Journal of Power Sources 85 Ž2000. 212–223
www.elsevier.comrlocaterjpowsour
Determination of reaction resistances for metal-hydride electrodes during
anodic polarization
Chunsheng Wang a , Manuel P. Soriaga
b
a,)
, Supramaniam Srinivasan
b
a
Department of Chemistry, Texas A & M UniÕersity, College Station, TX 77843, USA
Center for Energy and EnÕironmental Studies, Princeton UniÕersity, Princeton, NJ 08544, USA
Received 21 May 1999; accepted 14 June 1999
Abstract
The anodic polarization process in metal-hydride electrodes occurs via the following consecutive steps: Ži. diffusion of absorbed
hydrogen from the bulk to the surface of the electrode, Žii. hydrogen transfer from absorbed state to adsorbed state at the electrode
surface, and Žiii. electrochemical oxidation of the adsorbed hydrogen on the electrode surface. A theoretical treatment is presented to
account for the dependencies on these three consecutive steps, of the reaction resistances, anodic limiting current, cathodic limiting
current, and exchange current. The theoretical analysis predicts that the total resistances measured from linear micropolarization is the
sum of the charge-transfer, hydrogen-transfer, and hydrogen-diffusion resistances, which is in contradiction with the generally accepted
idea that the resistance measured from linear micropolarization is only the charge-transfer resistance. For a metal-hydride electrode with a
flat pressure plateau at a low state of discharge ŽSOD., the resistance measured from linear micropolarization is approximately equal to
the sum of three resistances measured from AC impedance, namely, charge-transfer, hydrogen-transfer and hydrogen-diffusion
resistances; however, when the SOD is high, the resistance measured from linear micropolarization is higher than total resistances
measured from AC impedance. q 2000 Elsevier Science S.A. All rights reserved.
Keywords: Metal-hydride electrodes; Anodic polarization; Hydrogen
1. Introduction
Nickelrmetal hydride ŽNirMH. batteries have been
developed and commercialized because of their high energy density, high rate capability, tolerance to overcharge
and overdischarge, freedom from poisonous heavy metals,
and the absence of consumption of electrolyte during
charge–discharge cycles w1a,1bx. The performance of a
metal-hydride electrode is determined by the diffusion of
absorbed hydrogen from the bulk to the electrode surface,
the rate of transfer hydrogen from the absorbed state to the
adsorbed state, and the kinetics of electrochemical oxidation of adsorbed hydrogen at the electrode surface w2x.
Extensive work has been carried out on the measurement of kinetic parameters of metal-hydride electrodes
)
Corresponding author. Tel.: q1-409-845-1846; fax: q1-409-8153523; E-mail: soriaga@chemvx.tamu.edu
w3–10x. The exchange current of metal-hydride electrodes
can be determined by linear micropolarization Žh - 10
mV. w3,4,6x, Tafel polarization w4x and electrochemical
impedance spectroscopy ŽEIS. w4,8x. Witham et al. w4x and
Ratnakumar et al. w10x found that exchange currents estimated from linear micropolarization for LaNi 5y x Sn x , and
LaNi 5y xGe x Ž0 F x F 0.5. alloys were in agreement with
those from EIS, but were quite different from values
obtained from Tafel polarization methods. From electrochemical investigations of bare and Pd-coated
LaNi 4.25 Al 0.75 electrode in alkaline solution, Zheng et al.
w8x believed that the resistances determined from linear
micropolarization are the sum of the electrochemical reaction and ohmic resistances, and the total resistance agrees
with the sum of three resistances obtained from AC
impedance measurements. In a previous paper, we showed
that the three resistances in the Nyquist plot represent the
charge-transfer reaction, hydrogen-transfer between absorbed and adsorbed states, and hydrogen diffusion in the
bulk of alloy, respectively w9x. Hence, the resistance for
0378-7753r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 7 8 - 7 7 5 3 Ž 9 9 . 0 0 3 3 6 - 5
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
hydrogen transfer at the surface and the resistance for
hydrogen diffusion in the alloy may be included in the
resistances determined from linear micropolarization
curves. The diversity of results from previous impedance
measurements may be caused by a neglect of the effect of
hydrogen transfer between the absorbed and adsorbed states
on the resistance in linear micropolarization and Tafel
polarization techniques.
In this paper, we describe results from our investigation
of the kinetics of the anodic polarization on a metal-hydride electrode with a flat pressure plateau. Apart from the
charge-transfer and hydrogen-diffusion steps, the present
treatment considers hydrogen transfer through the interface, a process neglected in previous studies.
2. Basic assumptions
The pressure plateau of a hydrogen storage alloy is flat
ŽFig. 1. and the fully charged hydride electrode with
hydrogen concentration Cb a is initially considered to be
uniformly discharged to a state of discharge ŽSOD.. The
hydrogen concentration C, expressed in moles of hydrogen
atoms stored in the volume unit of the host, is Cb a in the b
phase, and Ca b in the a phase. The molar quotient of
hydrogen concentration in the bulk can be expressed as
X s CrNm , where Nm is the total molar number available
for hydrogen per unite volume of host metal. The hydrogen distribution before and in anodic polarization are
shown in Fig. 2.
The anodic polarization process involves a charge-transfer step followed by the hydrogen transition from the
absorbed site in the near-surface to the adsorbed site on the
electrode surface, and then the diffusion of absorbed hydrogen from the bulk to the near surface. When the
hydrogen concentration of the metal Ž a . phase at the
interface between the a and b phases, X is lower than the
hydrogen concentration X a b , which is the hydrogen con-
213
Fig. 2. The hydrogen concentration distribution in a metal-hydride particle: Ža. initial state, Žb. under anodic polarization.
™
centration of the a phase equilibrating with that of the
hydride Žb . phase, the b a phase transition occurs.
MH ads q OHy
k1
y
Ž 1.
2
ky1
m MH
™ MH
™ MHŽ a .
MH absŽsurface.
MH absŽ bulk .
m MqH Oqe
k2
Ž 2.
ads
ky2
diffusion
Ž 3.
absŽsurface.
phase transition
MH Ž b .
Ž 4.
During EIS, linear polarization and Tafel polarization measurement, the SOD should not be changed, which means
that the phase transformation does not occur. Therefore,
steps Ž1. to Ž3. are assumed to be responsible for the
overpotential associated with the anodic polarization of a
metal-hydride electrode.
2.1. Diffusion of hydrogen in the bulk of metal-hydride
electrode with spherical symmetry
During anodic polarization, a steady state is assumed to
be established; it is therefore reasonable to consider that
the hydrogen diffusion is in a steady state and the SOD is
not changed during anodic polarization. From Fick’s first
law and the continuity of H flow in the a phase, the
diffusion of hydrogen in the bulk of the metal-hydride
electrode, reaction Ž3., can be expressed as follows w11x:
4 FDa Nm p r 2
dX
dr
s 4 FDa Nm p r 02
s 4p
Fig. 1. Typical curve of pressure vs. hydrogen content for a metal-hydride
electrode.
ro2 Ia
dX
dr
rs r o
Ž 5.
where F is the Faraday’s constant, ro is the average radius
of alloy particles, Ia is the anodic current density, and Da
is the diffusion coefficient of hydrogen in the a phase
which is taken as a constant.
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
214
Integrating Eq. Ž5. with the boundary conditions, r s ro ,
X s Xs , and r s rb , X s Xb gives:
Xs s X a b y
Ia ro
FDa Nm
ž
ro
rb
y1
gen-diffusion and hydrogen-transition currents., and substituting Xs from Eq. Ž10. into Eq. Ž12., one gets, at u s 0:
1
/
Ž 6.
1
1
s
I La
q
Fk 2 X a b
1
1
s
I Lbs
Ž 14 .
q
I Lbd
I Lbs
where rb is the radius of the unreacted hydride and Xs is
the hydrogen concentration in the near-surface of the
electrode. If SOD s QdrQmax , then rorrb can be expressed as w11x:
ro
y1r3
s Ž 1 y SOD .
Ž 7.
rb
where I Lbd Žs Fk 2 Xa b . is the limiting current imposed by
the hydrogen transfer from the absorbed to adsorbed state.
By substitution of Eq. Ž14. into Eq. Ž10., Xs can be
rewritten as:
Therefore, Eq. Ž6. can be rewritten as:
When Ia s 0, the Xs s X a b , which mean that the hydrogen
concentration of equilibrated a phase Žwith b phase. is
same when the hydrogen pressure plateau is flat as shown
in Fig. 1.
Combination of Eqs. Ž12., Ž13. and Ž15., and introduction of a new parameter w2x K t Žs ŽŽ k 2 y ky2 . X a b .rŽŽ k 2
y ky2 . Xa b q ky2 .., which is the interfacial hydrogentransfer coefficient, one gets:
Xs s X a b y
Ia ro
FDa Nm
Ž 1 y SOD.
y1r3
y1
Ž 8.
The anodic hydrogen diffusion Žfrom bulk to near-surface.
limiting current I Lbs , at which the absorbed-hydrogen surface concentration goes to zero, can be obtained by setting
in Xs s 0 in Eq. Ž8.:
I Lbs s
FDa Nm X a b
ro Ž 1 y SOD .
y1r3
y1
ž
Xs s X a b 1 y
I Lbs
/
us
1yKt
Ž 10 .
2.2. Transfer of hydrogen from the absorbed state in the
near-surface to adsorbed state on the electrode surface
By assuming first-order kinetics for the absorption reaction, the hydrogen transfer wreaction Ž2.x can be expressed
as w2x:
Ia s Fk 2 Xs Ž 1 y u . y Fky2 u Ž 1 y Xs .
Ž 11 .
where k 2 and ky2 are, respectively, the forward and
backward rate constants for the hydrogen-transfer reaction,
and u is the coverage of adsorbed hydrogen on the surface. From Eq. Ž11., one can obtain the dependence of the
surface coverage u on Xs w2x:
ž
ž
ž
1
1
/
y
I La
uo 1 y
Ž 9.
Substitution of Eq. Ž9. into Eq. Ž8., one obtains:
Ia
Xs s X a b 1 y
I Lbd
Ia
I La
1
/
Ž 15 .
ž
uo 1 y
s
1
y
I La
Ia
I Lbd
/
1yKt
Ia
Ia
I La
Ia
/
Ž 16 .
I Lbs
where K t is a parameter which characterizes the relative
difficulty of hydrogen transfer from the absorbed to the
adsorbed state. If K t ) 0, then k 2 ) ky2 , which means that
the transfer of hydrogen from the absorbed state occurs
more easily than the transfer from a desorbed to absorbed
state w2x. The reverse is true if K t - 0.
When Eq. Ž9. is substituted into Eq. Ž14., the anodic
limiting current I La is obtained:
I La s
Fk 2 Nm Da Xa b
k 2 ro Ž 1 y SOD .
y1r3
Ž 17 .
y 1 q Nm Da
Ž 12 .
From Eq. Ž17., it can be seen that I La increases with
increasing k 2 , Da , Nm , Xa b , but decreases with increasing
SOD and ro .
Although Eq. Ž16. is deduced from a metal-hydride
electrode with a flat pressure plateau and spherical symmetry, it can also be obtained in a more general case ŽAppendix A.. In other words, Eq. Ž16. is valid for all
metal-hydride electrodes.
At the equilibrium potential, Ia s 0, ha s 0, u s uo and
X s Xa b . The equilibrium condition gives w2x:
2.3. Electrochemical oxidation of hydrogen on the electrode surface
k 2 Xs y
us
uo s
Ia
F
Ž k 2 y ky2 . Xs q ky2
k 2 Xa b
Ž k 2 y ky2 . Xa b q ky2
Ž 13 .
where uo is the initial coverage of adsorbed hydrogen on
the surface.
Using I La to denote the anodic limiting current at which
the hydrogen coverage goes to zero Ži.e., limiting hydro-
The anodic current density for the charge-transfer reaction ŽEq. Ž1.. can be written as w7x:
u
Ia s Io
bF
½ ž /
uo
exp
RT
h y
1yu
1 y uo
exp y
Ž1yb . F
RT
h
5
Ž 18 .
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
215
while the exchange current Io can be expressed as w9x:
Io s Fk 1 aOH y uo exp
ž
bF
RT
wo
/
s Fky1 a H 2 O Ž 1 y uo . exp y
Ž1yb . F
RT
wo
Ž 19 .
where wo is the initial potential of the metal-hydride
electrode, h Žs w y wo . the overpotential, a H 2 O and aOH y
the activities of H 2 O and OHy, respectively, k 1 and ky1
the forward and reverse rate constants of the charge-transfer reaction, respectively, b the symmetry factor, R the
gas constant, and T is the absolute temperature.
2.4. The anodic polarization of metal-hydride electrode
2.4.1. Reaction resistance in AC impedance spectroscopy
In a previous paper w9x, a mathematical model was
developed for the EIS of a metal-hydride electrode. The
equivalent circuit for metal-hydride electrode is shown in
Fig. 3. In this figure, R s , R ct , and R ht are, respectively,
the solution, charge-transfer and hydrogen-transfer resistances; Cdl and C ht are the double-layer and absorption
capacitances, respectively; Zd is the impedance of hydrogen diffusion in the alloy. For a metal-hydride electrode
with a flat pressure Žpotential. plateau, Zd can be simplified as a Warburg impedance w Zw s s Ž1 y j .x in series
'2 v D
Fig. 4. Equivalent circuit for finite spherical hydrogen diffusion Zd : Ža. in
the high-frequency range, Žb. in the low-frequency range. Z w s s
'2 v D
a
Ž1 y j ., C d s
s
r o w1y Ž 1y SOD . 1r 3 x
,
3s
r o w1y Ž1ySOD . 1r 3 x s
Rd s
i s r o w1y Ž 1y SOD . 1r 3 x
,
R
Da Ž 1y SOD . 1r 3
.
5Da
a
with a capacitance w C1Žs ro Ž1 y Xa b . Xa b .Ž Nm F 2rRT .x in
the high-frequency range; at low frequencies, Zd can be
taken as a resistance Ž R d . in parallel with a resistance
Ž R . –capacitance Ž Cd . series ŽFig. 4..
Simulations based on the present model show that there
are usually three arcs in the Nyquist plot; these represent
charge-transfer, hydrogen-transfer, and hydrogen-diffusion
steps. The relative frequency range of these three steps in
the Nyquist plot depends on the difference of their respective time constants. The arc of the step with a small time
constant will appear in the high-frequency range, whereas
the arc at low frequency corresponds to the step with the
large time constant.
The parameters for the hydride electrode have been
expressed as follows w9x:
R ct s
R ht s
Rd s
C ht s
Cd s
RT
Ž 20 .
FIo
RT
2
F ky2
ž
1q
1
/
K1 A
s ro Ž 1 y SOD .
Ž 1 q K1 K 2 A.
y1 r3
Ž 21 .
y1
Ž 22 .
Da
F 2G
RT
(K
1A q
y2
1
(K
ro 1 y Ž 1 y SOD .
1
Ž 23 .
A
1r3
Ž 24 .
3s
s
Zd s
'jv D
tanh ro Ž 1 y Ž 1 y SOD .
Fig. 3. Equivalent circuit for metal-hydride electrodes.
D
y
1r3
.
(
jv
D
ro
Ž 25 .
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
216
where
ss
ž(
K1 K2 A q
K1 s
As
k1
, K2 s
ky1
aOH y
aH 2 O
exp
ž
(
K1 K2 A
ky2
RT
wo
bF
/
C ht s
RT
Nm F 2
Ž 27 .
/
Ž 28 .
Ž 29 .
Ž 31 .
FDa Nm Ž 1 y Xa b .
Rd s
F
RT
F
ž
ž
s I Lsb
y1
1
q
I Lbd
I Ldb
1
1
q
I Lbs
q
I Lbs
I Lsb
ž
1
/
1
I Lsb
I Lsb
/
/
(
jv
y
Da
Da
ro
(
) (
I Lbd
q
I Ldb
Ž 38 .
I Ldb
I Lbd
The total resistance measured from AC impedance is:
R ta s R ct q R ht q R d
RT
s
F
ž
1
1
q
Io
1
q
I Lbd
1
q
I Ldb
1
q
ILbs
ILsb
/
Ž 39 .
The time constants of three steps can be expressed as
follows:Charge-transfer reaction:
te s
RTCdl
Ž 40 .
FIo
Hydrogen-transfer reaction:
ta s R ht = C ht s
FG
Ž 41 .
I Lbd q I Ldb
Hydrogen diffusion:
td s
ro FNm 1 y Ž 1 y SOD .
1r3
Ž 42 .
3 Ž I Lsb q I Lbs .
It is more convenient to employ dimensionless parameters tarteŽs K ae . and tdrteŽs K de . to assess whether the
three arcs overlap. When Eqs. Ž40. – Ž42. are combined, the
following relationships are obtained:
K ae s
K de s
Ž 34 .
Ž 37 .
'jv D
F k 1 ky1 aH 2 O aOH
Ž 32 .
Ž 33 .
Ž 36 .
1r3
/
q
I Lbs
1r3 2
Ž 1 y SOD.
=coth w ro Ž 1y Ž 1ySOD. 1r3 . x
Combination of Eqs. Ž13., Ž21. – Ž32. yields:
1
ž
1
RT
Ž 30 .
Fky2 Ž 1 y Xa b . s I Ldb
RT
1
Ž 35 .
I Ldb
Fro2 1 y Ž 1 y SOD .
3 RTDa
Io s
where R d and Cd are the resistance and capacitor for
hydrogen diffusion in the bulk of alloy.
Simulation results indicate that: Ži. R d , not Ž R d q R ., is
the resistance for the equivalent circuit of Zd in the
low-frequency range ŽFig. 4b., and Žii. R d is approximately equal to the resistance calculated from the hydrogen-diffusion impedance Zd at a low SOD. However,
when the SOD is high, R d is a little larger than the
resistance calculated from Zd in Fig. 3 Žvide infra..
The limiting current of a step reflects its reaction rate;
hence, the dependence of the parameters on the limiting
current can be deduced. Similar to the definitions of the
anodic hydrogen-transfer limiting current Ž I Lbd s Fk 2 Xa b .,
the Fky 2 Ž1 y X a b . were considered as cathodic
hydrogen-transfer limiting current Žhydrogen transfer from
the adsorbed to absorbed state. in Ref. w2x ŽEq. Ž31...
Comparing the anodic hydrogen-diffusion limiting current
ŽEq. Ž9.., we can consider Ž FDa Nm Ž1 y Xa b ..rŽ ro wŽ1 y
SOD.y1 r3 y 1x. as a cathodic hydrogen-diffusion limiting
current although it is only approximately equal to the real
cathodic hydrogen-diffusion limiting current measured
from Tafel polarization.
R ht s
I Lbd
w
uo k 1
y1r3
q
I Lbd
Fro w Ž 1ySOD. y1r3 y1 x
Ž 1 y uo . ky1
ro Ž 1 y SOD .
( (
RT
Zd s
Combination of Eqs. Ž19. and Ž28. gives:
As
I Ldb
Ž 26 .
,
ž /
RT
K1 A q 1
Io s
2
Cd s
k2
F
Fk 1 aOH exp
1
y2
F 2G
F 2 Io G
Ž 43 .
RTCdl Ž I Lbd q I Ldb .
ro F 2 Nm Io 1 y Ž 1 y SOD .
3 RTCdl Ž I Lsb q I Lbs .
1r3
Ž 44 .
Eqs. Ž43. and Ž44. can be used to estimate the relative
frequency range of the three steps at a low SOD. The
simulation obtained using Eqs. Ž40. and Ž42. indicates that
the first arc that appears in the higher frequency range is a
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
charge-transfer reaction, the second arc at intermediate
frequencies range represents the hydrogen-transfer reaction, and the third arc at low frequencies corresponds to
the diffusion of absorbed hydrogen in the alloy. For example, a TiMn 1.5 H x electrode with a double-layer capacitance Cdl s 2 = 10y5 Frcm2 , size of a particle ro s 10y3
cm, concentration of absorbed hydrogen Nm s 0.02
molrcm3, maximum coverage of adsorbed hydrogen G H
s 7.5 = 10y1 0 molrcm2 , Faraday’s constant F s 96,487
Crmol, and a SOD of 10%, has K ae s Ž141.34 Io .rŽ I Lbd
q I Ldb . and K de s Ž4.4 = 10 4 Io .rŽ I Lsb q I Lbs .; this means
that a hydrogen-transfer and hydrogen-diffusion limiting
current larger than 141.34 and 4.4 = 10 4 times the exchange current, respectively, are needed for hydrogen
transfer and hydrogen diffusion steps appearing in higher
frequency range than that for the charge-transfer step. If
the values of the hydrogen-transfer and hydrogen-diffusion
limiting currents are similar to the anodic or cathodic
limiting current, and the anodic limiting current is 4.1–6.2
times larger than the exchange current w5x, the frequency
range of the three steps appear in decreasing order: charge
transfer) hydrogen transfer) hydrogen diffusion. If the
difference between time constants among the three steps
are large, and the difference between resistances are small,
the three arcs are well-resolved from one another. But if
the SOD is high, or the hydrogen diffusion coefficient is
low, the hydrogen-diffusion arc moves toward the lower
frequency range; in addition, the radius of the almost
circular low-frequency segment increases rapidly such that
the arcs of charge-transfer and hydrogen-transfer reactions
may overlap with the hydrogen-diffusion arc due to the
larger resistance of hydrogen diffusion w9x.
It can be seen from this equation that the Tafel polarization
includes electrochemical polarization, hydrogen-diffusion
polarization, and a third polarization associated with a
hydrogen-transfer and hydrogen-diffusion reaction. During
anodic Tafel polarization, the anodic current increases as
the overpotential is increased but, when strongly polarized,
a limiting current I La can be observed with its value
measurable from Tafel polarization curves.
2.4.3. Reaction resistances in linear micropolarization
For small polarization Žh - 10 mV., the anodic current
density for the charge-transfer reaction ŽEq. Ž18.. can be
linearized. The reaction resistances R lm can be obtained
from a complete Taylor expansion of Eq. Ž18. ŽAppendix
B.:
h
R lm s
hs
RT
bF
ln
Ia
RT
q
bF
Io
ln
uo
Ž 45 .
u
If Io ) I La , then Ia - Io , which will yield a negative electrochemical polarization in Eq. Ž45.. Hence, when Io ) I La ,
Eq. Ž45. cannot be employed to calculate the polarization
and exchange current.
If Io < I La , substitution of Eqs. Ž13. and Ž16. into Eq.
Ž45. yields:
hs
RT
bF
ln
RT
q
bF
Ia
Io
RT
q
bF
ln
ln 1 y K t
ž
ž
I La
I La y Ia
1
/
1
y
I La
I Lbd
/
Ia
Ž 46 .
RT
Ia
q
F
uo y u
1
s
Io
Ž 47 .
Ia uo Ž 1 y uo .
Substitution of Eqs. Ž13., Ž16., Ž31. and Ž32. into Eq.
Ž47. gives:
1
R lm s
RT
F
1
q
1
q
I Lbd
1
q
I Ldb
Io
1yKt
1
q
I Lbs
Ia
0
ILsb
I Lbs
Ž 48 .
In a linear micropolarization process, Ia is much lower
than I Lbs , and K t Ž IarILbs . can be neglected. Eq. Ž48. can
thus be simplified as:
R lm s
2.4.2. Reaction resistances in anodic Tafel polarization
During the anodic polarization process, a steady state is
characterized by Ži. an equality of the rates for the three
consecutive steps, and Žii. a constant in SOD. During
anodic Tafel polarization, when Io < I La , and h 4 RTrF,
Eq. Ž18. can be simplified in the following form w2x:
217
RT
F
ž
1
1
q
Io
1
q
I Lbd
1
q
I Ldb
1
q
ILbs
I Lsb
/
Ž 49 .
Upon comparison with Eq. Ž39., it can be seen that the
resistances measured from linear micropolarization are the
same as the sum of the charge-transfer, hydrogen-transfer
and hydrogen-diffusion resistances measured from AC
impedance techniques. That is,
R lm s R ta s R ct q R ht q R d
Ž 50 .
It can be readily seen from Eq. Ž50. that the resistance
measured from linear micropolarization is the sum of three
resistances in series: charge-transfer resistance R ct , hydrogen-transfer resistance R ht , and hydrogen-diffusion resistance R hd . This theoretical result is in contradiction with
the generally accepted idea that the resistance determined
from linear micropolarization arises solely from a chargetransfer reaction. When Io is much greater than the limiting currents, R ct < R ht q R hd and the overpotential, even
near Eeq is mass-transfer overpotential induced by hydrogen diffusion and hydrogen transfer. Only when Io is
much smaller than the limiting currents, will R ct 4 R ht q
R hd , and the overpotential near Eeq is due to charge-transfer activation.
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
218
When combined with Eqs. Ž20., Ž50. and Ž52. yields:
RT
F
ž
1
1
q
I La
I Lc
/
s R ht q R d
Ž 53 .
Eqs. Ž50., Ž52. and Ž53. reveal the relationships between
the parameters determined from linear micropolarization,
Tafel polarization and AC impedance; i.e., the dependencies of the total resistances measured from linear micropolarization and AC impedance on the parameters measured
from Tafel polarization Žexchange current Io , anodic limiting current I La and cathodic limiting current I Lc ..
It is worth mentioning that I Lc in the present model is
an apparent cathodic limiting current. Its value will be
approximately equal to the real cathodic limiting current
measured from Tafel polarization if the a b phase
transformation in cathodic Tafel polarization does not occur and the potential at limiting current is smaller than
y0.932 V vs. HgrHgO, i.e., no hydrogen recombinations
occur.
™
3. Discussion
Fig. 5. The simulated resistances of hydride electrodes at different states
of discharge with the parameters in Table 1 and X a b s 0.1, I lbd s 0.154
Arcm2 s, I ldb s 0.695 Arcm2 s. Ža. The hydrogen-diffusion resistances
are calculated from the hydrogen-diffusion impedance Zd ŽEqs. Ž9., Ž32.
and Ž37.., the equivalent circuit of hydrogen-diffusion impedance, Zd
Žparameters in Fig. 4b. and R d ŽEqs. Ž9., Ž32. and Ž34... The resistance
calculated from Zd is obtained from the real part of Zd at a low
frequency, 0.0005 Hz. Žb. The total resistances calculated, respectively,
from linear micropolarization at different K t values w Ia in Eq. Ž48. was
approximated as 0.01rŽ R ct q R ht q R d .x and from the real part of AC
impedance spectrum at a low frequency, 0.0005 Hz.
2.5. The relationship of the parameters measured from
linear micropolarization, Tafel polarization and AC
impedance
Similar to Eq. Ž14., the cathodic limiting current can be
expressed as
1
1
s
I Lc
1
q
I Lsb
I Ldb
™
Ž 51 .
I Lc is the cathodic limiting current if the a b phase
transition does not occur during cathodic polarization.
Substitution of Eqs. Ž14. and Ž51. into Eq. Ž50. and
rearrangement gives the following equation for R lm :
R lm s
RT
F
ž
1
1
q
Io
1
q
ILa
I Lc
/
Ž 52 .
Although Eq. Ž50. is deduced from a metal-hydride
electrode with flat pressure plateau, it is also applicable to
all types of metal-hydride electrodes regardless of the
morphology of its pressure plateau; this is because Eq. Ž50.
is derived from Eqs. Ž13. and Ž16., both of which can also
be deduced from a more general case ŽAppendix A..
Hence, the resistance of metal-hydride electrodes measured
from linear micropolarization curves is the total resistance
that includes contributions from charge-transfer, hydrogen-transfer and hydrogen-diffusion reactions. However,
most treatments of linear micropolarization assume that the
resistance measured is only a charge-transfer resistance
w3,4,7,10x; the contributions from hydrogen transfer and
hydrogen diffusion are simply neglected. As a matter of
fact, numerous experiments show that Ži. the discharge
process for metal-hydride electrodes is controlled by hydrogen diffusion within the bulk w12–14x, and Žii. the
Table 1
Parameters used in the simulations
Parameters
Values
aH 2 O
aOH
Cdl
D
F
k1
ky1
k2
ky2
Nm
G
ro
0.68
1.38=10 18
2=10y5
10y8
96 487
5.38=10y1 6
4.8=10y16
1.6=10y5
0.8=10y5
0.2
7.5=10y9
10y3
Unit
References
Frcm2
cm2 rs
A srmol
cmrs
cmrs
molrcm2 s
molrcm2 s
molrcm3
molrcm2
cmy3
w9x
Calculated from Ref. w15x
TiMn 1.5 alloy
w9x
w9x
Calculated from Ref. w15x
Calculated from Ref. w15x
w9x
w9x
fitted
fitted
w9x
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
Fig. 6. The time constant of hydrogen diffusion calculated from R d =Cd
ŽEq. Ž41.. and directly from Zd , respectively, with the same parameters as
in Fig. 5.
219
impedance Zd , and Žii. the neglect of the term in Eq. Ž48.
for linear micropolarization.
Fig. 5a shows the hydrogen-diffusion resistances at
different states of discharge calculated from hydrogen-diffusion impedance Zd , the equivalent circuit of Zd at lowfrequency range, and R d , along with the parameters in
Table 1 and: X a b s 0.1, I Lbd s 0.154 Arcm2 s, and I Ldb
s 0.695 Arcm2 s. The resistance calculated from Zd is
obtained from the real part of impedance at a low frequency v s 0.0005. It can be seen in Fig. 5a that R d is
equal to the resistance of the equivalent circuit of Zd , but
both are larger than the resistance calculated from the
hydrogen-diffusion impedance Zd with increasing SOD.
The reason is that the expansion of cothw ro Ž1 y Ž1 y
j
SOD.1r3 . Dv x of Zd at the point of ro Ž1 y Ž1 y
j
SOD.1r3 . Dv s 0 at low frequencies, only the first three
terms in the expansion are selected w9x. When the SOD is
j
low, ro Ž1 y Ž1 y SOD.1r3 . Dv is close to zero, and the
relative error is small. However, when the SOD is high,
j
ro Ž1 y Ž1 y SOD.1r3 . Dv is non-zero even in the lowfrequency range; this means that the use of only the first
three expansion terms leads to inaccuracies. Since the
fourth expansion term will be positive, the admittance of
hydrogen diffusion is smaller than actual value, while the
impedance of hydrogen diffusion would be higher than
real value. Hence, the resistance R d calculated from only
the first three expansion terms is larger than the actual
value when the SOD is high ŽFig. 5a.. Neglecting the
K t IarILbs term likewise causes the resistance determined
from linear micropolarization to be approximately equal to
R ct q R ht q R d ŽEq. Ž50.., but the difference in the resistances calculated from Eqs. Ž48. and Ž49. is smaller than
the difference between R lm and resistance calculated form
Zd ŽFig. 5b.. Therefore, the resistance determined from
linear micropolarization would be higher than the sum of
three resistances measured from AC impedance if the SOD
is high. Based on the same argument, the hydrogen-diffusion time constant calculated as R d = Cd would also be
higher than the time constant measured from Zd as shown
in Fig. 6.
The present theoretical analysis has uncovered a relationship between resistance, exchange current, anodic limiting current, and cathodic limiting current ŽEqs. Ž20. and
(
(
(
anodic overpotential is highly dependent on the surface
properties introduced by various surface pretreatments w2x.
Such experiments demonstrate the importance of a theoretical treatment that includes not only charge-transfer resistance but also hydrogen-transfer and hydrogen-diffusion
resistances during anodic polarization. Zheng et al. w8x
recently found that, for LaNi 4.25 Al 0.75 , the resistance measured from linear micropolarization is approximately equal
to the sum of three resistances measured from AC
impedance; this is in agreement with predictions based on
the present model. However, it was speculated that the
other two resistances were Ži. the resistance between the
current collector and the electrode pellet, and Žii. the alloy
particle-to-particle contact resistance. It is our contention
that the two arcs in the high-frequency range of the
Nyquist plot are due to the charge-transfer and hydrogentransfer reactions, with the arc at the low-frequency region
to be due to hydrogen-diffusion resistance w9x. From the
present model, it can be argued that the resistance measured from linear micropolarization is approximately equal
to the sum of three Žcharge-transfer, hydrogen-transfer and
hydrogen-diffusion. resistances measured from AC
impedance. The approximation arises from Ži. the assumption of R d as the resistance of hydrogen-diffusion
(
Fig. 7. Simulated Nyquist plot for the hydride electrode with flat pressure plateau at different states of discharge. The simulations are based upon the model
in Fig. 3 with the parameters given in Tables 1 and 2.
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
220
Fig. 8. Simulated Nyquist plot for the hydride electrode with flat pressure
plateau at different hydrogen concentrations. The simulations are based
upon the model in Fig. 3 with the parameters given in Tables 1 and 2.
Ž53.. as well as a dependence of the anodic and cathodic
limiting currents on the SOD and the hydrogen concentration X a b ŽEq. Ž17. and the combinations of Eqs. Ž31., Ž32.
and Ž51... The influence of the anodic and cathodic limiting currents on the Nyquist plot can be observed by
simulation of the Nyquist plot at different SOD and Xa b
as shown in Figs. 7 and 8, respectively. The parameters
used in the present simulation are shown in Table 1, with
the exchange current and limiting current values at different SOD and Xa b given in Table 2. It can be seen in Fig.
7 and Table 2 that the diameter of the low-frequency arc
Ž R d . increases with decreasing anodic and cathodic limiting currents; the SOD affects the arc only in the lowfrequency region. Since, in fact, the pressure plateau of
practical hydride electrode is not very flat, X a b will
decrease with increasing SOD. Fig. 8 shows the influence
of X a b on the AC impedance. From Fig. 8 and Table 2,
one can see that the hydrogen-transfer and hydrogen-diffusion resistances decreases with increasing X a b , but the
exchange current first increases and later decreases as Xa b
is increased; this suggests the existence of an optimum
X a b value for the exchange current. Differentiation of Eq.
Ž38. with respect to X a b allows one to determine the
optimum X a b value to yield a maximum exchange current:
Xa b p s
Fig. 9. Exchange currents of hydride electrodes at different K 2 values
calculated from Eq. Ž38. with the parameters of k 1 s 5.38=10y1 6 cmrs,
ky 1 s 4.8 = 10y 16 , a H 2 O s 0.68 molrcm 3 , a OH y s 1.38 = 10 18
molrcm3 , ky2 s 0.8=10y5 molrcm2 s, ky2 s 0.8=10y5 molrcm2 s,
X a s 0.1.
reaction. From this figure and Eq. Ž54., it can be ascertained that the optimum hydrogen concentration X a b decreases with increasing K 2 . When the latter is large, X a b
is small, and the decrease in exchange current with an
increase in X a b can easily be observed w8x. In actuality,
the value of K 2 is not very high; hence, the exchange
current usually increases with Xa b w15x.
Further, use of Tafel polarization curves to measure the
exchange current must be done with caution since the
Tafel Eq. Ž46. is valid only when Io < Ia F I La . Even in
such a case, the influence of the hydrogen-transfer reaction
on the polarization must be considered in the determination of exchange current. Usually, the exchange current is
determined by the following equation obtainable from Eq.
Ž46..
hs
RT
bF
ln
RT
1
Ž 54 .
K2 q1
Fig. 9 shows the relationship between exchange current
and K 2 , the equilibrium constant for the hydrogen-transfer
Table 2
Resistances, limiting current, and exchange current at different SOD and
Xa b
Resistances Ž V cm2 .
Current ŽArcm2 .
Rt
R ht
Rd
Io
I La
I Lc
1.397
1.397
1.397
1.397
1.143
1.081
1.143
0.203
0.203
0.203
0.203
0.124
0.103
0.009
0.053
0.165
0.304
1.157
0.03
0.023
0.019
0.018
0.018
0.018
0.018
0.022
0.024
0.022
0.12
0.077
0.05
5.123=10y3
0.24
0.36
0.6
0.608
0.462
0.341
0.045
0.54
0.473
0.338
SOD
Xa b
0.1
0.3
0.5
0.99
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.3
0.5
q
bF
I La
Io
RT
q
bF
ln 1 y K t
ln
ž
ž
Ia
I La y Ia
1
1
y
I La
I Lbd
/
/
Ia
Ž 55 .
According to Eq. Ž55., a plot of h-vs.-lnŽ IarŽ I La y Ia ..
should produce a straight line with slope and intercept,
respectively, being RTrb F and Ž RTrb F .lnŽ I La rIa . at
h 4 Ž RTrF . if K t s 0. In other words, the exchange
current Io can be calculated from I La and RTrb F data.
However, when K t is in positive w16x Ži.e., hydrogen is
transferred more easily from the absorbed to the adsorbed
state than in the reverse direction., the h-vs.-lnŽ IarŽ I La y
Ia .. plot shows a linear relationship between h and
lnŽ IarŽ I La y Ia .. only in the middle range of polarization;
negative deviations from linearity appear at the low-overpotential domain, while positive deviations from linearity
emerge at the large-overpotential polarization region ŽFig.
10a.. In this case, the exchange current density measured
from Tafel polarization is lower than actual value. When
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
221
sion and hydrogen-transfer effects. If the hydrogen-diffusion limiting current is smaller than the exchange current,
the resistance is not controlled by the charge-transfer
reaction, and the exchange current cannot be measured
directly from linear micropolarization. In this case, the
exchange current cannot also be determined from Tafel
polarization curves but it can be measured by AC
impedance. For metal-hydride electrodes with a flat pressure plateau and low SOD, the resistance measured from
linear micropolarization is approximately equal to the sum
of the charge-transfer, hydrogen-transfer and hydrogen-diffusion resistances, all three measurable from AC impedance
spectroscopy. If the SOD is high, the resistance measured
from linear micropolarization is larger than the sum of
three resistances measured from AC impedance.
5. List of symbols
aH 2 O , aOH y
Ca b , Cb a
Fig. 10. Overpotential as a function of lnŽ Ia rŽ I La y Ia .. for the hydride
electrode at different K t values calculated from Eq. Ž55. with the
parameters: Ža. K t s 0.9, I La s 0.27, Ž1r I La .yŽ1rI Lbd . s1r0.26; Žb.
K t sy0.9, I La s 0.113, Ž1r I La .yŽ1r I Lbd . s1r0.26.
K t - 0, the h-vs.-lnŽ IarŽ I La y Ia .. curves show a linear
relationship between h and lnŽ IarŽ I La y Ia .. throughout
the entire polarization range ŽFig. 10b.. Under these conditions, the exchange current is a little higher than the actual
value.
Cdl , Cd , C ht
Da
F
Ia , Io
4. Conclusions
I La , I Lc
Based on a three-step anodic polarization process ŽH
diffusion from the bulk to the surface, H transfer from
absorbed to adsorbed state, and electrochemical oxidation
of adsorbed H., a theoretical relationship between Ži. the
resistances in anodic linear micropolarization, Žii. the exchange currents and anodic and cathodic limiting currents
in Tafel polarization, and Žiii. the resistance in AC
impedance measurements has been obtained. Application
of the theory shows that the resistance measured from
linear micropolarization is the sum of three resistances: Ži.
charge-transfer, Žii. hydrogen-transfer, and Žiii. hydrogendiffusion resistances. When the charge-transfer reaction is
the rate-determining step, the exchange current can be
measured approximately from linear micropolarization or
Tafel polarization after correction for the hydrogen-diffu-
I Lbs , I Lsb
I Lbd , I Ldb
j
activity of H 2 O and OHy in
solution, respectively Žmolrcm3 .
hydrogen concentration of a
phase equilibrating with b phase
and hydrogen concentration of b
phase equilibrating with a
phase, respectively Žmolrcm3 .
double-layer capacitance, ca
pacitance associated with hydrogen diffusion, and hydrogen
transfer capacitance, respectively ŽFrcm2 .
diffusion coefficient of hydrogen in the a phase of metal-hydride particles Žcm2rs.
Faraday’s constant Ž96 487
Crmol.
anodic current and exchange
current, respectively ŽArcm2 .
anodic limiting current and cathodic limiting current, respectively ŽArcm2 .
limiting current of hydrogen
diffusion from the bulk to the
surface of particle and limiting
current of hydrogen diffusion
from the surface to the bulk of
particles, respectively ŽArcm2 .
limiting current of hydrogen
transfer from absorbed state to
adsorbed state, and limiting current of hydrogen transfer from
adsorbed state to absorbed state,
respectively ŽArcm2 .
'y 1
222
k 1 , ky1
k 2 , ky2
K 1Žs k 1rky1 .,
K 2 Žs k 2rky2 .
K ae , K de
Nm
R s , R ct , R ht , R d
r o , rb
SOD
X Žs CrNm .,
Xs , Xa bŽs Ca brNm .
Greek
b
u , uo , G
w , wo
h Žs w y wo .
v
ta , te , td
s
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
forward and backward rate constants of charge-transfer reaction, respectively Žcmrs.
forward and backward rate constants for hydrogen-transfer reaction, respectively Žmolrcm2 s.
equilibrium constant of chargetransfer reaction and hydrogen
transfer process, respectively
ratio of time constant of hydrogen-transfer reaction to time
constant of charge-transfer reaction, and ratio of time constant
of hydrogen diffusion to time
constant of charge-transfer reaction, respectively
available concentration of
absorbed hydrogen Žmolrcm3 .
solution resistance, chargetransfer reaction resistance, hydrogen-transfer reaction resistance, and hydrogen-diffusion
resistance, respectively Ž V cm2 .
radius of alloy particles and
radius of untransformed hydride-core phase Žcm.
state of discharge
relative concentration of hydrogen in a phase, relative concentration of hydrogen at the
surface of the electrode, and relative concentration of hydrogen
in a phase equilibrating with b
phase, respectively
symmetry factor in the oxidation direction
coverage degree of adsorbed
hydrogen, equilibrium Žinitial.
coverage of adsorbed hydrogen,
maximum coverage degree of
adsorbed hydrogen on the electrode surface Žmolrcm2 .
potential and equilibrium Žinitial. potential of hydride electrode, respectively ŽV.
overpotential of hydride electrode ŽV.
angular frequency of ac single
time constant of hydrogentransfer reaction, time constant
of charge-transfer reaction, and
time constant of hydrogen diffusion, respectively Žs.
impedance coefficient of hydrogen diffusion
Acknowledgements
The work was performed under the auspices of the US
Department of Energy, Division of Chemical Sciences,
Office of Basic Energy Science ŽContract No. DE-AC0276CH00016, BNL and DE-FG03-93ER1481.. The assistance of M.R. Marrero ŽTexas A & M University., D.
Barsellini ŽINIFTA, La Plata, Argentina. and Dr. H.G. Pan
ŽZhejiang University of China. are gratefully acknowledged. Support for the collaborative work with INIFTA
was made possible by the National Science Foundation
ŽINT-9813852..
Appendix A
In the absence of migration, the rate of mass transfer is
proportional to the concentration gradient at the electrode
surface, which can be approximated by Nm ŽŽ Xo y Xs .rd .,
where d is the thickness of the hypothetical diffusion layer
at the electrode surface, and X o is the initial concentration
of hydrogen in the bulk. The hydrogen diffusion rate can
be express as
X o y Xs
Ia s FD
Ž A-1.
d
When anodic current increases to limiting current I La , at
which the hydrogen coverage goes to zero, Xs would
decrease to XsU . The relationship between I La and XsU can
be obtained from Eq. Ž12. at u s 0
I La s Fk 2 XsU
Ž A-2.
Substitution of Eq. ŽA-2. into Eq. ŽA-1. at Ia s I La
FDXo Nm
I La s
Ž A-3.
DNm
dq
k2
Combination of Eqs. ŽA-1. and ŽA-3. along with the use of
I Lbd Žs Fk 2 X o . to denote the limiting current imposed by
the hydrogen transfer from the absorbed to the adsorbed
state yields:
Xs s X o 1 y
s Xo 1 y
ž
ž
1
1
y
I La
Fk 2 X o
1
1
y
I La
I Lbd
/
Ia
/
Ia
Ž A-4.
Substitution of Eq. ŽA-4. into Eq. Ž12. and introduction of
a new parameter, K t Žs ŽŽ K 2 y ky2 . X o .rŽŽ k 2 y ky2 . X o
q ky2 .., which is the interfacial hydrogen-transfer coefficient, results in the following equation:
uo 1 y
us
1yKt
ž
Ia
I La
1
1
y
I La
I Lbd
Ž A-5.
/
Ia
C. Wang et al.r Journal of Power Sources 85 (2000) 212–223
223
Eq. ŽA-5. is the same as Eq. Ž16.; X o in Eq. ŽA-5. and
Xa b in Eq. Ž16. both denote the initial concentration of
hydrogen in the alloy.
Substituting for the derivatives and evaluating at the
central point:
Ia
u y uo
F
s
q
h
Ž A-9.
Io
uo Ž 1 y uo .
RT
Appendix B
B.1. Linearizing the current density for charge-transfer
reaction
which is equivalent to Eq. Ž47.. By truncating the series at
j s 1, a simple linear approximation has been derived from
the more complex Eq. Ž18.. It is valid for small excursions
from the central point.
A function of two variables, f Ž u ,h ., can be expanded
about the point Ž uo ,ho . by the Taylor formula:
References
f Ž u ,h . s f Ž uo ,ho .
`
q
1
Ý
js1
ž
j
E
du
Eu
E
q dh
Eh
j
/
f Ž u ,h .
Ž u o ,h o .
Ž A-6.
where du s u y uo , and dh s h y ho . The expression in
parentheses is a differential operator that is raised to the
jth power. The various powers of ErEu , and ErEh are
symbols for repeated differentiation. After the operator
acts on f Ž u ,h ., the limits are taken for f Ž uo ,ho ..
Consider the expansion of the current-overpotential Eq.
Ž18.:
f Ž u ,h . s
Ia
Io
u
s
uo
exp
1yu
y
1 y uo
bF
ž /
RT
exp
h
yŽ 1 y b . F
RT
h
Ž A-7.
The central point is chosen at Ž uo ,0., where Ia s 0. If only
the terms for j s 1 are kept, one obtains:
Ia
Io
E
s du
Eu
E
q dh
Eh
f Ž u ,h .
where the derivative is evaluated at uo ,h s 0.
Ž A-8.
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