Mathematical Modelling of Primary Alkaline Batteries Jonathan Johansen Bachelor of Applied Science (Honours) Queensland University of Technology A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy March 2007 Principal Supervisor: Dr Troy W. Farrell Associate Supervisor: Professor Ian W. Turner Queensland University of Technology School of Mathematical Sciences Faculty of Science Brisbane, Queensland, 4001, AUSTRALIA
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Mathematical Modelling of
Primary Alkaline Batteries
Jonathan JohansenBachelor of Applied Science (Honours)Queensland University of Technology
A thesis submitted in partial fulfilment of the requirements for the degree ofDoctor of Philosophy
March 2007
Principal Supervisor: Dr Troy W. FarrellAssociate Supervisor: Professor Ian W. Turner
Queensland University of TechnologySchool of Mathematical Sciences
Faculty of ScienceBrisbane, Queensland, 4001, AUSTRALIA
the numerical scheme and the discretisation method employed.
The numerical simulation software starts with the initial distributions of all variables,
and solves the model equations iteratively to determine the distribution of each variable
at every time step. An overview of the algorithm is given below.
1. Set constants and read input data file.
2. Initialise all physical, chemical and electrochemical variables, all main loop’s logic
variables and open output files.
3. Enter main loop:
3.3 The Numerical Solution of the Simplified Equations 47
(a) Determine time step size.
(b) Initialise solver loop’s logic variables.
(c) Enter solver loop.
i. Calculate the coefficient values that correspond to the linearised form
of the model equations.
ii. Solve the matrix system based on the linearised system of equations.
iii. Exit the solver loop if any of the following conditions apply:
• The solution has converged,
• The solution has taken too many iterations without converging,
• A non-physical solution has been predicted. For example, a chemical
concentration that is negative.
iv. Otherwise, update variables and go to 3(c)i.
(d) If the solution converged, output data to files and update time.
(e) Exit the main loop if any of the following conditions apply:
• The maximum time is reached,
• The cell voltage is below the cutoff,
• The solver has failed to converge too many times.
(f) Otherwise, go to 3a.
4. Close output files and exit.
The simplified model equations are able to be solved in steps 3(c)i and 3(c)ii by dis-
cretising a linear version of the equations, formed by applying a combination of lin-
earisation and fixed-point techniques, to create a system of linear equations which may
be solved in matrix form. The model equations are discretised in dimensional form
using a control-volume approach (Patankar 1980), which has the benefit of implicitly
conserving the physical quantities. In this approach the model domain is divided into
48 Chapter 3. The Simplified Model
Figure 3.2: Schematic diagram of the control-volumes in one dimension. The variables areevaluated at the node points 1 to kmax.
a number of so called control-volumes and the model equations with spatial deriva-
tives are integrated over these representative control-volumes. A schematic diagram
of the control-volumes in one dimension is shown in Figure 3.2. The model equations
containing a time derivative are then integrated over the time step.
The discretisation, in time, of the model equations is achieved through a flexible time
weighting technique, for example,
t+∆t∫
t∗=t
f (t∗) dt∗ ≈ θf (t + ∆t) + (1 − θ) f (t) , (3.38)
where t (s) is the time, ∆t (s) is the time step, f (t) is an arbitrary function of time,
and θ is the time weighting parameter. Note that using Equation (3.38), we can choose
θ = 0 for fully explicit time stepping, θ = 1 for fully implicit time stepping, and θ = 1/2
for Crank-Nicolson time stepping. In our code, fully implicit time weighting (θ = 1)
is used in all simulations. Equation (3.29) is first differentiated with respect to time,
and then discretised as described above. In addition, the sum in Equation (3.29) is
evaluated to 47 terms to ensure accuracy.
As an example of an equation obtained using the control-volume approach, we present
the discretised form of Equation (3.36). It has been integrated over the kth control-
volume, as displayed in Figure 3.2, and integrated over the time step as described
in Equation (3.38). The dimensional form of Equation (3.36) is given in Chapter 4
in Table 4.1. We note that we have used the dimensional form of Equations (3.35)
and (3.37) (also given in Table 4.1) to eliminate the volume average velocity and solution
3.3 The Numerical Solution of the Simplified Equations 49
phase current, respectively. The discretised form is,
(
ε∼ +εEMD
εEMDpε∼p
)
R2k+ 1
2
− R2k− 1
2
2
(
COH− |t+∆tk − COH− |tk
)
=
∆tθ
[
B|mk+ 12
COH− |t+∆tk+1 − COH− |t+∆t
k
Rk+1 − Rk− B|mk− 1
2
COH− |t+∆tk − COH− |t+∆t
k−1
Rk − Rk−1
]
+ ∆t (1 − θ)
[
B|tk+ 12
COH− |tk+1 − COH− |tkRk+1 − Rk
− B|tk− 12
COH− |tk − COH− |tk−1
Rk − Rk−1
]
−(
V H2O − tK+V KOH
)
∆t
FRk+ 1
2×
θ
[
COH− |mk+ 12
κ∼∞|mk+ 12
η|t+∆tk+1 − η|t+∆t
k
Rk+1 − Rk+ COH− |t+∆t
k+ 12
κ∼∞|mk+ 12
η|mk+1 − η|mkRk+1 − Rk
− COH− |mk+ 12
κ∼∞|mk+ 12
η|mk+1 − η|mkRk+1 − Rk
]
+ (1 − θ) COH− |tk+ 12
κ∼∞|tk+ 12
η|tk+1 − η|tkRk+1 − Rk
+
(
V H2O − tK+V KOH
)
∆t
FRk− 1
2×
θ
[
COH− |mk− 12
κ∼∞|mk− 12
η|t+∆tk − η|t+∆t
k−1
Rk − Rk−1+ COH− |t+∆t
k− 12
κ∼∞|mk− 12
η|mk − η|mk−1
Rk − Rk−1
− COH− |mk− 12
κ∼∞|mk− 12
η|mk − η|mk−1
Rk − Rk−1
]
+ (1 − θ) COH− |tk− 12
κ∼∞|tk− 12
η|tk − η|tk−1
Rk − Rk−1
−3εEMD
(
1 − tOH−
)
εEMDproF
R2k+ 1
2
− R2k− 1
2
2
∆t(
θ i∼p|t+∆tk + (1 − θ) i∼p|tk
)
, (3.39)
where the superscript m denotes a trial, or “best guess”, value, subscript k denotes
the value at the kth node, and the subtraction or addition of a half to k denotes the
value at the inner or outer control-volume face, respectively. All variables evaluated
at the inner and outer control-volume faces are approximated by linearly interpolating
between the data at the node points. The parameter B (cm2.s−1) is defined as
B = R
[
DOH−∞ε∼ +2RgasT
(
V H2O − tK+V KOH
)
ε3/2∼
F 2κ∼∞COH−
×(
1 − tOH− +
COH−
CH2O
)
∂ ln aKOH
∂COH−
]
. (3.40)
The time step must be calculated at the beginning of each main loop iteration. This
is done in such a manner so as to maintain stability in the numerical algorithm, and
50 Chapter 3. The Simplified Model
is governed by several factors, including the maximum allowable time step size, the
previous time step size, whether the solution converged in the previous attempt, and
the proximity to data output times. In addition, during the initial stages of discharge,
the time step is chosen to be small, to accurately capture the rapid changes that occur
when the discharge starts.
In the simplified model equations, certain non-linear terms, namely, the advection
term in Equation (3.36), and all of Equation (3.4) are linearised on the basis of a trial
solution. The advection term is not upwinded because, as observed by Farrell et al.
(2000), the term only accounts for a very small proportion of the electrolyte flux and
does not cause instability in the numerical simulations. The remaining nonlinearities
are treated using a fixed-point iteration approach (Burden & Faires 2001).
After initialising the resulting linear system, using a trial solution, the system is solved.
The new solution is then used as the trial solution to obtain a second approximation.
This process is repeated until the trial solution does not change significantly, using the
criteria that,∥
∥dk∥
∥
2
(‖dk−1‖ − ‖dk‖) ‖xk‖ < tolerance, (3.41)
where
dk = xk+1 − xk. (3.42)
Here, xk represents a vector of the variables’ values on the kth iteration. This criterion is
based on the convergence of a first order process. We note that Expression (3.41) would
give convergence in the undesirable case when∥
∥dk∥
∥ >∥
∥dk−1∥
∥, which corresponds to
the divergence of the iterative process. If this is the case, Expression (3.41) is not used,
and it is assumed that the process is not converged. Upon convergence of a particular
series of iterations, the time is updated to t + ∆t and the linear system of equations is
again used to determine the values of the variables at the new time.
The simplifications carried out in the previous section provide a substantial reduction
in complexity of the model system, which now has only one partial differential equation,
as compared with three in the full model system. The ensuing reduction in the number
of numerical calculations needed to solve the system means that the software that
3.4 Results and Discussions 51
Time (h)
Ece
ll(V
)
0 1 2 3 4 50.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Figure 3.3: Comparison of model output (filled symbols) with experimental data (outlinedsymbols) (Williams 1995). Shown is experimental data for cathodes with radii in the ranges0-22.5 (), 38.5-53 (♦), 75-150 (⊳) and 150-250 (#) µm, and model output for cathodes withradii of 10 (H), 45 (), 100 () and 180 ( ) µm.
implements the numerical solution is easily run on a standard desktop PC. For example,
A comparison of the output of the simplified model with the experimental data of
Williams is given in Figure 3.3, and Tables 3.3 and 3.4 list the parameter values used
in the model to simulate the experimental data.
52 Chapter 3. The Simplified Model
From Figure 3.3, it is seen that the model results compare well with the experimental
data. The simplified model accurately captures the polarisation effects seen in the
experimental results with the cathodic discharge time increasing as the EMD particle
size is decreased. This is due to more uniform reaction distributions within the smaller
EMD particles that lead to a greater utilisation of active material within the cathodes
containing these particles.
In simulating the data of Williams, we have taken the value of CMn4+ that appears
in the EMD conductivity function (given by Expression (3.12)) to be that at 80% of
the radius (i.e., 0.8yo) of a given crystal. This corresponds to the value of CMn4+
at y = 0.8 appearing in Equations (3.11) and (3.32). The effect of choosing various
positions within the oxide crystals at which to take the value of CMn4+ in order to
calculate the EMD conductivity is shown in Figure 3.4. The simulations are for the
cathode manufactured by Williams that consists of EMD particles in the size fraction
150 µm ≤ 2ro ≤ 300µm (refer to Table 3.4). The corresponding experimental discharge
result is also given in Figure 3.4 for comparison.
The results in Figure 3.4 indicate that taking the value of CMn4+ at 0.8yo yields a
theoretical discharge that corresponds well with the experimental result. Farrell et al.
(2000) and Farrell & Please (2005) previously took the concentration value appearing
in the conductivity function to be that at the outer radius of the EMD crystals. As
Figure 3.4 demonstrates, however, this approach appears to overestimate the resistance
experienced by electrons moving within the oxide used by Williams and leads to shorter
discharge times in comparison with the experimental data. To obtain more accurate
predictions of the conductivity of non-uniformly reduced EMD, an in-depth study into
the current paths and the connectivity on a crystal scale would be needed.
Upon examination of Table 3.2, the assumption that α1 → 0, seems valid at low
to medium discharge rates. Furthermore, the second assumption, that the transfer
current does not change on the timescale of crystal diffusion, seems valid in the particle
setting, where the total discharge current of the particle may be directly specified (as
constant). However, within an operating primary alkaline battery there are diverse
conditions throughout the cathode, where solid phase conductivity and variations of
3.4 Results and Discussions 53
Time (h)
Ece
ll(V
)
0 1 2 30.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Figure 3.4: Comparison of experimental data (⊳) and model results (filled symbols) whenEMD conductivity is calculated at 0.6yo (), 0.8yo ( ) and yo (N). The experimental cathodecontains particles of the size fraction 150µm ≤ 2ro ≤ 300µm.
concentrations within the electrolyte contribute to non-uniformity. In this environment,
crystals may experience changes in their individual discharge rates at shorter time scales
than seen in individual particles.
We also note here that, for the case of constant particle discharge current, Equa-
tion (3.29) simplifies to the equivalent of Equation (3.44), and thus we recover the
more specific particle discharge result of Farrell & Please.
In Section 3.2.2, an expression was obtained for the concentration distribution of Mn4+
within an EMD crystal (i.e., Equation (3.29)) by applying the method of Laplace trans-
forms. This expression does not depend on any simplifying assumptions. Nevertheless,
if we are willing to admit assumptions, namely that α1 → 0, and that in
does not change
on the timescale of proton diffusion, then asymptotic methods can be applied to the
crystal-scale proton diffusion problem (defined by Equations (3.1) to (3.3) and (3.7))
in order to obtain approximate expressions for the distribution of Mn4+ within EMD
crystals. The analysis follows closely that reported by Farrell & Please (2005) for the
54 Chapter 3. The Simplified Model
discharge of porous EMD particles, and we find that the O(1) expression is given by,
CMn4+
(
y, t)
= 1 + 3
t∫
0
in
dt∗, (3.43)
and the O(α1) expression is given by,
CMn4+
(
y, t)
= 1 + 3
t∫
0
in
dt∗ + α1
(
y2
2− 3
10
)
in
− α12i0
n
y
∞∑
n=1
sin (λny)
λ2n sin (λn)
exp
(−λ2nt
α1
)
, (3.44)
where t∗ is a dummy variable.
The discharge results (given in terms of the fraction of the theoretical capacity of the
cathode that is used) of the simplified cathode model at various discharge rates are
presented in Figure 3.5. Either Equation (3.29) or (3.43) is used to model the distri-
bution of Mn4+ within EMD crystals. The cathode configuration for these simulations
is that of a cylindrical AA-cell, details of which are given in Table 3.5. At low dis-
charge rates, the use of either Equation (3.29) or (3.43) within the model yields very
similar discharge curves, however, as the current is increased, significant discrepancies
between the two models are observed. To understand why these discrepancies occur,
distributions of Mn4+ within an EMD crystal were obtained at R = Ri and r = ro as
given by Equations (3.29), (3.43) and (3.44) for discharge rates of 20 (Figure 3.6(a)),
50 (Figure 3.6(b)), and 100 (Figure 3.6(c)) mA.g−1 of EMD. At low discharge rates,
such as that shown in Figure 3.6(a), the distribution of Mn4+ within an EMD crystal
is essentially independent of crystal radius and the simplifying assumptions that con-
stitute the asymptotic solutions, namely, α1 → 0 and that in
does not change on the
timescale of proton diffusion, are well supported (in fact the α1 value corresponding to
Figure 3.6(a) is 0.17). Thus, the results predicted by Equations (3.29), (3.43) and (3.44)
correspond well in this discharge regime. When the discharge rate is increased, as in
Figure 3.6(b) and 3.6(c), the distribution of Mn4+ within an EMD crystal becomes more
non-uniform and the discrepancies between the predictions of Equations (3.29), (3.43)
and (3.44) become significant. Indeed, at a discharge rate of 100 mA.g−1 of EMD, the
3.5 Conclusions 55
Fraction of theoretical capacity used
Ece
ll(V
)
0 0.2 0.4 0.60.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Figure 3.5: Discharge results for simplified model using an asymptotic (filled symbols) (Equa-tion (3.43)) or Laplace (hollow symbols) (Equation (3.29)) crystal scale solution. The simula-tions correspond to 20 (), 50 (), 100 (#) and 200 (♦) mA.g−1 of EMD.
assumption that α1 → 0 can no longer be supported and the α1 value corresponding
to Figure 3.6(c) is 0.86. In this regime, Equations (3.43) and (3.44) become invalid.
3.5 Conclusions
In this chapter we have simplified an existing model of primary alkaline battery cathode
discharge (Farrell et al. 2000) to yield a smaller model that accounts for the important
written and used to provide validation and insight into the operation of the primary
alkaline battery system.
In particular, we presented the model equations of Farrell et al. (2000) in dimension-
less form and gave approximate sizes for the dimensionless constants appearing in the
equations. A simplified model was obtained by applying Laplace transform and pertur-
bation methods. In the analysis that ensued, it is shown that the three size scales used
by Farrell et al. to describe the porous EMD cathode can be reduced to two size scales
56 Chapter 3. The Simplified Model
Table 3.5: Discharge parameters and cell geometry for AA-cell cathode geometry asused in the simulations presented in Figure 3.5
Parameter Value
Discharge rate (mA.g−1of EMD) 20, 50, 100 and 200Particle radius (µm) 100Inner radius, Ri (cm) 0.45Outer radius, Ro (cm) 0.67
Height, H (cm) 4.04Total mass of cathode (g) 10.62
Mass of EMD in cathode (g) 9.24Mass of graphite in cathode (g) 0.8
y (cm)
CM
n4+
(mol
.cm
-3)
0 0.01 0.020
0.01
0.02
0.03
0.04
0.05
t = 5 h
t = 1 h
t = 3 h
t = 7 h
t = 9 h
t = 0 h
(a) Distributions during a 20 mA.g−1 of EMDdischarge.
y (cm)
CM
n4+
(mol
.cm
-3)
0 0.01 0.020
0.01
0.02
0.03
0.04
0.05
t = 1 h
t = 2 h
t = 3 h
t = 0 h
(b) Distributions during a 50 mA.g−1 of EMDdischarge.
y (cm)
CM
n4+
(mol
.cm
-3)
0 0.01 0.020
0.01
0.02
0.03
0.04
0.05
t = 15 min
t = 45 min
t = 0 min
(c) Distributions during a 100 mA.g−1 of EMDdischarge.
Figure 3.6: Mn4+ ion concentration distributions at various times within an EMD crystal atR = Ri, and r = ro, as given by Equation (3.29) (), Equation (3.43) (H) and Equation (3.44)( ).
3.5 Conclusions 57
without the loss of generality. In addition, the analysis demonstrates that the time
taken for electrolyte to diffuse into a porous EMD particle is fast compared with the
cathodic discharge time, and that ohmic losses within the graphite phase of the cathode
can be considered negligible. Furthermore, the simplified model incorporates a closed
form expression for the distribution of Mn4+ within an EMD crystal that is not reliant
on assumptions that may break down at high discharge rates. The simplified model of
primary alkaline battery cathode discharge extends the work of Farrell & Please into
the cathodic domain.
The simplified model equations are too complex to solve analytically, so a numerical
technique is used. Numerical solutions of the simplified model equations have been
easily be run on a standard desktop PC. In addition, the simplified model results
compare favourably with relevant experimental data.
CHAPTER 4
The Potentiostatic Model
4.1 Introduction
The reduction of EMD is a complex process, as evidenced by the number of research
articles that discuss its determination experimentally (see Chapter 2, Sections 2.2.1
and 2.2.2). Chabre & Pannetier (1995) give a comprehensive discription of the reduc-
tion process. They found that reduction consists of a mixture of heterogeneous and
homogeneous processes, some of which are irreversible. In addition, the crystal struc-
ture of the EMD changes during reduction. It is clear that writing down an accurate
mathematical description of the reduction process requires knowledge of the values
of the kinetic parameters. However, many of these values are not easy to measure
experimentally.
Chabre (1991) and Chabre & Pannetier (1995) used Step Potential Electrochemical
Spectroscopy (SPECS) to examine the reduction process. In a SPECS discharge, the
cell is subjected to a series of consecutive potentiostatic discharges, in which the cell
potential is decreased (or stepped) by a fixed amount in each discharge (usually starting
near the OCV), and the current response is recorded. A typical potential versus time
60 Chapter 4. The Potentiostatic Model
Time (h)
Ece
ll(V
)
0 50 100 1500.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Figure 4.1: The potential experienced by a cell during a SPECS discharge. In this SPECSdischarge the potential is stepped by 5 mV each hour.
curve is shown in Figure 4.1. As mentioned in Chapter 2, the advantage of SPECS over
other, continuous, discharge modes such as galvanostatic, constant load, or constant
power is that it emphasizes the electrochemical and physical responses of the cell.
Ideally, the cell is given time to equilibrate at each potential level, thus minimising
transport losses. However, this means that individual SPECS discharges can take
many days to complete, as can be seen from the time axis in Figure 4.1.
The multi-reaction nature of the reduction of EMD is very apparent in the SPECS
results. The minimum and maximum current or power experienced during each po-
tentiostatic discharge, versus potential, are frequently used formats to visualise results.
A typical plot of a SPECS discharge is shown in Figure 4.2. The figure shows sev-
eral clearly visible peaks in the power output (for example, at 1.46, 1.28 and 1.13
V). These peaks are attributed to the steps or reactions that constitute the reduction
process (Chabre 1991, Chabre & Pannetier 1995), and clearly show that it is not a
single-reaction process.
In contrast to the conclusions drawn by many researchers (see Chapter 2, Section 2.2.1)
4.1 Introduction 61
Ecell (V)
Min
and
Max
Pow
er(m
W.g
-1of
EM
D)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
2
4
6
8
10
Figure 4.2: The minimum and maximum power in each potential step of a typical EMDSPECS discharge. The minimum and maximum power values are usually found at the endand beginning of each potential step, respectively. Experimental data courtesy of Delta EMDAustralia Pty. Limited.
62 Chapter 4. The Potentiostatic Model
who have experimentally studied the reduction of EMD, the prevailing approach in
the mathematical modelling literature is to assume that reduction is described by a
single reaction, specifically, Reaction (2.1). Assuming this, a single Butler-Volmer like
expression (for example, see Chapter 3, Equation (3.4)) may be used to describe the
kinetics at the EMD/KOH interface, and a single Nernst like expression is adopted to
describe the zero current potential, or open circuit voltage (OCV).
In taking the above approach, many authors modify the Nernst like expression in order
to produce OCV curves that are closer to those determined experimentally. By doing
this, these authors are really modifying the kinetics of the reaction mechanism, and/or
the reaction mechansim itself. However, this is often done inconsistently, in that, the
Butler-Volmer equation is not modified to reflect these mechanistic changes. Farrell
et al. (2000) do make consistent modifications, based on the work of Chabre & Pan-
netier, however, (as we shall see) their linear approximation of the ion-ion interaction
term fails to account adequately for the true multi-reaction nature of the reduction
process.
In this chapter we aim to improve the treatment of this process in our mathematical
model. However, this is a difficult task because to accurately model the reduction of
EMD, the full reaction mechanism and associated parameter values, including exchange
current densities for each individual reaction at well characterised reference conditions,
should be known. Without the appropriate information, these become “free” param-
eters and there is no guarantee that the values of these parameters obtained by say,
fitting model simulations to experimental data, will be in any way realistic or unique. It
is conceivable that several sets of parameter values exist that give the same behaviour,
making the results and parameter values determined in such a way largely meaningless.
In addition, when there are many unknown parameters, the predictions of the model
may become simply a result of the choice of the values, so that no new information
may be extracted from the results. To avoid such ambiguity, the approach taken here
is to minimise the number of unknown parameters introduced into the model.
The question arises as to whether a modelling framework centred on a single-reaction
mechanism can display multi-reaction behaviour. To investigate this, we have chosen
4.2 Model Development 63
the zero current potential relationship proposed by Chabre & Pannetier (1995), given
by Equation (2.13) in Chapter 2 of this thesis. We note that this relationship was used
in Chapter 3, albeit with a linear approximation of the ion-ion interaction term. In
this work, however, we seek to find the best form for the ion-ion interaction term to
describe the multi-reaction reduction process. Furthermore, for this purpose, we will
attempt to model the stepped potential discharge of the alkaline battery cathode.
4.2 Model Development
Here we develop the model equations for the potentiostatic discharge of a primary al-
kaline battery in order to simulate SPECS discharges. The model is adapted from the
simplified model developed in Chapter 3. Only the changes to the previous model equa-
tions are detailed here. These are followed by two tables summarising the unchanged
equations (Table 4.1) and boundary conditions (Table 4.2) in dimensional form.
To account for the general form of the ion-ion interaction term, we use the modified
Butler-Volmer like expression (see Chapter 3, Equation (3.4)), namely,
in
= i00
(
CMn3+
C0Mn3+
)(
COH−
C0OH−
)
exp
[
(1 − αc)F
RgasT
(
ηp + Υ − Υ0)
]
−(
CMn4+
C0Mn4+
)(
CH2O
C0H2O
)
exp
[−αcF
RgasT
(
ηp + Υ − Υ0)
]
. (4.1)
The derivation of the general form of the above expression, as given by Farrell et al.
(2000), is reviewed in Appendix A. The process of determining a suitable form for the
ion-ion interaction term, Υ (CMn4+) (V), is detailed in Section 4.5.
In order to simulate potentiostatic discharge, we develop a boundary condition that
replaces Equation (3.24). By specifying the cell potential, Ecell(t) (V), we are fixing
the potential drop across the cathode, from the solid phase at the cathode/current
collector interface (R = Ro) to the solution phase at the cathode/bulk KOH interface
(R = Ri). This is designated by the expression,
φ|R=Ro− φ∼|R=Ri
= Ecell(t), (4.2)
64 Chapter 4. The Potentiostatic Model
where Ecell(t) is specified. We may rearrange the definition of the overpotential (see
Chapter 3, Equation (3.5)) to give an expression for the liquid phase potential on the
cathodic scale at R = Ri, namely,
φ∼|R=Ri= φ|R=Ri
− η|R=Ri− E0. (4.3)
Substituting this into Equation (4.2), we find that,
φ|R=Ro− φ|R=Ri
+ η|R=Ri+ E0 = Ecell(t). (4.4)
Because we assume that the graphite on the cathodic scale is well connected and con-
tinuous, the potential in the solid phase at the current collector, φ|R=Ro, is the same
as the potential in the solid phase near the bulk KOH interface, φ|R=Ri. Thus,
η|R=Ri= Ecell(t) − E0. (4.5)
The development of this potentiostatic boundary condition allows the simulation of
a SPECS discharge by choosing an Ecell(t) function that “steps” by an appropriate
voltage at regular intervals, such as shown in Figure 4.1.
Table 4.1 contains the governing equations used in this model that have not changed,
from Chapter 3. The boundary conditions associated with these equations are given in
Table 4.2.
4.3 The Numerical Solution
The numerical approach used to solve the potentiostatic model is similar to that used to
solve the simplified galvanostatic model as detailed in Chapter 3, Section 3.3, however,
there are some key differences and these are commented on here.
The major differences in the solution algorithm extend from the change in discharge
mode from galvanostatic to potentiostatic. By changing the discharge variable from
current to potential, and using the new boundary condition (4.5), no solution was
achievable with the previous code, because the fixed point iterative strategy diverged.
4.3 The Numerical Solution 65
Table 4.1: Additional governing equations used in the potentiostatic model. Theseequations are the dimensional equivalents of the simplified model equations developedin Chapter 3.
Governing Equations
Crystal Scale:
CMn4+ = C0Mn4+ +
t∫
t∗=0
3in
Fyodt∗
+∞∑
m=1
2 sin(
λmy
yo
)
Fyo sin(λm)
t∫
t∗=0
in
(t − t∗) e−λ2
mDH+ t∗
y2o dt∗ (3.29)
Particle Scale:
Fvp =
(
V H2O − tK+V KOH
)
i∼p (3.34)1r2
∂∂r
(
r2i∼p
)
= ApεEMDpin (3.8)
∂ηp
∂r = i∼p
(
1εEMDpσ0
EMD∞
(
CMn4+ |y=0.8yo
C0Mn4+
)−XMn4+
+ 1√ε3∼pκ∼∞
)
(3.32)
Cathode Scale:
Fv =(
V H2O − tK+V KOH
)
i∼ (3.35)1R
∂∂R (Ri∼) = 3εEMD
εEMDproi∼p|r=ro
(3.19)
∂η∂R = i∼√
ε3∼
κ∼∞
+2RgasT
F
(
1 − tOH−
+COH−cCH2Oc
)
∂ ln aKOHc∂COH−
∂COH−
∂R (3.37)(
ε∼ + εEMDεEMDp
ε∼p
)
∂COH−
∂t = 1R
∂∂R
R(
DOH−∞ε∼∂COH−
∂R − COH−v
)
−3εEMD(1−tOH−
)εEMDproF i∼p|r=ro
(3.36)
Table 4.2: Additional boundary and initial conditions used in the potentiostatic model.These conditions are the dimensional equivalents of those developed in Chapter 3 forthe simplified model.
Boundary Conditions∂ηp
∂r
∣
∣
∣
r=0= 0 (3.17)
ηp|r=ro= η (3.14)
COH− |R=Ri= C0
OH−(3.23)
∂COH−
∂R
∣
∣
∣
R=Ro
= 0 (3.27)
COH− |t=0 = C0OH−
(3.28)
i∼|R=Ro= 0 (3.25)
66 Chapter 4. The Potentiostatic Model
This divergence was overcome by reorganising the matrix structure to ensure that key
elements were on the diagonal, and that it was not as ill-conditioned.
A further difference in the solution procedure arises from the introduction of discrete,
or discontinuous steps in the cell potential. This necessitated a change in the adap-
tive time-stepping algorithm in which small time steps are used to maintain stability
immediately following a potential step.
As with that of Chapter 3, the numerical simulation software developed here is imple-
Here we present discussion and results in relation to determining the functional form of
the ion-ion interaction term, Υ, in Equation (4.1) that best describes the multi-reaction
reduction process of EMD. We then compare the output of our potentiostatic model
with experimental data.
The cell geometry and discharge parameters used in all simulations presented in this
chapter are based on that of the experimental configuration of the button cell cathodes
used by Delta EMD Australia Pty. Limited, and given in Table 4.3. The remaining
parameters are the same as found in Table 3.3 of Chapter 3. However, the value of the
diffusion coefficient of H+ in EMD crystals, DH+ , is chosen to be 1 × 10−16 cm2.s−1,
based on fits to experimental data performed in Section 4.5.2. From Table 4.3 we
also note that the method used to simulate planar button-cell cathode geometry is the
same as that used in Chapter 3. Namely, we increase the inner and outer radii of
the simulated cathodes, while maintaining the experimentally observed thickness and
4.4 Determining the Ion-ion Interaction Term 67
Table 4.3: Discharge parameters and cell geometry used to simulate the experimentaldata
Parameter Value
Potential step size (mV) 5Potential step time (h) 1Particle radius (µm) 25Inner radius, Ri (cm) 1000.0Outer radius, Ro (cm) 1000.0928
Height, H (cm) 2.81 × 10−4
Total mass of cathode (g) 0.5Mass of EMD in cathode (g) 0.3
Mass of graphite in cathode (g) 0.175
volume by decreasing the cathode “height”. This causes the curvature of the electrode
to decrease and the model equations converge to those of linear geometry.
As an initial test of the model, we simulated a SPECS discharge using the same linear
form of the ion-ion interaction term as used in Chapter 3. The comparison of the
model output with experimental data is shown in Figure 4.3. We observe that the
model output compares poorly with the experimental data. The model output does not
display any multi-reaction behaviour, as there is only one peak, while the experimental
data has at least two clearly evident peaks, one at 1.46 V and the other at 1.3 V.
Improving the approximation of the ion-ion interaction term to yield multi-reaction
behaviour is not a straightforward task. However, there are some constraints on the
choice of possible approximations. One constraint is that the domain of Υ (CMn4+) must
be within realistic Mn4+ ion concentrations. In addition, its range should be positive,
because a negative range would increase the zero current potential to voltages above
the predictions of the Nernst expression corresponding to the single reaction given by
Reaction (2.1), which already overpredicts the zero current potential. Furthermore, we
assume that the standard potential, E0 (V), in the modified Nernst equation proposed
by Chabre & Pannetier (1995), namely,
E = E0 − Υ (CMn4+) − RgasT
Fln
CMn3+
CMn4+
, (4.6)
takes into account the initial value of Υ when the initial open circuite voltage is mea-
sured. Based on this, the value of Υ(
C0Mn4+
)
is chosen to be zero. This simplifies
68 Chapter 4. The Potentiostatic Model
Ecell (V)
Min
and
Max
Pow
er(m
W.g
-1of
EM
D)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
2
4
6
8
10
Figure 4.3: Comparison of a simulated 5 mV/hr SPECS discharge (#) using a linear ion-ioninteraction term, with experimental data (). The model parameters used are described inTable 4.3.
Equation (4.1) when we choose the reference equilibrium state of the cell to be the
state that exists in the cathode immediately before discharge.
We note that the linear approximation of Υ (CMn4+) used in Chapter 3 and by Farrell
et al. (2000) has yielded realistic galvanostatic discharge behaviour. Thus, as a first
approximation we consider extending the linear form to a higher degree polynomial in
CMn4+ . An example of two different polynomial representations of Υ (CMn4+) is given in
Figure 4.4 and the corresponding 5 mV/hr SPECS results are displayed in Figure 4.5.
The results show very little similarity with the previous experimental data, however,
we do note, by comparison with Figure 4.3, the significant effect that changing the
form of Υ (CMn4+) has on the output of the SPECS simulation. Thus the choice of
the form of this function would seem crucial to successfully simulating multi-reaction
reduction behaviour in a single-reaction framework. However, its form is not obvious,
and guessing it is very unlikely.
To give an estimate of the form of the ion-ion interaction term, we consider using the
4.4 Determining the Ion-ion Interaction Term 69
CMn4+ (mol.cm-3)
Υ(V
)
0 0.02 0.040
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 4.4: Two possible representations of the ion-ion interaction term. The model resultsbased on these to functions are shown in Figure 4.5. The symbols used in this figure correspondto those used in Figure 4.5
experimental data provided by Delta EMD Australia Pty. Limited. The Butler-Volmer
expression, given by Equation (4.1) is ideal for this. However, to relate the experimental
SPECS data to Υ (CMn4+), we must make several simplifying assumptions, which are
detailed as follows.
Firstly, we must assume that the concentration distributions within the cathode are
close to uniform at the end of each potentiostatic discharge. This assumption may
be poor, especially if the time at which the potential is maintained constant is short.
Secondly, we assume that each potential step occurs instantaneously, so that the con-
centration distributions before and after the potential step are the same. Thirdly, we
assume that ohmic losses in both the solid and solution phases are negligible. This as-
sumption may also be poor, especially later in discharge on the particle scale, because
reduced EMD is not a high resistance. This assumption, however, is necessary because
it allows us to specify that any change in the cell potential is exactly reflected by a
change in the particle scale overpotential at all points in the cathode.
70 Chapter 4. The Potentiostatic Model
Ecell (V)
Min
and
Max
Pow
er(m
W.g
-1of
EM
D)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
2
4
6
8
10
12
14
16
18
Figure 4.5: Comparison of two 5 mV/hr SPECS simulations using the two different ion-ioninteraction terms shown in Figure 4.4. The symbols in this figure correspond to those used inFigure 4.4.
If we admit these assumptions, we may relate the increase in experimentally observed
current at each potential step to the difference between two Butler-Volmer like ex-
pressions, one for the transfer current before, and one after, the potential step. The
concentrations in the Butler-Volmer terms remain the same before and after the po-
tential step because of the first two assumptions, and we may determine the expected
change in the transfer current, ∆in
(A.cm−2), throughout the cathode based on the
experimental data. This yields the following expression, namely,
∆in
i00=
(
CMn3+
C0Mn3+
)(
COH−
C0OH−
)
exp
[
(1 − αc) F
RgasT(ηp − ∆Ecell + Υ)
]
−(
CMn4+
C0Mn4+
)(
CH2O
C0H2O
)
exp
[−αcF
RgasT(ηp − ∆Ecell + Υ)
]
−(
CMn3+
C0Mn3+
)(
COH−
C0OH−
)
exp
[
(1 − αc)F
RgasT(ηp + Υ)
]
+
(
CMn4+
C0Mn4+
)(
CH2O
C0H2O
)
exp
[−αcF
RgasT(ηp + Υ)
]
, (4.7)
4.4 Determining the Ion-ion Interaction Term 71
CMn4+ (mol.cm-3)
Υ(V
)
0 0.02 0.040
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Figure 4.6: A polynomial representation of Υ based on fitting the Butler-Volmer equation toexperimental 5 mV/hr SPECS discharge data
where ∆Ecell (V) is the size of the potential step, ηp (V) is the average particle scale
overpotential of the cell before the potential step, and each (non-reference) concentra-
tion variable is calculated by volume averaging its distribution over the whole cathode.
Solving Equation (4.7) for Υ at each potential step, we obtain a number of estimates
at different Mn4+ concentrations.
A polynomial approximation for Υ (CMn4+), based on the above process is shown in
Figure 4.6. The maximum and minimum power outputs for the corresponding 5 mV/hr
SPECS simulation are shown in Figure 4.7. The previously introduced experimental
5 mV/hr SPECS results are also shown in the figure. The model output does display
two prominent peaks, one at 1.52 V, and another at 1.3 V. However, overall, the model
output does not correspond well with the experimental data.
It should be noted that due to the very interconnected nature of the phenomena that
govern cathode discharge, the parameter values used in the model simulations impact,
sometimes significantly, on the prediction of Υ, when determined in the above manner.
For example we found that changing the value of the initial exchange current density,
i00, significantly affected the predicted Υ term. This behaviour makes it difficult to
72 Chapter 4. The Potentiostatic Model
Ecell (V)
Min
and
Max
Pow
er(m
W.g
-1of
EM
D)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
2
4
6
8
10
12
Figure 4.7: A comparison of the minimum and maximum power of a simulated 5 mV/hrSPECS discharge (#), obtained using the Υ depicted in Figure 4.6, with the correspondingexperimental data ().
extract useful predictions from the experimental data in the above manner, because
some cell parameters are not measured and there is some level of uncertainty in the
values of these parameters, and using wrong values in the model creates differences
between experimental and simulated cell operation that should not be attributed to Υ.
It is important to note, however, that the above analysis was not futile as it facilitates
two crucial observations in linking the form of Υ to the observed multi-reaction dis-
charge behaviour and the successful simulation of SPECS discharge. The first is that
plateaus in the ion-ion interaction term, for example those observed at Mn4+ concen-
trations of 0.042 and 0.012 mol.cm−3, correspond to peaks in the SPECS discharge at
1.52 and 1.3 V, respectively. We note that plateaus at high Mn4+ concentrations are
reached earlier in reduction and their effects appear in simulated SPECS discharges
at higher voltages than plateaus at low Mn4+ concentrations. The second is that the
width of each plateau corresponds to the size of the predicted peak in the simulated
SPECS discharge.
Given the above observations we attempted to modify the polynomial approximation
4.5 Results and Discussion 73
shown in Figure 4.6 in order to obtain improved SPECS simulations, however in doing
so it became evident that a polynomial cannot adequately represent the information
required. The oscillations found in higher order polynomials limit the amount of infor-
mation that can be represented. Upon investigation, the form of the ion-ion interaction
term that we choose as being better in representing the essential features of Υ that con-
vey accurate multi-reaction behaviour in the SPECS discharge simulations is the sum
of several inverse tan functions, namely,
Υ (CMn4+) =n∑
i=1
hi
π
[
arctan(
si
(
CMn4+ − CMn4+,i
))
− arctan(
si
(
C0Mn4+ − CMn4+,i
))]
,
(4.8)
where hi (V) controls the magnitude of the arctan terms, CMn4+,i (mol.cm−3) denotes
the approximate Mn4+ concentration at which the corresponding plateau occurs, and
si controls the slope of the arctan function and how quickly it flattens off to create
a plateau. The second arctan function ensures that the value of Υ(
C0Mn4+
)
is zero.
Equation (4.8) is able to naturally represent each plateau with a single term in the sum,
and does not cause unwanted numerical oscillations in our model output. In practice, a
satisfactory form for Υ can be obtained using only three terms in the above sum. Such
a form is shown in Figure 4.8. A corresponding 5 mV/hr SPECS discharge using this
ion-ion interaction term is compared to the relevant experimental data in Figure 4.9.
We note that the model output shows a main peak at 1.29 V, with a secondary peak
or shoulder at 1.45 V. These correspond well with the position, width and magnitude
of the peaks in the experimental data.
The values of the initial exchange current density and the diffusion coefficient of H+ in
EMD are also very important for obtaining the agreement seen in Figure 4.9. These
parameters have distinct, yet interconnected, influences on discharge behaviour. The
effects of these two parameters are discussed in the following two sections.
4.5 Results and Discussion
Here we present and discuss the results of the modelling work. In Sections 4.5.1
and 4.5.2 we discuss the effects two key parameters have on the simulation of SPECS
74 Chapter 4. The Potentiostatic Model
CMn4+ (mol.cm-3)
Υ(V
)
0 0.02 0.040
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 4.8: The non-linear form (♦) of the ion-ion interaction term used to produce thesimulated SPECS discharge in Figure 4.9 compared to the linear approximation () used inChapter 3.
Ecell (V)
Min
and
Max
Pow
er(m
W.g
-1of
EM
D)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
2
4
6
8
10
12
Figure 4.9: A comparison of the minimum and maximum power of a simulated 5 mV/hrSPECS discharge (♦) using the form of Υ given in Figure 4.8, with experimental data ().
4.5 Results and Discussion 75
tests.
4.5.1 The Initial Exchange Current Density
The initial exchange current density, i00 (A.cm−2), is found at the front of the Butler-
Volmer like expression (see Equation (4.1)). It describes the facility of charge transfer
at the EMD/KOH interface (Bard & Faulkner 2001). It directly affects the size of
the response of the current to changes in the potential and chemical concentrations
involved in the reduction of EMD. However, we note that a representative i00 is difficult
to obtain experimentally.
The effect of different i00 values on the current response, over three potential steps in
a 5 mV/hr SPECS simulation, is displayed in Figure 4.10, where the model output is
also compared with the relevant experimental data. We observe that the value of i00
significantly affects the current spike at each step in the potential. For small values
of i00, for example 5×10−9 A.cm−2, the resulting current spikes are small, while for
larger values of i00, for example 5×10−7 A.cm−2, the current response is much more
pronounced. In addition to this, the initial exchange current density also affects the rate
of relaxation. This is less intuitive than its effect on the initial current spike, however, it
may be explained by considering the crystal scale. When i00 is small, protons are inserted
at the surface of the EMD crystals at a slow rate, and are able to be transported away
from the crystal surface faster than they are inserted. This corresponds to a situation
that is kinetically limited, and leads to a very even, or flat, current response. For larger
values of i00, protons are able to be inserted into the EMD crystals faster than they can
diffuse from the surface. This corresponds to a situation where the process is diffusion
limited, and leads to larger current responses that diminish quickly. Based on this,
we observe that the two 5 mV/hr SPECS simulations with i00 values of 5×10−8 and
5×10−7 A.cm−2 are both diffusion limited. Interestingly, the experimental data seems
to match the model predictions for an i00 value of 5×10−7 A.cm−2 for the first half
of each potentiostatic discharge, and seems to match the model predictions for an i00
value of 5×10−9 A.cm−2 for the remainder of each potentiostatic discharge. This would
suggest that there are multiple i00 values in the experimental data. This is consistent
with our understanding of the multi-reaction reduction process, and may be why a
76 Chapter 4. The Potentiostatic Model
Time (h)
Cur
rent
(A)
49 50 51 520
0.001
0.002
0.003
0.004
0.005
Figure 4.10: A comparison of the current spike shape of several simulated 5 mV/hr SPECStests with i00 values of 5×10−9 ( ), 5×10−8 ( ) and 5×10−7 ( ) A.cm−2, with experimentaldata ().
better fit was not obtained using a single i00 value.
The effect of i00 on overall discharge behaviour for simulated 5 mV/hr SPECS discharges
is shown in Figure 4.11. The fluctuations seen in the data corresponding to the model
output for an i00 value of 5×10−7 A.cm−2 are caused by the difficulty the model has
in numerically capturing the extremely thin current spikes observed at large i00 values.
The primary effect of i00, displayed in Figure 4.11, is that increasing its value increases
the difference between minimum and maximum power. This is consistent with our
observations of the current response displayed in Figure 4.10. However, this effect is
diminished near the end of discharge, below approximately 1.15 V, because the EMD is
almost completely reduced and there is simply not enough there to produce a noticeable
current response.
Revisiting Figure 4.9, and considering in particular the comparison of the experimental
data with the model output using the form of Υ shown in Figure 4.8, we see that the
shoulder peak at 1.45 V in the experimental data has a larger difference between the
minimum and maximum power than it does at the main peak at 1.29 V. In the context
4.5 Results and Discussion 77
Ecell (V)
Min
and
Max
Pow
er(m
W.g
-1of
EM
D)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
5
10
15
20
Figure 4.11: A comparison of the minimum and maximum power of simulated 5 mV/hr SPECSdischarges with i00 values of 5×10−9 (♦), 5×10−8 (#) and 5×10−7 () A.cm−2.
of a multi-reaction reduction process, this suggests that the process occurring at 1.45
V has a larger i00 than the main process. This effect is unable to be reproduced using
our model, but it may be possible with a “variable” i00 which is dependent on CMn4+ .
Using this, i00 could have a higher value at the first peak, and a lower value at the main
peak.
4.5.2 The Diffusion Coefficient of Protons
The diffusion coefficient of protons in EMD crystals, DH+ (cm2.s−1), is found in Equa-
tion (3.29). It is a measure of the ability of inserted protons to move within the EMD
crystals, and more to the point, how fast these protons are able to vacate the reaction
sites at the EMD surface. Figure 4.12 shows the effect DH+ has on three individ-
ual current spikes in a 5 mV/hr SPECS simulation. We observe, as expected, that
changes in DH+ have little to no effect on the initial current spike height. However,
DH+ greatly influences the relaxation response. For small DH+ values, for example
1×10−17 cm2.s−1, the current almost immediately decreases to below half of its initial
value at the potential step. Following this, the current seems to maintain a steady
78 Chapter 4. The Potentiostatic Model
Time (h)
Cur
rent
(A)
49 50 51 520
0.001
0.002
0.003
Figure 4.12: A comparison of the current spike shape of simulated 5 mV/hr SPECS dischargewith DH+ values of 1×10−17 ( ), 1×10−16 ( ) and 1×10−15 ( ) cm2.s−1, with experimentaldata.
response. This corresponds to a situation where the process is diffusion limited. For
larger DH+ values, for example 1×10−15 cm2.s−1, the current does not experience a
large immediate decrease, but rather a gradual decline. This corresponds to a process
which is kinetically limited. We see that the amount of EMD utilised in a certain
time-frame (which is proportional to the area under the current response) is dependent
upon the value of DH+ . This is especially true when diffusion is the limiting process,
as seen in Figure 4.12, when DH+ takes the values 1×10−16 and 1×10−17 cm2.s−1. We
observe that a small DH+ value of 1×10−17 cm2.s−1 allows less EMD to be utilised than
observed when DH+ has the value 1×10−16 cm2.s−1. This is expected because DH+
directly influences the availability of reaction sites at the EMD surface.
The SPECS simulation shown in Figure 4.12 with a DH+ value of 1×10−16 cm2.s−1
seems to match the experimental data for the beginning of each potentiostatic dis-
charge, but none of the simulations match the experimental data at the end of each
potentiostatic discharge. The inability of the model to accurately predict the current
relaxation curve may be evidence for a variable DH+ . We note that this is consistent
with the multi-reaction reduction process of EMD.
4.5 Results and Discussion 79
Ecell (V)
Min
and
Max
Pow
er(m
W.g
-1of
EM
D)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
1
2
3
4
5
6
7
8
9
10
11
12
Figure 4.13: A comparison of the minimum and maximum power of simulated 5 mV/hr SPECSdischarge with DH+ values of 1×10−17 (♦), 1×10−16 (#) and 1×10−15 () cm2.s−1.
Figure 4.13 shows the effect of DH+ on overall discharge behaviour. We previously
observed in Figure 4.12 that DH+ primarily affects the current relaxation response, and
not the minimum and maximum current. This is seen to be true in the overall discharge
until approximately 1.25 V. Below this voltage, the simulated SPECS discharge with
a DH+ value of 1×10−17 cm2.s−1 has a larger power response. This difference occurs
because the cathodes with larger DH+ values have utilised, or exhausted, most of their
EMD, and cannot produce a sizeable current response.
It is thought that the shape of the minimum power response is evidence of a variable
DH+ (Delta 2005) because the current flowing at the end of each potentiostatic dis-
charge is thought to be related to DH+ . In our model framework, this is true, but the
relationship between the minimum power and DH+ seems to only appear later in the
SPECS simulations, below aprroximately 1.25 V, as noted above. In fact, the variable
Υ has a greater effect on the shape of the minimum power response. Furthermore, our
results clearly show that it is possible that a constant DH+ can produce a curve shape
that is consistent with the relevant experimental data.
80 Chapter 4. The Potentiostatic Model
4.6 Conclusions
A novel model for the potentiostatic discharge of primary alkaline battery cathodes
Here we present and discuss the results of the modelling work. In Sections 5.4.1 to 5.4.3
we discuss the effects of three key parameters on the simulation of primary alkaline
battery discharges. The parameters investigated are the bulk ZnO conductivity, the
initial KOH concentration, and the separator thickness. All simulations presented here
use the parameter values given in Table 5.1, which describe AA-cell geometry, unless
otherwise noted.
Figure 5.1 shows the external discharge current and the total discharge current of two
AA-cells discharged through 3.3 and 6.6 Ω loads. The external discharge current is
assumed to be the externally applied current which flows through the load resistance,
while the total discharge current is assumed to be the sum of the external current
and the internal self-discharge current. We note that the difference between the total
5.4 Results and Discussions 101
Time (h)
Cur
rent
(A)
0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 5.1: Comparison of self-discharge of two AA-cells under 3.3 () and 6.6 (#) Ω loads.Shown is the external cell discharge current (hollow symbols, ) and the total current (filledsymbols, ) (including short circuit current). The configuration of the cell is given in Table 5.1
and external current (as seen in the figure) essentially represents the self-discharge (or
short circuit) current. The amount of capacity lost to self-discharge does not seem
to be greatly influenced by the discharge rate. However, in both cases self-discharge
occurs at the beginning (of discharge). This is confirmed by examining the volume
fraction of the ZnO within the separator. Figure 5.2 shows the volume fraction of ZnO
within the separator for several times during the 3.3 Ω discharge. The formation of a
continuous ZnO phase within the separator occurs at very early times, and is dissolved
during the operation of the cell. As the ZnO phase grows in size, it allows an increasing
self-discharge current. However, after the first hour of discharge, the OH− ions from the
cathode begin to dissolve the ZnO in the separator at the separator/cathode boundary
(R = 0.45 cm), eventually severing the electrical connection, halting self-discharge and
not allowing it to reoccur later.
Interestingly, the information shown in Figures 5.1 and 5.2 suggest that the reduction
in overall cell capacity, shortening battery life, is a result of self-discharge that occurs
102 Chapter 5. The Precipitation Model
Radius (cm)
ε ZnO
0.435 0.44 0.445 0.450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 5.2: ZnO volume fraction within the separator of an AA-cell under a 3.3 Ω load. TheZnO volume fraction is shown for the times 0 (#), 15 (), 30 (♦), 45 (), 60 (), 120 (⊲) and180 (⊳) minutes.
during the initial moments of cell discharge, rather than as a result of a short circuit
event which occurs just prior to cell failure.
During intermittent discharges, self-discharge is seen to occur repeatedly. Figure 5.3
shows the external and total discharge current of an AA-cell undergoing a simulated
3.3 Ω constant load intermittent discharge. The load is applied for 4 minutes at the
beginning of each hour, for 8 hours at the beginning of each 24 hour period. Only the
first 8 hour period of this discharge is shown in Figure 5.3. It is seen that self-discharge
starts at the beginning of the first hour and lasts until after the second discharge has
started. Further self-discharge is observed, although to a lesser extent, at the beginning
of each of the following 6 hours. These small repeated losses also occur during the hourly
discharges of the next four 24 hour periods. Overall, more capacity is lost through self-
discharge in the simulated intermittent discharge than in the simulated continuous 3.3
Ω discharge.
Note that in the last 6 hours shown in Figure 5.3, the self-discharge current increases
5.4 Results and Discussions 103
Time (h)
Cur
rent
(A)
0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
Figure 5.3: External () and total () current for an intermittent 3.3 Ω discharge of an AA-cell. The load is applied for 4 minutes at the beginning of the first 8 hours of each 24 hourperiod. Only the first 8 hours of this discharge are shown.
while the load is applied, and decreases (almost immediately) after the load is discon-
nected. We note that during these rest periods, the OH− ions that dissolved the ZnO
at R = Rsc start to equilibrate, and this lets some ZnO precipate during each following
discharge pulse. Because the OH− ions do not have long enough to fully equilibrate,
less ZnO is able to precipitate, and it is dissolved in a shorter time. This results in the
small short circuits observed in the last six hours.
From this we conclude that self-discharge occurs when discharge is initiated and the
cell has almost spatially uniform electrolyte concentrations.
5.4.1 The Effect of Changing ZnO Bulk Conductivity
Here we present discussion and results in relation to the bulk ZnO conductivity, σZnO∞
(S.cm−1). It is found in Equation (5.45), which governs the solid phase potential
within the separator. It has been found that two types of ZnO may be formed during
the discharge of a primary alkaline battery (Powers & Brieter 1969). Because the
104 Chapter 5. The Precipitation Model
Time (h)
Cur
rent
(A)
0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 5.4: Comparison of model output for ZnO conductivities of 0.001 (), 0.005 (), 0.01(#) and 0.02 () S.cm−1. Shown is the external (hollow symbols) and the total (filled symbols)discharge current.
morphology of precipitated ZnO can be varied (Szpak & Gabriel 1979, Horn & Shao-
Horn 2003), the connectivity of the ZnO phase itself may be lower than expected, or
calculated, from the volume fraction, εZnO.
Figure 5.4 shows the effect of different ZnO conductivity values on discharge behaviour.
In Figure 5.4, the conductivity of ZnO has been given the values 10%, 50%, 100% and
200% of it’s bulk conductivity (0.01 S.cm−1). We see that larger ZnO conductivity in-
creases the amount of capacity lost to self-discharge in an almost linear fashion. It is not
expected to increase in an exactly linear fashion with respect to ZnO conductivity be-
cause extended discharge elevates the OH− ion concentrations at the separator/cathode
interface, which dissolves the ZnO connection in the separator. Thus, when the ZnO
conductivity increased, the self-discharge current is also increased, however the ZnO
connection is then dissolved at an earlier time.
5.4 Results and Discussions 105
5.4.2 The Effect of Changing Initial KOH Concentration
In this section we present discussion and results in relation to the initial KOH con-
centration and compare the model output of several discharges with different KOH
concentrations. In the simulations presented in this section, the initial Zn(OH)2−4 ion
concentration, C0KZn has been set at the saturation concentration, CKZn,eq (used in the
precipitation source terms that appear in the mass conservation equations for OH− and
Zn(OH)2−4 ions, and also the porosity equations in the cathode, separator and anode)
based on the initial KOH concentration. By doing this, the cell is in equilibrium before
discharge proceeds.
A definite relationship between the initial KOH concentration and the self-discharge
current is observed. Figure 5.5 shows the external and total discharge current for initial
KOH concentrations of 0.005, 0.007, 0.009 and 0.011 mol.cm−3 (the corresponding equi-
librium Zn(OH)2−4 ion concentrations are 0.000418, 0.000694, 0.001038 and 0.001450
mol.cm−3, respectively). The observed trend is that as the initial KOH concentration
is increased, the total self-discharge current decreases. This is expected, because the
precipitation reaction (5.3), shows that it will be more sensitive to high OH− con-
centrations than to high Zn(OH)2−4 concentrations. This means that at higher OH−
concentrations, less ZnO will precipitate within the separator.
We should note that the trend observed in Figure 5.5 may be dependent upon the
model of the anode. Since the anode is assumed to be uniform, it is as if the anode is
well mixed, and transport within the anode is fast. A non-uniform anode will change
the concentrations of OH− and Zn(OH)2−4 ions entering the separator from the anode,
influencing, and perhaps changing the trend observed in Figure 5.5. Kriegsmann &
Cheh (2000), while justifying their binary electrolyte model, note that in the anode,
precipitation is several orders of magnitude faster than diffusion. This means that
most of the Zn(OH)2−4 may precipitate out as ZnO before it gets the chance to leave
the anode. In addition, this directly contrasts with our model, and implies that the
amount of Zn(OH)2−4 leaving the anode will tend to be decreased. When the OH− ion
concentration is decreased, this may further decrease the amount of Zn(OH)2−4 entering
the separator, decreasing the total self-discharge current.
106 Chapter 5. The Precipitation Model
Time (h)
Cur
rent
(A)
0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 5.5: Comparison of model output for initial KOH concentrations of 5 (), 7 (#), 9 ()and 11 () M. Shown is external (hollow symbols) and total (filled symbols) discharge current.
5.4.3 The Effect of Changing Separator Thickness
Here we present discussion and results in relation to the separator thickness, and com-
pare the model output of several discharges with different separator thicknesses.
The discharge current of three 3.3 Ω constant resistance discharges with separators of
three different thicknesses, namely, 0.1, 0.15 and 0.2 mm, is shown in Figure 5.6. We
observe that decreasing the separator thickness greatly increases the total self-discharge
current. The thickness of the separator is expected to influence the self-discharge of
a cell, because the thickness directly influences the length of the conduction path.
However, the relationship between the separator thickness and the self-discharge current
is seen to be very nonlinear.
At the beginning of discharge, ZnO is observed to precipitate throughout the separator
in all cases (the precipitation behaviour in the separator is shown explicitly for a 0.15
mm thick separator in Figure 5.2). In addition, all of the simulations had less ZnO at
the separator/cathode interface than anywhere else in the separator (for example, see
5.5 Conclusions 107
Time (h)
Cur
rent
(A)
0 1 2 3 40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Figure 5.6: Comparison of model output for separator thicknesses of 0.1 (), 0.15 () and 0.2(#) mm. Shown is external (hollow symbols) and total (filled symbols) discharge current.
Figure 5.2). Thus, it is the thickness of the ZnO at the separator/cathode interface
that crucially determines the presence of self-discharge. To precipitate in the separa-
tor, Zn(OH)2−4 ions must diffuse from the anode, so decreasing the thickness of the
separator makes it easier for Zn(OH)2−4 to reach the separator/cathode interface before
precipitating out as ZnO. Thus, thin separators will have elevated levels of ZnO at the