing when the absolute value of applied potential increases. This trend can be explained from Fig. 9, which shows the time evolution of the diffusive contribution (Aq~d) and of the Donnan
.__!__.
0,5
0,25
c) -0.5
-1
,*0 I i
I
i
25
50
75
100
Fig. 8. Time evolution of the concentrations of ions H + (a), K + (b) and C1- (c) in baths 1 (- - -) and 2 ( ) for the externally applied electric potential q~,=0, - 0 . 5 and - 1 . The values r=-100 and c~,= 1 have been used.
191
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
Potential Difference
r
°°1
't
0a:l
AtPL
"',,.,
(Da__0.5
-
-0.5
"" - . . . .
0
25
50
75
100
-1 0
1
2
3
4
5
0.5
200
1;
Fig. 9. Temporal evolution of the diffusion potential AOd ( - - ) and Donnan potential contributions A~bLR ( - - - ) to the electric potential applied corresponding to the situation in Fig. 8. Donnan potentials do not change substantially with the application of the electric potential.
contributions (A~bL~) to the potential difference across the membrane. As can be seen, the extemal potential greatly influences the diffusion potential but not the Donnan potentials, which remain unaffected by the value of applied potential and, consequently, for ~ba=-0.5 and - 1 the diffusion potential favors the flux of cations towards the bath 2 and the flux of anions towards the bath 1 (see Fig. 8(a) and (c)). Finally, the time evolution of the electric charge density in the baths, p, and the electric current density through a membrane, L for different values of the applied potential are shown in Fig. 10(a) and (b), respectively. The results obtained for the charge density show that at short times there is an excess of cations in the bath 1, while in the bath 2 there is an excess of anions, thus resulting in a negative current density from the bath 2 to the bath I (Fig. 10(b)). This is due to, first, the higher mobility of H + and, secondly, to a diffusive contribution to flux of H + greater than its migrative contribution as a result of the initially smaller concentration of H + in bath 1 than in bath 2. When the time increases the situation
b)
~¢a_--O -200
-400
0
I
I
t
25
50
75
100
Fig, 10. (a) Time evolution of the electric charge density in baths 1 (- - -) and 2 ( ). (b) Total electric current density through the system. The conditions are the same as those in Fig. 8.
reverses and the migrative contributions to the fluxes prevail providing a positive current (towards the bath 2) which decreases to zero in longer times.
4. Conclusions The network method proposed can be easily used for the study of the uphill transport across ionexchange membranes. In this paper the ionic concentrations and electric potential profiles, as well as the ionic fluxes, the electric current density and the charge density, have been simulated as a function of time.
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J. Castilla et aL /Journal of Membrane Science 130 (1997) 183-192
The great influence that the parameters (namely, the charge density of the membrane, the concentration ratio of the driving electrolytes and an input electrical perturbation) have on the behaviour of a system consisting in a charged membrane bathed by two bulk solutions with three ionic species has thus been shown. It can be established that the uphill transport greatly increases as the charge density of the membrane and/ or the applied potential difference increase.
References [1] T. Teorell, Studies on the "diffusion effect" upon ionic distribution. I. Some theoretical considerations, Proc. Natl. Acad. Sci. Was., 21 (1935) 152. [2] K.H. Meyer and J.E Sierves, La permeabilit6 des membranes. I, Theorie de la permeabilit6 ionique, Helv. Chim. Acta, 19 (1936) 649. [3] M. Higa, A. Tanioka and K. Miyasaka, A study of ion permeation across a charged membrane in multicomponent ion systems as a function of membrane charge density, J. Membrane Sci., 49 (1990) 145. [4] M. Higa, A. Tanioka and K. Miyasaka, An experimental study of ion permeation in multicomponent ion systems as a function of membrane charge density, J. Membrane Sci., 64 (1991) 255. [5] J. Castilla, M.T. Garcia-Hemfindez, A. Hayas and J. Homo, Simulation of non-stationary electrodiffusion processes in charged membranes by the network approach, J. Membrane Sci., 116 (1996) 107. [6] E Helfferich, Ion Exchange, MacGraw-Hill, New York, 1962. [7] N. Lakshminarayanaiah, Transport Phenomena in Membranes, Academic Press, New York, 1969. [8] R. SchliJgl, Electrolytic separation of ions by ions by ionexchange membranes, Ber. Bunsenges. Phys. Chem., 82 (1978) 225. [9] R.P. Buck, Kinetics of bulk and interracial ionic motion: Microscopic bases and limits for the Nernst-Planck equation applied to membrane systems, J. Membrane Sci., 17 (1984) 1.
[10] W.D. Murphy, J.A. Manzanares, S. Mar6 and H. Reiss, A numerical study of equilibrium and non-equilibrium diffuse double layer in electrochemical cells, J. Phys. Chem., 96 (1992) 9983. [11] J.A. Manzanares, W.D. Murphy, S. Mafe and H. Reiss, Numerical simulation of the non-equilibrium diffuse double layer in ion-exchange membranes, J. Phys. Chem., 97 (1993) 8524. [12] M. Rudolph, Digital simulations with the fast implicit finitedifference (F1FD) algorithm. 4. Simulation of electrical migration and diffuse double layer effects, J. Electroanal. Chem., 375 (1994) 89. [13] 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. [14] M. Higa and A. Kira, Theory and simulation of ion transport in nonstationary states against concentratic gradients across ion-exchange membranes, J. Phys. Chem., 96 (1992) 9518. [15] P. Taskinen, K. Kontturi and A. Sipil~i, Validity of the Goldman constant field assumption in solution of the NernstPlanck equations for ionic flows in thin porous membranes, Finn. Chem. Lett., 4 (1980) 97. [16] S. Maf6, J. Pellicer and V.M. Aguilella, The Goldman constant field assumption: Significance and applicability conditions, Bet. Bunsenges Phys. Chem., 90 (1986) 476. [17] J. Homo, M.T. Garcfa-Hemfindez, C.E Gonzfindez and C.E Gonzfilez-Fem~indez, Digital simulation of electrochemical processes by network approach, J. Electroanal. Chem., 352 (1993) 83. [18] P.W. Tuinenga, SPICE: A Guide to Circuit Simulation and Analysis Using PSPICE, Prentice Hall, Englewood Cliffs, NJ, 1992. [19] J. Homo, J. Castilla and C.E Gonzhlez-Fem~ndez, A new approach to non-stationary ionic transport based on the network simulation of time dependent Nernst-Planck equations, J. Phys. Chem., 96 (1992) 854. [20] A.A. Moya, J. Castilla and J. Homo, Ionic transport in electrochemical ceils including electricai double-layer effects. A network thermodynamics approach, J. Phys. Chem., 99 (1995) 1292.
Journal of Membrane Science 130 (1997) 183-192
A study of the transport of ions against their concentration gradient across ion-exchange membranes using the network method J. Castilla, M.T. Garc~a-Hermindez, A.A. Moya, A. Hayas, J. Homo* Departamento de Fisica Aplicada, Universidad de Ja~n, Facultad de Ciencias Experimentales, Paraje Las Lagunillas s/n, 23071 Jadn, Spain Received 11 November 1996; received in revised form 2 January 1997; accepted 3 January 1997
Abstract The network method has been used to analyze the conditions that favour the uphill transport across ion-exchange membranes. A model for the Nernst-Planck-Poisson equations describing the ionic transport in such system is proposed, including the Donnan equilibrium relations at the membrane/solution interfaces. With this model and the electric circuit simulation program PSPICE, the transient response of the system under open circuit conditions (1=0) and the response of the system subject to an applied potential difference are simulated. The ionic concentrations and electric potential profiles, as well as the electric current density, the ionic fluxes and the charge density, have been obtained as a function of time.
Keywords: Network method; Uphill transport in membranes; PSPICE modeling
1. Introduction The uphill transport is a special type of transport that occurs when a species moves against its concentration gradient. This type of phenomenon has been studied by many authors [1-5] and includes a wide variety of experimental and theoretical studies. Although the uphill transport can appear in various systems, we will limit its study to charged membranes. When studying the ionic transport processes in charged membranes, the theoretical treatment based both on the Nernst-Planck flux equations and on the electrostatic Poisson's equation together with the Donnan equilibrium relations is one of the most widely used [6-8]. However, the nature of the electrodiffusion equations is such that their analytical *Corresponding author. 0376-7388/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI1 S 0 3 7 6 - 7 3 88 ( 9 7 ) 0 0 0 2 2 - 7
solution is nearly impossible in a great number of interesting physical situations, and it is preferable to use numerical methods [9-12]. In particular, the uphill transport in charged membranes has been numerically treated by Higa et al. [ 13,14] under the assumption of a constant field and current density equal to zero, despite the fact that these approximations are known to be valid only under certain conditions [15,16]. The main purpose of this paper is to propose a method to investigate the uphill transport in charged membranes, taking into account both the Poisson equation and an externally applied electric current or electric potential. This approach (hereafter referred to as the 'network method') has been described elsewhere [17] and consists in modeling a physicochemical process using a graphical representation analogous to circuit diagrams in the framework of the electrical network theory. The equivalent network
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J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
is thus analyzed using the electrical circuit simulation program PSPICE [18]. The network method can be used satisfactorily in the study of problems involving transport of charged particles in membranes [19,20], since it avoids serious difficulties of the mathematical analysis. Moreover, the network method permits to impose any condition on the electric current or the electric potential externally applied. This paper analyzes the transport of ions against their concentration gradient across ion-exchange membranes. To accomplish this, a network model is first proposed for the ionic transport in a charged membrane, including Donnan equilibrium effects. Using this model and the computation program PSPICE, we then study both the influence of charge in the membrane and the response of the system to an externally applied electric potential in a ternary electrolyte system. In particular, the ionic concentrations and the electric potential profiles, as well as the electric current density, the ionic fluxes and the charge density, have been obtained as a function of time. Some of our results confirm the ideas previously advanced by Higa et al. [14], despite the use of different approaches. Other results provide new physical insights into the problem of the influence of an input electrical perturbation on uphill transport.
Bath 1
Bath 2
Membrane
CAB=C*
CAB.~C*
Cc~=r.c*
x=-L,
x=O
x=L
x=L +L2
Fig. 1. A schematic diagram of the system considered.
transported ion and r stands for the initial concentration ratio of driving electrolytes at the bulk phases. On the assumption that volume flux is negligible, the basic dimensionless equations involved in this non-stationary transport problem are the NemstPlanck flux equations, Ji(~, T) : - O i <
O~
@ ZiCi(~, 7)
O~
i=A,B,C
]
(1)
the continuity equations, cgJi(~, T) __
O{
OCi(~, T)
Or
(2)
and the Poisson equation 2. Electrodiffusion processes in charged membranes
OE(~, r) _ ZxCx + ~ zici(x, t) _ p((, r) c9~ e e
(3)
2.1. Mathematical description Let us consider a planar charged membrane and the x-axis normal to the direction of the membrane surface, so that the plane x=0 coincides with the left boundary between the membrane and the solution. Fig. 1 shows a sketch of the membrane system under study. The ion-exchange membrane of thickness L is flanked by two bulk solutions that extend from x=--L1 to 0 and from x=L to x=L+L2. Both baths, which are assumed to be well-stirred (so that the boundary layer effect on membrane su.rface is negligible), include the same concentration, c , of a binary electrolyte ABH20, but in the right bath there is, also a binary electrolyte, CB-H20 of concentration r.c*. B and C are the ions driving uphill transport, A is the uphill
where Ji, Di, ci and zi stand for the ionic flux (in units of Doco/lo), the diffusion coefficient (in units of Do), the molar concentration (in units of Co) and the charge number of ion i, respectively. The electric potential is represented by ¢ (in units of RT/F), the dielectric permitivity by e (in units of F2col2/RT) and the electric field by E=-O¢/O~ (in units of RT/Flo). The letter r denotes the time (in units of 120~Do), denotes the position along the electrodiffusion region (in units of lo), R is the gas constant, T is the thermodynamic temperature, F is the Faraday constant and Z~x (in units of Co) the charge density of the membrane (zx = + 1 or - 1 according to the sign of the charge). Do, Co and lo are scaling factors with the dimensions of the diffusion coefficients, concentration and length,
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J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
respectively. In particular, we have taken lo=L=10 -2 cm, Co=C*=10 -6 mol/cm 3 and Do_10 -7 cm 2 s -1. On the other hand, it is convenient to introduce the overall electric current density, I (in FDoco/lo), as the sum of Faradaic components, If, and the displacement current density, Id
I(T) = Z
ziJi((, T) + e OE(~, T) = If + Ia cOT
GJ~ak(+)
I
GJo~(-)
GJo~(+)
I
J~ ~
rpk
E
c-.
: "v~, •
~
rpk
,~,:
(5)
where ci(0 +, z), ci(0-, r) are, respectively, the ionic concentrations in the phases membrane and solution in the surface (=0, and ci(1-,z), Ci(I+,T) for the surface (----1. KL and KR are the Donnan equilibrium constants, for ~--0 and ( = 1, respectively. Moreover, Donnan potentials (A~L and AxPR), which represent the electric potential difference when the ionic equilibrium at the interfaces (~=0 and ~--1) has been reached, can be connected with the Donnan equilibrium constants by the equations: Ak~L = ~b(0+) - ~b(0-) = --InKL A~R = ~b(1-) - ~b(1+) = --InKR
GJ~Bk(-) JB ~
ci(O +, ~-) ci(O-, 7-) - K~' /(~i
GJo~,(+)
(4)
In this work we have also assumed that the membrane surfaces are always at a state of Donnan equilibrium, in other words, the membrane is assumed to be thick so that the main resistance to the ion fluxes lies in the membrane proper and equilibria on the surfaces have enough time tO be practically established. The Donnan equilibrium for the ith ion can be represented as [6--8]:
Ci(1-,T) Ci(1 + , T)
k
GJ~Ak(-)
(6)
We have also assumed for every interface that all the ions have the same partition coefficients.
2.2. The network model According to network modeling procedures [17,19,20], the network model representative of any transport process can be obtained by dividing the physical region of interest, which we consider to have unit area, into N volume elements or compartments of width 6k (k= 1.... , N), sufficiently small for the spatial
Fig. 2. Network model for the Nernst-Planck and Poisson equations in a single compartment.
variations of the parameters within each compartment to be negligible. The network model for electrodiffusion processes in a volume element is shown in Fig. 2, and a complete explanation of it has been given elsewhere [20]. In this figure, the network elements are as follows: Rdik is the one-port resistor representing the dissipative effect of the diffusion of ion i in compartment k, GJeik(+) is the multivariable voltage-controlled current source modeling the electrical contribution to the ionic flux, Jeik(--) and Jeik(+) are the fluxes entering and leaving compartment k, respectively, Cd~ is the one-port capacitor representing the non-stationary effects of the electrodiffusion process in compartment k, and rpk and GJpk are, respectively, the one-port resistor and the voltage-controlled current source modeling the Poisson equation in compartment k. GJpk now, includes a density of static charge ZxCx. The relations between
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
186
these network elements and the system parameter are given by
being related by ci(O +, 7") = K~i ci(O- , T)
(16)
Rdik = - -
ci(1-, r) = K~ ci(1 +, r)
(17)
(7)
2Di
aJeik(-1- ) = -}-Dizici ( ~k "-}-
(8) (9)
c~k = ~k
6k
rpk = ~-
(10)
£ Any number N of circuit elements like those in Fig. 2 are connected in series to form a network model for the entire physical region undergoing a diffusionmigration process through a charged membrane, which is described by the Nernst-Planck and Poisson equations (Eqs. (1)-(3)). Likewise, to obtain the global network model it is necessary to model the external baths. If these external regions were smaller or if enough time were to elapse, the ionic concentration in each bath would then be like an electrical capacitor of capacitance equal to its volume and in our case, on considering baths of unit area, the capacitances equal the depths of the external baths (11 = L1/lo and 12 = L2/lo) (see Fig. 3). The next step is to include the initial and boundary conditions of system into the model. The initial conditions specified by
ci(~,O)=cio
0<~<
1
(12)
Ci(~, O) = CiL
-- ll < ~ < 0
(13)
Ci(~, O) : CiR
1 < ~ < 1 + 12
(14)
~(~, 0) = (riO
-- l 1 < ~ '~ 1 + 12
(15)
are included by means of the initial voltages at the appropriate nodes of the network (IC in Fig. 2). In order to avoid boundary layer effects, we have assumed that the solutions near the membrane surfaces are well-stirred. Moreover, our system has two membrane-solution interfaces and, consequently, four concentrations are needed, i.e., the external surface concentrations ci(O-, r), ci(1 +, r) and the internal surface concentrations ci(O +, 7-), ci(l-, 7-), all four
as it is given in Eq. (5). These relations between the concentrations at the interfaces (Eqs. (16) and (17)) must be synthesized, for network modeling purpose, by suitable linear voltage-controlled voltage sources of respective outputs: (1 K~i)ci(O -, 7-) and (1 - K ~ ) c i ( 1 +, 7-). Thus, each interface is linked to a controlled source which is the electrical analog of the physical interface. The boundary conditions for the electric potential are -
-
(18)
¢(--/1,7-) = 0
and in the case (a) of an externally applied electric current density, I (Eq. (4)), into account that I is not function of ~, it can be computed at any point in the system. Thus, at ~ = 1 + 12,
OE
e~(1
+ 12,7-) = I(r) -- EZiJi(1 + 12,7-)
= Id(1 + l> 7-)
(19)
Eq. (18) defines the left bath (( --- - I t ) as the origin of the electric potential and is introduced in the model by short-circuiting the node ¢ at ~ = - l l . Implementation of Eq. (19) into the model requires an additional subcircuit in the network: the electrical analogue of Eq. (19) is a capacitor with capacitance of value e. Thus, E(1 +/2, 7-) is the voltage through this capacitor in subcircuit-a of Fig. 3, where the displacement current, Id, has been modeled by a independent-current source of value 1(7-) in parallel with a multivariable current-controlled current source Fly = ~ ziJi(1 + 12, 7-). Now, E( 1 + 12, 7-)can be used as a boundary condition for the Poisson equation by means of the voltage-controlled current source GE----E(1 +12,7-). On the other hand, to consider the case (b) in which the current/is not know a priori, it is necessary to take into account the new boundary condition Eq. (20), instead of Eq. (19): ¢(1 + 12, r) = ¢(7-)
(20)
where ¢(7-) is an externally applied electric potential perturbation. For this purpose, Eq. (20) is modeled by
187
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
Bath 1
Membrane
Interface
Interface
Bath 2
i
l',x~CA~ 12
|1 T CALi(1-KL)CA(0,x) CB(0,~) :
11"~ CBL: (1-KLB)CB(0 "r) T
::
Cc(O,x)i I. >i
.... 1
I(1-K~)Ca(l+,x)iCBR=-~-- 12 ~
--o
11~ CCLi (1-K~c)Cc(Ox)
'
iCc(l÷,~) i ....
,.. X)io'~CR +-L [(1-KR~)Cc(1 T 12
E
.....
iAqJL=-lnKL ~=0-
~=0+
~(~)~ ~=1
E(I+12,x) I('[)+
~d
GE(I+12,x)
~=1÷
I('c) +FIf
®
HE~
TE
+Elf
®
Fig. 3. Network model for the electrodiffusion in charged membranes, assuming a Donnan equilibrium in the interfaces. Details of the structure of boxes 1-N are shown in Fig. 2. Subcircuit-a computes E(1 + 12, r) from the externally controlled electric current. Subcircuit-b computes the total electric current density.
an independent voltage source with a time-dependent value for transient analysis given by ~(r), as shown for -- 1 + 12 in Fig. 3. Given that the total electric current density is not know in this case, it will be necessary to obtain its value from any branch in the network. Eq. (4) at = 1 + 12 allows us to compute the current I through the membrane. Thus, the current I can be modeled by two current elements connected in parallel, a multivariable current-controlled current source modeling the function If(1 + 12, r) -= ~-~ziJi(1 + 12,'r), and the current across a capacitor with capacitance equal to as in subcircuit-b of Fig. 3. It should be noted that modeling the displacement current density, Ia, by the current across a capacitor requires considering E(1 +12, r) as a voltage-type variable which is obtained by a current-controlled voltage source HE = E(1 + 12, r) [20].
Finally, the membrane potential is computed from the diffusion potential in the membrane A ~ d , a s well as the Donnan potentials at the membrane solution interface A~b,~ = A ~ L + A~bd -- A~R
(21)
Donnan potentials in the interfaces (Eq. (6)) are synthesised by two voltage sources of values -lnKc and --lnKR at ~=0 and ~--1, respectively. All these initial and boundary conditions, as well as the transport equations, have been taken into account in the network of Fig. 3, which represents the global network model for the abovementioned electrodiffusion problem in charged membranes, both in the case of electric current (switch-a and subcircuit-a) and the drop of electric potential (switch-b and subcircuit-b) are used as the independent variables.
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
188
3. Results and discussion
from Eqs. (24) and (25):
The network model can be easily simulated with a circuit simulation program. We have found the PSPICE package to be very useful for this purpose. PSPICE is a sophisticated circuit simulation program for non-linear dc, non-linear transient and linear ac analysis, which can be run in personal computers. The format for entering a description of the network into the program is quite simple, and a complete explanation is given in the user's guide [18]. All the simulations presented here have been accomplished with a PC-486/66 by means of the network model of Fig. 3 with the appropriate numerical values for the system parameters. The electrolyte system chosen here (Fig. 1) was A = K +, B = C1- and C = H +. In order to investigate the possible uphill transport in the membrane, a cation-exchange membrane of thickness L = 10 -2 cm is considered. The volume of baths is L a = / ~ = 1 0 cm and c*=10-6mol/cm 3. The parameters ruling the problem are the fixed charge density of the membrane (cx), the concentration of HC1 in bath 2 (r.c*) and the externally applied electric current (/) or electric potential (~ba). The diffusion coefficients considered in the simulation are O i = Diw/(5Do ), where Diw is the diffusion coefficient at infinite dilution (DKw= 1.95×10 -5 cm 2 s -1, DHw--9.18×10 -5 cm 2 s -1 and Dclw=2.03 x 10 -5 cm 2 s 1) and e-- lOeoRT/(F2colo). The initial conditions for concentrations and potential used are CH(~, 0) = 0, CK(~, 0) = Cx, CCl(~, 0) = 0 and ~b(~, 0) = 0. We considered 6k=1/30, which provides excellent accuracy without excessively lengthening the computation time. From the electroneutrality condition in the membrane, that is to say:
E
Zici(O+' 7") -~- ZxCx = 0
(22)
E
ZiCi(1-, 7") + ZxCx = 0
(23)
and taking Eqs. (16) and (17) into account, we have
Z
ziK~ici(O-, 7-)+ ZxCx = 0
(24)
E
zig~i C,( l +, 7") -]- ZxCx = 0
(25)
and therefore, in this situation of three ionic species of charge numbers ZH ----ZK -------ZCl ----1, the Donnan equilibrium constants can be obtained analytically
( ZxCx,2
KL --
ZxCx 2ccl (0-, q-) +
l+t2cci-~,T)]
KR
ZxCx 2CCl(1 +, T) f-
1 + t : 2 c c l ~ .r)
-
-
(z
cx)
(26)
(27)
3.1. Case (a): The current through the system is controlled Let us first study the ionic transport across the charged membrane under open circuit conditions (I-0), a system that Higa et al. have thoroughly studied [3,4,13,14]. The network model in this case will be the one shown in Fig. 3 with switch-a and I=0. The simulated results for the time evolution of the concentration of K + ion in the baths as a function of both the membrane charge density (cx) and the initial concentration ratio of the driving electrolytes (r)are shown in Fig. 4(a) and (b), respectively. It can be observed in these figures that the concentration of K + ion in bath 2 increases to reach a maximum as the result of uphill transport; moreover, this maximum value increases with increasing Cx (Fig. 4(a)). However, the maximum ion concentration does not always increase with increasing r (Fig. 4(b)), as previously pointed out by Higa et al. [14]. Fig. 5, parts (a) and (b), show the temporal evolution of the flux of K + ion at ~= 1 as a function of Cx and r, respectively. As shown in Fig. 5 during the first times, the flux of K + ion is positive (flux entering bath 2) and thus the K + ion is transferred from bath 1 to bath 2 across the membrane against a concentration difference. After this the sign reverts (flux leaving bath 2) and the system evolves to its steady-state [5]. It can also be observed in Fig. 5(a) that the relaxation phenomenon is faster for low charged membranes than for high charged membranes. In Fig. 6 the time evolution of electric potential profiles from the initial value ~b(~,0)=0 are plotted for r=100 and Cx=100. As can be observed in this figure, the electric potential gradient is always negative, so that the migrative flux of the cations will always be positive, that is to say, will always move towards bath 2. Furthermore, the potential drop is greater in the first
189
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
CK 60
3r
2:I
3
(a)
1'5 t
30
(a)
1! . ..............................
0,5
__
0 0
25
50 17
75
0
1O0
I
4
8
12
CK 2
1,5
(b)
.......... ::. . . . . . . . . . . . . . . . . . . . . . . . .
1
, • • •. ,,;'..,,,,,...,
...... iii: ......
. 3
0¸
0'5I2'':':'''''''. ..........1........................... 0
/
0
,
L
I
25
50
_ _ 1
75
100
Fig. 4. Time evolution of the concentration of K + ion in baths ]
(---) and 2 ( ) under open circuit conditions (I=0), in the following two cases: (a) r=100 and Cx=10-z, 10 1, 1 (1), 10 (2), 102 (3) and 103 (4); (b) c~=100 and r=10 (1), 10a (2) and 10z (3).
times, this being the cause of uphill transport of the ion. The time evolution of electric charge density in the baths for r = 1 0 0 and c~=l is shown in Fig. 7. Although does not appear in this figure, we have also obtained that the membrane is locally electroneutral for 0+<(<1 -. Important deviations from electroneutrality can be observed in the baths, which is the origin of the negative value of the electric potential (Fig. 6) and, consequently, of the uphill transport.
0
25
50
Fig. 5. Simulatedtime evolutionof the flux of K+ ion as a function both of charge membrane density (a) and initial concentrationratio of driving electrolytes at the bulk phases (b), for the conditions: (a) r=100 and cx=l (1), 10 (2) and 102 (3); (b) cx=l and r=0.1 (1), 1 (2) and 102 (3).
The above results are in complete agreement with those of Higa et al. [14], despite the use of different approaches. 3.2. Case (b): The electric potential is controlled
We will now study the transient response of the system to an externally applied electric potential. The network in this case is the one shown in Fig. 3 with switch-b.
190
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
CH.10-2 0
11
4
[
-1
-2
-1
k._ M.
0,75
-0.5
0
0,5
a)
J I
-3
,,
.
-0.5
0,25
I
-1
-4
-0,5
i
J
i
0
0,5
1
O0
_ _
i
i
i
25
50
75
100
1,5
CK Fig. 6. Electric potential profiles for the time T=0.1 (1), 1 (2), 10 (3) and 100 (4) in the case I=0, r = 1 0 0 and Cx=100.
1,6
-1
1,3
9 L m ~
~""..
0,2
b)
-05
%
0,7
%
"-.... 0,4
,
i
-1 ,
25
I
,
50
I
,
75
1O0
"C
Ccf 10-2
-0,2 ¸
11 0
25
50
0,75 Fig. 7. Time evolution of the electric charge density in baths 1 (- - -) and 2 ( ) in the case I=0, r = 1 0 0 and cx=l.
The decay of the ionic concentrations in the baths is shown in Fig. 8 for the externally applied electric potential q~a ranging from 0 to - 1 , r--100 and cx=l. The results obtained show that a negative potential favors the uphill transport, the K + ions extraction of the bath 1 increasing when the absolute value of applied potential increases. This trend can be explained from Fig. 9, which shows the time evolution of the diffusive contribution (Aq~d) and of the Donnan
.__!__.
0,5
0,25
c) -0.5
-1
,*0 I i
I
i
25
50
75
100
Fig. 8. Time evolution of the concentrations of ions H + (a), K + (b) and C1- (c) in baths 1 (- - -) and 2 ( ) for the externally applied electric potential q~,=0, - 0 . 5 and - 1 . The values r=-100 and c~,= 1 have been used.
191
J. Castilla et al./Journal of Membrane Science 130 (1997) 183-192
Potential Difference
r
°°1
't
0a:l
AtPL
"',,.,
(Da__0.5
-
-0.5
"" - . . . .
0
25
50
75
100
-1 0
1
2
3
4
5
0.5
200
1;
Fig. 9. Temporal evolution of the diffusion potential AOd ( - - ) and Donnan potential contributions A~bLR ( - - - ) to the electric potential applied corresponding to the situation in Fig. 8. Donnan potentials do not change substantially with the application of the electric potential.
contributions (A~bL~) to the potential difference across the membrane. As can be seen, the extemal potential greatly influences the diffusion potential but not the Donnan potentials, which remain unaffected by the value of applied potential and, consequently, for ~ba=-0.5 and - 1 the diffusion potential favors the flux of cations towards the bath 2 and the flux of anions towards the bath 1 (see Fig. 8(a) and (c)). Finally, the time evolution of the electric charge density in the baths, p, and the electric current density through a membrane, L for different values of the applied potential are shown in Fig. 10(a) and (b), respectively. The results obtained for the charge density show that at short times there is an excess of cations in the bath 1, while in the bath 2 there is an excess of anions, thus resulting in a negative current density from the bath 2 to the bath I (Fig. 10(b)). This is due to, first, the higher mobility of H + and, secondly, to a diffusive contribution to flux of H + greater than its migrative contribution as a result of the initially smaller concentration of H + in bath 1 than in bath 2. When the time increases the situation
b)
~¢a_--O -200
-400
0
I
I
t
25
50
75
100
Fig, 10. (a) Time evolution of the electric charge density in baths 1 (- - -) and 2 ( ). (b) Total electric current density through the system. The conditions are the same as those in Fig. 8.
reverses and the migrative contributions to the fluxes prevail providing a positive current (towards the bath 2) which decreases to zero in longer times.
4. Conclusions The network method proposed can be easily used for the study of the uphill transport across ionexchange membranes. In this paper the ionic concentrations and electric potential profiles, as well as the ionic fluxes, the electric current density and the charge density, have been simulated as a function of time.
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J. Castilla et aL /Journal of Membrane Science 130 (1997) 183-192
The great influence that the parameters (namely, the charge density of the membrane, the concentration ratio of the driving electrolytes and an input electrical perturbation) have on the behaviour of a system consisting in a charged membrane bathed by two bulk solutions with three ionic species has thus been shown. It can be established that the uphill transport greatly increases as the charge density of the membrane and/ or the applied potential difference increase.
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