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Multiple steady states in ionic membranes with negative conductance
jourmmof MEMBRANE SCIENCE ELSEVIER
Journal of Membrane Science 99 (1995) 39-47
Multiple steady states in ionic membranes with negative conductance Michael G. Lee
1,*,
Jacob Jorn6
Department of Chemical Engineering, University of Rochester, Rochester, NY 14627, USA Received 9 May 1994; accepted in revised form 5 September 1994
Abstract Infinitesimal or negative polarization resistance can be induced by a negative ionic conductance in membranes. When the reactant or product have the "wrong" charge, they are respectively repelled or attracted, under the influence of the coulombic force, resulting in a negative conductance in the membrane, which in turn contributes to a negative component in the polarization resistance. The effects of diffusivity, concentration, supporting electrolyte, and fixed space charge on the conductance in the ionic membranes are also studied. An inverse concentration gradient may form as a result of the interaction between the ionic migration and the stoichiometric flux according to the reaction kinetics. Keywords: Concentration polarization; Diffusion; Electrochemistry; Membrane electrodes; Oscillations and excitable membranes; Multiple steady state
1. Introduction
The inverse relationship between current and voltage has been recognized in some chemical and biological systems, such as the anodic dissolution of iron and nickel [ 1,2], reduction of nitric acid, oxidation of tetrahydrofuran [3], and nerve excitation [4,5]. The existence of such an inverse relationship implies the system has multiple steady states in either galvanostatic or potentiostatic operation. An inverse current-voltage relationship may originate from a complex reaction network; however, negative conductance may also induce negative polarization resistance in current or potential as well. * Corresponding author. Present address: Fujitsu Computer Packaging Technologies, San Jose, CA 95134, USA. 0376-7388/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI0376-7388 (94)00205-3
Many biological transducers generate electrical pulses. As an external signal excites a molecular process, the chemical response is amplified, e.g., by catalysis, and finally the signal is converted into a sequence of pulses, by some mechanism similar to the one responsible for electrochemical oscillations. All electrically excitable membranes seem to exhibit N-shaped current-potential curves, and are therefore likely to have a region of negative resistance [ 6 ], and to exhibit multiplicities. Under negative conductance in membranes, current flows in a direction opposite to the electric field. Negative conductance is not usually detected in laboratory work, because the total overpotential and ohmic resistance in the bulk phase often dominate the global feature of the system, and make the negative conductance hard to be observed. The strong interactions between this negative conductance and the others can give rise
40
M. G. Lee, J. Jorn~ / Journal of Membrane Science 99 (1995) 39~t7
to some anomalous behaviors, and can explain some phenomena that cannot be possibly understood under positive conductance. Furthermore, only after the resistance polarization in the diffusion region and in the bulk phase are sorted out, activation overpotential can be obtained and utilized for the study of kinetic mechanism. The influence of negative conductance is thus of great importance and, hence, it is crucial to demonstrate when negative conductance can exist, what anomalous behaviors it may induce, and how it can be related to some experimental observations. The understanding of ion motion and ion distribution is important not only in electrodeposition but also in areas such as electrophoresis, nerve excitation, and chemically modified electrodes, in which the processes are primarily controlled by their electrochemical nature. Systems consisting of both ion transport and electrode reaction are usually analyzed by decomposing the total applied voltage into equilibrium potential, activation overpotential, concentration overpotential, ohmic drop in the bulk phase, and ohmic drop in the diffusion region. The distributions of potential, ionic strength, and concentration in the diffusion region can then be achieved by analytical or numerical methods for the cases with and without fixed space charge. Negative conductance and, thus, multiplicity are to be closely related to diffusivities and stoichiometric coefficients of the reacting species in the overall reaction. It will be shown that polarization resistance is also affected by the bulk resistance, bulk concentration, diffusion region thickness, supporting electrolyte, and space charge. Negative conductance has been demonstrated to be able to induce multiple steady states in both anodic and cathodic polarizations [7]. Since the conductivity in the bulk solution is always positive, the only source for negative conductivity is in the diffusion region. High currents are associated with high kinetic overpotentials, but not necessarily high applied voltages. The potential profile in the diffusion region is anomalous when the system has a negative effective conductance. In the case of cathodic deposition, if the effective conductance is negative, the potential can increase, rather than decrease, from the interface between the bulk phase and the diffusion region up to the electrode surface. The magnitude of the current density affects the slope of the potential profile across the region, and the kinetic overpotential determines the reaction rate and, conse-
quently, the current density. Therefore, both high current with high kinetic overpotential and low current with low kinetic overpotential are possible at the same applied voltage, and multiple steady states may be achieved in certain circumstances by coupling the reaction to the ionic transport process. This concept was first introduced by Lev and Pismen [7]; however, their work consists of only zeroth order reactions coupled with negative conductance, and the mass transfer overpotential was neglected for the region of high current densities. Intensive efforts have been made to study the nonlinear dependence of the current on the applied voltage in terms of the effective conductance in the diffusion region, which may be a diffusion boundary layer or a thin membrane. The theoretical work is being extended by including the mass transfer overpotential, kinetic overpotential, ohmic polarization over the diffusion region and the ohmic drop in the bulk phase for reactions with arbitrary reaction order. The results are outlined below, and the detailed procedure can be found in the Appendix.
2. Transport in diffusion region Unlike the case in the bulk solution, ion diffuse through membranes under the influence of a concentration gradient and an electric field. For one-dimensional geometry and when the convection is negligible, the ionic flux is reduced to the following form Nj= - 8 \ d s ~
~=x/8
RT --~)
(l)
is the dimensionless coordinate, where 8 is the thickness of the membrane, and the interface between the bulk solution and the diffusion region is located at ~= 0, while the electrode surface is at sc= 1. There are two driving forces, one is the concentration gradient for the diffusion contribution, the other is the potential gradient for the migration contribution under the electric field. This equation can be applied to polymeric membranes with high liquid content. Under steady state, the ion consumed or generated by the electrochemical reactions on the electrode surface must be commensurately supplied by or removed to the bulk solution. The flux of ions can be related to the current density by
M. G. Lee, J. Jorn~ / Journal of Membrane Science 99 (1995) 39--47
~,ji
N1=
(2)
nF
Eqs. ( 1 ) and (2) can be combined to give
vii Dj [dCj + z.iVCj d___._1~ RT
(3)
The potential gradient can be obtained when Eq. (3) is multiplied by zj/Dj and then summed up over all the reacting species, with the supplemental condition of electroneutrality,
Q + EzlCj = O
(4)
The space charge in the transport region, Q, is usually zero for aqueous solutions, but can be either positive, zero, or negative for membranes. Regardless of the nature of the space charge, the potential gradient is always
RT6i d-~=F2D.C .
de
(5)
where
l=l z j D*
n
Dj
negative, the potential profile behaves in an unusual way. For example, the potential may increase, not decrease, in the membrane for a cathodic process under negative effective conductance. This increased potential enhances the migration of ZnC14z-, the slow ion with the wrong sign of charge, so as to maintain a flux ratio relative to CI- in accordance with the stoichiometric coefficients as in Eq. (2). Recall that here the number of electron transferred, n, is positive for anodic reactions and negative for cathodic reactions and i is positive for anodic reactions and negative for cathodic reactions. Thus, i and n are always of the same sign. The concentration gradient can be obtained by substituting Eq. (5) into Eq. (3) and rearranging to give
dCj _( zFJ d'--~=
\D'C*
nDj] F
(8)
For an electrochemical system with k species, the concentration distribution for every species obeys Eq. (8) and the potential distribution obeys Eq. (5). Hence, there are k + l highly coupled ordinary differential equations to be solved simultaneously with the following boundary conditions,
(6) ~b= ~b°, Cj = C~'j,j= 1, k - 1 , @~=0
and
i=i( dp1, C)),j= 1, k @sc= 1
C* = 2zfCj
(7)
C* is twice the molar ionic strength, and the term FZD* C*/RT can be regarded as the effective conductivity, since Eq. (5) resembles Ohm's law. The effective conductivity is a complicated combination of diffusivities, valences, and stoichiometric coefficients of the reacting species in the overall reaction. The sign of the stoichiometric diffusivity, D*, depends on the relative magnitude and sign of the diffusivities, stoichiometric coefficients, and valences, and can be negative when there is a slow ion with a wrong sign of charge. Reduction of anions and oxidation of cations are examples for ions with wrong sign of charge. Generally speaking, repelled reactants and attracted products are example of ions with a wrong charge. For instance, the overall electrochemical reaction for zinc deposition in high chloride content may be [ 8] ZnC12- + 2e- ~
41
(10)
where superscripts 0 and 1 represent the membrane surfaces in contact with the bulk solution and the electrode, respectively.
3. Potential and concentration distributions
Since both ~band C/s depend on C*, we would first like to obtain the distribution of C* over the membrane. It is possible to obtain analytical solutions when the valences of all the ions are equal, Izjl =z, for allj Eq. (8) can be multiplied by z2 and then summed up over all species to give
dC* (z2Q
~Zn + 4C1-
Here ZnC12- is the slow repelled reactant. When D* is
(9)
Let
+
z2vj ~i
42
1
M.G. Lee, J. Jornd / Journal of Membrane Science 99 (1995)39-47
Z2_Vj
-~7 = nZ, Dj
(12)
then dC* { zZQ l "l 6i d~ =~D---~C.+~-7) ~
(13)
If there is little or no fixed space charge Q, then 1/ D' is much greater than z2Q/D" C*. On the other hand, when there is a strong fixed charge, z2Q/D" C* is much greater than 1/D'. These two cases have been extensively investigated in this study. The boundary condition for Eq. (13) is C* = C "° at ~=0 (bulk phase). The solution of the above differential equation can be algebraically solved to yield the potential and concentration distributions, as shown in the Appendix. The ohmic polarization over the diffusion region for membranes with weak or no space charge is
Adp=qb ~
RTD' [
- 4, ° = ~ l n / 1 Pt)
\
8i
"~
(14)
+ .~--~zz;~,o I rLn~ /
the local extrema with respect to the current. The turning points in an S-shaped curve of current versus potential are exactly the local extrema points in an N-shaped curve of potential versus current. The strategy is to find the corresponding applied voltage, V, for each current density, i, by calculating potential and concentration distributions, and consequently obtaining both the ohmic polarization from the potential profile and the overpotential on the electrode surface (including both mass transfer and kinetic terms) according to the surface concentrations. V = ( ff) l _ i f ) o ) + .qa + .t.lc+ ig s2
= A~b+ ~a + ~7¢+ iRa
(16)
Before the whole range of polarization is constructed, the existence of multiplicity may be analytically examined for some special cases. Extrema are, of course, located at the places where dV/di=0. The derivative of the above equation with respect to i is dV dA4~+drb + ~ i = ~- ~ -~/~+Ra
(17)
while for systems with strong space charge,
AqS=qSl--qS°= RTC*° [ / 1 + 2 z2Qri Fz2e k V FD* C *°-''---~
1)
4.1. Derivatives
(15)
4. Analysis of multiplicity When the ohmic polarization is negative, it is possible to admit multiple steady states even with a rather simple reaction. Multiplicity of current density versus the applied voltage is illustrated in Fig. 1, and the multiplicity of current versus voltage is analyzed by finding
>.. C o)
C3
'6 D O
/
The derivatives of the ohmic polarization in the diffusion region for the two cases in Section 3 are as follows. For weak or no space charge, dAq~= RT8 di F2D* C *°
1 8i 1 + ~ FD,C *°
(18)
and for strong space charge, daub= RT8 di F2D* C "° /
1 2zZQSi I +FD.C.O----------~
(19)
V
Note that dA~b/di=RTS/F2D*C *° as i ~ 0 for both cases. The derivative of the kinetic overpotential in the Tafel form is as follows. For cathodic processes,
•
[-[31nlF%
i= --/°exp~
RT
H/=2_I \C~j]
(20)
or equivalently,
Voltage Fig. 1. Schematic illustration of polarization curves.
Ba=.,,n,F I n ( - i ) - I n io-]~% In o
(21)
43
M.G. Lee, J. Jorng / Journal of Membrane Science 99 (1995) 39-47
where/3 is the symmetry factor for the cathodic reaction. Assuming that the difference between the surface concentration and the bulk concentration is small enough to be neglected, the mass transfer overpotential becomes negligible and the derivative of the kinetic overpotential is
RT 1 [31nlF i
d'rla = --
di
(22)
4.2. Analysis When multiplicity occurs, there must be at least one solution for dV/di = 0. If the surface concentrations are assumed constant, the following is obtained [9]. For weak or no space charge,
dV - R T 1 RT8 d--i=fllnlF7 +Ra+ F2D*C *°
1 6i 1-t-.~ FD,C .°
(23)
Because the first two terms are always positive, D" has to be negative to render dV/di = 0. By rearranging after setting dV/di = 0, one obtains
RTD'C *o - - = 0 /3lnl 6Ra
(24)
The solution for i is i=
-s + ~s2-4t 2
(25)
where
s=
RT RTD' D'FC'°'~ /3InlFRa ~ ' ~ + - - - - - ~ )
RTD'C "° /31nl6Ra
t= - -
dV - R T 1 RT6 d--7=~lnlF 7 + Rn4 F2D* C *o
(26) (27)
For cathodic processes, i should be negative. To have two such solutions for Eq. (25), it takes t > 0, s > 0 and s z - 4t > 0. When t > 0, D' must be negative, and when D' < 0, and s > 0, the middle term on the right hand side of Eq. (26) has to be positive, since the other two are definitely negative, and consequently it is shown
~
(28)
1 + 2zZQ6i
Again, D* has to be negative to have solutions, because both the first two terms are always positive. The analytical solution for this case can be obtained in a way similar to that in the case of weak or no space charge and is not presented here.
5. C o m p u t a t i o n
Whether or not analytical solutions are available, the potential and concentration distribution can be obtained numerically according to Eqs. (5) and (8) for any electrochemical system, even when the valences of ions are unequal. The polarization curve can be constructed according to Eq. (16), and the mass transfer overpotential (concentration overpotential), ~7c,is given by ~?c=
RrO' D'FC*°'i /31nlVRn F F--D-'~o+ " " ' - ~ - - ) i RT
iz +
again that D* must be negative to admit multiple steady states. For strong space charge,
RT C1 --- ~vjln( C-'C)~ InlF-~,~]
(29)
Usually, the mass transfer overpotential and the kinetic overpotential are combined together to give the total overpotential, r/. The reduction of chromate ion in aqueous solution is an example of an electrochemical system with a negative effective conductance [ 7 ]. CrO 2- + 4H20 + 3e- ~
~ 8 O H - + Cr 3+
Furthermore, quite frequently in organic synthesis, negative conductance has been observed [ 7 ]. The following reaction occurring in solvent (S), similar to that of chromate reduction, with somewhat different stoichiometry is used as an example. A-+S+3e-
~
~6B- + 2 C +
The physicochemical parameters of this example are given in Table 1. The diffusivities of both A - and C ÷ are significantly lower than that of B - . It was found that the multiplicity induced by the negative conductance for the cathodic process in a neutral diffusion region is likely to be over
44
M.G. Lee, J. Jorng / Journal o f M e m b r a n e Science 99 (1995) 3 9 - 4 7
Table 1 Physical parameters DA=0,015.
10 - 5
Da
ZA = -- 1 1)A=
= 1•10-s
Dc= 0.02.10-s
Ca ° = 0.1
DA ° = 1
ZB= va=6
-- l
yA=0.5
-- ]
YB = 0 . 5
=2.2 M D* = -0.0187D'=
-0.076.10
Q=0 R=0.1
10 - 5 cm2/s
D s = 1 . 1 0 - 5 cm2/s
Cc°=O.l
C s ° = 1M
zc=l
Zs = - 1
/JC~2
/)s=0
yc =0.5 F = 96487 Coul/equiv.
Ys=0
R = 8.314 J / t o o l K i o = 10 - 6 m A / c m 2
- 5 cm2/s
n= - 3 /3=0.5
g2cm 2
T= 300 K
30 A~
r,,
/
IR
2o
E •-
q
10
0
i 0.1
-0.1
,
~
, 0.3
v (volO F i g . 2. Cathodic polarization with negative conductance in the dif-
fusion layer. 50
R~0Qcm ~ -'-',
2O
E
and this ohmic polarization is the only factor which favors the occurrence of multiple steady states. Fig. 2. also presents the contribution of mass transfer and charge transfer overpotentials, and the mass transfer overpotential predominates the polarization, as always, near the limiting current where the multiplicity disappears. The characteristics of multiplicity are affected by several factors. When the ohmic drop in the bulk phase is high, the polarization depends strongly on the linearity of the ohmic drop, as shown in Fig. 3. When the ohmic resistance in the bulk solution is large, migration in the bulk phase becomes dominant and the overall multiplicity disappears, though multiplicity still holds over the diffusion region. The effect of the bulk concentration of the electrolyte is presented in Fig. 4. High bulk concentration results in high ionic strength in the bulk phase and in the membrane, and thus reduces the ohmic polarization, as shown in Eq. (14), and the multiplicity disappears. 80
•-
• - c*" = 66 tool&/
10 6O o
J J
0.25
f
i
0,30
0.35
0.40
v (volt) Fig. 3. The effects o f the ohmic drop in the bulk on polarization.
a wide range of current density, from zero to near the limiting current as shown in Fig. 2. This is attributed to the fact that the effect of the ohmic polarization due to the negative conductance becomes more important as the current increases, as can be seen from Eq. (14),
<<. 4o 20
/
~
S
"4
0
2.2tl
I 0.;5
0.20
0.2s
0.30
0.35
v (vo.) F i g . 4. T h e effects o f bulk concentration on polarization,
0.40
M.G. Lee, J. Jorn~ / Journal of Membrane Science 99 (1995) 39-47
accompanied by a limiting current density significantly higher than expected.
50 Q = 0.99 equlvff
40 E
# ~
45
0.5
3o
Appendix
.0.2 -0.5
20
The distributions of ionic strength, potential and ion concentration mentioned above are elaborated in the following sections. I
0.25
0.30
0.35
0.40
v (voH)
A.1. Ionic strength distribution Recalling Eq. (13), dC* [ z2Q l ~ ~i d~: - tD---:-~ + ~") F
Fig. 5. The effects of fixed space charge on polarization.
(13)
Let imJVcm_.___~2 = 7.5 -1
1
t~
12.5
z2 Q t~
a = ~-7-~, b=D. F then Eq. (13) becomes dC* bi =ai+-d~ C*
0.5 ¢..)
(A-l)
or
C*dC* = (aiC* +bi)d~
o o.o
0.5
1.o
Fig. 6. Concentration distribution in the diffusion layer at various current densities.
The same example was also used to study the effect of space charge. For the case of a fixed charge in the membrane, the polarity and the density of the charge have a profound effect on the occurrence of multiplicity. Fig. 5 shows multiplicity when a positive space charge is present, and Fig. 5 also indicates that a small negative charge is sufficient to destroy multiplicity, since this fixed charge requires some additional cations from the bulk solution to maintain electroneutrality in the membrane, and the interference of these cations diminishes the effect of the repelled reactant, A - . The concentration of the limiting species may unusually increase from the bulk phase to the electrode surface at low current density. As the current increases, the concentration profile for the limiting species within the membrane may reach a maximum as shown in Fig. 6. Such an abnormal concentration distribution is often
(A-2)
Let w = aiC* + bi, then
(ali 2 a~i2w~W = d sc
(A-3)
The solution for the above is
w t/2i 2
bi a2iilnw= ~+ K
(A-4)
or when substituting back w
aiC* + bi a2i 2
bi a2i~ln(aiC* + bi) =sC+K
(A-5)
By applying the boundary condition C* = C *° at sc= 0 C*-C
,o
b /aC*+b~ -aln~J-al~=0
.
(A-6)
Since
b z2QSD'F=z2QD ' a D*F ~ D* we have
(A-7)
M.G. Lee, J. Jorng/ Journalof MembraneScience 99 (1995)39-47
46
By applying the boundary condition,
, , t~ C..k_z2Qt~
eeD, I
-if:71
C. __ C . ° _ _ "if7 -m[~ 1 I 6-.o zQSI
t-B:-~c +-ff:-F)
C*=x/C*°2+2bi~
~;--,-~,i~= 0 t~r
~
C'°2 ~,~z2Q6i -~-2-~
(1-15)
Substituting the above into Eq. (5), (A-8)
A.2. Potential distribution The potential distribution is analyzed as follows
dqb RTSi 1 d~ FED * /,..o2_,z2QN~ t. ± ZFD *
(A-16)
V
The solution is
Case 1: For a>>b, Eq. (A-9) reduces to C* = C *° + A i s c
=
RT / .o2. ~z2Q6i. ~b=~--QVC ~-Z--~7"~+K
(A-17)
(A-9) By applying the boundary condition, we have
or it can be solved by starting from
,~= qb +725-_27, Fz Q C*°~+ Z~D-rg - C'°
dC* -ai d~
(A-18)
(A-10) and the potential difference is
Since
d49 RT6 i d--~= F 2 D'C*
(5)
(A-19)
by substituting C* into Eq. (5)
d49 RT$
-o RT [ /-.o2 -z2Q$i o~ A(b= ~bl- gb =w_--~-S-ff--~l~/C + 2---g-X~.-- C* rz Q \ V FD 1 or equivalently
i D* C*°+
A~b= (b 1 -
o RTC*°[ / + 2 zEQ6i
=
tV 1
i
1]
]
(15) The solution is + ~ ) 4,=4, o + PRTD'~ D ln(1 FDC
(A-12)
d_ff2C= _ ( ziCl __~._)6i d~ I,D*C* - n D J 7
and the potential difference is
RTD' [ ~ A~b=4,1- 4, o =-~--~ln~ 1 + ~ )
A.3. Concentration distribution Recalling Eq. (8),
(14)
(8)
let z__zA_8v. = ~j~ U:=FD*' J nFDj
Case 2: For b >> a, Eq. (A- 1 ) reduces to dC* d~
bi C*
therefore Eq. (8) becomes (A-13)
(A-20)
This is a linear first order equation, and it can be solved by applying the integrating factor
The solution is
1C'2 = bi~ + K 2
dC/= u.i - ~ . C i + vii d~
(A-14)
h=exp (S__% c . td~)
(A-21)
M.G. Lee, J. Jorng / Journal of Membrane Science 99 (1995) 39-47
Case 2:
to have the solution,
1 uji
47
For b >> a, the integrating factor is
K
Cj = ~ b-TviidsC+-~
(A-22) h=exp (-~)
(A-29)
For the case that a and b are comparable,
dC* d~
the concentration distribution is then
bi -ai+--
(A-23)
c*
Cj=Kexp(-~C*)+vJ--(C*-~)uj\
(A-30)
and
h = (aC* - b) "/~
(A-24)
with surface concentration
so the concentration distribution is
Cj=K(aC* +b)-"/n+
vj C*-VJ b u~+a uju~+a
(A-25)
and the surface concentration is
where K=expt~C
)'LC~-u,t C -~)J
C) = K(aC" + b) -'u/a + vj C*' vj b ua+a ujuj+a
(A-26) References
where
K=(aC.,,+b).da(Q -
v~ C.O+~ b ) uj+a ujuj+a
The constant K is determined by the boundary condition at ~ = 0, Cj = Cj° and C* = C *°.
Case I: For a >> b, Eqs.(A-25) and (A-26) can be reduced to give
Cj=K(aC.)_.,/. + vj C* uj+a
(A-27)
and
C)=K(aC.,)_.j~+
vj C*' uj+a
where
K= ( aC.O) _./~( C]_
v~ C.O) uj+a
(A-28)
[ 1] J. Osterwald and H. G. Feller, Periodic phenomena at a nickel electrode in sulfuric acid, J. Electrochem. Soc., 107 (1960) 473. [2] M. Boyer, I. Epelboin and M. Keddam, New potentiometric method for the investigation of fast electrochemical processes, Elecgochim. Acta, 11 (1966) 221. 13] K. J. Vetter, Electrochemical Kinetics, Theoretical and Experimental Aspects, Academic Press, New York, 1961. [4] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London), 117 (1952) 500. [5] K. S. Cole, H. A. Antosiewicz and P. Rabinowitz, Automatic computation of nerve excitation, J. Soc. Ind. Appl. Math., 3 (1955) 153. [6] K. S. Cole and R. F. Baker, Transverse impedance of the squid giant axon during current flow, J. Gen. Physiol., 24 ( 1941 ) 771. [7] O. Lev and L. M. Pismen, Steady state multiplicity of electrochemical reactions due to negative stoichiometric diffusivity, Electrochim. Acta, 31 (1986) 451. [ 8 ] J. Jorn6 and W.-T. Ho, Transference numbers of zinc in zincchloride battery electrolytes, J. Electrochem. Soc. 129 (1982) 907. [9] M. G. Lee and J. Jorn6, Multiple Steady States Induced by Ion Transport Coupled with Surface Reaction, Abstract No. 821, presented at the 177th Meeting of the Electrochemical Society, Montr6al, May 6-1 l, 1990.
Journal of Membrane Science 99 (1995) 39-47
Multiple steady states in ionic membranes with negative conductance Michael G. Lee
1,*,
Jacob Jorn6
Department of Chemical Engineering, University of Rochester, Rochester, NY 14627, USA Received 9 May 1994; accepted in revised form 5 September 1994
Abstract Infinitesimal or negative polarization resistance can be induced by a negative ionic conductance in membranes. When the reactant or product have the "wrong" charge, they are respectively repelled or attracted, under the influence of the coulombic force, resulting in a negative conductance in the membrane, which in turn contributes to a negative component in the polarization resistance. The effects of diffusivity, concentration, supporting electrolyte, and fixed space charge on the conductance in the ionic membranes are also studied. An inverse concentration gradient may form as a result of the interaction between the ionic migration and the stoichiometric flux according to the reaction kinetics. Keywords: Concentration polarization; Diffusion; Electrochemistry; Membrane electrodes; Oscillations and excitable membranes; Multiple steady state
1. Introduction
The inverse relationship between current and voltage has been recognized in some chemical and biological systems, such as the anodic dissolution of iron and nickel [ 1,2], reduction of nitric acid, oxidation of tetrahydrofuran [3], and nerve excitation [4,5]. The existence of such an inverse relationship implies the system has multiple steady states in either galvanostatic or potentiostatic operation. An inverse current-voltage relationship may originate from a complex reaction network; however, negative conductance may also induce negative polarization resistance in current or potential as well. * Corresponding author. Present address: Fujitsu Computer Packaging Technologies, San Jose, CA 95134, USA. 0376-7388/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI0376-7388 (94)00205-3
Many biological transducers generate electrical pulses. As an external signal excites a molecular process, the chemical response is amplified, e.g., by catalysis, and finally the signal is converted into a sequence of pulses, by some mechanism similar to the one responsible for electrochemical oscillations. All electrically excitable membranes seem to exhibit N-shaped current-potential curves, and are therefore likely to have a region of negative resistance [ 6 ], and to exhibit multiplicities. Under negative conductance in membranes, current flows in a direction opposite to the electric field. Negative conductance is not usually detected in laboratory work, because the total overpotential and ohmic resistance in the bulk phase often dominate the global feature of the system, and make the negative conductance hard to be observed. The strong interactions between this negative conductance and the others can give rise
40
M. G. Lee, J. Jorn~ / Journal of Membrane Science 99 (1995) 39~t7
to some anomalous behaviors, and can explain some phenomena that cannot be possibly understood under positive conductance. Furthermore, only after the resistance polarization in the diffusion region and in the bulk phase are sorted out, activation overpotential can be obtained and utilized for the study of kinetic mechanism. The influence of negative conductance is thus of great importance and, hence, it is crucial to demonstrate when negative conductance can exist, what anomalous behaviors it may induce, and how it can be related to some experimental observations. The understanding of ion motion and ion distribution is important not only in electrodeposition but also in areas such as electrophoresis, nerve excitation, and chemically modified electrodes, in which the processes are primarily controlled by their electrochemical nature. Systems consisting of both ion transport and electrode reaction are usually analyzed by decomposing the total applied voltage into equilibrium potential, activation overpotential, concentration overpotential, ohmic drop in the bulk phase, and ohmic drop in the diffusion region. The distributions of potential, ionic strength, and concentration in the diffusion region can then be achieved by analytical or numerical methods for the cases with and without fixed space charge. Negative conductance and, thus, multiplicity are to be closely related to diffusivities and stoichiometric coefficients of the reacting species in the overall reaction. It will be shown that polarization resistance is also affected by the bulk resistance, bulk concentration, diffusion region thickness, supporting electrolyte, and space charge. Negative conductance has been demonstrated to be able to induce multiple steady states in both anodic and cathodic polarizations [7]. Since the conductivity in the bulk solution is always positive, the only source for negative conductivity is in the diffusion region. High currents are associated with high kinetic overpotentials, but not necessarily high applied voltages. The potential profile in the diffusion region is anomalous when the system has a negative effective conductance. In the case of cathodic deposition, if the effective conductance is negative, the potential can increase, rather than decrease, from the interface between the bulk phase and the diffusion region up to the electrode surface. The magnitude of the current density affects the slope of the potential profile across the region, and the kinetic overpotential determines the reaction rate and, conse-
quently, the current density. Therefore, both high current with high kinetic overpotential and low current with low kinetic overpotential are possible at the same applied voltage, and multiple steady states may be achieved in certain circumstances by coupling the reaction to the ionic transport process. This concept was first introduced by Lev and Pismen [7]; however, their work consists of only zeroth order reactions coupled with negative conductance, and the mass transfer overpotential was neglected for the region of high current densities. Intensive efforts have been made to study the nonlinear dependence of the current on the applied voltage in terms of the effective conductance in the diffusion region, which may be a diffusion boundary layer or a thin membrane. The theoretical work is being extended by including the mass transfer overpotential, kinetic overpotential, ohmic polarization over the diffusion region and the ohmic drop in the bulk phase for reactions with arbitrary reaction order. The results are outlined below, and the detailed procedure can be found in the Appendix.
2. Transport in diffusion region Unlike the case in the bulk solution, ion diffuse through membranes under the influence of a concentration gradient and an electric field. For one-dimensional geometry and when the convection is negligible, the ionic flux is reduced to the following form Nj= - 8 \ d s ~
~=x/8
RT --~)
(l)
is the dimensionless coordinate, where 8 is the thickness of the membrane, and the interface between the bulk solution and the diffusion region is located at ~= 0, while the electrode surface is at sc= 1. There are two driving forces, one is the concentration gradient for the diffusion contribution, the other is the potential gradient for the migration contribution under the electric field. This equation can be applied to polymeric membranes with high liquid content. Under steady state, the ion consumed or generated by the electrochemical reactions on the electrode surface must be commensurately supplied by or removed to the bulk solution. The flux of ions can be related to the current density by
M. G. Lee, J. Jorn~ / Journal of Membrane Science 99 (1995) 39--47
~,ji
N1=
(2)
nF
Eqs. ( 1 ) and (2) can be combined to give
vii Dj [dCj + z.iVCj d___._1~ RT
(3)
The potential gradient can be obtained when Eq. (3) is multiplied by zj/Dj and then summed up over all the reacting species, with the supplemental condition of electroneutrality,
Q + EzlCj = O
(4)
The space charge in the transport region, Q, is usually zero for aqueous solutions, but can be either positive, zero, or negative for membranes. Regardless of the nature of the space charge, the potential gradient is always
RT6i d-~=F2D.C .
de
(5)
where
l=l z j D*
n
Dj
negative, the potential profile behaves in an unusual way. For example, the potential may increase, not decrease, in the membrane for a cathodic process under negative effective conductance. This increased potential enhances the migration of ZnC14z-, the slow ion with the wrong sign of charge, so as to maintain a flux ratio relative to CI- in accordance with the stoichiometric coefficients as in Eq. (2). Recall that here the number of electron transferred, n, is positive for anodic reactions and negative for cathodic reactions and i is positive for anodic reactions and negative for cathodic reactions. Thus, i and n are always of the same sign. The concentration gradient can be obtained by substituting Eq. (5) into Eq. (3) and rearranging to give
dCj _( zFJ d'--~=
\D'C*
nDj] F
(8)
For an electrochemical system with k species, the concentration distribution for every species obeys Eq. (8) and the potential distribution obeys Eq. (5). Hence, there are k + l highly coupled ordinary differential equations to be solved simultaneously with the following boundary conditions,
(6) ~b= ~b°, Cj = C~'j,j= 1, k - 1 , @~=0
and
i=i( dp1, C)),j= 1, k @sc= 1
C* = 2zfCj
(7)
C* is twice the molar ionic strength, and the term FZD* C*/RT can be regarded as the effective conductivity, since Eq. (5) resembles Ohm's law. The effective conductivity is a complicated combination of diffusivities, valences, and stoichiometric coefficients of the reacting species in the overall reaction. The sign of the stoichiometric diffusivity, D*, depends on the relative magnitude and sign of the diffusivities, stoichiometric coefficients, and valences, and can be negative when there is a slow ion with a wrong sign of charge. Reduction of anions and oxidation of cations are examples for ions with wrong sign of charge. Generally speaking, repelled reactants and attracted products are example of ions with a wrong charge. For instance, the overall electrochemical reaction for zinc deposition in high chloride content may be [ 8] ZnC12- + 2e- ~
41
(10)
where superscripts 0 and 1 represent the membrane surfaces in contact with the bulk solution and the electrode, respectively.
3. Potential and concentration distributions
Since both ~band C/s depend on C*, we would first like to obtain the distribution of C* over the membrane. It is possible to obtain analytical solutions when the valences of all the ions are equal, Izjl =z, for allj Eq. (8) can be multiplied by z2 and then summed up over all species to give
dC* (z2Q
~Zn + 4C1-
Here ZnC12- is the slow repelled reactant. When D* is
(9)
Let
+
z2vj ~i
42
1
M.G. Lee, J. Jornd / Journal of Membrane Science 99 (1995)39-47
Z2_Vj
-~7 = nZ, Dj
(12)
then dC* { zZQ l "l 6i d~ =~D---~C.+~-7) ~
(13)
If there is little or no fixed space charge Q, then 1/ D' is much greater than z2Q/D" C*. On the other hand, when there is a strong fixed charge, z2Q/D" C* is much greater than 1/D'. These two cases have been extensively investigated in this study. The boundary condition for Eq. (13) is C* = C "° at ~=0 (bulk phase). The solution of the above differential equation can be algebraically solved to yield the potential and concentration distributions, as shown in the Appendix. The ohmic polarization over the diffusion region for membranes with weak or no space charge is
Adp=qb ~
RTD' [
- 4, ° = ~ l n / 1 Pt)
\
8i
"~
(14)
+ .~--~zz;~,o I rLn~ /
the local extrema with respect to the current. The turning points in an S-shaped curve of current versus potential are exactly the local extrema points in an N-shaped curve of potential versus current. The strategy is to find the corresponding applied voltage, V, for each current density, i, by calculating potential and concentration distributions, and consequently obtaining both the ohmic polarization from the potential profile and the overpotential on the electrode surface (including both mass transfer and kinetic terms) according to the surface concentrations. V = ( ff) l _ i f ) o ) + .qa + .t.lc+ ig s2
= A~b+ ~a + ~7¢+ iRa
(16)
Before the whole range of polarization is constructed, the existence of multiplicity may be analytically examined for some special cases. Extrema are, of course, located at the places where dV/di=0. The derivative of the above equation with respect to i is dV dA4~+drb + ~ i = ~- ~ -~/~+Ra
(17)
while for systems with strong space charge,
AqS=qSl--qS°= RTC*° [ / 1 + 2 z2Qri Fz2e k V FD* C *°-''---~
1)
4.1. Derivatives
(15)
4. Analysis of multiplicity When the ohmic polarization is negative, it is possible to admit multiple steady states even with a rather simple reaction. Multiplicity of current density versus the applied voltage is illustrated in Fig. 1, and the multiplicity of current versus voltage is analyzed by finding
>.. C o)
C3
'6 D O
/
The derivatives of the ohmic polarization in the diffusion region for the two cases in Section 3 are as follows. For weak or no space charge, dAq~= RT8 di F2D* C *°
1 8i 1 + ~ FD,C *°
(18)
and for strong space charge, daub= RT8 di F2D* C "° /
1 2zZQSi I +FD.C.O----------~
(19)
V
Note that dA~b/di=RTS/F2D*C *° as i ~ 0 for both cases. The derivative of the kinetic overpotential in the Tafel form is as follows. For cathodic processes,
•
[-[31nlF%
i= --/°exp~
RT
H/=2_I \C~j]
(20)
or equivalently,
Voltage Fig. 1. Schematic illustration of polarization curves.
Ba=.,,n,F I n ( - i ) - I n io-]~% In o
(21)
43
M.G. Lee, J. Jorng / Journal of Membrane Science 99 (1995) 39-47
where/3 is the symmetry factor for the cathodic reaction. Assuming that the difference between the surface concentration and the bulk concentration is small enough to be neglected, the mass transfer overpotential becomes negligible and the derivative of the kinetic overpotential is
RT 1 [31nlF i
d'rla = --
di
(22)
4.2. Analysis When multiplicity occurs, there must be at least one solution for dV/di = 0. If the surface concentrations are assumed constant, the following is obtained [9]. For weak or no space charge,
dV - R T 1 RT8 d--i=fllnlF7 +Ra+ F2D*C *°
1 6i 1-t-.~ FD,C .°
(23)
Because the first two terms are always positive, D" has to be negative to render dV/di = 0. By rearranging after setting dV/di = 0, one obtains
RTD'C *o - - = 0 /3lnl 6Ra
(24)
The solution for i is i=
-s + ~s2-4t 2
(25)
where
s=
RT RTD' D'FC'°'~ /3InlFRa ~ ' ~ + - - - - - ~ )
RTD'C "° /31nl6Ra
t= - -
dV - R T 1 RT6 d--7=~lnlF 7 + Rn4 F2D* C *o
(26) (27)
For cathodic processes, i should be negative. To have two such solutions for Eq. (25), it takes t > 0, s > 0 and s z - 4t > 0. When t > 0, D' must be negative, and when D' < 0, and s > 0, the middle term on the right hand side of Eq. (26) has to be positive, since the other two are definitely negative, and consequently it is shown
~
(28)
1 + 2zZQ6i
Again, D* has to be negative to have solutions, because both the first two terms are always positive. The analytical solution for this case can be obtained in a way similar to that in the case of weak or no space charge and is not presented here.
5. C o m p u t a t i o n
Whether or not analytical solutions are available, the potential and concentration distribution can be obtained numerically according to Eqs. (5) and (8) for any electrochemical system, even when the valences of ions are unequal. The polarization curve can be constructed according to Eq. (16), and the mass transfer overpotential (concentration overpotential), ~7c,is given by ~?c=
RrO' D'FC*°'i /31nlVRn F F--D-'~o+ " " ' - ~ - - ) i RT
iz +
again that D* must be negative to admit multiple steady states. For strong space charge,
RT C1 --- ~vjln( C-'C)~ InlF-~,~]
(29)
Usually, the mass transfer overpotential and the kinetic overpotential are combined together to give the total overpotential, r/. The reduction of chromate ion in aqueous solution is an example of an electrochemical system with a negative effective conductance [ 7 ]. CrO 2- + 4H20 + 3e- ~
~ 8 O H - + Cr 3+
Furthermore, quite frequently in organic synthesis, negative conductance has been observed [ 7 ]. The following reaction occurring in solvent (S), similar to that of chromate reduction, with somewhat different stoichiometry is used as an example. A-+S+3e-
~
~6B- + 2 C +
The physicochemical parameters of this example are given in Table 1. The diffusivities of both A - and C ÷ are significantly lower than that of B - . It was found that the multiplicity induced by the negative conductance for the cathodic process in a neutral diffusion region is likely to be over
44
M.G. Lee, J. Jorng / Journal o f M e m b r a n e Science 99 (1995) 3 9 - 4 7
Table 1 Physical parameters DA=0,015.
10 - 5
Da
ZA = -- 1 1)A=
= 1•10-s
Dc= 0.02.10-s
Ca ° = 0.1
DA ° = 1
ZB= va=6
-- l
yA=0.5
-- ]
YB = 0 . 5
=2.2 M D* = -0.0187D'=
-0.076.10
Q=0 R=0.1
10 - 5 cm2/s
D s = 1 . 1 0 - 5 cm2/s
Cc°=O.l
C s ° = 1M
zc=l
Zs = - 1
/JC~2
/)s=0
yc =0.5 F = 96487 Coul/equiv.
Ys=0
R = 8.314 J / t o o l K i o = 10 - 6 m A / c m 2
- 5 cm2/s
n= - 3 /3=0.5
g2cm 2
T= 300 K
30 A~
r,,
/
IR
2o
E •-
q
10
0
i 0.1
-0.1
,
~
, 0.3
v (volO F i g . 2. Cathodic polarization with negative conductance in the dif-
fusion layer. 50
R~0Qcm ~ -'-',
2O
E
and this ohmic polarization is the only factor which favors the occurrence of multiple steady states. Fig. 2. also presents the contribution of mass transfer and charge transfer overpotentials, and the mass transfer overpotential predominates the polarization, as always, near the limiting current where the multiplicity disappears. The characteristics of multiplicity are affected by several factors. When the ohmic drop in the bulk phase is high, the polarization depends strongly on the linearity of the ohmic drop, as shown in Fig. 3. When the ohmic resistance in the bulk solution is large, migration in the bulk phase becomes dominant and the overall multiplicity disappears, though multiplicity still holds over the diffusion region. The effect of the bulk concentration of the electrolyte is presented in Fig. 4. High bulk concentration results in high ionic strength in the bulk phase and in the membrane, and thus reduces the ohmic polarization, as shown in Eq. (14), and the multiplicity disappears. 80
•-
• - c*" = 66 tool&/
10 6O o
J J
0.25
f
i
0,30
0.35
0.40
v (volt) Fig. 3. The effects o f the ohmic drop in the bulk on polarization.
a wide range of current density, from zero to near the limiting current as shown in Fig. 2. This is attributed to the fact that the effect of the ohmic polarization due to the negative conductance becomes more important as the current increases, as can be seen from Eq. (14),
<<. 4o 20
/
~
S
"4
0
2.2tl
I 0.;5
0.20
0.2s
0.30
0.35
v (vo.) F i g . 4. T h e effects o f bulk concentration on polarization,
0.40
M.G. Lee, J. Jorn~ / Journal of Membrane Science 99 (1995) 39-47
accompanied by a limiting current density significantly higher than expected.
50 Q = 0.99 equlvff
40 E
# ~
45
0.5
3o
Appendix
.0.2 -0.5
20
The distributions of ionic strength, potential and ion concentration mentioned above are elaborated in the following sections. I
0.25
0.30
0.35
0.40
v (voH)
A.1. Ionic strength distribution Recalling Eq. (13), dC* [ z2Q l ~ ~i d~: - tD---:-~ + ~") F
Fig. 5. The effects of fixed space charge on polarization.
(13)
Let imJVcm_.___~2 = 7.5 -1
1
t~
12.5
z2 Q t~
a = ~-7-~, b=D. F then Eq. (13) becomes dC* bi =ai+-d~ C*
0.5 ¢..)
(A-l)
or
C*dC* = (aiC* +bi)d~
o o.o
0.5
1.o
Fig. 6. Concentration distribution in the diffusion layer at various current densities.
The same example was also used to study the effect of space charge. For the case of a fixed charge in the membrane, the polarity and the density of the charge have a profound effect on the occurrence of multiplicity. Fig. 5 shows multiplicity when a positive space charge is present, and Fig. 5 also indicates that a small negative charge is sufficient to destroy multiplicity, since this fixed charge requires some additional cations from the bulk solution to maintain electroneutrality in the membrane, and the interference of these cations diminishes the effect of the repelled reactant, A - . The concentration of the limiting species may unusually increase from the bulk phase to the electrode surface at low current density. As the current increases, the concentration profile for the limiting species within the membrane may reach a maximum as shown in Fig. 6. Such an abnormal concentration distribution is often
(A-2)
Let w = aiC* + bi, then
(ali 2 a~i2w~W = d sc
(A-3)
The solution for the above is
w t/2i 2
bi a2iilnw= ~+ K
(A-4)
or when substituting back w
aiC* + bi a2i 2
bi a2i~ln(aiC* + bi) =sC+K
(A-5)
By applying the boundary condition C* = C *° at sc= 0 C*-C
,o
b /aC*+b~ -aln~J-al~=0
.
(A-6)
Since
b z2QSD'F=z2QD ' a D*F ~ D* we have
(A-7)
M.G. Lee, J. Jorng/ Journalof MembraneScience 99 (1995)39-47
46
By applying the boundary condition,
, , t~ C..k_z2Qt~
eeD, I
-if:71
C. __ C . ° _ _ "if7 -m[~ 1 I 6-.o zQSI
t-B:-~c +-ff:-F)
C*=x/C*°2+2bi~
~;--,-~,i~= 0 t~r
~
C'°2 ~,~z2Q6i -~-2-~
(1-15)
Substituting the above into Eq. (5), (A-8)
A.2. Potential distribution The potential distribution is analyzed as follows
dqb RTSi 1 d~ FED * /,..o2_,z2QN~ t. ± ZFD *
(A-16)
V
The solution is
Case 1: For a>>b, Eq. (A-9) reduces to C* = C *° + A i s c
=
RT / .o2. ~z2Q6i. ~b=~--QVC ~-Z--~7"~+K
(A-17)
(A-9) By applying the boundary condition, we have
or it can be solved by starting from
,~= qb +725-_27, Fz Q C*°~+ Z~D-rg - C'°
dC* -ai d~
(A-18)
(A-10) and the potential difference is
Since
d49 RT6 i d--~= F 2 D'C*
(5)
(A-19)
by substituting C* into Eq. (5)
d49 RT$
-o RT [ /-.o2 -z2Q$i o~ A(b= ~bl- gb =w_--~-S-ff--~l~/C + 2---g-X~.-- C* rz Q \ V FD 1 or equivalently
i D* C*°+
A~b= (b 1 -
o RTC*°[ / + 2 zEQ6i
=
tV 1
i
1]
]
(15) The solution is + ~ ) 4,=4, o + PRTD'~ D ln(1 FDC
(A-12)
d_ff2C= _ ( ziCl __~._)6i d~ I,D*C* - n D J 7
and the potential difference is
RTD' [ ~ A~b=4,1- 4, o =-~--~ln~ 1 + ~ )
A.3. Concentration distribution Recalling Eq. (8),
(14)
(8)
let z__zA_8v. = ~j~ U:=FD*' J nFDj
Case 2: For b >> a, Eq. (A- 1 ) reduces to dC* d~
bi C*
therefore Eq. (8) becomes (A-13)
(A-20)
This is a linear first order equation, and it can be solved by applying the integrating factor
The solution is
1C'2 = bi~ + K 2
dC/= u.i - ~ . C i + vii d~
(A-14)
h=exp (S__% c . td~)
(A-21)
M.G. Lee, J. Jorng / Journal of Membrane Science 99 (1995) 39-47
Case 2:
to have the solution,
1 uji
47
For b >> a, the integrating factor is
K
Cj = ~ b-TviidsC+-~
(A-22) h=exp (-~)
(A-29)
For the case that a and b are comparable,
dC* d~
the concentration distribution is then
bi -ai+--
(A-23)
c*
Cj=Kexp(-~C*)+vJ--(C*-~)uj\
(A-30)
and
h = (aC* - b) "/~
(A-24)
with surface concentration
so the concentration distribution is
Cj=K(aC* +b)-"/n+
vj C*-VJ b u~+a uju~+a
(A-25)
and the surface concentration is
where K=expt~C
)'LC~-u,t C -~)J
C) = K(aC" + b) -'u/a + vj C*' vj b ua+a ujuj+a
(A-26) References
where
K=(aC.,,+b).da(Q -
v~ C.O+~ b ) uj+a ujuj+a
The constant K is determined by the boundary condition at ~ = 0, Cj = Cj° and C* = C *°.
Case I: For a >> b, Eqs.(A-25) and (A-26) can be reduced to give
Cj=K(aC.)_.,/. + vj C* uj+a
(A-27)
and
C)=K(aC.,)_.j~+
vj C*' uj+a
where
K= ( aC.O) _./~( C]_
v~ C.O) uj+a
(A-28)
[ 1] J. Osterwald and H. G. Feller, Periodic phenomena at a nickel electrode in sulfuric acid, J. Electrochem. Soc., 107 (1960) 473. [2] M. Boyer, I. Epelboin and M. Keddam, New potentiometric method for the investigation of fast electrochemical processes, Elecgochim. Acta, 11 (1966) 221. 13] K. J. Vetter, Electrochemical Kinetics, Theoretical and Experimental Aspects, Academic Press, New York, 1961. [4] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London), 117 (1952) 500. [5] K. S. Cole, H. A. Antosiewicz and P. Rabinowitz, Automatic computation of nerve excitation, J. Soc. Ind. Appl. Math., 3 (1955) 153. [6] K. S. Cole and R. F. Baker, Transverse impedance of the squid giant axon during current flow, J. Gen. Physiol., 24 ( 1941 ) 771. [7] O. Lev and L. M. Pismen, Steady state multiplicity of electrochemical reactions due to negative stoichiometric diffusivity, Electrochim. Acta, 31 (1986) 451. [ 8 ] J. Jorn6 and W.-T. Ho, Transference numbers of zinc in zincchloride battery electrolytes, J. Electrochem. Soc. 129 (1982) 907. [9] M. G. Lee and J. Jorn6, Multiple Steady States Induced by Ion Transport Coupled with Surface Reaction, Abstract No. 821, presented at the 177th Meeting of the Electrochemical Society, Montr6al, May 6-1 l, 1990.