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Transient behaviour of porous electrodes with high exchange current densities
TRANSIENT BEHAVIOUR OF POROUS WITH HIGH EXCHANGE CURRENT
ELECTRODES DENSITIES
R. POLLAKD and J. NEWMAN Materials and Molecular Research Division; Lawrence Berkeley Laboratory and Department of Chemical Engineering, University of California, Berkeley, California 94720, U.S.A. (Received 7 June 1979) Ahstraet - A theoretical analysis of the non-steady-state reaction distribution in a porous electrode with a high exchange current density is made by application of a simplified macrohomogeneous model for porous electrodes. The dimensionless transfer current for short times is given as an expansion in time, and two terms in a moderate time solution are also presented. At moderate times the current is split into timez-dependentand time-independent parts, but this distinction is not apparent in the short time solution. For the limitingcase of reversible kinetics, the analysis specifies the fraction of the superficial current density that will be distributed through the electrode. Treatments for the PbOz and LiAl electrodes are presented as examples.
NOME.NCLATURE
Y
interfacial area per unit electrode volume (cm-‘) variable in s domain defined by (20) parameter in (4) concentration of species i (mol cme3) total ionic concentration (mol cmu3) molecular diffusion coefficient of pore electrolyte (cmZ s- 1) variable in s domain defined by (21) Faraday’s constant (96487 C equiv-‘) parameter defined by (39) exchange current density (A cm-‘) superficial current density in pore phase (A cm-‘) superficial current density to an electrode (Acm-‘) = V. i,, transfer current per unit electrode volume (A cmm3) dimensionless transfer current dimensionless transfer current in s domain penetration depth (cm) molality of electrolyte (mol kg-‘) symbol for the chemical formula of species i number of electrons transferred in electrode reaction superficial flux of species i (mol crne2 s-‘) parameter defined by (22) parameter defined by (6) parameter defined by (13) universal gas constant (8.3143 J moi-‘. K-l) variable defined by (18) stoichiometric coefficient of species i in electrode reaction time (s) transference number of species i relative to solvent velocity transference number of species i relative to common ion velocity transference number of species i relative to molar average velocity absolute temperature (K) molar average velocity (cm’ s- ‘) superficial volume average velocity (cm2 s ‘) partial molar volume of species i (cm3 mol- ‘) molar volume of electrolyte (cm’ mol- ’ ] parameter defined by (7) mole fraction of salt A 315
y Zl
distance through electrode measured from separator (cm) dlmensionless distance through porous electrode measured from separator valence or charge number of species i
Greek lerters transfer c#zff&ent in anodic direction transfer coefficient in cathodic direction mean molal activity coefficient exponent in (3) activity coeflicient for salt A porosity or void volume fraction tortuosity defined by (6) outer perturbation parameter defined by (25) dimensionless surface overpotential dimensionless concentration efiective solution conductivity (n-l cm-‘) parameter defined by (24) chemical potential of electrolyte (J mol - ‘) chemical potential of salt A (J mol-‘) total number ofions produced by dissociation of a mole of electrolyte number of moles of speries i produced by dissociation of a mole of electrolyte density of electrolyte (gcmeJ) effective matrix conductivity (SY’ cm-l) dimensionless time electric potential in the matrix (V) electric potential in the solution (V) Subscripts A B e 0 0s oe R m
lithium chloride potassium chloride electrolyte solvent in separator at electrode-separator in electrode at electrode-separator reference condition bulk solution property
Superscripts A I3 0
lithium chloride potassium chloride initial value
boundary boundary
R. POLLARD urn J.NEWMAN
316
lNTRODUCTlON Many porous electrodes in commercially important batteries have high values for the product of the electrochemically active specific interfacial area and the exchange current density. This permits operation at high superficial current densities without excessive surface overpotentials which would lower the energy efficiency of the system. Previously, the distributions of current and electrochemical reaction in porous electrodes have been considered under various special conditions[ 1,2]. Mass transport of reactants and the effective conductivities of the matrix and solution phases have a strong influence on electrode behaviour. The current distribution becomes non-uniform for large values of active surface area, exchange current density, electrode length, and current density, and for small conductivities. Particular attention has been paid to the electrodes of the lead-acid battery[3-61, and several of the proposed models make the assumption of reversible kinetics[& 63. With this assumption, it has been shown that highly non-uniform reaction distributions can be obtained, especially for high rate discharges[?+ This can result’ in an increase in internal cell resistance because local changes in electrolyte composition lower the electrolyte conductivity. Under some circumstances the electrode could become totally passivated. It has also been shown that, in principle, a fraction of the total reaction will be restricted to the electrode-separator interface[5]. Another study allows all the current to penetrate the electrode, and concludes that models with reversible kinetics are inferior because they cannot take proper account of transport of external acid into the electrode pores[6]. This paper considers the nature of the current and reaction distributions for electrode processes with high, but not infinite, exchange current densities. A simplified model is developed in order to give an analytic solution to the governing differential equations. The results obtained illustrate the important features that would be observed with a more sophisticated approach. The method used in this paper complements a computer solution which is able to account for many complications not considered here, but which cannot easily clairfy the highly non-uniform reaction distribution at moderately short times that results from a high value of the exchange current density. The model can also be used to assess quantitatively the validity of the assumption of reversible kinetics for a particular electrode system.
ANALYSIS
The system considered is a porous electrode in contact with an inert porous separator. The electrode is treated as two superposed continua which represent the solution and matrix phases independently. The actual geometric details of the pores bave been disregarded and only variations of parameters in a direction normal to the electrode face are considered. Each phase is taken to b-e electrically neutral, and consequently the divergence of the total current density is zero.
The analysis is restricted to a single electrode reaction with stoichiometry represented by :
(1)
7 siMp+ne-. A polarization equation of the form: j = ai,[@(%*&‘Rx
- e-@t’%-%r/RrJ
(2)
can be used to express the dependence of the reaction rate at any point in the electrode on the local potential jump at the matrix-solution interface. The potential of the pore solution a2 is measured with a reference electrode of the same kind as the working electrode. The composition dependence of the exchange current density can be assumed to have the form:
where i ranges over the ionic species. Solid phase activities are taken as constant, and initial concentrations are used for the reference condition, R. The possibility of homogeneous chemical reactions is not considered, and double layer charging effects are ignored. To simplify the analysis further, it is assumed that transport properties and pore structure do not change as the reaction proceeds. Under these circumstances, the major concentration changes are caused by electrochemical reaction and by diffusion of electrolyte. Therefore, a non-steady state material balance for species i becomes:
aci B d2C, = Fj + &D7. af dY
E-
For times over which (4) is to be applied, the parameter B is taken as a constant, dependent on the system considered. Relationships for B, for tbe positive plate of a lead-acid battery and for the negative electrode in a high temperature LiAl-LiCl, KCl-FeS, battery, are derived in Appendices I and II, respectively. The molecular diffusion coefficient D can be characterized by D = 0,/C=.
(5)
Also, it is assumed that the tortuosity factor c is directly related to porosity by[7]: < = e(l-9)/Zt
(6)
where the constant q is taken as 1.5. The driving force for tbe current can be specified in terms of the potential in the pore solution a2 measured with a reference electrode. With this definition, a modified Ohm’s taw for the pore electrolyte is: i2 -= K
-V@,+m
RTW F
Vinci,
(7)
where the second term on the right includes the diffusion potential. In this analysis, the parameter W is regarded as a constant, dependent on the electrode studied (see Appendices 1 and II). The effective conductivity K is estimated from K =
&Km/~‘,
(8)
together with (6). The movement of electrons in the matrix phase is
317
Transient behaviour of porous electrodes governed by Ohm’s law. However, it is assumed that the matrix conductivity is large (U/K >> 1) so that the matrix potential can be taken to be uniform. With this condition, the initial electrode reaction will be confined to a region adjacent to the electrode-separator interface. This reaction zone can be characterized with a penetration depth L which, for linear kinetics and constant matrix potential, is defined as[8] :
where i,, is evaluated at the initial composition. When the discharge process begins there is a rapid and substantial change in current distribution, caused by changes in composition. Nevertheless, the concentrations have only altered significantly from the standpoint of concentration overpotential, and the ratio ci/ciR has not changed enough to affect appreciably tbe exchange current density given by (3). In this restricted range ofinvestigation it is appropriate to define a set of dimensionless variables r = Dt/L2,
J = jL/I,
rl = WI
- W/IL,
Bi = F6D(ci - c;)/ILB, which have values of order unity. After these definitions have been substituted into the governing equations, it becomes possible to identify those terms which can be neglected as i, becomes large. When the exchange current density, or the product ato, is large, the local surface overpotential for a specified supcrficia.l current density I is expected to be low. Consequently, it is reasonable to reduce (2) to the form :
a0 -=J+ay2.
J = 4.
(11)
It might appear that (11) can be obtained directly from (2) provided that (@r - &)F/RT << 1. However, the analysis above emphasises that it is necessary to verify that the composition dependence of the exchange current density does not alter this result. For small concentration changes, it becomes valid to linearize the concentration dependent term in (7). Combination of (7) and (11) then gives, on differentiation :
J=g;+Qg,
e(Y,o)
D&F2
’
= 0,
(i) fI(Y,r)--, 0 as Y -+ 00, (ii) at the electrode-separator (a) i, = --_I,
(15)
interface (Y = 0);
(b) 0 continuous,
(16)
For condition (i), the electrode is assumed to be considerably thicker than the penetration depth so that its length can be regarded as infinite. The last boundary condition assumes that the reaction rate is not a Dirac &function at the origin. Equations (12)-( 16) define the dependence of concentration and transfer current on time and distance through the porous electrode for a constant current discharge. The reaction zone is restricted to the region immediately adjacent to the separator, and this limits the applicability of the results, as discussed below. The equations refer to systems with high exchange current densities or small penetration depths, but it should be emphasized that the macrohomogeneous model on which the analysis is based will break down if the reaction xone is significantly smaller than the characteristic dimensions of the microscopic porous structure. AND
DISCUSSION
In terms of Laplaos transforms, the solution to the stated problem may be written as:
J(Y,s) = AemEY f-
(17)
in which the variable s is defined by:
Jam e-nF(r)dr.
f(s) =
The parameters A and E are related by the imposed requirement that integration of (17) across the electrode gives the superficial current density, viz:
s m
0
&(Y,s)dY
= ‘. s
(19)
Equations (5). (6), (1 S), (16). and (19) may be combined to specify the coefficient A and E as: A
Q=
(12) and (14),
and the boundary conditions :
where Y = y/L and IC,RTBW
(14)
The governing differential equations are subject to the initial condition:
RESULTS
As the exchange current density is raised, the pens tration depth decreases and, for Br = O(l), the right side of (10) approaches unity. Under these circumstances, (10) can be simplified further to give:
a’8
at
=
s
(13)
In the appendices, it is shown that only one concentration variable is required for (7), and, therefore, the subscript on the dimensionless variable 19,can be dropped. The material balance, (4), can be rewritten in dimensionless form :
_A
and
[(~--EZ)u+P~/E)_ 1 ’ (20) 1 -I
(1 -s/E’)(l+PE)
2s 1+s+Q+[(1+s+Q)2-4~]1’2
112 >
’
(21)
where p = (c,/e)=~-
1’.
(22)
R. POLLARD AND J. NEWMAN
318
Equation (18), together with the definitions of T and L, indicates that the variables is inversely proportional to the exchange current density. Consequently, inversion of (17) in the limit as S-B 0 should give a moderate time solution for the dimensionless transfer current. Two terms in the inverse Laplace transform for (17) are:
+PbOz + 2H+ + $SO:-
l--1
J=,/~Ie-Ji+Qy+
Jme-*‘(23)
+ e= +PbS04
+ H20.
(26)
The stoichiometric coefficients defined in accordance
where
A=
Two examples have been chosen to illustrate these concepts; the positive plate of a lead-acid battery and the negative electrode of a LiAI-FeS, high temperature cell with molten LiCl, KCI electrolyte. For the lead-acid system, it is assumed that the electrolyte dissociates completely. The reaction for discharge of the lead dioxide electrode is then:
with (1) are, s+ = -2, s- = -3, s,, = 1, and n = 1. With the electrode parameters given in Table 1, the
&/l-I-Q+1 &/_+1+Q’
and t] = y/24m.
(25)
The variables Y and 9 can be regarded as inner and outer perturbation parameters, respectively. In the region Y 5 O(l), (23) describes the reaction zone immediately adjacent to the separator, whereas in the region q 2 O(1) it describes the progressive penetration of the reaction through the electrode. The first term in the equation represents a time-independent spike in J which is largest at the separator and which penetrates a relatively short distance into the electrode. The factor in the exponent reflects the influence of da on the spike shape. The fraction of the total current associated with the spike, II, is an important parameter in the description of an electrode process with a high exchange current density. If rZ.is small, most of the current will be progressively distributed through the electrode where as, as 1 approaches unity, it becomes more acceptable to disregard the time-dependent term in (23) and to assume that the reaction is restricted to the spike aione. The spike is sharper for larger exchange current densities, and in the limit of infinite exchange current density it will become a Dirac 8 function (with area U) at the electrode-separator interface. Consequently, in electrode models which neglect the surface overpotential (reversible kinetics, i, = a)), only a fraction (1 - .I) of the total current should be distributed through the electrode.
model proposed indicates that 39% of the superficial current density is associated with the timeindependent spike in (23). Alternatively, if incomplete dissociation of electrolyte were assumed, it would be necessary to change the stoichiometric coefficients for the electrode reaction, the cation transference number, and the activity coefficient for the electrolyte in a consistent manner. The reaction at the LiAl electrode of the high temperature battery is LiAl -+ Al + Lit
cm-‘) D, X lO~(cm~s-‘) T(K) &z E” a, = 01,
K~(R-~
103c~(molcm-“) &cm-“) 1+
din?
and LiAl electrodes
1.75 3.33 723.15 0.75 0.20 0.50 0.58 33.92
2.30
1.92
to p’
0.73
0.29
Q a
6.17
d In m
L(‘W
Parameter
LiAl 0.9 1 3.03 298.15 0.56 0.52 0.50 5.00 I .29
1.10 0.39 0.42
(27)
and therefore, s+ = -1, s-=0, s,=Oand n=l. Table 1 lists system parameters for discharge with an electrolyte of eutectic composition. The analysis shows that 85% of the total current is in the spike. Figure 1 gives the dependence of the dimensionless transfer current on the inner perturbation parameter Y for the LiAl electrode. For small values of Y, the major contribution to J is from the timeindependent term in (23), and the distance over which the spike is dominant increases with time for T 2 10. At progressively larger times the reaction penetrates further into theelectrode, and J falls locally in response to the requirement of constant total current. Eventually the material closest to the separator will become depleted of reactant, and the narrow reaction zone will then move through the electrode as the available material isconsumed. At this stage, the analysis presented here will no longer be valid. Additional complications can arise if local changes in composition significantly affect the exchange current density through (3). For example, if i, is
Table 1. Typical parameters for PbO,
Parameter
+ e-
5.22 1.60 0.85 1.32 x lo-’
K,(R-“%I-‘)
D,
x IO5 (cm’ se’)
T(K) E” : E cc. = a.
5:
V(cm3molm1) 1 + dlny, dlnx, r: P
Q R Wm)
Transient behaviour of porous electrodes
-4
319
-4
In tJ1 In(J) -6
-6
-\ -I 2
0
I I
L 2
I 3
I 4
I 5
I 6
-12 7
0
1
I
/
I
2
3
,‘\ 4
, 5
I 6
7
Y
Fig. 1. Solution in the inner region for the dimensionless transfer current at different times, 5, for the LiAl electrode,as
Fig. 2. Solution in the inner region for the dimensionless transfercurrent at different times, +, for the PbO, electrode, as predictedwith (23). The.solution at r = 0 is from (28). -time-independentspike.
most sensitive to the reactant concentration, it is expected that the spike will become more diffuse as the reaction proceeds. This is not accounted for by the model, although concentration changes are included in the non-steady-state material balance, (14). The zero time soiution of (17),
associated with the spike is negligible. The dimensionless distance within which half the current that pens trates is to be found is given by 0.913~~. This dependence indicates that the process is dominated by diffusion but that interactions between difision and reaction modify the current distribution. The reaction rate distribution at very short times is
predictedwith (23).The solution at 1 = 0 is from (28).- - - time-independentspike.
J = e-‘,
(28)
is also shown in Fig. 1 in order to emphasize how the reaction distribution becomes separated into two components corresponding to the two terms in (23). The apparent agreement between the zero time solution and the reaction distribution at 5 = 1 is coincidental; the moderate time solution cannot be extended to T < 1 and, moreover, an assessment of the influence of higher order terms not shown in (23) is needed to establish its validity at r = 1. Figure 2 shows the inner solution at moderate times for the Pb03 electrode. The different values for the parameters Q and d, in comparison with the LiAl electrode, both make the penetration of the spike smaller and hence reduce the contribution of this term to the total transfer current at a given position in the electrode. The penetration depth for the PbO, system, as calculated with (9), is approxirnately0.4 cm, which is considerably larger than the corresponding value for the LiAl electrode (see Table 1). Since the assumption, L CC1, is used in the high current density analysis for the PbO, electrode, the results should be treated with caution and, in particular, the assumption of reversible kinetics that has been used previously[5,6] requires some justification. The outer solution for J for the LiAl electrode at moderate times is given in Fig. 3 in terms of the outer perturbation parameter, 8. The incomplete curves show the contribution of the time-dependent term in (23). The intercepts of these parabolae represent inner limits of the outer expansion, given by (1 -2)/x,/m. This is also seen in Fig. 1 as the magnitude of J in the region where the current
Fig. 3. Solution in the inner region for the dimensionless transfer current at Merent times, 7, for the LiAl electrode, as
predictedwith (23). The solutionat T = 0 is from (28). - - - time-dependentterm in (23).
R. POLLARD AND J. NEWMAN
320
obtained as a power series expansion in r, by inversion of (17) in the limit as s -+ 00. It is found that:
regions. A more general inversion of the solution in Laplace space, (17), is needed to be able to predict this intermediate behaviour. CONCLUSIONS
*(;y’zz3,2(a,]
-
+ (s)r
[(I +2z’)erfc(z)
- 5
(29)
eCzz]
+ ($&3,‘[?-+2
where 01= m, z = ,/mq, and Z,(a) is a modified Bessel function of the first kind and of order v. For short times, eg r -z O(lO-‘1, J can be predicted adequately by the zero time solution, (28). At first sight, the e-’ term in (29) would seem to be equivalent to the spike observed in the moderate time solution. Numerically, however, the clear distinction between the two contributions at moderate times has not developed at the short times for which (29) is valid. This range of validity can be assessed by a comparison of the first order, short-time approximation for .I with the approximations accurate to orders T and z3’2(JI,Jz,JS reSpeCtiV&). Figure 4 shows that, for the LiAl electrode, the validity of (29) becomes questionable for T z 0.1. At r = 0.1, inclusion of the r3” term causes a relatively minor shift in Jz. At 7 = 0.5, the difference between .J2and .I, indicates that higher order terms are needed to obtain a satisfactory solution. For the PbO, electrode, the range of validity of (29) is restricted to even smaller values for 7 be-cause the parameter Q is significantly larger than for the LiAl system (see Table 1). The distinct differences in short and moderate time behaviour predicted by (23) and (29) preclude the formation of a combined perturbation expansion that would provide an acceptable link between the two time 7,
I
I
I
I
I
I
1
6
A simplified macrohomogeneous model can be used to elucidate the transient behaviour of porous electrodes with high exchange current densities. The highly non-uniform initial reaction distribution changes at moderately short times due to the influence of concentration overpotential. The analysis presented can be used to predict the magnitude of these changes and to characterize the time-dependent and timeindependent components of the transfer current at a given position within the electrode. The results show clearly the association between high exchange current densities and non-uniform reaction distributions which can, in turn, have a significant impact on battery performance. Acknowledgement -This work was supported by the Division of Solar, Geothermal, Electric and Storage Systems, Office of the Assistant Secretary for Energy Technology, U.S. Department of Energy under contract No. W-74QS-Eng-48. REFERENCES
1. J. Newman and C. W. Tobias. J. electrochem. Sm. 109, 1183 (1962). 2. I. Newman, Internal tech. report, NOLC project, Dept. Chem. Engng, University of California, Berkeley (1962). 3. K. Micka and I. Rousar, Electrochim. Acta. 21, 599 (1976). 4. D. Simonsson, J. appl. Electrochem. 3, 261 (1973). 5. W. Stein, Ph.D. thesis, Rheinisch-Westfalischen Tech. Hochschule, Aachen, %‘eqt Germany (1959). Actn 18, 629 6. K. Micka and I. Rousar, Electrochim. (1973); ibid. 19, 499 (1974). in Elec7. J. Newman and W. Tiedemann, Advances trochemfsrry and Elecrrochemical Engineering (Edited by H. Gerischer and C. W. Tobiasf. Vol. 11. WileyInters&ace, New York (1977). 8. J.Newmanand W.Tiedemann, A.I.Ch.E.JL 21,25 (1975). 9. J. Newman and T. W. Chapman, A.1.Ch.E. J1. 19, 343 (1973). 10. R. Pollard and J. Newman, J. electrochem. Sot. 126,1713 (1979). 11. M. Blander in Molten Salt Chemistry (Edited by M. Blander). Interscience, New York, pp, 127-238 (1964). APPENDKXS
Derivation
of parameters
for porous electrode
model
Appendix I. Positive plate of lead-acid battery. In the PbOl electrode, a non-steady state material balance for species i may be combined with Faraday’s law to give:
4 eY J 3
J(%) -= at
-V.Nf
(30)
--$j
for a single electrode reaction[8]. For a concentrated biiy electrolyte, the superficial flux of species i can be expressed in terms of the superficial volume average velocity uw [8,9],
I
0 0
I
I 2
I
I
I
I
3
4
5
6
N, = - EDVC, + 5 7
Y Fig. 4. Reaction rate distribution for Lil electrode at short times, as predicted with (29). J,, J2, J,, are first, second, and third order approximations, respectively.
1
i, + c@.
Substitution into (30) and comparison Parameter B as :
(31)
with (4) defines the
v+ Y->I (32)
s+t”_ ---++
s-t’,
.
Transient behaviour of porous electrodes Ohm’s law can be used in the form
321
:
where az is the potential in the pore solution measured with a reference electrode of the same kind as the working electrode. The chemical potential of the electrolyte can be defmed with : ,=~(1+~)v1,.,.
provided that the molar volume of electrolyte v is constant. Ohm’s law for the pore electrolyte can be written in a form analogous to (33) as:
(34) where
where Y is the total number ofcations and anions produced by dissociation of one molecule. Consequently, the parameter W in (7) is @ven by :
Appendix II. Negative elecrrode of LiAl/FeS, battery. This high temperature system can use an electrolyte that comprises two binary molten salts with a common ion, such as the LiCl, KC1 eutcctic. In the derivation below, subscripts 1, 2, and 3 refer to Li+, K+, and Cl- ions, respectively, and scripts A and B refer, in turn, to the LiCl and KC1 salts. The superficial flux of lithium ions baaed on the molar average reference velocity is[ lo] : N, = -
&(CI +
c,)+Dv:v’ CT
vx, +
5 +c,v*, 1
(36)
where xA = cA/(cA + ca) is the mole fraction of LiCl in the electrolyte and cr is the total concentration defined by cl. = ct + ct f cs. Combination of (30) and (36) gives, on simplification, (4) with :
G+$+~)(l+J@~.
(39)
The activity coefhcient of a salt in a binary molten salt electrolyte mixture is often defined without regard for dissociation and in terms of ion equivalent fractions or mole fractions[l 11. (These composition variables are equivalent when v< = vg.) The gradient of chemical potential can be written as[ 10) : r:+s)Vlne,,
(40)
and with this definition the parameter W in (7) becomes:
W= -G
(41)
In the absence of experimental data for transference numbers it is assumed that transference numbers are directly proportional to the electrolyte mole &action, viz: t: = ?&Iv*> t$ = vfxB/vB,and t; = x”.
ELECTRODES DENSITIES
R. POLLAKD and J. NEWMAN Materials and Molecular Research Division; Lawrence Berkeley Laboratory and Department of Chemical Engineering, University of California, Berkeley, California 94720, U.S.A. (Received 7 June 1979) Ahstraet - A theoretical analysis of the non-steady-state reaction distribution in a porous electrode with a high exchange current density is made by application of a simplified macrohomogeneous model for porous electrodes. The dimensionless transfer current for short times is given as an expansion in time, and two terms in a moderate time solution are also presented. At moderate times the current is split into timez-dependentand time-independent parts, but this distinction is not apparent in the short time solution. For the limitingcase of reversible kinetics, the analysis specifies the fraction of the superficial current density that will be distributed through the electrode. Treatments for the PbOz and LiAl electrodes are presented as examples.
NOME.NCLATURE
Y
interfacial area per unit electrode volume (cm-‘) variable in s domain defined by (20) parameter in (4) concentration of species i (mol cme3) total ionic concentration (mol cmu3) molecular diffusion coefficient of pore electrolyte (cmZ s- 1) variable in s domain defined by (21) Faraday’s constant (96487 C equiv-‘) parameter defined by (39) exchange current density (A cm-‘) superficial current density in pore phase (A cm-‘) superficial current density to an electrode (Acm-‘) = V. i,, transfer current per unit electrode volume (A cmm3) dimensionless transfer current dimensionless transfer current in s domain penetration depth (cm) molality of electrolyte (mol kg-‘) symbol for the chemical formula of species i number of electrons transferred in electrode reaction superficial flux of species i (mol crne2 s-‘) parameter defined by (22) parameter defined by (6) parameter defined by (13) universal gas constant (8.3143 J moi-‘. K-l) variable defined by (18) stoichiometric coefficient of species i in electrode reaction time (s) transference number of species i relative to solvent velocity transference number of species i relative to common ion velocity transference number of species i relative to molar average velocity absolute temperature (K) molar average velocity (cm’ s- ‘) superficial volume average velocity (cm2 s ‘) partial molar volume of species i (cm3 mol- ‘) molar volume of electrolyte (cm’ mol- ’ ] parameter defined by (7) mole fraction of salt A 315
y Zl
distance through electrode measured from separator (cm) dlmensionless distance through porous electrode measured from separator valence or charge number of species i
Greek lerters transfer c#zff&ent in anodic direction transfer coefficient in cathodic direction mean molal activity coefficient exponent in (3) activity coeflicient for salt A porosity or void volume fraction tortuosity defined by (6) outer perturbation parameter defined by (25) dimensionless surface overpotential dimensionless concentration efiective solution conductivity (n-l cm-‘) parameter defined by (24) chemical potential of electrolyte (J mol - ‘) chemical potential of salt A (J mol-‘) total number ofions produced by dissociation of a mole of electrolyte number of moles of speries i produced by dissociation of a mole of electrolyte density of electrolyte (gcmeJ) effective matrix conductivity (SY’ cm-l) dimensionless time electric potential in the matrix (V) electric potential in the solution (V) Subscripts A B e 0 0s oe R m
lithium chloride potassium chloride electrolyte solvent in separator at electrode-separator in electrode at electrode-separator reference condition bulk solution property
Superscripts A I3 0
lithium chloride potassium chloride initial value
boundary boundary
R. POLLARD urn J.NEWMAN
316
lNTRODUCTlON Many porous electrodes in commercially important batteries have high values for the product of the electrochemically active specific interfacial area and the exchange current density. This permits operation at high superficial current densities without excessive surface overpotentials which would lower the energy efficiency of the system. Previously, the distributions of current and electrochemical reaction in porous electrodes have been considered under various special conditions[ 1,2]. Mass transport of reactants and the effective conductivities of the matrix and solution phases have a strong influence on electrode behaviour. The current distribution becomes non-uniform for large values of active surface area, exchange current density, electrode length, and current density, and for small conductivities. Particular attention has been paid to the electrodes of the lead-acid battery[3-61, and several of the proposed models make the assumption of reversible kinetics[& 63. With this assumption, it has been shown that highly non-uniform reaction distributions can be obtained, especially for high rate discharges[?+ This can result’ in an increase in internal cell resistance because local changes in electrolyte composition lower the electrolyte conductivity. Under some circumstances the electrode could become totally passivated. It has also been shown that, in principle, a fraction of the total reaction will be restricted to the electrode-separator interface[5]. Another study allows all the current to penetrate the electrode, and concludes that models with reversible kinetics are inferior because they cannot take proper account of transport of external acid into the electrode pores[6]. This paper considers the nature of the current and reaction distributions for electrode processes with high, but not infinite, exchange current densities. A simplified model is developed in order to give an analytic solution to the governing differential equations. The results obtained illustrate the important features that would be observed with a more sophisticated approach. The method used in this paper complements a computer solution which is able to account for many complications not considered here, but which cannot easily clairfy the highly non-uniform reaction distribution at moderately short times that results from a high value of the exchange current density. The model can also be used to assess quantitatively the validity of the assumption of reversible kinetics for a particular electrode system.
ANALYSIS
The system considered is a porous electrode in contact with an inert porous separator. The electrode is treated as two superposed continua which represent the solution and matrix phases independently. The actual geometric details of the pores bave been disregarded and only variations of parameters in a direction normal to the electrode face are considered. Each phase is taken to b-e electrically neutral, and consequently the divergence of the total current density is zero.
The analysis is restricted to a single electrode reaction with stoichiometry represented by :
(1)
7 siMp+ne-. A polarization equation of the form: j = ai,[@(%*&‘Rx
- e-@t’%-%r/RrJ
(2)
can be used to express the dependence of the reaction rate at any point in the electrode on the local potential jump at the matrix-solution interface. The potential of the pore solution a2 is measured with a reference electrode of the same kind as the working electrode. The composition dependence of the exchange current density can be assumed to have the form:
where i ranges over the ionic species. Solid phase activities are taken as constant, and initial concentrations are used for the reference condition, R. The possibility of homogeneous chemical reactions is not considered, and double layer charging effects are ignored. To simplify the analysis further, it is assumed that transport properties and pore structure do not change as the reaction proceeds. Under these circumstances, the major concentration changes are caused by electrochemical reaction and by diffusion of electrolyte. Therefore, a non-steady state material balance for species i becomes:
aci B d2C, = Fj + &D7. af dY
E-
For times over which (4) is to be applied, the parameter B is taken as a constant, dependent on the system considered. Relationships for B, for tbe positive plate of a lead-acid battery and for the negative electrode in a high temperature LiAl-LiCl, KCl-FeS, battery, are derived in Appendices I and II, respectively. The molecular diffusion coefficient D can be characterized by D = 0,/C=.
(5)
Also, it is assumed that the tortuosity factor c is directly related to porosity by[7]: < = e(l-9)/Zt
(6)
where the constant q is taken as 1.5. The driving force for tbe current can be specified in terms of the potential in the pore solution a2 measured with a reference electrode. With this definition, a modified Ohm’s taw for the pore electrolyte is: i2 -= K
-V@,+m
RTW F
Vinci,
(7)
where the second term on the right includes the diffusion potential. In this analysis, the parameter W is regarded as a constant, dependent on the electrode studied (see Appendices 1 and II). The effective conductivity K is estimated from K =
&Km/~‘,
(8)
together with (6). The movement of electrons in the matrix phase is
317
Transient behaviour of porous electrodes governed by Ohm’s law. However, it is assumed that the matrix conductivity is large (U/K >> 1) so that the matrix potential can be taken to be uniform. With this condition, the initial electrode reaction will be confined to a region adjacent to the electrode-separator interface. This reaction zone can be characterized with a penetration depth L which, for linear kinetics and constant matrix potential, is defined as[8] :
where i,, is evaluated at the initial composition. When the discharge process begins there is a rapid and substantial change in current distribution, caused by changes in composition. Nevertheless, the concentrations have only altered significantly from the standpoint of concentration overpotential, and the ratio ci/ciR has not changed enough to affect appreciably tbe exchange current density given by (3). In this restricted range ofinvestigation it is appropriate to define a set of dimensionless variables r = Dt/L2,
J = jL/I,
rl = WI
- W/IL,
Bi = F6D(ci - c;)/ILB, which have values of order unity. After these definitions have been substituted into the governing equations, it becomes possible to identify those terms which can be neglected as i, becomes large. When the exchange current density, or the product ato, is large, the local surface overpotential for a specified supcrficia.l current density I is expected to be low. Consequently, it is reasonable to reduce (2) to the form :
a0 -=J+ay2.
J = 4.
(11)
It might appear that (11) can be obtained directly from (2) provided that (@r - &)F/RT << 1. However, the analysis above emphasises that it is necessary to verify that the composition dependence of the exchange current density does not alter this result. For small concentration changes, it becomes valid to linearize the concentration dependent term in (7). Combination of (7) and (11) then gives, on differentiation :
J=g;+Qg,
e(Y,o)
D&F2
’
= 0,
(i) fI(Y,r)--, 0 as Y -+ 00, (ii) at the electrode-separator (a) i, = --_I,
(15)
interface (Y = 0);
(b) 0 continuous,
(16)
For condition (i), the electrode is assumed to be considerably thicker than the penetration depth so that its length can be regarded as infinite. The last boundary condition assumes that the reaction rate is not a Dirac &function at the origin. Equations (12)-( 16) define the dependence of concentration and transfer current on time and distance through the porous electrode for a constant current discharge. The reaction zone is restricted to the region immediately adjacent to the separator, and this limits the applicability of the results, as discussed below. The equations refer to systems with high exchange current densities or small penetration depths, but it should be emphasized that the macrohomogeneous model on which the analysis is based will break down if the reaction xone is significantly smaller than the characteristic dimensions of the microscopic porous structure. AND
DISCUSSION
In terms of Laplaos transforms, the solution to the stated problem may be written as:
J(Y,s) = AemEY f-
(17)
in which the variable s is defined by:
Jam e-nF(r)dr.
f(s) =
The parameters A and E are related by the imposed requirement that integration of (17) across the electrode gives the superficial current density, viz:
s m
0
&(Y,s)dY
= ‘. s
(19)
Equations (5). (6), (1 S), (16). and (19) may be combined to specify the coefficient A and E as: A
Q=
(12) and (14),
and the boundary conditions :
where Y = y/L and IC,RTBW
(14)
The governing differential equations are subject to the initial condition:
RESULTS
As the exchange current density is raised, the pens tration depth decreases and, for Br = O(l), the right side of (10) approaches unity. Under these circumstances, (10) can be simplified further to give:
a’8
at
=
s
(13)
In the appendices, it is shown that only one concentration variable is required for (7), and, therefore, the subscript on the dimensionless variable 19,can be dropped. The material balance, (4), can be rewritten in dimensionless form :
_A
and
[(~--EZ)u+P~/E)_ 1 ’ (20) 1 -I
(1 -s/E’)(l+PE)
2s 1+s+Q+[(1+s+Q)2-4~]1’2
112 >
’
(21)
where p = (c,/e)=~-
1’.
(22)
R. POLLARD AND J. NEWMAN
318
Equation (18), together with the definitions of T and L, indicates that the variables is inversely proportional to the exchange current density. Consequently, inversion of (17) in the limit as S-B 0 should give a moderate time solution for the dimensionless transfer current. Two terms in the inverse Laplace transform for (17) are:
+PbOz + 2H+ + $SO:-
l--1
J=,/~Ie-Ji+Qy+
Jme-*‘(23)
+ e= +PbS04
+ H20.
(26)
The stoichiometric coefficients defined in accordance
where
A=
Two examples have been chosen to illustrate these concepts; the positive plate of a lead-acid battery and the negative electrode of a LiAI-FeS, high temperature cell with molten LiCl, KCI electrolyte. For the lead-acid system, it is assumed that the electrolyte dissociates completely. The reaction for discharge of the lead dioxide electrode is then:
with (1) are, s+ = -2, s- = -3, s,, = 1, and n = 1. With the electrode parameters given in Table 1, the
&/l-I-Q+1 &/_+1+Q’
and t] = y/24m.
(25)
The variables Y and 9 can be regarded as inner and outer perturbation parameters, respectively. In the region Y 5 O(l), (23) describes the reaction zone immediately adjacent to the separator, whereas in the region q 2 O(1) it describes the progressive penetration of the reaction through the electrode. The first term in the equation represents a time-independent spike in J which is largest at the separator and which penetrates a relatively short distance into the electrode. The factor in the exponent reflects the influence of da on the spike shape. The fraction of the total current associated with the spike, II, is an important parameter in the description of an electrode process with a high exchange current density. If rZ.is small, most of the current will be progressively distributed through the electrode where as, as 1 approaches unity, it becomes more acceptable to disregard the time-dependent term in (23) and to assume that the reaction is restricted to the spike aione. The spike is sharper for larger exchange current densities, and in the limit of infinite exchange current density it will become a Dirac 8 function (with area U) at the electrode-separator interface. Consequently, in electrode models which neglect the surface overpotential (reversible kinetics, i, = a)), only a fraction (1 - .I) of the total current should be distributed through the electrode.
model proposed indicates that 39% of the superficial current density is associated with the timeindependent spike in (23). Alternatively, if incomplete dissociation of electrolyte were assumed, it would be necessary to change the stoichiometric coefficients for the electrode reaction, the cation transference number, and the activity coefficient for the electrolyte in a consistent manner. The reaction at the LiAl electrode of the high temperature battery is LiAl -+ Al + Lit
cm-‘) D, X lO~(cm~s-‘) T(K) &z E” a, = 01,
K~(R-~
103c~(molcm-“) &cm-“) 1+
din?
and LiAl electrodes
1.75 3.33 723.15 0.75 0.20 0.50 0.58 33.92
2.30
1.92
to p’
0.73
0.29
Q a
6.17
d In m
L(‘W
Parameter
LiAl 0.9 1 3.03 298.15 0.56 0.52 0.50 5.00 I .29
1.10 0.39 0.42
(27)
and therefore, s+ = -1, s-=0, s,=Oand n=l. Table 1 lists system parameters for discharge with an electrolyte of eutectic composition. The analysis shows that 85% of the total current is in the spike. Figure 1 gives the dependence of the dimensionless transfer current on the inner perturbation parameter Y for the LiAl electrode. For small values of Y, the major contribution to J is from the timeindependent term in (23), and the distance over which the spike is dominant increases with time for T 2 10. At progressively larger times the reaction penetrates further into theelectrode, and J falls locally in response to the requirement of constant total current. Eventually the material closest to the separator will become depleted of reactant, and the narrow reaction zone will then move through the electrode as the available material isconsumed. At this stage, the analysis presented here will no longer be valid. Additional complications can arise if local changes in composition significantly affect the exchange current density through (3). For example, if i, is
Table 1. Typical parameters for PbO,
Parameter
+ e-
5.22 1.60 0.85 1.32 x lo-’
K,(R-“%I-‘)
D,
x IO5 (cm’ se’)
T(K) E” : E cc. = a.
5:
V(cm3molm1) 1 + dlny, dlnx, r: P
Q R Wm)
Transient behaviour of porous electrodes
-4
319
-4
In tJ1 In(J) -6
-6
-\ -I 2
0
I I
L 2
I 3
I 4
I 5
I 6
-12 7
0
1
I
/
I
2
3
,‘\ 4
, 5
I 6
7
Y
Fig. 1. Solution in the inner region for the dimensionless transfer current at different times, 5, for the LiAl electrode,as
Fig. 2. Solution in the inner region for the dimensionless transfercurrent at different times, +, for the PbO, electrode, as predictedwith (23). The.solution at r = 0 is from (28). -time-independentspike.
most sensitive to the reactant concentration, it is expected that the spike will become more diffuse as the reaction proceeds. This is not accounted for by the model, although concentration changes are included in the non-steady-state material balance, (14). The zero time soiution of (17),
associated with the spike is negligible. The dimensionless distance within which half the current that pens trates is to be found is given by 0.913~~. This dependence indicates that the process is dominated by diffusion but that interactions between difision and reaction modify the current distribution. The reaction rate distribution at very short times is
predictedwith (23).The solution at 1 = 0 is from (28).- - - time-independentspike.
J = e-‘,
(28)
is also shown in Fig. 1 in order to emphasize how the reaction distribution becomes separated into two components corresponding to the two terms in (23). The apparent agreement between the zero time solution and the reaction distribution at 5 = 1 is coincidental; the moderate time solution cannot be extended to T < 1 and, moreover, an assessment of the influence of higher order terms not shown in (23) is needed to establish its validity at r = 1. Figure 2 shows the inner solution at moderate times for the Pb03 electrode. The different values for the parameters Q and d, in comparison with the LiAl electrode, both make the penetration of the spike smaller and hence reduce the contribution of this term to the total transfer current at a given position in the electrode. The penetration depth for the PbO, system, as calculated with (9), is approxirnately0.4 cm, which is considerably larger than the corresponding value for the LiAl electrode (see Table 1). Since the assumption, L CC1, is used in the high current density analysis for the PbO, electrode, the results should be treated with caution and, in particular, the assumption of reversible kinetics that has been used previously[5,6] requires some justification. The outer solution for J for the LiAl electrode at moderate times is given in Fig. 3 in terms of the outer perturbation parameter, 8. The incomplete curves show the contribution of the time-dependent term in (23). The intercepts of these parabolae represent inner limits of the outer expansion, given by (1 -2)/x,/m. This is also seen in Fig. 1 as the magnitude of J in the region where the current
Fig. 3. Solution in the inner region for the dimensionless transfer current at Merent times, 7, for the LiAl electrode, as
predictedwith (23). The solutionat T = 0 is from (28). - - - time-dependentterm in (23).
R. POLLARD AND J. NEWMAN
320
obtained as a power series expansion in r, by inversion of (17) in the limit as s -+ 00. It is found that:
regions. A more general inversion of the solution in Laplace space, (17), is needed to be able to predict this intermediate behaviour. CONCLUSIONS
*(;y’zz3,2(a,]
-
+ (s)r
[(I +2z’)erfc(z)
- 5
(29)
eCzz]
+ ($&3,‘[?-+2
where 01= m, z = ,/mq, and Z,(a) is a modified Bessel function of the first kind and of order v. For short times, eg r -z O(lO-‘1, J can be predicted adequately by the zero time solution, (28). At first sight, the e-’ term in (29) would seem to be equivalent to the spike observed in the moderate time solution. Numerically, however, the clear distinction between the two contributions at moderate times has not developed at the short times for which (29) is valid. This range of validity can be assessed by a comparison of the first order, short-time approximation for .I with the approximations accurate to orders T and z3’2(JI,Jz,JS reSpeCtiV&). Figure 4 shows that, for the LiAl electrode, the validity of (29) becomes questionable for T z 0.1. At r = 0.1, inclusion of the r3” term causes a relatively minor shift in Jz. At 7 = 0.5, the difference between .J2and .I, indicates that higher order terms are needed to obtain a satisfactory solution. For the PbO, electrode, the range of validity of (29) is restricted to even smaller values for 7 be-cause the parameter Q is significantly larger than for the LiAl system (see Table 1). The distinct differences in short and moderate time behaviour predicted by (23) and (29) preclude the formation of a combined perturbation expansion that would provide an acceptable link between the two time 7,
I
I
I
I
I
I
1
6
A simplified macrohomogeneous model can be used to elucidate the transient behaviour of porous electrodes with high exchange current densities. The highly non-uniform initial reaction distribution changes at moderately short times due to the influence of concentration overpotential. The analysis presented can be used to predict the magnitude of these changes and to characterize the time-dependent and timeindependent components of the transfer current at a given position within the electrode. The results show clearly the association between high exchange current densities and non-uniform reaction distributions which can, in turn, have a significant impact on battery performance. Acknowledgement -This work was supported by the Division of Solar, Geothermal, Electric and Storage Systems, Office of the Assistant Secretary for Energy Technology, U.S. Department of Energy under contract No. W-74QS-Eng-48. REFERENCES
1. J. Newman and C. W. Tobias. J. electrochem. Sm. 109, 1183 (1962). 2. I. Newman, Internal tech. report, NOLC project, Dept. Chem. Engng, University of California, Berkeley (1962). 3. K. Micka and I. Rousar, Electrochim. Acta. 21, 599 (1976). 4. D. Simonsson, J. appl. Electrochem. 3, 261 (1973). 5. W. Stein, Ph.D. thesis, Rheinisch-Westfalischen Tech. Hochschule, Aachen, %‘eqt Germany (1959). Actn 18, 629 6. K. Micka and I. Rousar, Electrochim. (1973); ibid. 19, 499 (1974). in Elec7. J. Newman and W. Tiedemann, Advances trochemfsrry and Elecrrochemical Engineering (Edited by H. Gerischer and C. W. Tobiasf. Vol. 11. WileyInters&ace, New York (1977). 8. J.Newmanand W.Tiedemann, A.I.Ch.E.JL 21,25 (1975). 9. J. Newman and T. W. Chapman, A.1.Ch.E. J1. 19, 343 (1973). 10. R. Pollard and J. Newman, J. electrochem. Sot. 126,1713 (1979). 11. M. Blander in Molten Salt Chemistry (Edited by M. Blander). Interscience, New York, pp, 127-238 (1964). APPENDKXS
Derivation
of parameters
for porous electrode
model
Appendix I. Positive plate of lead-acid battery. In the PbOl electrode, a non-steady state material balance for species i may be combined with Faraday’s law to give:
4 eY J 3
J(%) -= at
-V.Nf
(30)
--$j
for a single electrode reaction[8]. For a concentrated biiy electrolyte, the superficial flux of species i can be expressed in terms of the superficial volume average velocity uw [8,9],
I
0 0
I
I 2
I
I
I
I
3
4
5
6
N, = - EDVC, + 5 7
Y Fig. 4. Reaction rate distribution for Lil electrode at short times, as predicted with (29). J,, J2, J,, are first, second, and third order approximations, respectively.
1
i, + c@.
Substitution into (30) and comparison Parameter B as :
(31)
with (4) defines the
v+ Y->I (32)
s+t”_ ---++
s-t’,
.
Transient behaviour of porous electrodes Ohm’s law can be used in the form
321
:
where az is the potential in the pore solution measured with a reference electrode of the same kind as the working electrode. The chemical potential of the electrolyte can be defmed with : ,=~(1+~)v1,.,.
provided that the molar volume of electrolyte v is constant. Ohm’s law for the pore electrolyte can be written in a form analogous to (33) as:
(34) where
where Y is the total number ofcations and anions produced by dissociation of one molecule. Consequently, the parameter W in (7) is @ven by :
Appendix II. Negative elecrrode of LiAl/FeS, battery. This high temperature system can use an electrolyte that comprises two binary molten salts with a common ion, such as the LiCl, KC1 eutcctic. In the derivation below, subscripts 1, 2, and 3 refer to Li+, K+, and Cl- ions, respectively, and scripts A and B refer, in turn, to the LiCl and KC1 salts. The superficial flux of lithium ions baaed on the molar average reference velocity is[ lo] : N, = -
&(CI +
c,)+Dv:v’ CT
vx, +
5 +c,v*, 1
(36)
where xA = cA/(cA + ca) is the mole fraction of LiCl in the electrolyte and cr is the total concentration defined by cl. = ct + ct f cs. Combination of (30) and (36) gives, on simplification, (4) with :
G+$+~)(l+J@~.
(39)
The activity coefhcient of a salt in a binary molten salt electrolyte mixture is often defined without regard for dissociation and in terms of ion equivalent fractions or mole fractions[l 11. (These composition variables are equivalent when v< = vg.) The gradient of chemical potential can be written as[ 10) : r:+s)Vlne,,
(40)
and with this definition the parameter W in (7) becomes:
W= -G
(41)
In the absence of experimental data for transference numbers it is assumed that transference numbers are directly proportional to the electrolyte mole &action, viz: t: = ?&Iv*> t$ = vfxB/vB,and t; = x”.