Mathemathical modelling of lithium ion batteries.
UNIVERSITY OF SOUTHAMPTON
FACULTY OF SOCIAL AND HU'.\IA:'-J SCIEXCES
School of
ｾｉ｡ｴｨ･ｭｩ｣ｳ＠
Mathematical Modelling of Lithium Ion Batteries
by
Rahifa Ranom
Thesis for the Jegree of Doctor of Philosophy
October 201-!
UNIVERSITY OF SOUTHAMPTON
ABSTRACT
FACULTY OF SOCIAL AND HUl\IAN SCIENCES
School of Mathematics
Doctor of Philosophy
MATHEl\fATICAL l\IODELLING OF LITHIU1'1 ION BATTERIES
by Rahifa Ranom
In this study, we discuss a lithium battery model based on dilute electrolyte theory
and fast diffusion of lithium in the electrode particle and calculate some novel solutions
tu the model. 'Ve then discuss moderately concentrated electrolyte theory and outline
how homogenisation techniques can be applied to this theory, in combination with a
microscalc model for lithium transport in the electrode particles in order to derive a
Newman type model of the battery [59]. We formulate a numerical method, based on
the method of lines in order to solve this model, and apply it to the cases of a half cell
graphite anode and a half cell LiFePO.i cathode. In both scenarios, the results show
very good agreement to experimental discharge curves.
Contents
DECLARATION OF AUTHORSHIP
xv
List of Publications
xvii
Acknowledgements
xix
1
Introduction
1.1 Lithium batteries as energy storage solution
1.2 Battery materials for Lithium ion batteries . . . . .
1.2.1 Desirable electrode and electrolyte properties
1.2.2 The cathode material
1.2.3 The anode material
1.2.4 The electrolyte .
1.3 Charge-transfer reaction
1.4 Battery Terminology
1.5 The half-cell . . . . . . .
1.6 Battery modelling . . .
1.6.1 Statement of originality
2 Dilute electrolyte modelling of battery
2.1 Introduction . . . . . . . . . . . . . . . .
2.2 Derivation of a model fur a dilute electrolyte
2.2.l Charge neutrality. . . . . . . . . . . .
2.3 Reaction kinetics on the electrode particle surfaces
2.-1 The electrode particles . . . . . . . . . . . . . . .
2.5 Homogenisation of model accounting for microstructure on electrode particle scale . . . . . . . . . . .
2.5.1 The current collectors
2.5.2 The separator . . . . .
2.5.3 The initial conditions
2.5.4 The relation between current and global reaction rate
2.5.5 Summary of the battery model and comparison to other models.
2.6 Numerical and analytical solutions for the full cell model
Equilibrium solution . . . . . . . .
2.6.1 Noudimensio11alizatio11 . . . . . . . . . . . . . . . .
2.6.1.1 Size of dimensionless parameters . . . . .
2.6.2 The Tafel equation approximation for r!" < < 1 and fl" < < 1
2.6.3 The quasi steady approximations for r < < 1
\'
1
1
3
3
-1
-1
5
5
6
7
9
10
13
13
13
14
15
17
19
21
21
21
22
22
23
2-1
2-1
26
CONTESTS
2.6.-±
Solution for flat discharge curYes . . . . . . . . . . . . .
Solution before the particles arc fully discharged.
Numerical procedure
2.6.5 Results and Discussion . . . . .
2. 7 The half cell cathode model . . . . . .
2. 7.1 Quasi-steady state limit r -7 0
2. 7.2 Flat discharge curve approximation for LiFePO .t cat ho de
2. 7.3 Analytic solutions . . . . . . . . . . . . . . . . . . .
2.7.3.1 Before the development of a free boundary
Numerical solution procedure . . . . .
2. 7.3.2 After development of free boundary
Numerical solution procedure
2.7.3.3 Results and discussion.
2.8 Summary . . . . . . . . . . . . . . . . . . .
3
4
Modelling moderately concentrated electrolytes
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Stefan-Maxwell equations . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Chemical potential (µ) a11d electrochemical potential (µ) of the
electrolyte at constant pressure and tcmpcratnrc
Chemical potential . . . . . . . . . . . . . . .
Electrochemical potential . . . . . . . . . . .
3.3 The Stefan Maxwell equations for the binary 1:1 electrolyte
A vera.ged approximation to Poisson's equation
Non-dimensionalising Poisson's equation
Charge neutrality . . . . . . . . . : . . . .
Equations for the current density j . . . . . . .
Derivation of the ion velocities in terms of the electrolyte
chemical potential p.,., and j. . . . . . . .
Diffusion equation for the electrolyte concentration .
3.3.1 Summary of model for moderately concentrated electrolyte
3.3.2 An ideal solution . . . . . . . . . . . . . . . . . . . . . .
3.3.3 How might we deal with the electric potential . . . . . . .
3.3.4 The potential measured with respect to Lithium electrode
Remarks . . . . .
3.-± Thermodynamic fitting to data
3 ..5 Summary . . . . . . . . . . . .
Review of homogenisation technique for moderately concentrated electrolyte model
4.1 Introdnct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The cell scale electrolyte equations by homogenisation technique . . . . .
Boundary conditions on the surface of the electrode particles
General set of microscalc electrolyte equations. . . . . . .
The asymptotic expansions. . . . . . . . . . . . . . . . . .
The solution to the moderately concentrated electrolyte
model. . . . . . . . . . . . . . . . . . . . . . .
28
29
30
30
33
35
36
37
37
38
38
39
40
.u
45
45
-Hi
46
46
47
47
48
49
50
50
52
53
53
54
54
55
56
56
58
59
59
60
GO
61
62
6!
CONTEXTS
4.3
4.4
4.5
5
6
7
vii
Butler-Volmer reaction equations . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The general version of Butler-Volmer equations for insertion material
4.3.2 The Butler-Volmer equations of electrode materials for lithium
battery . . . . . . . . . . . . . . . . . . . . . .
LiC5 anode material . . . . . . . . . . .
LiFeP04 and LiCo02 cathode materials
Summary of the resulting model . . . . . . . . . . .
4.4.1 Bonndary conditions for the foll cell battery .
Summary . . . . . . . . . . .
64
64
67
67
67
68
69
70
Models for electrode particles
5.1 Introduction . . . . . . . . . .
5.2 Two phase Lithium insertion/extraction
5.2.1 ''Shrinking-core" model . . .
Nondimensionalisation ..
5.2.l.1 Solution Procedure
5.2.1.2 A8ymptotic 8olution of 8hrinking core diff1rnion
Summary. .
5.2.2 Phase-field model . . . . . . . . . . . .
5.3 1fore than two pha8es . . . . . . . . . . . . .
5.-1 Diffusion equation in the spherical coordinate
Current density in the electrode .
5.5 Summary . . . . .
71
71
71
73
74
76
Numerical Procedure
6.1 Introduction . . . .
6.2 11ethod of Lines
6.3 Development of sparse matrix for the system
6.3.1 The development of the solution vector u
6.3.2 The development of matrices for electrolyte concentration, c
Sun1111ary . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 The development of matrices for electrolyte potential, ¢ . . .
6.3.4 The development of matrices for the electrode potential, /Ps .
6.3.5 The development of matrices for concentration iu the electrode
particles, r's . . • • . . . . . . . . .
6.3.6 The development of A, JVI and f .
6.4 ode15s ..
Convergence .
6.5 Summary
83
83
85
The
7.1
7.2
7.3
97
7.4
Half cell Anode
Introduction . . . .
The half cell anode model
Nondimensionalisation . .
Remarks.
Discussion of dimensionless parameters .
:\lode! - experimental compri.risons for the 1rnt11ral graphite electrode
7.-Ll Results and Discussions . . . . . . . . . . . . . . . . . . . . .
77
78
79
80
81
81
82
86
87
88
89
89
90
91
93
93
94
96
97
97
99
99
101
102
. 10-1
CONTESTS
viii
7.-!.l.1
7.5
8
9
An approximation solution . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . .
7.-!.l.2 Concentration-dependent of diffusion coefficient
Summary . . .
Half cell cathode
8.1 Iutroductiou . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Transport data and parameter values used in the simulation
8.2.l Nondimensionalisation . . .
Parameter Values . .
Numerical Procedure
8.3 l\fodel-experimental comparison . .
8.4 The effects of parameter variations on the discharge of a uauostructured
half-cell cathode
8.5 Summary . . . . . . . . . . . .
.
.
.
.
107
108
108
112
115
. 115
. 116
. 118
. 119
. 121
. 121
. 123
. 136
Conclusions and Future Works
137
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
. 137
9.2 Future works . . . . . . . . . . . . . . . . . . . . . .
. 139
The effect of different sizes of particles
. 139
The effect of changes in particle shape and packing upon
cell perfonnauce . . . . . . . . . . . . . . . . . . 139
List of Figures
1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.-1
2.5
2.G
2. 7
A schematic diagram of the Lithium Ion Battery during discharge [59].
The current flowing out of the positive electrode drives the extraction of
lithium ion from negative electrode (anode) particles to the electrolyte
across the porous separator (by diffusion and advection) into the positive
electrode (cathode) and insert into the positive electrode (cathode). The
charge of electrons arc moving from the negative electrode particles to
the negative electrode current collector and from the positive electrode
current collector to the positive electrode particles. . . . . . . . . . . . .
Structure of the electric double layer near a solid electrolyte interface
when external electric field is applied. The electric drops linearly from
the electrode potential
FACULTY OF SOCIAL AND HU'.\IA:'-J SCIEXCES
School of
ｾｉ｡ｴｨ･ｭｩ｣ｳ＠
Mathematical Modelling of Lithium Ion Batteries
by
Rahifa Ranom
Thesis for the Jegree of Doctor of Philosophy
October 201-!
UNIVERSITY OF SOUTHAMPTON
ABSTRACT
FACULTY OF SOCIAL AND HUl\IAN SCIENCES
School of Mathematics
Doctor of Philosophy
MATHEl\fATICAL l\IODELLING OF LITHIU1'1 ION BATTERIES
by Rahifa Ranom
In this study, we discuss a lithium battery model based on dilute electrolyte theory
and fast diffusion of lithium in the electrode particle and calculate some novel solutions
tu the model. 'Ve then discuss moderately concentrated electrolyte theory and outline
how homogenisation techniques can be applied to this theory, in combination with a
microscalc model for lithium transport in the electrode particles in order to derive a
Newman type model of the battery [59]. We formulate a numerical method, based on
the method of lines in order to solve this model, and apply it to the cases of a half cell
graphite anode and a half cell LiFePO.i cathode. In both scenarios, the results show
very good agreement to experimental discharge curves.
Contents
DECLARATION OF AUTHORSHIP
xv
List of Publications
xvii
Acknowledgements
xix
1
Introduction
1.1 Lithium batteries as energy storage solution
1.2 Battery materials for Lithium ion batteries . . . . .
1.2.1 Desirable electrode and electrolyte properties
1.2.2 The cathode material
1.2.3 The anode material
1.2.4 The electrolyte .
1.3 Charge-transfer reaction
1.4 Battery Terminology
1.5 The half-cell . . . . . . .
1.6 Battery modelling . . .
1.6.1 Statement of originality
2 Dilute electrolyte modelling of battery
2.1 Introduction . . . . . . . . . . . . . . . .
2.2 Derivation of a model fur a dilute electrolyte
2.2.l Charge neutrality. . . . . . . . . . . .
2.3 Reaction kinetics on the electrode particle surfaces
2.-1 The electrode particles . . . . . . . . . . . . . . .
2.5 Homogenisation of model accounting for microstructure on electrode particle scale . . . . . . . . . . .
2.5.1 The current collectors
2.5.2 The separator . . . . .
2.5.3 The initial conditions
2.5.4 The relation between current and global reaction rate
2.5.5 Summary of the battery model and comparison to other models.
2.6 Numerical and analytical solutions for the full cell model
Equilibrium solution . . . . . . . .
2.6.1 Noudimensio11alizatio11 . . . . . . . . . . . . . . . .
2.6.1.1 Size of dimensionless parameters . . . . .
2.6.2 The Tafel equation approximation for r!" < < 1 and fl" < < 1
2.6.3 The quasi steady approximations for r < < 1
\'
1
1
3
3
-1
-1
5
5
6
7
9
10
13
13
13
14
15
17
19
21
21
21
22
22
23
2-1
2-1
26
CONTESTS
2.6.-±
Solution for flat discharge curYes . . . . . . . . . . . . .
Solution before the particles arc fully discharged.
Numerical procedure
2.6.5 Results and Discussion . . . . .
2. 7 The half cell cathode model . . . . . .
2. 7.1 Quasi-steady state limit r -7 0
2. 7.2 Flat discharge curve approximation for LiFePO .t cat ho de
2. 7.3 Analytic solutions . . . . . . . . . . . . . . . . . . .
2.7.3.1 Before the development of a free boundary
Numerical solution procedure . . . . .
2. 7.3.2 After development of free boundary
Numerical solution procedure
2.7.3.3 Results and discussion.
2.8 Summary . . . . . . . . . . . . . . . . . . .
3
4
Modelling moderately concentrated electrolytes
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Stefan-Maxwell equations . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Chemical potential (µ) a11d electrochemical potential (µ) of the
electrolyte at constant pressure and tcmpcratnrc
Chemical potential . . . . . . . . . . . . . . .
Electrochemical potential . . . . . . . . . . .
3.3 The Stefan Maxwell equations for the binary 1:1 electrolyte
A vera.ged approximation to Poisson's equation
Non-dimensionalising Poisson's equation
Charge neutrality . . . . . . . . . : . . . .
Equations for the current density j . . . . . . .
Derivation of the ion velocities in terms of the electrolyte
chemical potential p.,., and j. . . . . . . .
Diffusion equation for the electrolyte concentration .
3.3.1 Summary of model for moderately concentrated electrolyte
3.3.2 An ideal solution . . . . . . . . . . . . . . . . . . . . . .
3.3.3 How might we deal with the electric potential . . . . . . .
3.3.4 The potential measured with respect to Lithium electrode
Remarks . . . . .
3.-± Thermodynamic fitting to data
3 ..5 Summary . . . . . . . . . . . .
Review of homogenisation technique for moderately concentrated electrolyte model
4.1 Introdnct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The cell scale electrolyte equations by homogenisation technique . . . . .
Boundary conditions on the surface of the electrode particles
General set of microscalc electrolyte equations. . . . . . .
The asymptotic expansions. . . . . . . . . . . . . . . . . .
The solution to the moderately concentrated electrolyte
model. . . . . . . . . . . . . . . . . . . . . . .
28
29
30
30
33
35
36
37
37
38
38
39
40
.u
45
45
-Hi
46
46
47
47
48
49
50
50
52
53
53
54
54
55
56
56
58
59
59
60
GO
61
62
6!
CONTEXTS
4.3
4.4
4.5
5
6
7
vii
Butler-Volmer reaction equations . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The general version of Butler-Volmer equations for insertion material
4.3.2 The Butler-Volmer equations of electrode materials for lithium
battery . . . . . . . . . . . . . . . . . . . . . .
LiC5 anode material . . . . . . . . . . .
LiFeP04 and LiCo02 cathode materials
Summary of the resulting model . . . . . . . . . . .
4.4.1 Bonndary conditions for the foll cell battery .
Summary . . . . . . . . . . .
64
64
67
67
67
68
69
70
Models for electrode particles
5.1 Introduction . . . . . . . . . .
5.2 Two phase Lithium insertion/extraction
5.2.1 ''Shrinking-core" model . . .
Nondimensionalisation ..
5.2.l.1 Solution Procedure
5.2.1.2 A8ymptotic 8olution of 8hrinking core diff1rnion
Summary. .
5.2.2 Phase-field model . . . . . . . . . . . .
5.3 1fore than two pha8es . . . . . . . . . . . . .
5.-1 Diffusion equation in the spherical coordinate
Current density in the electrode .
5.5 Summary . . . . .
71
71
71
73
74
76
Numerical Procedure
6.1 Introduction . . . .
6.2 11ethod of Lines
6.3 Development of sparse matrix for the system
6.3.1 The development of the solution vector u
6.3.2 The development of matrices for electrolyte concentration, c
Sun1111ary . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 The development of matrices for electrolyte potential, ¢ . . .
6.3.4 The development of matrices for the electrode potential, /Ps .
6.3.5 The development of matrices for concentration iu the electrode
particles, r's . . • • . . . . . . . . .
6.3.6 The development of A, JVI and f .
6.4 ode15s ..
Convergence .
6.5 Summary
83
83
85
The
7.1
7.2
7.3
97
7.4
Half cell Anode
Introduction . . . .
The half cell anode model
Nondimensionalisation . .
Remarks.
Discussion of dimensionless parameters .
:\lode! - experimental compri.risons for the 1rnt11ral graphite electrode
7.-Ll Results and Discussions . . . . . . . . . . . . . . . . . . . . .
77
78
79
80
81
81
82
86
87
88
89
89
90
91
93
93
94
96
97
97
99
99
101
102
. 10-1
CONTESTS
viii
7.-!.l.1
7.5
8
9
An approximation solution . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . .
7.-!.l.2 Concentration-dependent of diffusion coefficient
Summary . . .
Half cell cathode
8.1 Iutroductiou . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Transport data and parameter values used in the simulation
8.2.l Nondimensionalisation . . .
Parameter Values . .
Numerical Procedure
8.3 l\fodel-experimental comparison . .
8.4 The effects of parameter variations on the discharge of a uauostructured
half-cell cathode
8.5 Summary . . . . . . . . . . . .
.
.
.
.
107
108
108
112
115
. 115
. 116
. 118
. 119
. 121
. 121
. 123
. 136
Conclusions and Future Works
137
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
. 137
9.2 Future works . . . . . . . . . . . . . . . . . . . . . .
. 139
The effect of different sizes of particles
. 139
The effect of changes in particle shape and packing upon
cell perfonnauce . . . . . . . . . . . . . . . . . . 139
List of Figures
1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.-1
2.5
2.G
2. 7
A schematic diagram of the Lithium Ion Battery during discharge [59].
The current flowing out of the positive electrode drives the extraction of
lithium ion from negative electrode (anode) particles to the electrolyte
across the porous separator (by diffusion and advection) into the positive
electrode (cathode) and insert into the positive electrode (cathode). The
charge of electrons arc moving from the negative electrode particles to
the negative electrode current collector and from the positive electrode
current collector to the positive electrode particles. . . . . . . . . . . . .
Structure of the electric double layer near a solid electrolyte interface
when external electric field is applied. The electric drops linearly from
the electrode potential