1670
J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
3. C. F. Holmes and P. Keister, Abstract 39, p. 61, The
Electrochemical Society Extended Abstracts,
Vol. 86-2, San Diego, CA, Oct. 19-24, 1986.
4. C. F. Holmes, R. L. McLean, and C. C. Liang, in "Cardiac Pacing," G. A. Feruglio, Editor, p. 1193, Piccin
Medical Books, Padua, Italy (1982).
5. M. J. Brookman and R. L. McLean, Abstract 303,
p. 489, The Electrochemical Society E x t e n d e d Abstracts, Vol. 82-2, Oct. 17-21, 1982.
6. M. J. Brookman and R. L. McLean, Abstract 90, p. 142,
7.
8.
9.
10.
The Electrochemical Society Extended Abstracts,
Vol. 83-2, Washington, DC, Oct. 9-14, 1983.
Cricket Graph, Version 1.3, Cricket Software, Inc.,
Malvern, PA (1988).
Mathematica, Version 1.1, Wolfram Research, Inc.,
Champaign, IL (1988).
K.M. Abraham and M. Alamgir, This Journal, 134, 2112
(1987).
E. S. Takeuchi, K. J. Takeuchi, C. Bessel, and E. Cads,
J. Power Sources, 24, 229 (1988).
Nucleation of Lead Sulfate in Porous Lead-Dioxide Electrodes
Dawn M. Bernardi*
Physical Chemistry Department, General Motors Research Laboratories, Warren, Michigan 48090-9055
ABSTRACT
A one-dimensional mathematical model of a porous lead-dioxide electrode is described and used to investigate leadsulfate nucleation and growth during discharge. Derivation of a nucleation rate expression that is based on classical, heterogeneous nucleation theory is outlined. An electrochemical kinetic expression is derived based on a reaction mechanism involving elementary steps, and concentrated ternary electrolyte theory is used in formulating material-transport
equations. Nucleation and electrochemical kinetic parameters are estimated by comparison of model results with experimental results available in the literature. The interplay of nucleation and growth kinetics of lead sulfate is responsible for
the initial m i n i m u m in the voltage-time curve that is c o m m o n l y observed during constant-current discharge. The model
simulates the voltage minimum, which is referred to as the coup defouet, and calculates the degree of lead-ion supersaturation, the n u m b e r density of lead-sulfate particles, and the free energy of formation as well as the size of critical nuclei. The
model also predicts a disappearance of the voltage m i n i m u m with the addition of seed particles for lead-sulfate nucleation, which is experimentally observed. The satisfactory agreement between model and experimental results confirms
that the voltage dip is caused by a temporary oversaturation of lead ions during discharge and supports the proposed
theoretical approach.
During constant-current discharge of a fully charged
lead-acid cell, a m i n i m u m in voltage at the beginning of
the voltage-time curve is observed (1), which is often referred to as the coup de fouet. The shape of the discharge
curve has led to this French terminology, which translates
to English as "stroke of a whip." The coup de fouet is believed to be caused by a supersaturated solution of lead
ions occurring temporarily in the lead-dioxide electrode
during discharge. The duration of this voltage transient is
rather short in comparison to the full discharge time of the
battery, and the magnitude of the dip is small compared to
the terminal voltage. The effect, however, is of consequence in the design of batteries for the operation of sensitive switches (2), and a better understanding of the phenomena will fill one of the gaps in our knowledge about
these batteries.
The observation of a voltage dip during the discharge of
lead-acid batteries was first recorded in the open literature
in the early 1900's and soon after postulated to be caused
by a temporary supersaturation of lead ions of u n k n o w n
origin (2). The mechanism associated with this phenomenon has since been subject to several other explanations.
In 1955, Vinal (3) attributed it to ohmic resistance and a
sudden drop in acid concentration, which would reduce
the electrode potential. Later, in a series of papers, the
coup de fouet was shown to be associated with PbO2 electrode (4) and in particular the B-modification of the oxide
(5). Voltage minima were also demonstrated in the discharge of MnO~ and T1203 electrodes, and the coup de fouet
was suggested as being connected with expansion of the
oxide lattice (6). B6rndt and Voss (2) in 1965 presented a
relatively detailed study of the voltage dip observed in the
discharge of porous, commercial-size PbO2 electrodes,
with the supposition that the effect is associated with leadion supersaturation and PbSO4 nucleation and crystal
growth, which is now the generally accepted mechanism
(1). Slowness in the elucidation of the cause of the coup de
fouet was closely associated with uncertainties in the discharge mechanism of the PbO2 electrode; for example, solution-phase lead ions must be a reaction intermediate if
* Electrochemical Society Active Member.
their supersaturation is the cause of the voltage dip. A
solid-state diffusion discharge mechanism has been proposed (7) which would conflict with the aforementioned
cause of the coup de fouet. Recently, H~meenoja et. al. (8)
reported rotating ring-disk electrode measurements that
proved the existence of solution-phase lead ions and substantiated that they were an intermediate in the PbO2 discharge mechanism and the cause of the coup de fouet.
The reason that a voltage drop should be the consequence of an oversaturated solution of lead ions can be
seen through simple thermodynamic arguments. During
normal discharge and charge conditions, there are two
solid phases present in the lead-dioxide electrode system
(PbSO4 and PbO2, see Fig. 1) along with the electrolytic solution phase. The Gibbs phase rule can be applied at constant temperature, pressure, and H2SO4 composition to
show that the state of the system is fixed under these conditions; that is, the thermodynamic open-circuit potential
(relative to a reference electrode such as Hg/Hg2SO4) is a
constant. In a system without any PbSO4 present, one degree of freedom exists and we can expect the potential to
vary with the lead-ion concentration in the electrolyte. Of
course, our model is not formulated as an equilibrated system, but the basic principles apply in understanding the
phenomena. Early in the discharge of a PbO2 electrode
which contains virtually no lead sulfate, the acid ,concentration remains relatively constant, while lead ions, which
are the product of the overall electrochemical reaction
PbO2(~) + 4H + + 2e- --> Pb 2+ + 2H20
[I]
can reach large concentrations within the pores of the electrode. We expect the electrode potential (relative to a reference electrode) to fall in accordance with this concentration increase until the PbSO4 nucleation and growth
processes
Pb 2+ + HSO4 ---->PbSO4cs) + H +
[II]
(which are accelerated by increases in Pb z+ concentration)
can overcome the electrochemical reaction process and
thereby stabilize potential by reducing the Pb 2+ concentration. In the formulation of a mathematical description of
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this scenario, t h e processes d e s c r i b e d a b o v e m u s t be characterized along w i t h t r a n s p o r t p r o c e s s e s of diffusion, migration, and c o n v e c t i o n of electrolyte species.
T h e d e v e l o p m e n t of a m a t h e m a t i c a l m o d e l of the PbO2
e l e c t r o d e that can s i m u l a t e the c o u p de f o u e t is t h e n e x t
logical step in substantiating an u n d e r s t a n d i n g of t h e phen o m e n a - t h i s is t h e objective of our work. P r e v i o u s researchers h a v e c o n c e n t r a t e d m o d e l i n g efforts on t h e leadd i o x i d e e l e c t r o d e (9, 10) and the lead-acid cell (11, 12).
T h e i r m o d e l s h a v e n o t c o n s i d e r e d t h e p r e c i p i t a t i o n process; rather, t h e y h a v e a s s u m e d that PbSO4 was a direct
p r o d u c t of t h e d i s c h a r g e reaction. As yet, a m a t h e m a t i c a l
m o d e l of t h e effects of n u c l e a t i o n on d i s c h a r g e b e h a v i o r of
l e a d - d i o x i d e electrodes has not a p p e a r e d in the literature.
R e c e n t l y , Tsaur and Pollard (13) p r e s e n t e d a m a t h e m a t i c a l
m o d e l that i n c l u d e d empirically b a s e d e x p r e s s i o n s for
precipitate n u c l e a t i o n and g r o w t h kinetics and p r e d i c t e d a
v o l t a g e m i n i m u m w h e n the m o d e l was u s e d to s i m u l a t e
l i t h i u m / t h i o n y l - c h l o r i d e cell behavior.
T h e m o d e l p r e s e n t e d h e r e can p r e d i c t t h e n u m b e r of
PbSO4 particles n u c l e a t e d as a f u n c t i o n of d i s c h a r g e rate,
w h i c h can u l t i m a t e l y d e t e r m i n e t h e e l e c t r o d e capacity
(1, 14). The n u m b e r density is a f u n c t i o n of initial acid conc e n t r a t i o n and t e m p e r a t u r e , as well as d i s c h a r g e c u r r e n t
d e n s i t y (1); h o w e v e r , t h e scope of this w o r k will be limited
to t h e effect of d i s c h a r g e rate.
Model Description
F i g u r e 1 gives a s c h e m a t i c description of a porous PbO2
electrode. T h e porous PbO2 active material contacts a curr e n t collector, and v o i d space is o c c u p i e d by a concentrated sulfuric-acid solution that is saturated in lead sulfate. T h e p o r o u s electrode is s u r r o u n d e d by a w e l l - m i x e d
r e s e r v o i r of the solution. The m o d e l is o n e - d i m e n s i o n a l in
t h e direction p e r p e n d i c u l a r to the current collector, w h i c h
w e d e s i g n a t e as t h e x direction. The m o d e l is m a c r o h o m o g e n e o u s - - v o l u m e a v e r a g e d quantities that are continuous f u n c t i o n s of t i m e and space are u s e d to d e s c r i b e t h e
e l e c t r o d e (15), and it is d e v e l o p e d for c o n s t a n t - c u r r e n t operation. T h e m o d e l e q u a t i o n s are g i v e n for the case of a
single overall e l e c t r o c h e m i c a l and p r e c i p i t a t i o n reaction.
T h e s t o i c h i o m e t r i c coefficients of the overall e l e c t r o c h e m ical r e a c t i o n (reaction I) and precipitation reaction (reaction II) are defined by
~ S i , e M i zi = n e i
[1]
~Si,sMi zl = 0
[2]
1671
I f w e designate, a = Pb, b = PbSO4, 1 = H S O ( , 2 = H +,
3 = P b +2, and 0 = H20, t h e n the s t o i c h i o m e t r i c coefficients
are SO,e= 2, SLe = 0, S2,~= - - 4 , S3,e = 1, Sa,~=--1, Sb,e = 0,
S0,~ = 0, S~,~= 1, S2,~ = --1, S3.~ = 1, Sa,~ = 0, Sb,~ = --1, and n = 2.
T h e ionic c o m p o n e n t s m a k e up two electrolytes A and B,
and their c o n c e n t r a t i o n s are related by
C2
CA
C3
= - Pa
- A and
cB = - y3
- B
[3]
In writing Eq. [3] it is a s s u m e d that the ionic c o m p o n e n t
that is c o m m o n to A and B is species 1.
M o d e l e q u a t i o n s . - - T h e m a t h e m a t i c a l m o d e l consists of
eight e q u a t i o n s w i t h eight u n k n o w n s : v o l u m e fraction of
electrolyte (e), potential in the solution (qS~ot~),potential in
t h e solid (~ond), c o n c e n t r a t i o n of electrolyte A (cA), concentration of electrolyte B (CB), superficial c u r r e n t density in
t h e electrolyte (i~o~), superficial v o l u m e - a v e r a g e v e l o c i t y
(v~), and the n u m b e r density of lead-sulfate particles (nb).
T h e eight following relations constitute the m o d e l equations. T h e first e q u a t i o n
porosity variation
06
C3isoln
at
Ox
1 (~aSa, e 4- YbSb,e) 4-
Rb s -' (VaSa,s 4- YbSb,s) = 0
[4]
sb,s
nF
is a v o l u m e balance for the two solid phases, a and b, and
indicates h o w the e l e c t r o d e porosity c h a n g e as a result of
the e l e c t r o c h e m i c a l and p r e c i p i t a t i o n reactions. A p p e n d i x
B details an e x p r e s s i o n for t h e rate of t h e p r e c i p i t a t i o n reaction, Rb, in t e r m s of t h e lead-ion c o n c e n t r a t i o n and t h e
PbSO4 surface area available for growth.
The m o v e m e n t of electrons in an electronically c o n d u c t ing p h a s e is g o v e r n e d by
O h m ' s law in t h e solid m a t r i x
tgt~solid
isoln -- EoexmEr - -- I = 0
[5]
ox
T h e electronic c o n d u c t i v i t y is in t e r m s of an effective cond u c t i v i t y for a p o r o u s m e d i u m , ea~"a (16).
T h e species c o n s e r v a t i o n e q u a t i o n s
and
material balance for electrolyte A
{
e -
respectively, w h e r e M~ is a s y m b o l for t h e c h e m i c a l form u l a of species i, and z~ is the charge n u m b e r of species i.
-- - -
3t
+
Ox
O/~ol~ 1 [
--
Ox /
CA(SaeVa + S ~ , ~ )
Ox n F
Porous
Electrode
'
+ v ~ OCA + C~ - c3X
c3X
+ S~,o +
v2A
n(t2~
z2v~A
Rbs[
C A ( S . ~ + Sb,~Yb) +
Sb,s
J
= 0
[6]
1)2 _1
and
'Z."(,/
Current
Collector
+ eeXDAB
eexD~
Ox \
u,-,-,,
~/D:_W
H+
,m~
5,s
material balance for electrolyte B
\
OCB
0
at
aX
eeXDBA~ X + eeXDBB
+ -Oisoin
- - - 1 [ CB(Sa,eV,+ Sb.eVb) + S3,e + n(t~ ~ - z3csQ)_]
2L,~
ox
'_,'Y-,"
/4 ~(/k
....
X~L
nF
vsB
0vD
+ %--
+ v[]
0X
0X
Rb.s [
Sb, s [
zavs B
J
S3.~]
cB(s.,s~a + sb,~Yb) + - - - ' / = 0
[7]
p3BJ
I L "~ f**"~
I
)
X
Fig. 1. Schematic of a lead-dioxide porous electrode
are b a s e d on c o n c e n t r a t e d t e r n a r y electrolyte t h e o r y and
h a v e b e e n d e v e l o p e d b y S u n u (17, 18). T h e y are o b t a i n e d
by p l a c i n g an e x p r e s s i o n for species flux, Ni, in t h e general
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J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
1672
material-balance expression Oeci/Ot = -ONi/Ox - si,J(nF)Oi~old
ox + R~. S u n u gives the e x p r e s s i o n s for the fluxes N2 and
N3 (and No) relative to the v o l u m e - a v e r a g e v e l o c i t y ~ in
t e r m s of chemical-potential gradients, and A p p e n d i x A
gives results of the diffusion coefficients and t r a n s f e r e n c e
n u m b e r s (and Q) as functions of the binary-interaction par a m e t e r s u s e d in the c o n c e n t r a t e d - s o l u t i o n theory. In dev e l o p i n g Eq. [6] and [7], it was a s s u m e d that the partial
m o l a r v o l u m e s , transference n u m b e r s , and activity coefficients of t h e electrolyte species are i n d e p e n d e n t of t h e solution concentration, and Eq. [4] was u s e d to eliminate
OdOt.
A n overall material balance on the solution phase
Rate expression for the nucleation of lead s u l f a t e . -
electrolyte v e l o c i t y
Ovcl 0iso[~ 1 [
+ --V~sa,~+ VbSb,~
Ox
Ox nF l
--
(s2,~+ nt2~
(s3.,~+ nt3~
]
Z~V2A] + ~VB \v3
Z3v3B] + VoSo.~ - nQ
+ VA \v2A
Rbs [--
83s
--
S2,s
--
sb,~' \~VB~u3 + Vh--v2n + VoSo~, + Y~Sa~. + Vbsu,~) = 0 [9]
p r o v i d e s us w i t h an e x p r e s s i o n that describes c h a n g e s in
t h e v o l u m e - a v e r a g e velocity. This e q u a t i o n is d e r i v e d by
m u l t i p l y i n g the material-balance e x p r e s s i o n s for species
A, B, and 0, by VA, VB, and ~d0, respectively, and a d d i n g the
r e s u l t i n g e q u a t i o n s together.
T h e sixth m o d e l e q u a t i o n
O h m ' s law in solution
RT [ s2,~
t2~
__i~~ + 0~o1~ +(vlA + " 2 A ' ) _ [_-TXZ + - EexK
~X
F
\122 n
R T [ s~,e
ta~
+ (vl s + v~~ ) - |
~ +
F
\1~3 n
z21J2A
So,~CA~0 In (ca)
Con ]~
Ox
So,~CB~0 in (CB)
---i--con /
z31~3B
0 [lO]
Ox
relates gradients in solution potential and c h e m i c a l potential of t h e electrolytes to local c u r r e n t density. Here, we
h a v e c h o s e n to define potential in t h e solution (Go~) relative to a r e f e r e n c e electrode that is the s a m e k i n d as the porous electrode, and t h e solution c o n d u c t i v i t y is w r i t t e n in
t e r m s o f an effective c o n d u c t i v i t y for a porous m e d i u m
eexK. A p p e n d i x A gives an e x p r e s s i o n for ~ as it is defined
by t h e c o n c e n t r a t e d - s o l u t i o n theory.
The relationship of local solution c u r r e n t to the overp o t e n t i a l is g o v e r n e d by
e l e c t r o d e kinetics
0isoln0x A~i~
e x p [~--~
[ a " F ($sona - d~o,,)]
]}
- e x p [ - ~-~(Go,ia - ~)soln)
=
0 [11]
In A p p e n d i x B, e x p r e s s i o n s for the e x c h a n g e c u r r e n t density i0 and t h e transfer coefficients a~ and ~c for the leadd i o x i d e e l e c t r o d e are d e r i v e d f r o m a possible reaction
m e c h a n i s m i n v o l v i n g e l e m e n t a r y steps.
T h e n u c l e a t i o n rate e x p r e s s i o n is g i v e n h e r e as
n u c l e a t i o n kinetics
Ot - 4 L~-RT)
ea(n~
n b. ) e x p.
.
3. ( R T ) a l n 2eq~]\j( ~C3
Classical, h e t e r o g e n e o u s - n u c l e a t i o n rate t h e o r y (20) is
b a s e d largely on c o n c e p t s similar to t h o s e that u n d e r l i e the
h o m o g e n e o u s rate t h e o r y first p r o p o s e d by V o l m e r and
W e b e r (21) in 1926 and d e v e l o p e d further by B e c k e r and
D o e r i n g (22). The description p r e s e n t e d involves a spherical c a p - s h a p e d PbSO4 critical n u c l e u s m a k i n g a characteristic c o n t a c t angle O w i t h the PbO2 substrate and h a v i n g a
radius of c u r v a t u r e rcrit. The n u c l e u s grows by m e a n s of
lead ions u n d e r g o i n g diffusion j u m p s f r o m the solution
a d j a c e n t to the nucleus. (HSO4- ions are c o n s i d e r e d to be
in large excess). In the following, t h e e x p r e s s i o n for t h e
h e t e r o g e n e o u s n u c l e a t i o n rate will be outlined based on
e x p r e s s i o n s g i v e n by Moazed and H i r t h (23). T h e nucleation rate Onpbso4/Otis f o r m u l a t e d as the a d d i t i o n f r e q u e n c y
of single lead-sulfate m o l e c u l e s w i t h clusters of critical
size, w h i c h is t h e p r o d u c t of t h e n u m b e r d e n s i t y of
critically sized nuclei located on a substrate (i.e., npbso4crit,
t h e n u m b e r of nuclei per u n i t area of substrate) and t h e
p r o b a b i l i t y in unit t i m e of p r e c i p i t a t e - f o r m i n g ions imp i n g i n g on the e m b r y o and successfully a d d i n g to t h e ass e m b l y (i.e., ~, t h e n u m b e r of PbSO4 m o l e c u l e s a d d e d per
n u c l e u s per second)
0npbso 4
Ot
= npbsO4Critto
[13]
A n e x p r e s s i o n for t h e free e n e r g y of f o r m a t i o n of a
critically sized n u c l e u s AGcrit is r e q u i r e d b e c a u s e t h e n u m b e r density npbso4crit is a s s u m e d to reside in a B o l t z m a n n
d i s t r i b u t i o n a b o u t AGcrit (see Eq. [21]). The q u a n t i t y AGcritis
d e p e n d e n t on the contact angle 0 and the d e g r e e of supersaturation. We m a y e x p e c t that for relatively large contact
angles t h e substrate will h a v e a greater p o t e n c y as a nucleation catalyst (24). Let us take, for e x a m p l e , the case of a
cluster of lead-sulfate m o l e c u l e s in the shape of a h e m i s pherical cap (90 ~ contact angle) that is located on a certain
site of PbO2 substrate. The Gibbs free e n e r g y of f o r m a t i o n
of t h e cluster is c o m p o s e d of surface and v o l u m e contributions and m a y be written as
2
AG = 2~r2~/+ -~rr~hGv
3
[14]
w h e r e -] is t h e PbSO4 solid-liquid specific interfacial free
e n e r g y 4 and t h e free e n e r g y of solidification per unit volu m e of PbSO4, AGv, is e x p r e s s e d as
-t~T
AG~- ~r
I n \ceb2+~q/
[15]
F o r a g i v e n v a l u e of AGv (i.e., d e g r e e of supersaturation),
w e can see that t h e surface e n e r g y c o n t r i b u t i o n (first t e r m
on the right side of Eq. [14]) is d o m i n a n t at small r and for
large r t h e v o l u m e c o n t r i b u t i o n to t h e free e n e r g y of the
cluster is dominant. T h e r e exists a m a x i m u m v a l u e of AG
~,-c'9 vexp - ~ - ~ - /
.
T h e e q u a t i o n s applied at t h e c u r r e n t collector b o u n d a r y
(x = 0) are OCA/OX= 0, OCdOX = O, 04~sodOX = 0, Eq. [4], Eq.
[11], v D = 0, i~o~ = 0, and Eq. [12]. T h e e q u a t i o n s cA = CA~f,
CB = CBref, 0~solid/0X = 0, ~soln = '+'solnA"
ref ---- -,n ~sol,
~ -- I, and nb = 0
are applied at t h e electrode/reservoir interface (x = L). 2
T h e s e c o n d i t i o n s r e p r e s e n t a w e l l m i x e d reservoir of electrolyte a d j a c e n t to t h e e l e c t r o d e 3 and an i m p e r m e a b l e curr e n t collector, respectively. T h e initial c o n d i t i o n s are
CA(X) = CAref, CB(X ) = CBref, and e(x)= e~ The eight m o d e l
e q u a t i o n s along w i t h t h e b o u n d a r y c o n d i t i o n s are cast into
finite difference forms and s o l v e d iteratively u s i n g t h e
m e t h o d of N e w m a n (19).
[12]
D e r i v a t i o n of this e q u a t i o n is outlined in t h e n e x t section.
' The flux expressions are most conveniently used in our porous
electrode model if they are written relative to the volume-average
velocity
vc] = N~V~ + N2V2 + N3V3+ N0V0
[8]
2 The equation nb= 0 is a consequence of the boundary condition
that the solution is saturated in lead ions at this point--since the solution is not oversaturated nucleation and precipitation of PbSO4
cannot occur.
3 The experimental setup of Bemdt and Voss (2) approaches this
boundary condition because they used a large excess of electrolyte
that was circulated between the PbO2 electrode and the counter
electrode during the tests.
4 For a spherical cap, ~/is isotropic and the three interfaeial free
energies of consequence (nucleus-solution, solution-substrate, and
nucleus-substrate) can be related by the Young equation. See Ref.
(24) for the general case of any 0.
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d. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
c o r r e s p o n d i n g to a c e r t a i n critical size o f t h e c l u s t e r Veto,
a b o v e w h i c h a n i n c r e a s e i n size m a y o c c u r w i t h a d e c r e a s e
i n free e n e r g y . We c a n m a x i m i z e Eq. [14] w i t h r e s p e c t to r
to o b t a i n t h e free e n e r g y o f f o r m a t i o n of t h e critical c l u s t e r
8,.~,y3
AGcrit -
[16]
3AG~ ~
Cpb2+
npb2+ = - -
T h e s u b s t i t u t i o n o f Eq. [15], [19], a n d [21]-[23] i n t o Eq. [13]
r e s u l t s i n t h e e x p r e s s i o n for t h e n u c l e a t i o n r a t e
Onpbso+ _ Vpbso4 (__~_~ ~ m (
-2~
rcrit --
T h e n u m b e r o f m o l e c u l e s i n t h e critical n u c l e u s ~ n t is exp r e s s e d as
2
16
&r
~mt = ~ ~ r ~ ~ - -
-
~r
3
/ \ a
~ t~-G-~)
9P~so~
16~r7s
hGcrit = 3 - ~ v ~ f ( 0 )
[19]
2 - 3 cos 0 + cos a 0
[20]
4
I t c a n b e s e e n t h a t f ( 0 ) = 1/2 for t h e c a s e of a 90 ~ c o n t a c t
angle, a n d Eq. [19] r e d u c e s to Eq. [16]. We c a n n o w e x p r e s s
nebso+crit i n Eq. [13] as
AGcri~-~
RT /
- npbso4) e x p
[21]
I n t h i s e q u a t i o n , t h e initial n u m b e r of a c t i v e sites less t h e
c u r r e n t n u m b e r o f g r o w i n g c e n t e r s (n~
npbso4) exp r e s s e s t h e c u r r e n t n u m b e r o f sites a v a i l a b l e for n u c l e ation, n+~t+. T h e q u a n t i t y Z is t h e Z e l d o v i c h n o n - e q u i librium factor
Z - - 2"~rcrit2
~'
~i+ (0) -t]+
[22]
This factor corrects an equilibrium distribution of nuclei
to a c c o u n t for critical n u c l e i t h a t d e c o m p o s e b y loss o f
m o l e c u l e s to b e c o m e s u b c r i t i c a l c l u s t e r s (23).
T h e g r o w t h f r e q u e n c y 00 i n Eq. [1] is e x p r e s s e d as t h e
product of the number of lead ions in contact with unit
a r e a o f n u c l e u s , npb2+, t h e s u r f a c e a r e a o f t h e n u c l e u s , a n d
t h e j u m p f r e q u e n c y o f a s i n g l e l e a d i o n i n s o l u t i o n to t h e
P b S O , surface, v e x p ( - A G a / R T )
r = npb2+
(1 - cos O)
2
~rcrit 2 v e x p
(
\ CHSO+-/
f ( 0 ) 112
_ ~Ga\
o
- - ~ - ) Cpb2+(n +ite - npbso+)
e +
161r73~PbSO42f(0)X
[26]
[18]
for t h e free e n e r g y o f f o r m a t i o n o f t h e critical n u c l e u s ,
w h e r e t h e q u a n t i t y f (0) is a f u n c t i o n o f c o n t a c t a n g l e a n d is
given by ~
nebso4crit = Z(n~
\~(RT/
N ~+ ~,v+(I = c o s O)
3r
F o r t h e g e n e r a l c a s e of a n y c o n t a c t a n g l e w e c a n w r i t e (23)
f(0) -
4
exp
[17]
hGv
[25]
(J~CH2sO4)2/3
CI~2SO4
Ot
a n d t h e r a d i u s of t h e critical n u c l e u s
1673
AOdl
]
P a r a m e t e r s a n d p r o p e r t i e s . - - T a b l e I gives d a t a u s e d i n
t h e calculations. Initially, all b u t t h e last+ six q u a n t i t i e s
w e r e t a k e n as k n o w n a n d n o t a d j u s t a b l e . (The t w o s e c t i o n s
f o l l o w i n g will d i s c u s s t h e i r d e t e r m i n a t i o n . ) T h e e l e c t r o d e
h a l f t h i c k n e s s is a p p r o x i m a t e d as h a l f t h e t h i c k n e s s o f t h e
dry, p r e f o r m e d e l e c t r o d e s ( i n c l u d i n g t h e grid) u s e d b y
B e r n d t et al. (2) a n d Wales et al. (14). T h e initial p o r o s i t y
w a s c h o s e n to b e 0 . 5 : P b O 2 e l e c t r o d e p o r o s i t i e s u s u a l l y fall
i n t h e r a n g e o f 0.5-0.6 (1), a n d t h e m o d e l r e s u l t s s h o w n are
n o t v e r y s e n s i t i v e to t h e v a l u e o f C i n t h i s r a n g e . T h e a c t i v e
i n t e r f a c i a l a r e a of PbO2 e l e c t r o d e s is u s u a l l y a r o u n d
2 • 105 cm2/cm 3 (1) (bm2/g). T h e v a l u e i n T a b l e I w a s
c h o s e n to c o r r e s p o n d w i t h t h e d a t a of Wales et al. (14) be-
Table I. Input parameters and properties
Quantity
[23]
w h e r e h ~ is t h e a c t i v a t i o n a l free e n e r g y p e r m o l e for leadi o n s o l u t i o n diffusion.
I n t h e following, w e s h a l l d e v e l o p a n a p p r o x i m a t e exp r e s s i o n for nvb~+. T h e v o l u m e o c c u p i e d b y a s u l f u r i c - a c i d
m o l e c u l e a l o n g w i t h its a c c o m p a n y i n g w a t e r m o l e c u l e s is
(CH~so4~)-~, t h e d i m e n s i o n o f w h i c h is o n t h e o r d e r o f
(Ca~so,~) - ~ , w h i c h is also t h e v o l u m e o f a u n i t - a r e a layer.
Multiplication of this by the molecular concentration
~CH~so4 r e s u l t s i n
-l]~(WCH2so4) = (~'Cg2so4)2/3
nH2so4 = (XC~I~SO4)
T h e n u c l e a t i o n e q u a t i o n g i v e n i n t h e p r e v i o u s s e c t i o n is
t h e f o r m of t h i s e q u a t i o n for a n u c l e u s w i t h a 90 ~ c o n t a c t
angle. T h e r e are t h r e e u n k n o w n q u a n t i t i e s i n t h e n u c l e ation rate expression: the PbSO4-solution interfacial tens i o n 7, t h e j u m p f r e q u e n c y v e x p (-hGd/RT), a n d t h e initial
n u m b e r d e n s i t y of n u c l e a t i o n sitesLn%~t~. All o t h e r q u a n t i ties i n Eq. [26] are k n o w n (N, R, T, Vpbso4) or c a l c u l a t e d b y
t h e m o d e l (npbSO4, Cpb2+ , CI-ISO4-). L a t e r w e will give e s t i m a t e s
of the surface tension and frequency by comparing model
c a l c u l a t i o n s of n u m b e r d e n s i t y (at v a r i o u s d i s c h a r g e rates)
to v a l u e s o f n u m b e r d e n s i t y s e e n i n e x p e r i m e n t a l electrodes.
T h e t h e o r y d e s c r i b e d a b o v e r e s u l t e d i n w h a t is r e f e r r e d
to as t h e " s t e a d y - s t a t e n u c l e a t i o n r a t e " (Eq. [26]). U p o n
c h a n g i n g s u p e r s a t u r a t i o n c o n d i t i o n s , t h e r e is a finite period o f t i m e r e q u i r e d for t h e s y s t e m to e x h i b i t a n e w ,
s t e a d y n u c l e a t i o n r a t e (25). I n o u r study, t h i s t i m e p e r i o d is
s h o r t c o m p a r e d to t h e t i m e p e r i o d of o b s e r v a t i o n , a n d t h e
steady rate equations provide a good description of the nucleation process.
[24]
T h e q u a n t i t y nH2so4 c a n t h e n b e m u l t i p l i e d b y t h e ratio o f
l e a d i o n s to s u l f u r i c - a c i d m o l e c u l e s to o b t a i n
5 Physically, the function f(0) represents the volume of the spherical cap divided by the volume of a sphere with the same radius of
curvature (24).
Electrode half thickness
Initial porosity (charged state), C
Initial volume fraction of PbSO4, e~
Specific area of PbO2, Apbo~
Initial acid concentration, C~a+so4 CH~ ref
Saturation concentration, c~
Cpbso4r~f
Transference number of Pb § species, tpb2+~
Transference number ofH + species, ta+~
H2SO4 diffusion coefficient (25~ DAA
Diffusion coefficient, DAB
Diffusion coefficient, DBA
Diffusion coefficient, DBB
Partial molar volume of H2SO~ VA
Partial molar volume of H20, V0
Partial molar volume of Pb(HSO4)2, VB
Quantity defined by Eq. [20], Q
Lead dioxide conductivity,
Electrolyte conductivity, K
Reference exchange current density, i0ref
Symmetry factor of the slow reaction,
Initial number density of active sites, n~,~
Surface tension,
Frequency, v exp (-hGjRT)
Precipitation rate constant, k,pt
Value for
PbO2 Model
0.09 cm
0.5
0
1.6 x l0 s cm2/cm 3
4.9 x 10 3 mol/cm3
1.1 x 10-8 mol/cm3
4 • 10-G
0.72 (1)
3.02 x 10-5 cm2/s
-1.22 x 10-10 cm2/s
-1.22 x 10-1~cm2/s
6.04 x 10-5 cm2/s
45 cm3/mol (1)
17.5 cm3/mol (1)
93 cm3/mol
0
80 S/cm (26)
0.79 S/cm (1)
7.8 x 10-7 A/cm ~
0.31
1.6 • 107 cm -2
8.01 x 10-6 J/cm 2
0.3 • 102 S-I
0.8 • 10-3 cm/s
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J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
1674
cause we compare the model calculations of the n u m b e r of
PbSO4 particles per unit PbO2 area obtained from their
data. The acid concentration is the value used by Berndt
and Voss (and Wales et al.). The equilibrium concentration
of lead ions is a function of acid concentration
I0a
107 ]
c3eq = Cpb+2~q = 2.34 • 10 -8 -- 7.30 X 10-7[CH2soj
-
3.73 x 10-4[C~2so4]2 [27]
This equation is based on the data given by Bode (1) for
sulfuric acid concentrations between 1 and 6M. The reference electrode is chosen to be a PbO2/PbSO4 placed at the
electrode/reservoir interface with concentration given by
c~
in Table I: the electrode voltage before the onset of
current is zero in the model calculations. The transport
properties are discussed in Appendix A, and the six remaining parameters were estimated by comparing model
and experimental results: the details are discussed in the
following two sections.
Estimation of electrochemical kinetic parameters.--The
evaluation of initial polarization data that do not exhibit
nucleation behavior (i.e., no voltage dip) can allow us to
make independent estimates of the two electrochemical
kinetic parameters A~
~efand ~RDS (see Eq. [11] and Appendix B). (As discussed later, voltage dips do not occur if
there are sufficient PbSO4 seed particles for nucleation).
The initial polarization is defined as the absolute value of
the voltage (relative to a PbO2 reference electrode) at zero
current minus the initial voltage at a specified discharge
current. Wales et al. (14) provide experimental data for
lead-dioxide electrodes that do not exhibit voltage dips;
presumably because their charging procedure allowed a
sufficient amount of PbSO4 to remain. Table II shows the
comparison between the model and experimental initial
polarization as a function of discharge current density for a
specific choice of kinetic parameters. The experimental
values are for the first discharge of four different electrodes. In these model calculations we assume that the initial, active interfacial area, initial porosity, and thickness
are the same at each discharge current density. These approximations are valid at the first discharge because all
four electrodes were formed at the same conditions. The
input data given in Table I were used in these model calculations with the exception kppt->
(Nucleation parameters are not required if kppt-~ ~: PbSO4 becomes a direct
product of the electrochemical discharge reaction.) The
values of A%bo2io~ and ~RDS given in Table II were obtained by examining model results for many combinations
of these parameters and minimizing the difference between the calculated initial voltage and the experimentally
observed value. The uncertainty given for the experimental results reflects how well we could read the data from
the figure in Ref. (14). Since calculated polarizations are
within 5 mV of the experimental results, we regard
A~
ref and ~BDSas non-adjustable parameters in subseque n t m o d el calculations. The satisfactory comparison of
model and experimental results also supports the assumption that the first electron-transfer reaction is slow relative
to the second, because it is not possible to fit the model to
the experimental results if the second electrochemical reaction is assumed to be slow relative to the first (see Appendix B).
0o.
Estimation of nucleation parameters and growth rate
constant.--In the following, we describe the estimation of
n~
~, u exp (-hGd/RT), and kppt. The first three parame-
~,'~1~ I
mO103
].02|I
9
, I ,r,I
i
io
i r I rlrll
WALESs
,
lOO
AL.. 1983.
, r
i
, r r ,
Iooo
Discharge Current Deasity, mA/cm2
Fig. 2. Calculated PbS04 number density after the nucleation period
and estimates of experimental npbso4as a function of discharge current
density. See Appendix C for explanation of dashed line.
function of discharge rate derived from experimental electrodes. The shaded area of Fig. 2 shows estimates of number density obtained from scanning electron micrographs
of fully discharged experimental electrodes (14). Appendix
C gives a detailed description of the procedure used to estimate the n u m b e r densities shown in Fig. 2 from the experimental data given by Wales. These npbso~ results indicate
the expected trend of increasing npbso4 with increasing discharge rate. The leveling-off of npbso4 after 100 mAJcm 2 is
perhaps an indication of a finite n u m b e r of nucleation sites
available on the PbO2 substrate--increasing the discharge
rate further will not lead to further increases in npbso4. In
light of this we choose the available n u m b e r of active nucleation sites, n~
to be the m a x i m u m n u m b e r density
shown in this figure.
The value of npbso4 calculated by the model is very sensitive to the values of the two nucleation parameters ~ and
v exp ( - A G j R T ) . Preliminary estimates of these parameters were made first by assuming that the growth rate constant, kppt was very small (i.e., ~ 10 -l~ cm/s) so that very little growth and a m a x i m u m amount of nucleation could
occur. In this case, the voltage drops at the onset of current
and does not rise to form a m i n i m u m (see Fig. 4). The
quantities ~ and v exp ( - A G J R T ) were then chosen so that
the n u m b e r densities calculated after the expected nucleation period (after approximately 800 m A - s/cm 2 delivered
capacity) for the three current densities examined by
Berndt and Voss fell in the shaded area of Fig. 2. It should
be noted that the surface energy ~ could be changed by approximately a factor of two and the frequency by an order
of magnitude and the results would still fall within the
shaded region. Figures 3a and b compare calculated electrode voltage and experimental results given by Berndt
and Voss (2) for three discharge current densities. The
value of these two parameters (given in Table I) were then
kept constant, and kppt was increased until a rise in voltage
occurred that agreed qualitatively with the experimental
results shown in Fig. 3b. The three large points indicate
the steady-state n u m b e r density at the back portion of the
electrode (see Fig. 7) that is calculated by the model.
ters are determined by examining n u m b e r density as a
Results and Discussion
Table II. Initial polarization of the lead-dioxide electrode
Voltage-dip behavior during constant current discharge.--The agreement between model and experimental
Experiment (11), V
-+0.005V
Model, V
~RDS= 0.31
~0ref = 7.8 • 10-7
A/cm2
Current, mA/cm2
0.260
0.120
0.035
0.000
0.260
0.130
0.026
0.002
200
50
12.5
1.8
results shown in Fig. 3a and 3b is satisfactory: the model is
able to predict the shape of the voltage dip. As the discharge rate is decreased, the voltage m i n i m u m occurs
sooner and becomes more shallow relative to the starting
voltage, as shown in the experimental results and predicted by the model. The nucleation rate calculations that
will be discussed later indicate that most of the nucleation
occurs between 250 and 1000 m A 9 s/cm 2 (i.e., near the mini m u m in voltage), which we term the nucleation period.
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J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
~>
a. EXPERIMENT
'Berndt and Voss, 1965)
-002
l
c~
~\
- o o~
I = 4.8 m ~ / c m 2
I
9.6
1675
I = 9.6 m A / c m 2
-0.02
r~
-o.o~
mA/cm 2
g
~
-o.o~
-o.o5
J
Q
~
-o.oa
-o.os
-o,lo
-0.10
I
I
[
500
1000
1500
no PBSO,
formation
I
2000
2500
500
.~,ooo
1000
1500
2000
2500
3000
Discharge Capacity, m A . s / c m 2
b.
Fig. 4. Demonstration of model results with (solid curve) and without
(dashed curve) nucleation and growth of PbS04 for a discharge rate of
9.6 mA/cm2.
MODEL
;>
3,02
c~
d
~>
-0,04
I = 4.8 m A /c m 2
-0.00
-0.00
b
\
I=9.6mA/em'
I *~ ,~
9
,s
[ = 19.1 m k / c m 2
-o.~o
-0.12
i
soo
i
tooo
i
15oo
.....
r
i~
2ooo
2s o
3ooo
Discharge Capacity, mA.s/cm ~
Fig. 3. Initial voltage minima obtained for a) experimental (2), and
b) model discharges of lead-dioxide electrodes at various discharge
rates (discharge capacity = It). Model parameters are given in Table I.
Relatively little nucleation occurs after 1000 m A . s/cm a of
discharge capacity is delivered; time from this point on is
considered the growth period. The solid curve in Fig. 4 repeats the model results shown in Fig. 3 for the case of
9.6 m/Ucm ~. The dashed curve illustrates the behavior of
the voltage if nucleation and subsequent precipitation are
not allowed to occur. In this case, the lead ion concentration builds continuously to a m a x i m u m value. The fact
that the results for these two cases are the same up to
500 m A 9 s/cm 2 indicates that this initial voltage behavior is
not very sensitive to nucleation. With this in mind, let us
e x a m i n e the difference between model and experimental
results in the behavior just after the onset of current and
before the voltage minimum. The model predicts the voltage to drop m u c h more suddenly, with essentially infinite
slope, than experimental voltage, which clearly shows a
finite slope. Since Fig. 4 demonstrated that this behavior
may not be associated with nucleation, we suspect that
some process other than nucleation is determining the behavior in this region. Berndt and Voss (2) have addressed
this initial slope and propose that it is characteristic of
charge transfer in the electrical double layer. They claim to
have estimated the double-layer capacity from the slope
and obtained values on the order of those reported from
different methods. Since the model does not account for
double-layer charging, we are essentially assuming that
the double layer is charged before t h e onset of current. It is
therefore possible that the observed discrepancy is a consequence of our neglect of the charging of the electrical
double layer in the model formulation.
One other point worthy of discussion is the recovery of
the voltage after the dip. Model results show that as the
discharge rate decreases, the voltage rise is more sluggish
after the voltage minimum. This is not the case with the
experimental results--the recovery of the voltage appears
more rapid with decreasing discharge rate, and the dip appears to be shorter, in terms of capacity, at the lower current densities than at the high discharge current density.
Since this voltage recovery occurs during the growth period, the formulation of the growth kinetics (Appendix B)
is probably the cause of the discrepancy. The quantities
kppt and Apbso4 should be discussed. As mentioned earlier,
Apbso4 in Eq. [8]-[14] was formulated in an approximate
m a n n e r and may be responsible for the discrepancy. If we
were to increase the precipitation rate constant kvpt, then
we could obtain m u c h better agreement between experimental and model results at the low discharge current density; however, the agreement at the higher discharge current densities would suffer. It is possible that kppt is
d ep en d en t on the electric potential or surface composition, which have not been considered. One final possible
reason for the discrepancy in voltage recovery lies in the
assumption that the lead-ion concentration at a given position within the electrode cross section is uniform throughout a pore. The lead-ion concentration near the P b S O , surface may not be the same as the average pore
concentration Cpb2§ especially at the higher discharge
rates? This could cause us to underestimate the value of
kppt at the higher discharge rates, which when used at the
lower discharge rates results in the calculated sluggish recovery of the voltage that was discussed earlier.
s An estimate of the diffusion rate [assuming a Pb 2+ diffusion
layer thickness 8pb2+ of 1 micron (1)] relative to the precipitation
rate indicates that this is a valid approximation: Dpb2+/~ph2+> kppt.
0.020
N
~ o.ol5
o
o.oTo
§
o.oos
:c=L
o.ooo
~oo
~ooo
|5oo
2000
250o
5000
Discharge Capacity, m A . s / e m a
Fig. 5. Calculated lead-ion concentration at various positions within
the electrode during discharge at a rote of 9.6 mA/cm2.
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J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
1676
It should also be mentioned that the vertical positioning
of the experimental results on the voltage axis of Fig. 3a is
subject to error because Berndt and Voss reported their results on an arbitrary voltage scale; that is, they did not report the potential at the start of the experiment (which
would be zero, relative to our chosen reference electrode).
We extrapolated their voltage data back to zero time for
each discharge rate in order to place all their data on a single plot for comparisons.
Figure 5 shows the calculated lead-ion concentration as
a function of discharge capacity at various positions
throughout the electrode cross section. These results support the proposal of Berndt and Voss (2) that the m i n i m u m
in potential corresponds to a m a x i m u m in lead-ion concentration (cf. Fig. 3b). The model also indicates that the
concentration in the back portions of the electrode is
higher than in the front. We can see that a m a x i m u m degree of supersaturation Cpbg+/Cpb2+eq of approximately 2000
is reached at 500 mA 9s/cm 2 discharge capacity. The maxim u m degree of supersaturation is a function of discharge
rate and is calculated as 3600 and 800 for the 19.1 and
4.8 mA/em 2 current densities, respectively.
Berndt and Voss (2) estimated a m a x i m u m degree of
supersaturation of approximately 5, independent of discharge rate, from the Nernst equation. They used the
depth of the voltage dip relative to the voltage plateau obtained at long times in the Nernst equation and incorrectly
assumed that the resulting calculated supersaturation corresponds to the m a x i m u m degree of supersaturation. If
Berndt and Voss, however, had instead used in the Nernst
equation the depth of the dip relative to their open-circuit
voltage, then their estimate would be more on the order of
our calculated results for supersaturation.
Number-density behavior during constant-current discharge.--Figure 6 shows the n u m b e r density of PbSO4 particles that is obtained for electrode discharge capacity
greater than approximately 1500 mA 9s/cm 2. At the electrode/reservoir interface (x = L) the n u m b e r density is zero
because of the boundary condition that the solution is saturated in lead ions at this point. It can be seen that the
n u m b e r density is distributed quite uniformly throughout
most of the electrode cross section for these particular conditions. Figure 7 gives the n u m b e r density npbso4 (X = 0) VS.
discharge capacity for the three discharge rates. The leveling-off of the n u m b e r density mentioned previously can be
seen after 1500 mA 9s/cm 2. The slopes of the curves are
proportional to the nucleation rate, C3npbsO4Crit/63t; i n all
cases, the rate at initial time and long times are zero, and
the m a x i m u m rate occurs just after 500 m A . s/cm 2 delivered capacity. Actually, the n u m b e r density would remain constant only for a limited amount of t im e - - f ur t h er
on in discharge, the substrate area will begin to diminish
because PbO2 is being consumed by the electrochemical
I0 ~
I = 19.1 r a i / c m 2
l0 s
/
I = 9.6 mA/cm2
/
i04
I = 4.8 m A / c m 2
Z
10 2
c.
10 2
T
0.1
0.2
0.3
o.4
r
r
p
0.5
0.6
0.7
Dimensionless d i s t a n c e ,
0,8
0.9
x/L
Fig. 6. Calculated PbSO4 number density as a function of position x
within the electrode after 1500 mA 9s/cm2 delivered capacity.
i=
x
20
_~
-15
---'~o
10
~-,
.'~
//
I = 19.1 m A / c m 2
[ = 9.6 m A / c m 2
I = 4.8 mA/cm2
z
o~
-~
500
1000
1500
2o00
2500
5000
Discharge Capacity, m A . s / c m 2
Induction
I
I
Nucleation
Fig. 7. Calculated PbSO4 number density at the back of the electrode
(x = 0) as a function of delivered discharge capacity for various discharge rates.
reaction, and this will result in a slight increase in npbso4.
Throughout the range of discharge capacity studied in this
work, the PbO2 substrate area (Apm2) decreases less than
1%; therefore, changes in npbso4 are attributed to nucleation. If the n u m b e r of PbSO4 particles per unit electrode
v o l u m e NPbSO4 w e r e plotted throughout deep discharge,
it
would
remain
constant
after
approximately
1500 m A . s/cm 2.
The estimate of the PbSOdsolution interracial tension ~/
(Table I) allows us to use classical nucleation theory to estimate properties of the critical nuclei formed during certain
supersaturation conditions. The degree of supersaturation
in our results ranges from about 3000 to 300--the largest
value occurring at the voltage m i n i m u m for the 19 mA/cm 2
discharge rate, and the smaller value is attained after the
nucleation period. Equation [18] can be used to estimate
the size of the critical nuclei, and results in rent ranging
from 4 to 6 • 10 -4 I~m, with the critical nuclei containing 2
to 5 lead sulfate molecules (Eq. [17]). The free energy of formation of critical nuclei can be calculated from Eq. [4] and
ranges from 3 to 5 x 10 -20 J.
It should be mentioned that for the relatively low discharge rates examined in this study the nucleation rates
were low enough that the (n~
-- npbso4) term in Eq. [12] remained approximately constant and therefore depletion of
active sites did not play an active role. At larger discharge
rates, a s n p b s o 4 increased, this term would decrease and result in a leveling-out of the steady n u m b e r density, as seen
experimentally at the higher discharge rates (see Fig. 2).
At this point one may challenge the use of properties described in terms of macroscopic thermodynamic parameters for such small clusters of PbSO4 molecules. Classical
nucleation theory is based on the premise that nuclei may
be assigned the properties of bulk phases regardless of
their size--the bulk molar volume Vpbso4 characterizes the
critical nuclei and the surface free energy can be expressed in terms of the macroscopic flat-film surface tension. The latter assumption (also k n o w n as the capillary
approximation) is the most crucial and has continually
plagued the application of classical theory. In reality, -/will
not be constant but will decrease with decreasing size of
critical nuclei (27) and will not necessarily bear any simple
relation to the usual macroscopic free energy defined for a
plane interface. The value of -/= 8 x 10 .6 J/cm 2 for this
work should be interpreted, therefore, as a value associ-
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J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
A.
M o d e l results for 1-micron barium-sulfate particles
(n~
~ 105 c m -2) are s h o w n in Fig. 8b. I n a g r e e m e n t
w i t h e x p e r i m e n t a l results, the m o d e l predicts t h e voltage
dip will d i s a p p e a r w i t h a d d i t i o n of 3% BaSO4 seed particles. F o r this case, the m a x i m u m d e g r e e of supersaturation attained is Cpb~+/Cpb~eq <--80, w h i c h is too small to
cause any significant a m o u n t of nucleation. T h e b e h a v i o r
o b s e r v e d in this case reflects only that of g r o w t h of PbSO4
on t h e surfaces of the seed particles in c o n t a c t w i t h t h e
electrolyte. This t y p e of b e h a v i o r e x p l a i n s w h y t h e coup de
f o u e t is not s e e n in t h e discharge of only partially c h a r g e d
electrodes and electrodes that h a v e self-discharged after
long periods of t i m e b e t w e e n charge and d i s c h a r g e (1)--in
s u c h cases seed particles are available for PbSO4 precipitation.
T h e smaller v a l u e of n~
~ 102, w h i c h c o r r e s p o n d s to
a p p r o x i m a t e l y 10 ~ m particles, also results in m o d e l predictions w i t h o u t a voltage dip; h o w e v e r , m o r e polarization
at 3000 m A 9 s/cm 2 is seen b e c a u s e the d e g r e e of supersaturation at this delivered capacity is larger--cpb2+/
Cpb2+eq ~ 500 (cf. 80 for 1 ~ m particles). Similarly, v a l u e s of
n~
larger t h a n 105 (l~
< 1 ~m) result in less polarization (due to less supersaturation) and also do n o t e x h i b i t a
v o l t a g e dip.
EXPI'2RIM ENT
(Berndt and Voss, 1965)
>
d
g,
~a
0% BaSO,
b.
1677
MODEL
-o,o:
:>
u.~
Conclusions
-o.o4
-o.c~
~,
3% BaSO4
aSO.l
[
-0.~2 ]
'
$oo
*
~ooo
~oo
2oo0
I
2.~o~
3ooo
DiscI~arge Capacity, mA-s/cm 2
Fig. 8. Initial voltage minima obtained for a) experimental (2), and
b) model discharges at a rate of 19.1 mA/cm2 and various additions of
BaSO4.
ated only w i t h the size r a n g e of nuclei for the particular
c o n d i t i o n s of our calculations. T h e r e are few e x p e r i m e n t a l
e s t i m a t e s of ~PbSO4available in the literature. R e f e r e n c e (28)
reports ~PbSO4~ 100 • 10 -6 and cites o t h e r m e a s u r e m e n t s
of ~PbSO4 ~ 10 • 10 -6. I n v i e w of t h e s e results our v a l u e appears reasonable.
It s h o u l d be n o t e d that t h e v a l u e of t h e f r e q u e n c y parameter v e x p (AGa/RT) is v e r y small relative to t h e vibrational
f r e q u e n c y of water, w h i c h is a p p r o x i m a t e l y 10 ~3 s-L This
suggests that it is relatively difficult for a lead ion to shed
its salvation shell and c o n t r i b u t e to the g r o w t h of a nucleus.
Effect of addition of seed particles.--Berndt and Voss (2)
p e r f o r m e d e x p e r i m e n t s on positive electrodes w i t h inc r e a s i n g a m o u n t s of BaSO4 in order to confirm further
their p r o p o s e d t h e o r y for the voltage m i n i m u m . T h e comp o u n d BaSO4 is i s o m o r p h o u s to PbSO4 and should serve
as seed nuclei for PbSO4 precipitation. B e r n d t et al. obs e r v e d that additions of 0.1% and 1% BaSO4 d e c r e a s e d the
d e p t h of the voltage m i n i m u m , and the a d d i t i o n of 3%
BaSO4 r e s u l t e d in d i s a p p e a r a n c e of t h e m i n i m u m , w h i c h
is s h o w n in Fig. 8a. This b e h a v i o r s h o u l d s e r v e as a further
t e s t of o u r m o d e l as well. In o r d e r to s i m u l a t e this situation
in t h e m o d e l w e m u s t obtain e s t i m a t e s of t h e initial volu m e fraction of seed particles, e~
and t h e m a n n e r in
w h i c h it is distributed, i.e. n~
T h e f o r m e r can be easily
e s t i m a t e d f r o m k n o w l e d g e of the density of BaSO4 (4.5 g/
c m 3 (29)) to obtain e~
= 0.03 for the case of 3% BaSO4. I f
t h e size of t h e BaSO4 particles w e r e k n o w n t h e n w e c o u l d
estimate
n~
-
~~
A~176
4
[28]
)3
The p r o p o s e d theoretical a p p r o a c h satisfactorily describes PbSO4 n u c l e a t i o n in porous l e a d - d i o x i d e electrodes. The b e h a v i o r of the calculated PbSO4 n u m b e r density for various discharge rates follows w h a t is e x p e c t e d
b e h a v i o r of e x p e r i m e n t a l electrodes. T h e b u i l d - u p and relaxation of lead-ion c o n c e n t r a t i o n d u r i n g d i s c h a r g e has
also b e e n e x a m i n e d . The a d d e d insight o b t a i n e d f r o m t h e
analysis has allowed r e i n t e r p r e t a t i o n of e x p e r i m e n t a l data,
and lead-ion s u p e r s a t u r a t i o n is n o w seen to be m u c h larger
t h a n p r e v i o u s l y reported. T h e calculated b e h a v i o r of the
voltage dip agree w e l l w i t h e x p e r i m e n t a l results and upholds t h e solution-diffusion m e c h a n i s m of d i s c h a r g e as
w e l l as t h e postulates that t h e coup de fouet is c a u s e d b y
t e m p o r a r y lead-ion supersaturation. A d d i t i o n of seed particles for PbSO4 n u c l e a t i o n in m o d e l calculations results in
d i s a p p e a r a n c e of the coup de fouet, w h i c h has b e e n obs e r v e d e x p e r i m e n t a l l y by o t h e r researchers. R e l a t e d w o r k
c o u l d focus on the PbSO4 n u m b e r density as a f u n c t i o n of
acid c o n c e n t r a t i o n or t e m p e r a t u r e , rather t h a n current
d e n s i t y as w e h a v e d o n e in this paper.
M a n u s c r i p t s u b m i t t e d April 17, 1989; revised m a n u script r e c e i v e d Dec. 2, 1989.
General Motors Research Laboratories assisted in meeting the publication costs of this article.
APPENDIX A
Transport Properties for the Concentrated Ternary Electrolyte
T h e m u l t i c o m p o n e n t diffusion e q u a t i o n (19)
~Kij(vj - vi) = ciV~j
J
[A-l]
is c u s t o m a r i l y u s e d to d e s c r i b e t r a n s p o r t in c o n c e n t r a t e d
electrolytes. T h e friction coefficients K~ are defined in
t e r m s of binary-interaction coefficients ~ij by t h e relation
K~ = R T - -
cicj
- Kji
[h-2]
CT@ij
w h e r e t h e q u a n t i t y CT is t h e s u m of all t h e concentrations,
i n c l u d i n g t h e solvent.
T h e flux of a species i is Ni a n d is t h e p r o d u c t of its conc e n t r a t i o n and average velocity, civi. E q u a t i o n [A-l] is not a
direct e x p r e s s i o n for the flux. It can be i n v e r t e d to obtain
e x p l i c i t e x p r e s s i o n s for the fluxes in t e r m s of t h e driving
forces, namely, chemical-potential gradients V~h. S u n u
(17, 18) derives e x p r e s s i o n s for t h e fluxes. We r e p e a t the results of t h e t r a n s p o r t properties in light of the t y p o g r a p h ical errors p r e s e n t in S u n u ' s text, and w e use the s a m e nom e n c l a t u r e . Results for the six i n d e p e n d e n t t r a n s p o r t
properties L ~ , LAs, LBB, t2~ t3~ and K in t e r m s of friction
coefficients and c o n c e n t r a t i o n s are
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1678
L~ = ~
(C~2Ma + 2C~C~K~3 + Ca2M,)
[A-3]
CACB
LAB = LBA -- ~
(CI~(K~ + K2a)
+ C~Ca(2Kla + Kto) + Ca2M~) [A-4]
2
LBB = ~ - ( C ~ ( K l a + K~a + K~o + K~o)
+ 2C~C~(K~3 + K~o) + Ca2M1) [A-5]
v o l u m e s . (This goes a l o n g w i t h t h e r e a s o n i n g t h a t t h e
l a r g e r t h e ion, t h e l o w e r its m o b i l i t y , a n d t h e l o w e r f r a c t i o n
o f c u r r e n t it will carry.) F o r t h e p r e s e n t c a s e w i t h tl ~ ~ 0.28
a n d t2~ ~ 0.72 w e are m a k i n g t h e a p p r o x i m a t i o n t h a t t h e
p a r t i a l m o l a r v o l u m e o f HSO4- i o n is 2.57 t i m e s l a r g e r t h a n
t h e p a r t i a l m o l a r v o l u m e o f H § ion. I t is i n t e r e s t i n g to n o t e
t h a t w e c a n d e f i n e bo_unds for t h e v a l u e of Q b y a s s u m i n g
t h a t e i t h e r Vl_Or Vz is d o m i n a n t i n t h e e x p r e s s i o n
VA = viA'J1 + v2hV2. F o r t h e p r e s e n t case, w e h a v e
- 1 3 -< Q -< 32.
T h e flux e x p r e s s i o n s t h a t are g i v e n b y S u n u (17, 18) a n d
n o t r e p e a t e d h e r e p r o v i d e d e f i n i t i o n s o f d i f f u s i o n coeffic i e n t s i n t e r m s of t h e a b o v e p r o p e r t i e s as
(vl A + v2A)RT
C2
t2~ = ~-(C~(MaKlo + KaoK~a) + C~(K2~Klo - K12K~0)) [A-6]
DAA --
( L ~ - CA(VALA* + VBLAB)) [A-20]
CA
(vl B + v3B)RT
C3
ta~ = - ~ (C~(M1K2o + K~0K21) + CI(KlaK2o - K~K~o))
DAB --
[A-7]
(VlA + v2A)RT
F2D
[A-8]
DBA --
Ci = zici
[A-9]
DaB --
M0 = K10 + K20 + K30
[A-10]
MI = Klo + K12 + KI3
[A-11]
/142 = K20 + K21 + K23
[A-12]
M3 = K30 + K31 + K32
[A-13]
-
(LAB -- cA(VALAB + VBLBB)) [A-21]
CB
(LAB -- cB(VAL** + VbLAB)) [A-22]
CA
E
(vlB + v3B)RT
where
and
[A-23]
OVA = ( h A + v2A)RT 0 in (~12c,) ~ (via + v2A) __RT __OCA [A-24]
Ox
Ox
CA OX
-- (Vl B +
+ C~2(M,Mo - K,o 2) [A-14]
-- c B ( V A L A B + VBLBB))
w h e r e t h e f o l l o w i n g a p p r o x i m a t i o n s w e r e utilized i n t h e
derivation
0~B
D = C12(MzMo - K302) + 2CIC3(MoK,~ + KloK3o)
(LBB
CB
Ox
v~B)RT
0 In
('Y13CB)
Ox
R T OcB
(v, B + v3~ ) -cB Ox
[A-25]
T h e t r a n s f e r e n c e n u m b e r of species 1 is d e f i n e d as
T h e s e d i f f u s i o n coefficients h a v e b e e n m u l t i p l i e d b y eex i n
Eq. [6] a n d [7] to a c c o u n t for t h e p o r o u s s t r u c t u r e . We c a n
e s t i m a t e t h e d i f f u s i o n coefficients for o u r c a s e w i t h t h e
s a m e a p p r o x i m a t i o n s t h a t w e r e utilized a b o v e i n e v a l u a t i n g t h e t r a n s f e r e n c e n u m b e r s . T h e r e l a t i o n s h i p s of t h e DAA
to t h e o t h e r d i f f u s i o n coefficients are t h e n
C,
t, ~ = 1 - t2~ - t3~ = --~ (CI(M~K20 + KaoK2~)
and
E = Mo(K12K13 + K12K23 + KI~K2~) + K~oK2o(K~ + K2~)
+ K2oK~o(KI~ + K,3) + K~oK~o(KI2 + KI~) + K~oK2oK~o [A-15]
+ Ca(K~aK2o - Ki2Ka0))
[A-16]
a n d t h e q u a n t i t y Q is d e f i n e d as
Q = t~~
V~+
Zl
t2~
V2+
Z2
oV~
t~ - -
[A-17]
Z3
S i m p l i f i c a t i o n s o f transport properties.--In t h e following, w e s h a l l d i s c u s s c e r t a i n s i m p l i f i c a t i o n s o f t h e s e e q u a t i o n s t h a t w e r e utilized in t h e m o d e l . T h e t r a n s f e r e n c e
n u m b e r of t h e P b +2 s p e c i e s is a p p r o x i m a t e d b y
Z32 CB
t~~ -
[A-18]
Zl2 2CA
T h i s e q u a t i o n is d e r i v e d from~Eq. [A-7] b y a s s u m i n g t h a t
t h e b i n a r y i n t e r a c t i o n coefficients, ~ , are e q u a l a n d t h a t
c~ < < Cl so t h a t c~ ~ c2. T h e v a l u e at t h e r e f e r e n c e c o n c e n t r a t i o n s CB~ a n d CA ref is t~~ = 4 • 10 -~. E q u a t i o n [A-16] sugg e s t s t h a t w e m a y u s e for t h e p r e s e n t c a s e t h e e x p e r i m e n t a l l y d e t e r m i n e d v a l u e s o f t~~ a n d t2~ for a p u r e
s u l f u r i c - a c i d / w a t e r b i n a r y electrolyte.
T h e v a l u e o f Q is n o t easily e s t i m a t e d b e c a u s e t h e p a r t i a l
m o l a r v o l u m e s o f i n d i v i d u a l ions are n o t easily o b t a i n a b l e .
A c o n v e n i e n t c h o i c e for Q is zero (30). We c a n i n v e s t i g a t e
t h e a p p r o p r i a t e n e s s of t h i s choice. First, w e n e g l e c t t h e
t h i r d t e r m o n t h e r i g h t of t h e e q u a l sign i n Eq. [A-17] bec a u s e t30 for o u r c a s e is five o r d e r s of m a g n i t u d e s m a l l e r
t h a n t~~ a n d w e d o n o t e x p e c t t h e m o l a r v o l u m e o f P b +2 to
b e l a r g e e n o u g h to r e n d e r t h i s t e r m significant. I f Q = 0,
then
V2t2 ~
--
Z2
-
Yltl ~
[A-19]
DAB = DBA ~ DAA X (--4 • 10 -6)
[A-26]
DBB ~ D** x 2
[A-27]
I f w e a s s u m e t h e d i f f u s i o n coefficient DhA to b e t h e e x p e r i m e n t a l l y m e a s u r e d v a l u e for t h e s u l f u r i c - a c i d e l e c t r o l y t e
w i t h o u t lead ions, t h e n w e o b t a i n t h e v a l u e s g i v e n i n
T a b l e I for DAB, DBA, a n d DBB. F o r t h e p r e s e n t case, t h e diff u s i o n coefficients DBA a n d DAB are n e g a t i v e . T h i s is bec a u s e t h e c o n c e n t r a t e d - s o l u t i o n t h e o r y a c c o u n t s for all
s p e c i e s i n t e r a c t i o n s , n o t o n l y t h e i n t e r a c t i o n s w i t h t h e solv e n t (as in d i l u t e - s o l u t i o n theory).
M o d e l c a l c u l a t i o n s s h o w t h a t it is a p p r o p r i a t e to a s s u m e
t h a t DAB is zero i n Eq. [6] b e c a u s e t h e t r a n s p o r t of t h e m a j o r
c o m p o n e n t s H § a n d H S O ( are r e l a t i v e l y u n a f f e c t e d b y t h e
d i f f u s i o n of P b § ion: h o w e v e r , it is n o t a p p r o p r i a t e to ass u m e t h a t DBA is zero b e c a u s e t h e t w o c o n c e n t r a t i o n gradie n t t e r m s i n Eq. [7] are e q u a l i n m a g n i t u d e , t h e t r a n s p o r t of
P b +2 is a f f e c t e d b y t h e t r a n s p o r t o f s u l f u r i c acid. T h e elect r o l y t e c o n d u c t i v i t y K i n t h e m o d e l is a s s u m e d to b e c o m p o s i t i o n - i n d e p e n d e n t a n d e q u a l to t h e e x p e r i m e n t a l l y det e r m i n e d v a l u e g i v e n i n T a b l e I.
APPENDIX B
Electrochemical Kinetic Expression
H a m p s o n et al. (1, 31) p r o p o s e d t h e f o l l o w i n g m e c h a n i s m for t h e overall e l e c t r o c h e m i c a l d i s c h a r g e r e a c t i o n of
t h e PbO2 e l e c t r o d e (see r e a c t i o n II)
PbO2 + 2H + k~ PbO2(H+)2 (*)
kc2
PbO2(H+)2 r + e- ~ Pb(OH)2 +(*) (RDS)
k~
kc3
Pb(OH)2 +r + e- ~ Pb(OH)2 (*)
Zl
I n o t h e r w o r d s , t h e t r a n s f e r e n c e n u m b e r s of t h e i o n s are
a s s u m e d to b e i n v e r s e l y p r o p o r t i o n a l to t h e i r p a r t i a l m o l a r
Pb(OH)2 r + 2H + ~kf4 pb+2 + 2H20
~b4
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d. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
w h e r e (*) refers to an a d s o r b e d species and R D S indicates
t h e p r o p o s e d rate-limiting step.
We can write
i2
w h e r e the activity of PbSO4 is t a k e n as unity, and Apbso4 is
t h e surface area of the precipitate e x p o s e d to t h e electrolyte. A t e q u i l i b r i u m
kf
-- ka2[FPb(OH)2+(*)]e x p ((1 -- ~)fV)
(am eq)
-
kb
F
k~[FP~O2(H§
-
1679
e x p (-~ffV)
-
K
[B-11]
(apb+2eq)(a~iso4-eq)
w h e r e K is an e q u i l i b r i u m constant. The rate e x p r e s s i o n
can t h e n be written as
[B-l]
and
RpbSO4.~ = A pbso4kf[ ( apb +2)(aHso4- )
- k~[Fp~(on)~(.)] e x p ((1 - ~)fV)
F
- k~[re~(o~)2§
e x p ( - ~al~)
to e x p r e s s t h e c u r r e n t densities of the s e c o n d and third reactions in this m e c h a n i s m , respectively. F o l l o w i n g t h e
s a m e a p p r o a c h outlined b y N e w m a n (19), an e x p r e s s i o n
for the p o t e n t i a l - d e p e n d e n t e q u i l i b r i u m surface concentration F~b(om~§ can b e o b t a i n e d f r o m Eq. [A-27] b y ass u m i n g t h e third reaction is fast relative to t h e s e c o n d reaction. The e x c h a n g e c u r r e n t d e n s i t y t h u s o b t a i n e d is
( k ~ ka~
_\ p~/2
i0 = 2Fk~2 ~ ~c3 [FPD(OH)2(*)J)/ ([FPbO2(H+)2(*)])
(2-p2)/2 [B-3]
T h e e l e c t r o c h e m i c a l kinetic e x p r e s s i o n for the lead-dioxide e l e c t r o d e is
i = i0 {exp [(2 - ~ ) f ~ ] - e x p [(- ~)f~]}
[B-4]
I f w e a s s u m e that t h e rate constants kf,, kb~, kfl, and k~ for
t h e c h e m i c a l reaction steps are large e n o u g h that their reactions are in equilibrium, t h e n w e can write
[rPb(Om2(*)] -
[B-5]
[cm] 2
and
k~
= ~bi [CH+]2
[B-6]
We can substitute t h e s e e x p r e s s i o n s into Eq. [B-2] to
obtain
k.
io
2Fkr
/ - -
|
\kc2 kc3 kb4
[CH+]2
]
kfl
2\ (2-~2)i~
T h e e x c h a n g e current d e n s i t y at a r e f e r e n c e c o n c e n t r a t i o n
can be defined, and t h e e x c h a n g e d e n s i t y at any concentrat i o n can be w r i t t e n as
\ [Cpb+2ref]/
\[CH2oref]/
\[cH+ref]/
E q u a t i o n [B-3] s h o w s that
a~ = (2 - P2) and ~ = ~2
[B-9]
I n Eq. [11], t h e e x c h a n g e c u r r e n t density i0 is m u l t i p l i e d b y
Apbo~, w h i c h m o d e l calculations s h o w r e m a i n s approximate2y c o n s t a n t d u r i n g the short d i s c h a r g e t i m e s that w e
i n v e s t i g a t e in this work.
R a t e e x p r e s s i o n f o r the g r o w t h o f lead-sulfate p a r t i c l e s . O n c e a stable n u c l e u s is formed, it will g r o w w i t h a rate dep e n d e n t on its area e x p o s e d to t h e solution and t h e conc e n t r a t i o n of lead ions in t h e solution. We write t h e precipitation reaction as
T h e n e t v o l u m e t r i c rate of p r o d u c t i o n of lead sulfate R~bso
(i.e., m o l e s of lead sulfate precipitated p e r u m t e l e c t r o d e
v o l u m e per second) can be e x p r e s s e d as
.
S i n c e t h e sulfuric-acid c o n c e n t r a t i o n r e m a i n s large relat i v e to c~
and a p p r o x i m a t e l y c o n s t a n t at c~
it is reas o n a b l e to a s s u m e that the activities of H + and H S O ( are
i n d e p e n d e n t of the P b +2 concentration, i.e., a~+ eq = aK+
and asso4_ ~ = aHso4-. It follows that t h e activity coefficient
of lead ion should be relatively i n d e p e n d e n t of lead-ion
concentration. T h a t is, ~pb+2eq = ~pb+2. E q u a t i o n [B-11] can
be simplified to yield
RpbsO4,s = Apbso4kf(aHSO4-eq)~Ipb+2([Cpb +2] -- [Cpb+2eq])
= Apbso4kpvt([epb+2] -- [Cpb+2eq]) [B-13]
w h e r e w e h a v e defined
k,pt = kf(aHso4-eq)~pb+2
,
Rp~so~.~ = Apbso~[k~(aeb~+)(aHso~-) -- k~(am)]
4
[B-10]
[B-14]
I n this w o r k w e e s t i m a t e t h e q u a n t i t y Apbso4 (the total
surface area of lead sulfate particles in c o n t a c t w i t h solution, p e r u n i t e l e c t r o d e v o l u m e ) w i t h t h e e x p r e s s i o n
)1/3
[B-15]
This e x p r e s s i o n was f o r m u l a t e d w i t h t h e a s s u m p t i o n that
all t h e lead sulfate particles at a g i v e n position x w i t h i n t h e
e l e c t r o d e h a v e v o l u m e s g i v e n b y the c u b e of a characteristic d i m e n s i o n lPbSO4 g i v e n by
l so,
= l
ePbSO4
~/3
[B-16]
Realistically, at any g i v e n t i m e d u r i n g d i s c h a r g e t h e r e exists a distribution of particle sizes at each position w i t h i n
t h e electrode: t h e larger particles h a v i n g b e e n b o r n earlyon in d i s c h a r g e and t h e smaller ones b o r n m o r e recently.
T h e a p p r o x i m a t i o n is m o s t valid w h e n the birth of nuclei
occurs over a v e r y short period of time, a p p r o a c h i n g w h a t
is d e s c r i b e d in s o m e treatises as i n s t a n t a n e o u s n u c l e a t i o n
at a critical d e g r e e of s u p e r s a t u r a t i o n (32).
It should be m e n t i o n e d that, in principle, t h e rate constant kp,t can be related to the f r e q u e n c y r e x p ( - A G J R T ) .
H o w e v e r , in v i e w of the a p p r o x i m a t e n a t u r e of the g r o w t h
rate expression, w e h a v e c h o s e n to n e g l e c t any relationship b e t w e e n this f r e q u e n c y and kppt.
APPENDIX C
Determination of npbso4from Particle Size Data
T a b l e C-I s h o w s sizes of particles g i v e n b y Wales et aL
(14) at four current densities at t h e e n d of discharge. As w e
m e n t i o n e d earlier, d u r i n g d i s c h a r g e npbSO4 r e m a i n s relatively c o n s t a n t after t h e n u c l e a t i o n p e r i o d and t h e n eventually decreases d u e to a s h r i n k i n g of interfacial area
c a u s e d b y utilization of PbO2. I f t h e total e l e c t r o d e v o l u m e
does n o t c h a n g e substantially, t h e n t h e n u m b e r d e n s i t y
b a s e d on unit e l e c t r o d e v o l u m e , Npbso4, r e m a i n s c o n s t a n t
after t h e n u c l e a t i o n period t*. The end-of-discharge range
of particle sizes g i v e n by Wales et aL (14) (Table C-I) can b e
u s e d to e s t i m a t e Npbso4(t > t*). T h e n u m b e r d e n s i t y b a s e d
on u n i t i n t e r n a l PbO2 area, npbso4(t = t*), w h i c h is p l o t t e d
in Fig. 2, is related to Npbso4(t > t*) and can b e calculated
f r o m t h e relation 7
Npbso4(t > t*) = npbSO4(t = t*)A~
P b § + H S O ( ~k~ PbSO~(~) + H +
kb
9
[B-12]
ApbsO4 = 5ePbSO42~(npbso4A~
k~ [Cpb+2]CCH20]2
k~
(an+) (apb§
(am eq)
[B-2]
[C-l]
L e t us suppose, for illustrative purposes, that t h e PbSO4
particles at the end of d i s c h a r g e are all t h e s a m e size w i t h a
single characteristic d i m e n s i o n I and that particle v o l u m e
can be a p p r o x i m a t e d by t h e c u b e of t h e characteristic di-
7 As we mentioned earlier Apb02 does not change significantly
during the nucleation period: Apbo2(t = t*) ~ A~
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J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
1680
Table C-I. Data needed to estimate experimental number density
Discharge current Experimental (14) Capacity delivered
density, m.A/cm~ particle size range
q, mA 9s/era~
ev~so~
~~
1.8
12.5
50
200
10-20 i~m
2-6 I~m
0.5-3 ~m
0.5-3 ~m
2100
1700
1200
700
0.53
0.43
0.30
0.18
kppt
ka,1
k~.l
kf,l
kb,1
K
K~
m e n s i o n . I f the v o l u m e fraction of lead sulfate is k n o w n ,
then
s
e~
Npbso4 -- (lpbSO4eod)3
/PbSO4
LAA
[C-2]
LAB
T h e lead-sulfate v o l u m e fraction s
e~ is related to t h e
c a p a c i t y d e n s i t y delivered d u r i n g discharge, q, by
LBB
M~
evbso4 =
qYPbsO4
2F
[C-3]
T h i s relation as applied to a w h o l e e l e c t r o d e m u s t be realized as a p p r o x i m a t e if the e l e c t r o d e is n o t u n i f o r m l y utilized d u r i n g discharge. This is not a bad a p p r o x i m a t i o n for
t h e two l o w e r d i s c h a r g e rates s h o w n in Table C-I (1). The
l o w e r r a n g e of n u m b e r densities plotted in Fig. 2 is the result of a p p l y i n g Eq. [18]-[20] at the h i g h end of t h e particle
size r a n g e g i v e n in Table C-I. Conversely, the h i g h e r range
of n u m b e r densities plotted in Fig. 7 results f r o m a p p l y i n g
t h e s e e q u a t i o n s at t h e low e n d of the particle size range.
T h e s h a d e d area s h o w n in Fig. 2 is the region b e t w e e n
t h e s e t w o e x t r e m e s in particle sizes, and w e can see that
t h e actual e x p e r i m e n t a l values of npbso4(t = t*) will fall
w i t h i n this region.
If t h e distribution of particle sizes w i t h i n t h e r a n g e g i v e n
b y Wales et al. w e r e k n o w n , t h e n a single v a l u e of
npbso4(t = t*), rather t h a n a range of values, c o u l d be estimated. As a s e c o n d d e g r e e of a p p r o x i m a t i o n , we can ass u m e that t h e particles w i t h i n the g i v e n r a n g e are unif o r m l y distributed. In this case, the e x p r e s s i o n relating
particle sizes to n u m b e r density b a s e d on e l e c t r o d e volume becomes
NpbSO4 =
EpbSO4
cod / 4(lhigh -- /low)'~
cod ('hlgh
JIlow dl __
;lh,gh ~3dl EpdSO4 ~ ~high4 -- ~]ow4 )
[C-4]
Mi
nppt
n
Ni
N
n
N
q
Q
r
R~
R
si
t~0
t
T
Vi
Y~
Vi
V
X
zi
Z
rate c o n s t a n t for the precipitation reaction, cm/s
rate c o n s t a n t for reaction l in t h e anodic direction
rate c o n s t a n t for reaction I in the cathodic direction
rate c o n s t a n t for reaction l in t h e forward direction
rate c o n s t a n t for reaction l in t h e b a c k w a r d direction
e q u i l i b r i u m c o n s t a n t (Eq. [B-10])
friction coefficient for interaction of species i
and j, J - s / c m 5
characteristic d i m e n s i o n o f a PbSO4 particle, c m
m u l t i c o m p o n e n t transport parameter, mo12/
J-cm-s
m u l t i c o m p o n e n t transport parameter, mol2/jcm-s
m u l t i c o m p o n e n t transport parameter, mol2/Jcm-s
s y m b o l for t h e c h e m i c a l f o r m u l a of species i
q u a n t i t y defined by Eq. [A-10]-[A-13]
n u m b e r density of precipitate particles, c m -2
n u m b e r of electrons i n v o l v e d in an e l e c t r o d e reaction
superficial flux of species i, moYcm2-s
A v o g a d r o ' s n u m b e r , 6.0225 x 102~ m o l -~
surface n u m b e r d e n s i t y of particles, nuclei, or
sites, c m -2
v o l u m e n u m b e r density of particles, nuclei, or
sites, cm -3
capacity density delivered by an electrode,
C/cm 3
q u a n t i t y defined by Eq. [A-17]
radius of c u r v a t u r e of spherical-cap nucleus, c m
v o l u m e t r i c p r o d u c t i o n rate of species i by reaction, mol/cm3-s
universal gas constant, 8.3143 J / m o l - K
stoichiometric coefficient of species i in an elect r o d e reaction
transference n u m b e r of species i relative to species 0
time, s
absolute t e m p e r a t u r e , K
partial m o l a r v o l u m e of species i, cm~/mol
m o l a r v o l u m e of species i, cm~/mol
v e l o c i t y of species i, cm/s
e l e c t r o d e potential, V
d i s t a n c e f r o m the current collector, c m
c h a r g e n u m b e r of species i
Z e l d o v i c h n o n - e q u i l i b r i u m factor (Eq. [22])
Jllow
T h e d a s h e d line in Fig. 2 r e p r e s e n t s t h e results of a p p l y i n g
t h e a b o v e e q u a t i o n along w i t h Eq. [18] to e s t i m a t e npbso4.
A
ch
Ci
CT
Ci
D
DAA
DAB
DBA
D~B
Dpb2+
AG
2~G~
AGd
F
f
f(e)
isoh~
isoad
ioref
I
LIST OF SYMBOLS
interfacial area per unit e l e c t r o d e v o l u m e , cm2/
cm 3
activity of species i
c o n c e n t r a t i o n of species i, m o l / c m 3
total solution concentration, moYcm 3
c o n c e n t r a t i o n defined by Eq. [A-9] m o l / c m 3
q u a n t i t y defined by Eq. [A-13]
diffusion coefficient, cm2/s
diffusion coefficient, cm2/s
diffusion coefficient, cm2/s
diffusion coefficient, cm2/s
lead-ion diffusion coefficient, cm2/s
binary-interaction diffusion coefficient, cm2/s
q u a n t i t y defined by Eq. [A-15]
free e n e r g y of f o r m a t i o n for a PbSO4 cluster, J
free e n e r g y of solidification per unit PbSO4 volume, J/cm 3
m o l a r activational free e n e r g y for P b 2+ diffusion,
J/mol
F a r a d a y ' s constant, 96,487 C / e q u i v
F/RT, V -1
v o l u m e ratio of a spherical cap to its sphere
(Eq. [20])
superficial current d e n s i t y in the solution phase,
A/cm 2
superficial current density in t h e solid phase,
A/cm2
e x c h a n g e c u r r e n t density at ~
2
total c u r r e n t density, A / c m 2 c H2so4, A / c m
Greek Symbols
aa, ac
anodic and cathodic transfer coefficient, respectively
s y m m e t r y factor
~vb2+
lead-ion diffusion layer thickness, c m
e
porosity or v o l u m e fraction of electrolyte
ei
v o l u m e fraction of solid species i
7
specific, solid-liquid interfacial free energy,
J/cm 2
7~
m e a n m o l a r activity coefficient of species i and j
ri
surface c o n c e n t r a t i o n of species i, m o l / c m 2
K
electrolyte conductivity, m h o / c m
~A, ~B
c h e m i c a l potential of electrolyte A or B, J / m o l
~cnt
n u m b e r of m o l e c u l e s in a critical n u c l e u s
v
f u n d a m e n t a l j u m p frequency, s -1
vi
the n u m b e r of species i into w h i c h a m o l e c u l e of
an electrolyte dissociates
~o
f r e q u e n c y of PbSO4 a d d i t i o n per critical nucleus, s -1
~soln
electric potential in the solution, V
~soHd
electric potential in the solid, V
c o n d u c t i v i t y of t h e solid matrix, m h o / c m
0
contact angle of PbSO4 on PbO2
Subscripts
a,b
solid-phase species a and b
A,B
electrolytes A and B
crit
refers to a cluster of critical size
e
refers to t h e e l e c t r o c h e m i c a l reaction
cod
e n d of d i s c h a r g e
ex, e x m e x p o n e n t s on porosity (for tortuosity correction)
high
refers to the larger end of a particle-size r a n g e
low
refers to the smaller end of a particle-size r a n g e
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J. Electrochem. Soc., Vol. 137, No. 6, June 1990 9 The Electrochemical Society, Inc.
1
PbSO4
ppt
PbO2
RDS
refers to a reaction
refers to PbSO4 particle, nucleus, or species
refers to precipitate
refers to PbO2 species
refers to the rate-determining step in a reaction
mechanism
site
refers to an active nucleation site on PbO2
S
refers to the precipitation reaction
refers to a solid compound
(s)
0
species 0, the neutral solvent
1,2
species 1, 2 which make up electrolyte A
1,3
species 1, 3 which make up electrolyte B
[]
relative to the volume average velocity
Superscripts
crit
refers to cluster of critical size
eq
refers to an equilibrated concentration
(*)
refers to an adsorbed species
*
signifies the value at the end of the nucleation
period
o
refers to initial condition (charged state)
ref
refers to the reference condition
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Performance of the Low-Current-Density-Synthesized
Polypyrrole in Lithium Cells Containing Propylene Carbonate
Petr Novdk 1 and Wolf Vielstich*
Institute of Physical Chemistry, University of Bonn, D-5300 Bonn 1, Germany
ABSTRACT
The properties of polypyrrole (PPy) films synthesized at very low current densities (1-250 ~A/cm2) were investigated.
Potentiodynamic cycling, in situ FTIR spectroscopy, and in situ differential electrochemical mass spectroscopy were employed. The use of propylene carbonate (PC) based electrolytes for the synthesis results in an incorporation of PC fragments (arising by the electro-oxidation of PC) into the grown polypyrrole film. During subsequent cycling of P P y films in
0.5M LiC104/PC electrolyte, the electrochemical oxidation of PC proceeds parallel with the doping/undoping process. The
decrease in current density during polymer growth has the same effect as an addition of small amounts of water into the
electrolyte for synthesis--the performance of the polymer in secondary lithium cells is improved.
Since the first reports on electrochemical synthesis of
good-quality polypyrrole (PPy) films published in 1979 (1),
n u m e r o u s papers on both the P P y preparation and properties have appeared (2). Recently, the most publicized appli* Electrochemical Society Active Member.
1 Present address: J. Heyrovsk:~ Institute of Physical Chemistry
and Electrochemistry, Czechoslovak Academy of Sciences,
Dolej~kova 3, CS-182 23 Prague 8, Czechoslovakia.
cation of P P y is the rechargeable battery having polymeric
positive and lithium negative electrodes (3, 4).
Many articles have been devoted to the experimental
conditions during P P y synthesis. It has become clear that
variations of the supporting electrolyte and solvent are of
crucial importance to the properties of the resulting polymer (5). I n Li/PPy batteries, the best results reported to
date were achieved on polymers synthesized from the
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