Simulation of Amorphous Silicon Anode in Lithium-Ion Batteries
By
Miao Wang

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering - Doctor of Philosophy
2017

ABSTRACT
SIMULATION OF AMORPHOUS SILICON ANODE IN LITHIUM-ION BATTERIES
By
Miao Wang
The energy density of the current generation of Li-ion batteries (LIBs) is only about 1% of
that of gasoline. Improving the energy density of the LIBs is critical for vehicle electrification.
Employing high capacity electrode materials is a key factor in this endeavor. Silicon (Si) is one
of the high capacity anode materials for LIBs. However, Si experiences large volume variation
(up to 300%) during battery cycling, which affects the structural integrity of battery and results
in rapid capacity fading. It has been shown that the cycle life of Si anode can be improved
greatly through novel electrode designs. So far, such work is conducted through experiments.
Numerical simulations have the potentials for design optimization of LIBs, as demonstrated
in multiphysics models for LIBs with graphite anode. This research extends a previously
developed microstructure-resolved multiphysics (MRM) model to LIBs with amorphous Si
anode. The MRM model considers the electrochemical reactions, Li transport in electrodes and
electrolyte, lithiation induced volume change, mechanical strains and stresses, material property
evolution with lithiation, and the chemo-mechanical coupling. The model is solved using finite
element package COMSOL Multiphysics.
The major challenges in this work are the large deformation of the Si, and the uncertainty in
parameters and the coupling relation. To simulate the large deformation of Si, a large strain
based formulation for the concentration induced volume expansion was used. The electrolyte
was modeled as fluid. A method to simulate the galvanostatic charge/discharge of a finite
deformation electrode with moving boundary was developed.

Important model parameters were determined one by one by correlating the simulation to
appropriate experiments. For example, the Li diffusivity in Si reported in literature varies from
10-13 to 10-19 m2/s. To estimate this parameter, the in-situ transmission electron microscope
experiment of two-phase lithiation of a-Si nanospheres was simulated. The diffusivity was found
at the order of 2×10-17m2/s for the lithium poor phase in first lithiation and 2×10-15 m2/s for
lithium rich phase and in subsequent cycles. The reaction rate constant and the apparent transfer
coefficient are determined in a similar way using different experiments.
In literature, different forms of chemo-mechanical coupling theories have been proposed for
Li diffusion in Si. The coupling relationship and parameters were often derived based on one
type of experiment even though the process is highly coupled. In this work, the chemomechanical coupling was investigated by simulations of two geometries: a thin film and a sphere.
A strong asymmetric rate behavior between lithiation and delithiation has been observed in thin
film a-Si anode but not in other geometries. The results reveal that the rate behavior is affected
by the geometry and the constraint of the electrode, the chemo-mechanical coupling, and the
prior process. A substrate-constrained film has a relatively low surface/volume ratio and a
constant surface area. Its lithiation has a great tendency to be hindered by surface limitation. The
chemo-mechanical coupling plays an important role in the specific rate behavior of a geometry.
Finally, an MRM model was built for a half cell with a-Si nanowalls as anode. The specific
and volumetric capacities of the cell as a function of size, length/size ratio, spacing of the
nanostructure, and the Li+ concentration in electrolyte were investigated. The results show that
the factors reducing the concentration polarization can enhance the maximum achievable SOC of
the cell. However, the cell with the highest SOC does not necessarily lead to the highest capacity.

ACKNOWLEDGEMENT

First of all, I would like to express my deepest appreciation to my advisor, Prof. Xinran Xiao.
After spending a few years with her for my Ph.D. study, I learned a lot and progress greatly. Her
attitude towards research and life inspired me greatly. She is always very informative and
respectful. She patiently guided me through difficulties in my research, not only on specific
problems, but also on the research methodologies. She trained me repeatedly to become an
independent researcher. Moreover, she always encouraged me to become a better person, full
with courage and hope towards my life. Her integrity, hardworking and ethnic set an excellent
example for me to follow.
I would like to express my deep thanks to my committee members, Professors Alejandro
Diaz, Peter Lillehoj and Jason Nicholas for their helpful directions and suggestions on my
researches. Additionally, special thanks to our collaborator, Dr. Xiaosong, Huang, at General
Motor R&D for his helpful discussions and technical contributions to this project. It has been a
great pleasure to work with him.
I also want to thank all my friends at Michigan State University I made through these years:
Wei Zhang, Wu Zhou, Shutian Yan, Danghe Shi, Christopher Cater and Andy Vanderklok, etc. It
has been a pleasure to know you and spent such a wonderful time with you.
The most importantly, I want to thank my beloved ones for their truthful care, endless love
and support. This work is dedicated to my parents. I would like to thank my husband, Fang Hou
for his being with me.
This research is supported by NSF CMMI 1030821, the start-up funding to Dr. Xiao from
Michigan State University and support from Department of Mechanical Engineering.

iv

TABLE OF CONTENTS

LIST OF TABLES ....................................................................................................................... viii
LIST OF FIGURES ........................................................................................................................ x
KEY TO ABBREVIATIONS ....................................................................................................... xv
Chapter 1
Introduction .............................................................................................................. 1
1.1 Lithium-Ion Battery .......................................................................................................... 2
1.2 Si Anode ........................................................................................................................... 3
1.3 Scope of the Work ............................................................................................................ 8
1.4 Outline of the Dissertation................................................................................................ 9
Chapter 2
Literature Review ................................................................................................... 11
2.1 Lithiation of Silicon ........................................................................................................ 11
2.1.1 Two-Phase and Single-Phase Lithiation .................................................................. 11
2.1.2 Lithiation Induced Volume Expansion .................................................................... 13
2.2 Chemical-Mechanical Two Way Coupling .................................................................... 14
2.2.1 Experimental Observations ...................................................................................... 14
2.2.2 Theories for Diffusion Induced Stress ..................................................................... 16
2.2.3 Theories for Stress Influenced Li Diffusion ............................................................ 18
2.2.3.1 Stress Effect on Li Diffusivity ............................................................................... 20
2.2.3.2 Stress Effect on Chemical Potential ...................................................................... 20
2.3 Material Property Variations during Si Lithiation .......................................................... 22
2.3.1 Li Diffusivity in Si ................................................................................................... 23
2.3.2 Electrochemical Kinetics with Si Anode ................................................................. 23
2.3.3 Mechanical Properties .............................................................................................. 23
2.4 Simulation of Si Particles and Li-Ion Cells .................................................................... 24
2.4.1 Si Anode................................................................................................................... 25
2.4.2 Li-Ion Cell ................................................................................................................ 26
2.4.3 Modeling of Li Transport......................................................................................... 27
Chapter 3
Microstructural-Resolved Multiphysics Model for Li-ion Battery ........................ 29
3.1 “Battery” Sub-Model ...................................................................................................... 31
3.1.1 Mass Transport......................................................................................................... 31
3.1.2 Electron Transport ................................................................................................... 34
3.1.3 Electrochemical Kinetics ......................................................................................... 34
3.1.4 Boundary Conditions ............................................................................................... 36
3.2 “Stress” Sub-Model ........................................................................................................ 36
3.2.1 Deformation of Si .................................................................................................... 36
3.2.2 Coupling between Li Transport inside Si and Volume Expansion .......................... 38
3.2.3 Stress Generation in Si ............................................................................................. 41
3.2.4 Fluid Electrolyte....................................................................................................... 41
3.2.5 Current Collector ..................................................................................................... 42

v

3.2.6 Boundary Conditions ............................................................................................... 42
3.3 Modeling Parameters ...................................................................................................... 43
3.3.1 Parameters in “Battery” Sub-Model ........................................................................ 43
3.3.1.1 Electrode Properties ............................................................................................. 43
3.3.1.2 Electroly Properties .............................................................................................. 44
3.3.2 Mechanical Properties .............................................................................................. 45
3.3.3 Source of Parameter and Form of Si Used............................................................... 46
Chapter 4
Study of Lithium Diffusivity in Amorphous Silicon via Finite Element Analysis 47
4.1 Introduction .................................................................................................................... 47
4.2 Model Development and Implementation ...................................................................... 47
4.3 Li Diffusivity in a-Si ...................................................................................................... 49
4.4 Results and Discussion ................................................................................................... 53
4.4.1 Two-Phase Lithiation ............................................................................................... 53
4.4.2 Single-Phase Lithiation ............................................................................................ 57
4.4.3 Concentration Dependency of Diffusivity ............................................................... 59
4.5 Conclusion ...................................................................................................................... 60
Chapter 5
Investigation of the Chemo-Mechanical Coupling in Lithiation of Amorphous Si
through Simulations of Si Thin Films and Si Nanospheres .......................................................... 62
5.1 Introduction .................................................................................................................... 62
5.2 Model Setup.................................................................................................................... 64
5.3 FE Models and Simulation Conditions........................................................................... 64
5.3.1 Amorphous Si Thin Film Model .............................................................................. 65
5.3.2 Si Nanosphere Model ............................................................................................... 68
5.4 Results and Discussion ................................................................................................... 70
5.4.1 The Rate Behaviors .................................................................................................. 70
5.4.2 Electrode Geometry ................................................................................................. 73
5.4.3 Chemo-Mechanical Coupling .................................................................................. 73
5.4.3.1 Thin Film .............................................................................................................. 74
5.4.3.2 Nanosphere ............................................................................................................ 76
5.4.4 Effect of Prior Process ............................................................................................. 78
5.5 Conclusion ...................................................................................................................... 81
Chapter 6
A Microstructure-Resolved Multiphysics Model for Li-ion Battery Si Anode ..... 83
6.1 Introduction .................................................................................................................... 83
6.2 Model Description .......................................................................................................... 84
6.3 Model Parameters ........................................................................................................... 86
6.3.1 Chemical Potential ................................................................................................... 86
6.3.2 Li Diffusivity in a-Si ................................................................................................ 87
6.3.3 Electrolyte Properties ............................................................................................... 87
6.3.4 Determine the Reaction Rate Constant and Apparent Transfer Coefficient ............ 88
6.4 Model Validation ............................................................................................................ 93
6.5 Cell Design with Si Nanowalls....................................................................................... 94
6.5.1 Design Parameter Study ........................................................................................... 95
6.5.2 Concentration Polarization....................................................................................... 97
6.5.3 Potential Evolution................................................................................................... 97

vi

6.5.4 Cell Capacity.......................................................................................................... 101
6.6 Conclusion .................................................................................................................... 104
Chapter 7
Conclusion and Outlook ....................................................................................... 105
7.1 Conclusion and Contributions ...................................................................................... 105
7.1.1 Identify the Range of Li Diffusion for a-Si Anode ................................................ 105
7.1.2 Identify the Chemical-Mechanical Two-Way Coupling Mechanism for Si under
Large Deformation Framework ........................................................................................... 105
7.1.3 Identify the Range of the Electrochemical Kinetic Parameters ............................. 106
7.1.4 A Design Tool for the Optimal Design of Battery Cells with a-Si Anode ............ 106
7.2 Outlook of Future Work ............................................................................................... 107
7.2.1 Interfacial Interplay between Si and Other Components ....................................... 107
7.2.2 Thermal Management of Li-Ion Cell ..................................................................... 107
7.2.3 Extension of Model to Other High Energy Density Electrode Material Li-Ion Cell ...
…………………………………………………………………………………….108
APPENDICES ............................................................................................................................ 109
Appendix A .. Effect of Li Transport Distance Change and Geometrical Constraint on Different
Si Geometries ......................................................................................................................... 110
A.1
Si Thin Film .......................................................................................................... 110
A.2
Si Nanosphere ....................................................................................................... 111
Appendix B Transport Phenomena in Si-Li Cells .................................................................. 114
Appendix C Electrical Conductivity....................................................................................... 118
Appendix D Si-Current Collector Interaction ........................................................................ 120
Appendix E Stress State in Si Nanosphere ............................................................................. 125
E.1
Si Nanosphere with Two Phase Lithiation............................................................ 125
E.2
Si Nanosphere with Single Phase Lithiation ......................................................... 126
BIBLIOGRAPHY ....................................................................................................................... 127

vii

LIST OF TABLES

Table 1.1 Specific and areal capacities of Si nanostructures. ......................................................... 7
Table 3.1 Parameters used in the MRM model............................................................................. 45
Table 3.2 Source of parameter used for Si .................................................................................... 46
Table 4.1 Governing equations for Si model in Chapter 4 (Symbols are explained in
Nomenclature)............................................................................................................................... 49
Table 4.2 Literature data on Li diffusivity in Si. .......................................................................... 51
Table 4.3 Diffusivity study test cases. .......................................................................................... 53
Table 4.4 Comparison of simulations with experimental results of two-phase lithiation. ........... 54
Table 5.1 Governing equations for Si model in Chapter 5 (Symbols are given in Nomenclature).
....................................................................................................................................................... 64
Table 5.2 Two conditions used in simulations.............................................................................. 65
Table 5.3 The specific lithiation/delithiation capacity at 100C in respect of the capacity at 0.1C
for the thin film and nanosphere under Conditions 1 and 2. ......................................................... 72
Table 6.1 Governing equations for cell model in Chapter 6 (Symbols are in Nomenclature)...... 85
Table 6.2 Electrolyte systems used in chosen experiments .......................................................... 87
Table 6.3 Comparison of reaction rate constant, exchange current density and apparent transfer
coefficient between literature and current work. .......................................................................... 90
Table 6.4 Model dimensions used in the Si nanowire model ....................................................... 91
Table 6.5 Model dimensions used in the Si thin film model ........................................................ 93
Table 6.6 Model dimensions used in the baseline model ............................................................. 96
Table 6.7 Half cells with SiNWs of different design parameters and the maximum SOC at 1C . 96
Table 6.8 The influences of design parameters on c 2 polarization,  2 ,  ,  1 and SOC ......... 100
Table 6.9 Comparison of the specific and volumetric capacity of half cell with SiNWs under
0.25C, 1C and 4C ........................................................................................................................ 102
Table 6.10 Comparison of the SOC and the capacity retention of half cell with SiNWs under
0.25C, 1C and 4C ........................................................................................................................ 102
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Table B.1 Results in baseline test at t=130s (SOC=0.03) ........................................................... 116
Table D.1 Preliminary study on Si-Ni interface ......................................................................... 123

ix

LIST OF FIGURES

Figure 1.1 Comparison of energy density for various chemistries [3] ........................................... 1
Figure 1.2 Inside a battery. Schematics are from referenced source[4–6] and self-drawn changes
are made according to needs. .......................................................................................................... 2
Figure 1.3 Gravimetric and volumetric capacities for pure metals as anodes for Li-ion batteries,
in comparison with graphite [14]. ................................................................................................... 4
Figure 1.4 In situ observation of a string of a-Si beads during lithiation [16]. ............................... 4
Figure 1.5 Capacity fading mechanism in Si system: (a) particle pulverization; (b) whole Si
electrode morphology change and active material losing contact to current collector [18]. .......... 5
Figure 1.6 Nanostructured Si anodes: (a) 1000 nm a-Si lm after 5 cycles [39]; (b) structure of
c–Si nanocomposite [40] B1–B2, TEM images recorded at different magnifications. The black
arrows point to spherical a-Si nanoparticles. (d) morphology evolution of an a-Si/CNF in the rst
two cycles [41]. Nanocracks nucleated during the second delithiation process (red dots); (d)
cycling of an a-Si sphere [42]; (e) typical cross-sectional SEM image of hollow Si nanospheres
at different magnifications E1-E2 [36]; (f) images of Si NWs after carbon coating F1-F2 [43]. .. 6
Figure 1.7 Schematics of the modeling tasks with their relations. ................................................. 8
Figure 2.1 Summary of the first lithiation and subsequent cycling of a-Si and c-Si[42]. ............ 12
Figure 2.2 Stress history during cycling of a-Si thin film along X and Y direction in thin film
plane [60]. ..................................................................................................................................... 15
Figure 2.3 Lithiation of c-Si nanoparticles showed in lithiated thickness changes vs. time plot of
multiple nanoparticles[61]. ........................................................................................................... 16
Figure 2.4 Comparison of stress in a-Si thin film during lithiation and delithiation cycle and DFT
calculation of yield stresses for LixSi[65]. .................................................................................... 17
Figure 2.5 (a) Inelastic deformation of Si during lithiation; (b) yield function Ψ defined in the
space of deviatoric stress Sij and driving force Îś from chemical potential[65]. ............................ 17
Figure 2.6 Free energy diagram[80]. Diffusion process is influenced by (a) free energy
differences between the ground state and product, and (b) activation energy. ............................. 19
Figure 2.7 Experimental measurement and simulation prediction of variation of stress with time
during cycles with a-Si thin film electrode[60] ............................................................................ 22
Figure 2.8 Mechanical properties generally decreases with Li concentration (state of charge): (a)
Young’s modulus measured in in situ[102] and ex situ[103] experiments, and calculated in first

x

principle simulation[108]; (b) Poisson’s Ratio calculated in first principle simulation[108]; and
(c) yield stress calculated in first principle simulation[105]. ........................................................ 24
Figure 2.9 Multiscale studies of Li-ion battery and their corresponding simulation
technique[108–115]. ..................................................................................................................... 25
Figure 3.1 A schematic presentation of the FE model, the governing equations and boundary
conditions for a half cell with Si anode and Li reference electrode. (a) "Battery" sub-model, and
(b) “stress” sub-model. The schematics were not drawn to scale. ................................................ 30
Figure 3.2 The ratio of c 1 to c as functions of volume ratio V/V0. .............................................. 39
Figure 3.3 Verification of the mass correction coefficient with (a) a free expanding Si sphere and
(b) a constrained thin film Si on a substrate, expanding only in the through thickness direction. (c)
The mass rate ratio as a function of volume ratio V/V0. The results showed that the mass
correction coefficient follows the relation of mc=V/V0. (d) The mass conservation was
maintained for both thin film and sphere geometries. .................................................................. 40
Figure 3.4 The stress-strain curve for (a) Cu substrate[157], and (b) stainless steel substrates[158]
....................................................................................................................................................... 42
Figure 3.5 Measurement of electronic conductivity in amorphous Si thin film held at various
potentials between 350 and 0 mV where the Li-Si alloy phase form [107]. ................................ 44
Figure 3.6 Mechanical properties of LixSi used: (a) elastic modulus, (b) Poisson’s ratio, and (c)
yield stress. .................................................................................................................................... 46
Figure 4.1 Axisymmetric model for modeling lithiation of a-Si nanoparticle. (a) Experimental
setup by McDowell et al.[30]; (b) two-phase lithiation in the first lithiation observed in
experiment [30]; and (c) simulating two-phase lithiation of a-Si nanospheres using an
axisymmetric 2D model. ............................................................................................................... 48
Figure 4.2 Constant diffusivity was assigned separately for Li-rich and Li-poor phases with a
small transition at around Li2Si..................................................................................................... 52
Figure 4.3 Concentration profile comparison: (a) The image intensity during lithiation of the 570
nm diameter sphere during lithiation in experiments[42]. Red circles label the concentration
Figure 4.3 (cont’d) drop at the reaction front. (b) Lithiation of Si in Tests 3 to 6 in simulations:
concentration profile vs. distance to particle center...................................................................... 56
Figure 4.4 Lithiated thickness as a function of time during lithiation of a-Si spheres, including: (a)
Two-phase lithiation comparison between results from experiments and simulations.
Experimental curves (black) #1-#4 are from McDowell et al.[42] out of 26 nanospheres with
unspecified diameter range. Simulation curves (red) are a-Si spheres of 400 nm and 570 nm. (b)
Comparing lithiation speed between single-phase lithiation and two-phase lithiation. ................ 58

xi

Figure 4.5 Simulations of lithiation of Si sphere of 570 nm diameter with a concentration
dependent diffusivity. (a) Diffusivity functions; (b) Lithiated-thickness versus time, comparing to
experimental data reported in Ref. [30] as shown in Fig. 4.4(a). ................................................. 60
Figure 5.1 Asymmetric rate performance of 70nm a-Si thin film observed in experiment[173]. 66
Figure 5.2 Schematic illustration of the 2D model for a-Si thin film with boundary conditions
(BCs), and the FE mesh. The schematic is not drawn to scale. .................................................... 67
Figure 5.3 Biaxial stress in thin film during cycling of a-Si thin film of 250 nm thickness under
C/4 rate were plotted, (a) the stress history vs. SOC was compared with the experimental result
was from Sethuraman et al.[60]. In simulations, the lithiation started with a residual stress from
delithiation at the same rate; (b) the stress profiles inside the thin film showed that the stress
distribution was almost uniform in the film during lithiation. ...................................................... 68
Figure 5.4 (a) The anode with Si nanospheres encapsulated inside graphite shells[174] and (b) its
capacity measured under various cycling rates, reproduced from Fig. S8 in Ref. [174]. (c) The
FE model and its boundary conditions used in simulations. ......................................................... 69
Figure 5.5 The predicted specific capacities vs. C rate for both 70nm a-Si thin film and a single
a-Si nanosphere with ( =0.01) and without (=0) the stress effect (a) thin film under Condition 1;
(b) thin film under Condition 2; (c) nanosphere under Condition 1 and (d) nanosphere under
Condition 2.................................................................................................................................... 71
Figure 5.6 Profiles of Li concentration, hydrostatic stress and Li flux vs. the distance to the
bottom of the thin film at 100C under Condition 1. Solid line for without ( =0) and dash line for
Figure 5.6 (cont’d) with the stress effect ( =0.01). (a) and (b) plot the Li concentration; (c) and
(d) plot the hydrostatic stress; (e) and (f) plot the Li flux 𝑵𝒛. (a)(c)(e) are under 100C lithiation
and (b)(d)(f) are under 100C delithiation...................................................................................... 75
Figure 5.7 Profiles of Li concentration, hydrostatic stress and Li flux vs. the radial distance for
the sphere at 100C under Condition 1. Solid line for without ( =0) and dash line for with the
Figure 5.7 (cont’d) stress effect ( =0.01). (a) and (b) plot the Li concentration; (c) and (d) plot
the hydrostatic stress; (e) and (f) plot the Li flux 𝑁𝑧. (a)(c)(e) are under 100C lithiation and
(b)(d)(f) are under 100C delithiation. ........................................................................................... 77
Figure 5.8 Profiles of Li concentration and hydrostatic stress vs. the distance to the bottom of the
thin film at 100C with =0.01 under Condition 1 (dashed line) and Condition 2 (solid line). (a)
and (b) are Li concentration; (c) and (d) are hydrostatic stress. (a) and (c) are under 100C
lithiation; (b) and (d) under 100C delithiation. ............................................................................. 80
Figure 6.1 (a) The relaxation potential of lithiated Si during lithiation and delithiation from
Sethuraman et al.[118] and the OCP curve used in the current work. (b) The chemical potential
derived from extrapolated OCP curve using Eq. 3.6. ................................................................... 86
Figure 6.2 The ionic conductivity k2 as a function of the salt concentration of LiPF 6 in EC:DMC
(2:1 vol.%)[131]. ........................................................................................................................... 88

xii

Figure 6.3 (a) The schematic of cell model with Si nanowire and (b) the mesh. (c) Comparisons
of the cell voltage vs. SOC curves obtained by simulations with k0 = 8×10-15 (m/s) (mol/m3)-0.7,
Îąa=0.7 and Îąc=0.3 (solid lines) and the experimental results (markers) by Zhang et al.[30]. ...... 92
Figure 6.4 Simulation of the experiment of the thin film a-Si electrode by Sethuraman et al. [60].
The FE model used was depicted in (a) and the mesh used in (b). The comparison between
simulations and experiments were given for both (c) cell voltage and (d) biaxial stress vs. SOC
during cycling at C/4 rate. ............................................................................................................. 94
Figure 6.5 Schematic of the model used for the parameter study for Si half cell. Line ① and
Point A are for the data analysis in later discussion. .................................................................... 95
Figure 6.6 (a) The cell voltage changes with SOC at various test cases, and (b) the Li+
concentration along the Si-electrolyte interface (line ① in Fig. 6.5) at the end of lithiation in each
test case. ........................................................................................................................................ 97
Figure 6.7 The evolution of potentials at point A for (a) the baseline test; (b) Thickness = 400 nm,
and (c) 3M LiPF6 electrolyte......................................................................................................... 99
Figure 6.8 The profiles of (a) electrolyte potential and (b) overpotential along line ① in Fig. 6.5
at SOC=0.03 for various test cases. ............................................................................................ 100
Figure 6.9 Si nanowalls of 200 nm thickness with various AR and SR in 1 M LiPF 6. (a) The
specific capacity and (b) the volumetric capacity at 0.25C rate, (c)(d) at 1C, and (e)(f) at 4C. . 103
Figure A.1 Control elements at electrode surface in Si geometry (a) thin film and (b) sphere .. 110
Figure B.1 (a) Schematic of the model used for the parameter study for Si half cell. Line ① and
Points A-C are for the data analysis in later discussion; (b)-(f) are profiles at the end of charging
at 1C for the case of 200nm SiNWs, with SR=1, AR=15, in 1M LiPF 6 in EC/DMC. The original
sizes and shapes are shown in dark lines. (b) Li concentration in Si; (c) Li+ in electrolyte; (d) the
hydrostatic stress in Si and in current collector; (e) the magnitude of fluid velocity in the
electrolyte. ................................................................................................................................... 115
Figure C.1 The investigation of electronic conductivity in Li-Si alloy by plotting the Si potential
along ① in (a) at same time (t=150s) when (b) k1=33 S/m; (c) k1=0.33 S/m; (d) k1=0.033 S/m.
..................................................................................................................................................... 119
Figure D.1Lithiation and delithiation of a-Si pillars on Ni substrate: (a) lithiation of pillars with
diameter of 2100 nm and height of 2300nm; (b) delithiation of pillars with diameter of 2600 nm
and height of 2300nm, in which crack is observed at Si surface near Si-Ni interface; (c)
delithiation of pillar with diameter of 3150 nm and height of 2100nm, in which the crack
propagates toward center of Si pillar[167].................................................................................. 121
Figure D.2Mechanical properties for the bulk Si and the interface region are modeled separately.
(a) Young’s Modulus as a function of concentration; (b) yield stress as a function of
concentration. .............................................................................................................................. 122

xiii

Figure E.1 Hydrostatic stress along particle radius at selected times during (a) two phase
lithiation, and (b) single phase lithiation..................................................................................... 126

xiv

KEY TO ABBREVIATIONS

𝑐
𝑐+ , 𝑐−

Li concentration in Si based on its initial volume, mol m-3
Concentration of positive and negative ions in electrolyte, mol m-3

𝑐1

Li concentration in Si based on its current volume, mol m-3

𝑐2

Li+ concentration in fluid electrolyte, mol m-3

𝑐𝑡ℎ𝑒𝑜𝑟𝑦 Theoretical maximum Li concentration based on initial Si volume, mol m-3
𝐂

Elastic right Cauchy-Green deformation tensor

𝐷1

Li diffusivity in LixSi, m2 s-1

𝐷2

Li+ diffusivity in fluid electrolyte, m2 s-1

𝐷𝑝𝑜𝑜𝑟

Li diffusivity in Li poor phase in first lithiation of Si, mol m-3

𝐷𝑟𝑖𝑐ℎ

Li diffusivity in Li rich phase in first lithiation of Si, mol m-3

𝐄

Elastic Green-Lagrange strain tensor

𝑓±

Average molar activity coefficient

F

Faraday constant, 96485 C mol-1

𝐅

Deformation gradient tensor

𝐅𝑒𝑙
𝐅𝑖𝑛𝑒𝑙

Elastic deformation gradient tensor
Inelastic deformation gradient tensor

𝐅𝑝

Plastic deformation gradient tensor

𝐅𝑐

Li concentration induced deformation gradient tensor

i

Current density, A m-2

i1, 2

Transfer current per unit of interfacial area at Si-electrolyte interface, A m-2

i0 , 2

Transfer current per unit of interfacial area at Li-electrolyte interface, A m-2

iSi , i Li

Exchange current density at Si-electrolyte/Li-electrolyte interface, A m-2

xv

𝐈

Identity matrix

𝐽

Determinant of deformation gradient

𝑘0

Reaction rate constant, (m s-1) (mol m-3)-Îąa

K

Electrical conductivity, S m-1

𝑚𝑐

Mass correction coefficient

𝑛
𝑁+ , 𝑁−

Number of charges carried by Li+
Molar flux of positive and negative ions, mol m-2

𝑅

Gas constant, 8.314 J (K mol)-1

𝐒

Second Piola-Kirchhoff stress tensor, Pa

𝑡+ , 𝑡−

Transport number of the positive and negative ions

𝑇

Temperature, K

𝐮

Displacement vector, m

𝑈

Open-circuit potential, V

v fluid

Electrolyte fluid velocity, m s-1

V

Current volume of Si

V0

Initial volume of Si

Ws

Strain energy density, J m-3

𝑧+ , 𝑧−

Charge numbers of the positive and negative ions

Greek Alphabets
𝛼

a c
𝛽

Stress effect coefficient
Apparent transfer coefficient
Thermal expansion coefficient, K -1

𝛽(𝑐𝑒𝑓𝑓 ) Mass expansion coefficient, m3 mol-1

xvi

𝛿𝑖𝑗

Kronecker delta

𝜀𝑖𝑗𝑇

Thermal strain



Overpotential, V

𝜆

Stretch ratio

𝛬

Lamé constant, 𝛬 = 𝜈𝐸/((1 + 𝜈)(1 − 2𝜈)), Pa

𝜇𝑓𝑙𝑢𝑖𝑑

Electrolyte viscosity, Pa s

𝜇𝐿𝑖𝑥𝑆𝑖

Li chemical potential, J mol-1

𝜇𝐿𝑖
𝜇𝐿𝑖 (𝛔)

Li chemical potential the compound is in its standard state, J mol-1
Li chemical potential including stress contribution, J mol-1

𝛭

Lamé constant, 𝛭 = 𝐸/(2(1 + 𝜈)), Pa

𝛔

Cauchy stress tensor, Pa

𝜎ℎ

Hydrostatic stress, Pa

𝜎𝑖

Principal stresses, subscript i =1, 2, 3, Pa

𝜎𝑖𝑖𝑎𝑣𝑒

Averaged in-plane stress, subscript i =x, y, Pa

𝜎𝑉𝑀

Von Mises stress, Pa

𝜎𝑌

Yield stress, Pa

𝜙

Electrical potential, V

Ί

Partial molar volume of Li, m3 mol-1

Subscripts and superscripts
1

Solid phase Si

2

Fluid electrolyte

c

Current collector

surf

Surface of Si

xvii

Chapter 1

Introduction

Secondary (also called rechargeable) batteries are the most convenient form of energy
storage devices. Among all commercially available secondary batteries, lithium-ion (Li-ion)
batteries offer the highest energy density, as shown in Fig. 1.1. The light weight and compact
volume of Li-ion batteries has made them the primary choice in many consumer electronics,
such as portable computers and cell phones. Nevertheless, current Li-ion battery technology is
still far from satisfactory in meeting the power demands for electric or hybrid-electric vehicles
(EV or HEV)[1,2]. To improve the range of EVs, the specific power and energy density of the
batteries must be improved drastically.

Figure 1.1 Comparison of energy density for various chemistries[3]

1

1.1 Lithium-Ion Battery
A typical Li-ion battery can be in the form of a coin cell, a cylindrical cell, a prismatic cell,
or a pouch cell. It is usually composed of multiple basic cells, stacking together, as shown
schematically in Fig. 1.2.
A basic Li-ion battery cell consists of two electrodes, namely cathode and anode, two current
collectors and a separator. The electrodes in commercial cells consist of active materials, binder
and additives. The binder holds active electrode materials together and attaches them to current
collector. Carbon blacks are common additives used to improve the electrical conduction of the
electrode. The separator is a porous thin film which provides the electrical insulation between the
cathode and anode while allows ionic conduction. Fluid electrolyte fills the battery cell,
providing Li+ pathway in between electrodes.

Figure 1.2 Inside a battery. Schematics are from referenced source[4–6] and self-drawn changes
are made according to needs.

2

Li-ion batteries are closed systems where the electrical energy is provided by the active
electrodes via redox reactions. For example, for a cell with LiCoO2 as cathode material and
graphite as anode, the charge and discharge processes are as follows:
Discharge
Anode: Li x C 6  6C  xLi   xe 
Cathode: Li1-yCoO2  yLi  ye  LiCoO 2
Charge
Anode: 6C  xLi   xe   Li x C 6
Cathode: LiCoO 2  Li1-yCoO2  yLi  ye
1.2 Si Anode
The energy density of a battery is measured by the specific gravimetric energy (mAh/g) and
specific volumetric energy (mAh/cm3). These are determined by the total Li-ion storage capacity
of the cell divided by the total mass or volume of all components, including the non-active
materials such as the current collectors, separator, binder and additives, etc.
A very lucrative way to increase the specific energy density of Li-ion batteries is to replace
the graphite, the anode material employed since the first commercial Li-ion battery was
introduced in 1991, with high capacity materials. Graphite has a gravimetric capacity of 372
mAh/g and a volumetric capacity of 843 mAh/cm3. A number of materials offer a much higher
Li storage capacity[7–9], as shown in Fig. 1.3. Among all Li storage materials, silicon (Si) is the
most promising[10,11]. Si has a theoretical gravimetric capacity of 3579 mAh/g and volumetric
capacity of 8339 mAh/cm3, ten times of that of graphite[12–14].

3

Figure 1.3 Gravimetric and volumetric capacities for pure metals as anodes for Li-ion batteries,
in comparison with graphite [14].

However, the major obstacle in the use of large capacity anode materials like Si in Li-ion
batteries is the large volume variation accompanied with lithiation and delithiation cycles. Fig.
1.4 presents the dramatic volume expansion of a string of amorphous Si beads during lithiation.
It is reported that when Si is in fully lithiated state, the volume of the particle expands to about
400% of its original size (300% volume change)[15,16].

Figure 1.4 In situ observation of a string of a-Si beads during lithiation[16].

4

The large volume variation can induce high stresses inside the active materials. Li moves
inside the host material through diffusion. The Li concentration gradient results in a non-uniform
volume variation in the material, which induces the so-called diffusion induced stresses (DIS),
similar to that induced by a temperature gradient. The large diffusion induced stress can cause
fracture and even pulverize the Si particles. Moreover, the large volume variation of the active
particles also strains the binder, which provides the electrical connectivity throughout the
electrode. The capacity loss due to binder or binder/particle interface failure is common in
electrodes with high capacity active particles. The above problems, as shown in Fig. 1.5, have
been attributed to the short cycle life[10,11,17] and rapid capacity fading observed in Li-ion
batteries with Si anode.

Figure 1.5 Capacity fading mechanism in Si system: (a) particle pulverization; (b) whole Si
electrode morphology change and active material losing contact to current collector[18].

Numerous strategies have been proposed to mitigate the problems related to the large volume
variation in Si. Reducing the size of the active materials has been proven to be an effective way
to reduce the stress[10,11]. Nanostructured Si anodes also reduce the tendency of crack
formation. The size dependent fracture behavior has been explained from the viewpoint of

5

fracture mechanics[18–21]. In a smaller structure, the stress-relief volume accompanying crack
growth is not significant enough to overcome the surface energy penalty associated with the
crack growth and hence a smaller size offers a greater resistance to crack initiation and
propagation[18]. This has led to the use of various nano-sized Si including Si
nanoparticles[22,23] and their composites[10,11,24,25], ultrathin Si films[26–29], Si
nanowires[30,31], Si whiskers[32], Si coated carbon nanofiber (CNF)[33,34], Si coated metal
fiber[35], Si hollow spheres[36], nanostructured porous Si[37], Si NWs on CNF film[38] etc. Fig.
1.6 presents some typical nanostructured Si anodes reported in literature. Table 1.1 summarizes
the state-of-the-art performance of various nanostructured Si anodes. As shown, their cycle lives
are generally low. The nanostructured Si anodes have improved the capacity retention from less
than 10 cycles[10,11] to 40~100 cycles or more, but they still fall short to meet the target of 3000
cycles for EV or HEV applications.

Figure 1.6 Nanostructured Si anodes: (a) 1000 nm a-Si film after 5 cycles[39]; (b) structure of c–
Si nanocomposite[40] B1–B2, TEM images recorded at different magnifications. The black
arrows point to spherical a-Si nanoparticles. (d) morphology evolution of an a-Si/CNF in the rst
two cycles[41]. Nanocracks nucleated during the second delithiation process (red dots); (d)
cycling of an a-Si sphere[42]; (e) typical cross-sectional SEM image of hollow Si nanospheres at
different magnifications E1-E2[36]; (f) images of Si NWs after carbon coating F1-F2[43].
6

Table 1.1 Specific and areal capacities of Si nanostructures.

Geometry
Baseline graphite
Si film of 50nm[28]
Si Nanowire on
CNF[38]
C-Si core-shell tube[33]
a-Si on CNF[34]
Cu-Si NW array[35]
Si Nano
Particles[22,23]
Si Hollow sphere[36]
3D Porous Si
Particles[37]
*Estimated value

Loading
mg/cm2

comment

370
3500/3500 (2C, 200 cycles)

Area
capacity
mAh/cm2
1.48*
0.087*

4
0.039755*

-

1800/1500 (C/10, 40 cycles)

-

3.6

collector free

2000/1600 (C/5, 55 cycles)
1500/1000 (100 cycles)
883/839* (C/10, 40 cycles)
2500/1400 (600 cycles)
2200/2200 (C/10, 30 cycles)
2725/1500 (C/2, 700 cycles)

4
0.16
0.42
3.3
-

2.4
0.19

0.56~0.76*

conductive
binder
-

2840/2800 (C/5, 100 cycles)

-

-

-

Specific capacity (mAh/g)
Initial/after # cycles

0.3*

The large volume variation of the Si anode also presents a great challenge in the dimensional
control of the battery. The volume change of the anode and cathode is opposite in sign during a
battery cycle. The dimensional variation of the Si anodes is an order higher than the common
cathode materials. How to make a dimensional stable Li-ion battery with Si anode is still an open
question.
Si in Li-ion batteries presents tremendous technical challenges far beyond the boundary of
electrochemistry. The physical phenomena in a Li-ion battery include the species and charge
transports in electrodes and electrolyte, reaction at the electrochemical interface, heat generation
and heat transfer, and stresses and deformations caused by lithiation, thermal and mechanical
loads. These phenomena are coupled. To improve the design a system with such complexity,
computer simulations would be an imperative tool. As it is now, computational models for LIBs
with high capacity battery electrodes are still in its infancy.

7

1.3 Scope of the Work
The objective of this work is to develop a computation model for Li-ion battery cell with a-Si
anode. This model will be used for design optimization of Si anode and battery cell performance.
Towards this end, a microstructure-resolved multiphysics (MRM) model is required.
In the previous work, such model has been developed for LixC6/LiPF6/LiyMn2O4 cell as
shown in the center panel of Fig. 1.7. To implement the model for Li-ion batteries with a-Si
anode, there are a number of challenges. In the proposed work, we will focus on the following
two aspects: (1) the lack of the key material properties of lithiated silicon ( Li x Si ); and (2) the
chemical-mechanical coupling relationship between Li diffusion and stress. These problems will
be investigated through systematically conducted numerical simulations and correlations of
numerical simulations with experiments, as illustrated in Fig. 1.7. The outline of the thesis is
given in following section 1.4.

Figure 1.7 Schematics of the modeling tasks with their relations.
8

1.4 Outline of the Dissertation
This thesis is organized as follows.
Chapter 1 introduces the problem and defines the scope of work.
Chapter 2 provides a literature review on current understanding about Si anode, including the
lithiation/delithiation process, the dimensional changes during lithiation, the theories on chemomechanical coupling, and the mechanical properties of Si and lithiated Si.
Chapter 3 provides a detailed description of the Multiphysics and their couplings considered
in the MRM model, including the electrochemical kinetics, mass transport, charge balance, the
Li intercalation induced strain, the mechanical strain, and the stresses in the active particles and
other battery components.
The lack of key material properties for Si/LixSi, especially the Li diffusivity in Si, has
prevented the efficient use of MRM numerical tools in the design of lithium-ion batteries with Si
anodes. Therefore, in Chapter 4, a finite element (FE) analysis is used to study the Li diffusivity
in a-Si anodes.
Chapter 5 investigates the chemo-mechanical coupling of a-Si through numerical simulations
of the rate behavior of two geometries (thin film and sphere). An asymmetric rate behavior has
been observed in thin film a-Si anode but not in a-Si anode of other shapes. The results show that
the geometry, the constraint of the electrode, the chemo-mechanical coupling, and the prior
process influence the rate behavior.
Chapter 6 presents an MRM model for a half cell with a-Si anode. The model is used in a
parameter study for LIB with Si nanowalls. The influences of the size and the length/size ratio of
the Si nanowalls, the spacing in between, and electrolyte Li+ concentration on the specific and
volumetric capacities of the cell are investigated.

9

Final conclusions, contributions and proposed future work are presented in Chapter 7.

10

Chapter 2

Literature Review

This chapter provides a literature review on the current understandings, the problems and the
research development relevant to simulating Si in Li-ion batteries with MRM model.
2.1 Lithiation of Silicon
2.1.1

Two-Phase and Single-Phase Lithiation

Si can assume one of the three phases at room temperature: amorphous Si (a-Si),
monocrystalline Si (c-Si) and polycrystalline Si. In Li-ion battery applications, a-Si and c-Si are
commonly used.
During the first lithiation, a-Si exhibits isotropic swelling whereas c-Si suffers anisotropic
swelling[44,45]. The shape and volume changes are reversible for a-Si but not for c-Si.
Moreover, the phase transformation is observed during the first lithiation. The equilibrium Li-Si
phase diagram[46] shows that, when reacting with Li, c-Si goes through a series of phase
changes at high temperature. In Li-ion batteries at room temperature, the phase changes were not
observed. It has been proven that upon lithiation, both c-Si and a-Si transform to metastable
amorphous Si-Li alloys (a-LixSi). Anodes made of c-Si quickly become disordered. This
phenomenon was referred to as electrochemically-driven solid-state amorphization (ESA)[47].
Towards the fully lithiated stage, a-LixSi crystallizes to either Li15Si4 (Li3.75Si) or Li22Si5
(Li4.4Si)[34]. Crystalline Si eventually transformed into a-Si after the first cycle[48].
To understand the lithiation mechanism and stress evolution of Si anodes, the morphology
evolutions of various Si nanostructures have been investigated experimentally inside the
transmission electron microscopes (TEM) with ongoing electrochemical reactions[41,42,49–53].
TEM images revealed a sharp phase boundary instead of a diffused band between the reactant (cSi) and the product (a-LixSi), indicating that the first lithiation occurs via a two-phase

11

mechanism[42]. The sharp phase boundary is called reaction front. It was observed
experimentally and confirmed by the first principle simulation that a group of Li atoms
collectively weaken the Si-Si covalent bonding near the edge of {111} atomic facets at the
reaction front first and caused easy break of other Si-Si bonds after to form LixSi[54]. This result
has important consequences for mechanical stress evolution during lithiation[41,42]. The sharp
phase boundary between the c-Si core and a-LixSi shell coupled with anisotropic swelling has
been related to crack formation in c-Si NWs[51].
The two-phase phenomenon was also observed in the TEM images of a-Si anodes at the first
lithiation[42]. Since a-Si network is continuous random, the Si-Si bond breaking by Li atom is
isotropic[54]. The phase front was not visible in the delithiation and second lithiation as long as
LixSi did not crystallize into Li3.75Si at the end of lithiation[42,50], the so-called single-phase
lithiation. The lithiation of both c-Si and a-Si is shown in Fig. 2.1. Si anodes with an initial
amorphous structure have shown to improve the cycle life over crystalline Si[26–28]. Therefore,
this work will focus on study a-Si structure in Li-ion battery.

Figure 2.1 Summary of the first lithiation and subsequent cycling of a-Si and c-Si[42].

12

2.1.2

Lithiation Induced Volume Expansion

As mentioned earlier, Si experiences volume change about 300% when it is fully lithiated. To
simulate this process, the finite deformation theory[55,56] is invoked to consider the large
deformation. In experimental observation reviewed in 2.1.1, the lithiation in a-Si always
introduces isotropic volume expansion and its reversible, while the c-Si experiences anisotropic
volume expansion in its first lithiation and this process is not reversible. Since we are only
interested in the a-Si structure in this study, the finite deformation theory for isotropic
expansion[57,58] will be discussed.
Freestanding Si geometries, such as nanospheres and nanowires, do not experience constraint
imposed by other battery components. Therefore, the Si particle undergoes free expansion. The
total deformation includes elastic deformation, plastic deformation and lithiation induced
deformation [57].
To describe the behavior of solids with large deformation, such as elastomers, the stretch
ratio instead of strain is often used. This treatment has also been used for Si anode[57,58]. The
stretch ratio is defined as[55,59]


length in current state
l

length in reference state L

(2.1)

The volume change induced by elastic deformation Ve /V0 is

Ve
 1e e2 e3
V0

(2.2)

where the subscripts 1, 2, 3 correspond to the three material directions; the subscript/superscript
e donates the change induced by elastic deformation.
Plastic deformation is volume preserving[56], therefore,

13

Vp
V0

 1p 2p 3p  1

(2.3)

where subscript/superscript p donates the change induced by plastic deformation
The lithiation induced volume change VLi /V0 is proportional to the change in the Li
concentration[57,58],

VLi
 1  c
V0

(2.4)

where  is the partial molar volume of Li, which defines the volume increase due to one mole
of Li guest atom insertion in the Si host material, and c is the lithium concentration increment.
Li
Li
Since the deformation is isotropic in a-Si, the lithiation induced stretches 1 , 2 and 3Li

along all directions are assumed to be equal in the stress-free Si network

 

isotropic
VLi
 1  c  1Li 2Li 3Li  Li
V0

3

(2.5)

The total volume change should be





 

V
 1e e2 e3 1p 2p 3p Li
V0

3

(2.6)

2.2 Chemical-Mechanical Two Way Coupling
2.2.1

Experimental Observations

Lithium insertion can induce high stresses in the host material. The stress accumulation
during electrochemical lithiation and delithiation has been measured in situ for a-Si thin film [60],
as shown in Fig. 2.2. The measurement was based on the Stoney equation which determined the
stress in a thin film on a thick substrate by measuring its curvature using a Multi-beam Optical
Sensor (MOS) system [60]. In literature, this is the only method to measure the in situ stress in Si
electrode.

14

Figure 2.2 Stress history during cycling of a-Si thin film along X and Y direction in thin film
plane [60].

The Li diffusion process is also influenced by stress. This has been observed in lithiation
process of c-Si nanospheres [61,62]. As shown in Fig. 2.3, the lithiated thickness (thickness of
LixSi shell) growth rate reduced after the Si has been lithiated for some time. It was observed that
the sharp interface between c-Si and a- LixSi could not proceed towards nanospheres center as
fast as the beginning of lithiation [61]. The lithiation in small and large particles slowed on a
similar time scale. c-Si cores remained in the smaller particles even after much thicker regions
have been lithiated in larger particles. This indicated that the slowing of the reaction was not due
to diffusion limitation [61,62]. The phenomenon was attributed to the high compressive stress at
the reaction front that eventually prohibited Li further from insertion [61].

15

Figure 2.3 Lithiation of c-Si nanoparticles showed in lithiated thickness changes vs. time plot of
multiple nanoparticles[61].

2.2.2

Theories for Diffusion Induced Stress

The stress associated with the lithiation process is diffusion induced stress (DIS). It arises
inside Si due to compositional inhomogeneity during ion insertion and removal processes,
similar to the stresses caused by non-homogenous expansion in a solid particle subjected to a
non-uniform temperature field. Accompanied by lithiation, both elastic and inelastic stresses
could be generated. The elastic stress was generally calculated using Hooke’s Law[63]. The
plastic stress may be calculated by considering the process as plastic[64] and viscoplastic[63].
Recent studies showed that stress generated inside LixSi in experiment is much lower than yield
strength calculated in first principle simulated based on density functional theory (DFT) for
LixSi[65], as shown in Fig. 2.4. Therefore, purely mechanical plasticity/viscoplasticity
consideration may not be enough to capture the lithiation-induced stress because that the
lithiation process is driven by both chemical potential of Li and by the stress (hydrostatic and
non-hydrostatic stresses) generated during Li insertion[65], as shown in Fig. 2.5(a).

16

Subsequently, a chemo-mechanical yield function based on chemical potential and deviatoric
stress was defined, as shown in Fig. 2.5(b)[65].

Figure 2.4 Comparison of stress in a-Si thin film during lithiation and delithiation cycle and DFT
calculation of yield stresses for LixSi[65].

Figure 2.5 (a) Inelastic deformation of Si during lithiation; (b) yield function Ψ defined in the
space of deviatoric stress Sij and driving force Îś from chemical potential[65].
17

Furthermore, the experimentally measured ultimate tensile strengths for Si/LixSi[66] was also
much lower than the yield stresses calculated[65]. The ultimate tensile strength measured in a
nanowire with initial diameter of 130 nm[66] was 3.6GPa for pure c-Si and 0.72GPa for Li3.75Si,
which were in a similar range with the yield stresses measured in the thin film experiment[60,65].
This confirms that the yield strength predicted in DFT simulation was indeed higher than the real
yield strength.
The abovementioned theories[63,65,67] were proposed based on the observation of
experimentally measured stress from a-Si thin film electrode under lithiation[63,65].
2.2.3

Theories for Stress Influenced Li Diffusion

The Li diffusion influences the stresses generated inside Si, and vice versa.
Fick’s laws are usually adopted when considering the diffusion problem in ideal mixture[68]:

c
J  0
t

(2.7)

where J is the Li flux.
However, in solid-state diffusion problems, the Fick’s laws in Eq. 2.11 and 2.12 do not
always hold because the local flux considered through Fick’s first law may be influenced by
other factors, such as stress[63,64,69,70] and temperature[71,72]. Since the heat generation
inside Si during lithiation is not considered at this stage, we focus on the stress effect on
diffusion of Li in Si. Even without externally applied stresses, such as freestanding silicon, the
diffusion induced stress may affect the lithiation of Si. So the general Fick’s laws using chemical
potential as the driving force are used to consider the non-Fickian behavior introduced by
stress[73]:

18

c
 D

   J     
c 
t
 RT


(2.8)

where R is gas constant, with a value at 8.314 J/ mol  K  ; T is temperature;  is chemical
potential; D is Li diffusivity in Si.
In literature, two theories have been proposed for the diffusion of guest atoms into a host
material under the stress effect related to the free energy diagram in Fig. 2.6: (a) Diffusivity is
modulated by the stress induced activation energy barrier shift[69,74,75]. The rate of reaction
depends on a free energy between the ground state and transition state (i.e. activation energy). (b)
The chemical potential of Li in Si changes because the free energy difference between substrate
and product is altered by stress[63,64,76–79]. This free energy difference is the driving force for
reaction. Therefore, both effects should be considered for Li diffusion in stress Si.

Figure 2.6 Free energy diagram[80]. Diffusion process is influenced by (a) free energy
differences between the ground state and product, and (b) activation energy.

19

2.2.3.1 Stress Effect on Li Diffusivity
Stress is found to influence the diffusivity through the activation energy. The activation
energy is stress dependent[81]. The Arrhenius equation has been used to calculate the diffusivity
with the consideration of the changes on activation energy by stress[69,74,75]:

  E A  
D  D0 exp 
RT



  
Li
  D0 exp 


 RT 

(2.9)

  EA 
Li
where D0  D0 exp 
 is the diffusivity of Li in Si under zero stress; E A is the activation
 RT 
energy barrier;  is the partial molar volume of Li in Si;  is the hydrostatic stress[69,74,75,82]
calculated by

1   2   3
3

, which involves only pure tension or compression;  is a positive

dimensionless constant.
Theories proposed had been used to study both freestanding electrode geometry[69,74] and
constrained thin film electrode[75,82] in Li-ion battery. However, since the lack of experiments
and theoretical studies, the only validation was made through results comparison with atomistic
model[82]. Moreover, the determination of  is still an open question.
2.2.3.2 Stress Effect on Li Chemical Potential
The chemical potential change during lithiation is still argumentative given multiple different
treatments in literature. One of the treatments is to consider the system as an ideal solid solution
under hydrostatic stress only[57,64,82–84]. The chemical potential can be written as
Li
   0  RT ln C   h    SF
  h

(2.10)

Li
where  SF
is the chemical potential of Li atoms in Li-Si alloy under zero stress;  is the Li

partial molar volume and  h is the hydrostatic stress. The assumption is that Li in lithiated Si
could be viewed as a mobile component in a stressed solid. This could be reasoned based on MD

20

simulation showing that Li mobility is 3-4 orders higher than Si mobility in LixSi[85]. Therefore,
this equation considers the solid solution as homogeneous and the only stress applied here is
hydrostatic. Up to now, Eq. 2.15 was only used for qualitative analysis of the stress effect on
diffusion process[57,64,82–84].
Another treatment involved an expression of the chemical potential in multicomponent solid
under non-hydrostatic stress to reach equilibrium. This was originally developed by LarchĂŠ and
Cahn[86–88] and was used to study solid-state diffusion problem in geology[89] and
metallurgy[90].
It was recently adopted to consider Li diffusion in Li-ion battery electrode[60,63,91] in the
form of:
Li
   SF


 B
 Si
 kk  Si ijkl  ij  ij
3
2

(2.11)

where  Si is the molar volume of silicon,  is the rate of volumetric strain change due to
lithiation and Bijkl  dsijkl dc where sijkl is the elastic compliance tensor. It was used for a-Si thin
film electrode[63,92] in the form of:
Li
   SF


2 Si  Li
  1 
   Si 2  
31   Li c 
c  M 

(2.12)

where    yy   zz is the Cauchy stress in a-Si thin film and M is the biaxial modulus. Eq. 2.12
was used to predict the stress effect on Li diffusion in Si thin film and is validated with in situ
stress-strain curve[60], as shown in Fig. 2.7.

21

Figure 2.7 Experimental measurement and simulation prediction of variation of stress with time
during cycles with a-Si thin film electrode[60]

Since LarchĂŠ and Cahn theory was developed based on the small strain framework under an
elastic field, whether it is valid under finite deformation and yield of Si or not remains unclear.
Recent theories were developed to consider driving force change under finite deformation
framework[77–79]. These theories are yet to be validated.
Moreover, the stress free term of chemical potential (first term) in Eq. 2.12 was also based on
dilute solution in LarchĂŠ and Cahn theory. The chemical potential in Li x Si concentrated solution
must be determined through correlation of experiments[63,93].
2.3 Material Property Variations during Si Lithiation
The structural change in Si during lithiation results in changes in the material properties,
including those important to the performance and durability of the anode such as the Li
diffusivity and the mechanical properties. One of the major obstacles in the modeling of Si anode
is the uncertainty in these material properties. The reported ranges for the key material properties
are reviewed as follows.

22

2.3.1

Li Diffusivity in Si

The Li diffusivity in Si is a key parameter determining the lithiation time and Li
concentration gradient, the latter in turns influences the intercalation stresses. The literature data
on Li diffusivity in Si varies over a wide range[63,85,94–99].
2.3.2

Electrochemical Kinetics with Si Anode

The electrochemical reaction at the Si-electrolyte interface is determined by the ButlerVolmer equation. Without knowing the reaction parameters, such as the reaction rate constant
and the apparent transfer coefficient, the electrochemical reactions happen in cell with Si anode
could not be correctly studied.
2.3.3

Mechanical Properties

Mechanical properties involves here are Young’s modulus, Poisson’s ratio and yield strength
of Si/LixSi. The evolution of the above properties along with lithiation has been studied through
experiments[100–104] and the first principle simulations[85,105–107].
The Young’s modulus of the a-Si is 94 GPa. The c-Si has a modulus of 130-185 GPa
depending on crystallographic directions. The Young’s modulus of LixSi as a function of Li
fraction has been investigated using ex situ[103,104] and in situ[100–102] experimental methods.
The Young’s modulus of the LixSi phase was reported to decrease with increasing Li
concentration, as shown in Fig. 2.8(a). The reported Young’s modulus decreased from 95 GPa
for a-Si to 10 GPa for Li3.75Si[102].
The first principle simulations[85,105–107] have been used to calculate the mechanical
properties of the LixSi phase. The stress-strain response of the a- LixSi has been calculated[105].
The elastic modulus, Poisson’s ratio and yield stress were found to decrease with Li
concentration, as shown in Fig. 2.8.

23

(a)

(b)

(c)

Figure 2.8 Mechanical properties generally decreases with Li concentration (state of charge): (a)
Young’s modulus measured in in situ[102] and ex situ[103] experiments, and calculated in first
principle simulation[108]; (b) Poisson’s Ratio calculated in first principle simulation[108]; and
(c) yield stress calculated in first principle simulation[105].

2.4 Simulation of Si Particles and Li-Ion Cells
It always involves multiple scales in modeling the Li-ion batteries. As shown in Fig. 2.9, the
scale of modeling tools in literature ranges from microscopic level to macroscopic level
depending on the problems and system studied.

24

Given the focus of this work, two levels are reviewed: modeling of Si particle and Li-ion
cells models.

Figure 2.9 Multiscale studies of Li-ion battery and their corresponding simulation
technique[108–115].

2.4.1

Si Anode

In literature, there are some works investigated the deformation and stress in single Si
particles using analytical[53,116–121] or numerical models[51,63,100,103,105,122–127] by the
coupled diffusion-stress approach.

25

The analytical analysis was done on Si with various forms, like nanowires[53,116], thin
film[118,121] and spheres[58]. For example, diffusion-induced stress with[116] or without[105]
consideration of the change in mechanical properties based on either small deformation
theory[116,119,120] or large deformation theory[105]. The analytical studies guided the further
continuum modeling of Si using FEA codes like COMSOL and ABAQUS.
The continuum modeling of Si on multiple problems were investigated, usually involves one
of the following topics: the anisotropic growth of crystalline Si[51]; the crack propagation in
Si[123]; the interaction with other cell components like binder[124], and current collector[128];
the chemo-mechanical coupling relationship[64,127,129]; the electrochemistry involved in cell
with Si anode in 1D[63] and in 3D[128]. The questions that interest researcher the most were
targeted and resolved, respectively.
2.4.2

Li-Ion Cell

To simulate and predict performance of the Li-ion battery cell, multiple works has been
published on graphite system since 1993. Doyle et al. first developed the Li|polymer|composite
cathode cell[130] and further to polymer LiXC6|LiyMn2O4 cell[131] in 1D, considering the
mass/charge transport and the electrochemical reactions inside the cell. This model was further
developed to a 3D model with porous cathode matrix was constructed by Wang and Sastry[132].
By considering the porous structural of cathode, they investigated the influence of particle size
on cell performance. The model demonstrates that the microstructure-resolved model was a good
design tool for prediction of electrochemical cell performance.
Furthermore, the basic electrochemical cell was further developed to incorporate physics
other than pure electrochemistry, including: (1) the coupled thermal-electrochemical model[133–
135], which was used to understand the temperature influence on cell potential and predict the

26

heat generation under various cell design; (2) the coupled electrochemical-stress model[136,137],
which was used to predict the structural changes inside the cell during operation, and the DIS
generated in electrode and the stresses in other cell components due to the change in electrode.
Si as the emerging potential high energy density anode material to Li-ion cell and thus many
simulation works has been done for Si as in Section 2.4.1. However, rarely any cell model was
developed for Li-ion cell with Si. In 2011, Chandrasekaran and Fuller[138] developed a 1D
electrochemical cell model with silicon composite electrode in reference to Li counter electrode.
Effective parameters were used to take account of the electrode porosity. They showed that
porosity of the electrode vary a great deal during cycling and may severely affect the battery
performance. This work demonstrated the importance of the model of the cell with Si anode and
posed the requirement of development of the MRM model.
2.4.3

Modeling of Li Transport

The prediction of Li transport in Li-ion cell is one of the most important parts in simulation
the Li-ion cell. The Li/Li+ transport inside the Li-ion cell models should be considered for both
electrode and the electrolyte. Here is how they are considered in literature.
The Li+ transport in electrolyte solution is determined by the electrochemical potential
difference between the two electrodes, which in turns depends on the conditions such as solution
composition and the electrical state, etc. Li+ transport in electrolyte is often described by the
Nernst-Planck equation[131,132,138,139]. The electrochemical reaction is considered as
happening at the electrode-electrolyte interface, where Li++e-=Li. Li then transports further into
the electrode. The electrochemical potential includes both the chemical potential and the electric
potential. In the solid-state electrode, the driving force for Li transport is commonly considered
through concentration gradient[130–132,138] or chemical potential gradient[63,128,140]. With

27

the assumption of neutral charge of Li, the electric potential contribution is not considered in the
solid-state electrode.
This work follows the same assumptions as stated above.

28

Chapter 3

Microstructural-Resolved Multiphysics Model for Li-ion Battery

The model framework of the MRM model for Si anode in Li-ion battery is introduced in this
chapter. The model is composed of two sub-models: a “battery” sub-model and a “stress” submodel. The “battery” computes electrochemical kinetics, mass transport, and charge balance.
The “stress” sub-model computes the Li intercalation induced strain, the mechanical strain, and
the stresses in the active particles and other battery components. The two sub-models share the
same FE mesh and are coupled through Li concentration field and deformation/stress field.
Most of the available experiments for Si anodes are in a half cell configuration with Li metal
as the reference electrode. Therefore, the model was developed for LixSi|fluid electrolyte|Li cell.
It includes the current collector, the Si anode, the fluid electrolyte, and the Li metal counter
electrode. Fig. 3.1(a) and (b) present the schematics of the half cell model with the governing
equations and boundary conditions for the “battery” and “stress” sub-models, respectively. The
details of each sub-model are listed in the following sections.

29

(a)

(b)
Figure 3.1 A schematic presentation of the FE model, the governing equations and boundary
conditions for a half cell with Si anode and Li reference electrode. (a) "Battery" sub-model, and
(b) “stress” sub-model. The schematics were not drawn to scale.

30

3.1 “Battery” Sub-Model
In “battery” sub-model, the mass transport, charge balance and electrochemical reaction
kinetics are considered.
3.1.1

Mass Transport

The mass transport includes the Li transport in Si anode and the electrolyte.
The Li transport in Si is considered through either Fick’s second law in Eq. 3.1 with
concentration gradient as lithiation driving force
𝜕𝑐1
+ ∇ ∙ (−𝐷∇𝑐1 ) = 0
𝜕𝑡

(3.1)

or considered through general form diffusion equation in Eq. 3.2 with the chemical potential
gradient as driving force[141,142]
𝜕𝑐1
𝐷1
(𝛔)) = 0
+ ∇ ∙ (−
𝑐 ∇𝜇
𝜕𝑡
𝑅𝑇 1 𝐿𝑖𝑥𝑆𝑖

(3.2)

where 𝑐1 is Li concentration in Si, 𝐷 is the apparent Li diffusivity and 𝐷1 is the chemical
diffusivity of Li in LixSi, 𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) is the chemical potential of LixSi under stress influence, 𝑅 is
gas constant, and 𝑇 is the temperature.
The chemical potential under stress 𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) is defined as[141,142]
𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) = 𝜇𝐿𝑖𝑥𝑆𝑖 + ∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔)

(3.3)

where 𝜇𝐿𝑖𝑥𝑆𝑖 is the chemical potential in LixSi under zero stress, ∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) is the change in
chemical potential due to stress.
𝜇𝐿𝑖𝑥𝑆𝑖 in Eq. 3.3 can be mathematically determined under various assumptions, such as the
ideal solution assumption[57,83,138,143,144], where the driving force for Li-Si mixing is the
entropy with zero enthalpy so that it allows components miscibility in all proportions; the regular
solution assumption[145–147], which includes both the entropic and enthalpic contributions to

31

Li-Si mixing; and the non-ideal solid solution assumption[63,82,148,149], where Li occupies the
interstitial or substitutional lattice sites with a saturation limit in the host material. Moreover, it
could be extrapolated from the experimental measurement of the open circuit potential.
Depending on the different focuses and needs in each study presented in this work, various
theories for Li diffusion in Si is adopted in Chapter 4-6.
For example, when investigating the range of Li diffusivity in Chapter 4, the Eq. 3.1 is used
with a simplification of constant 𝐷 neglecting the contribution from stress.
When studying the chemo-mechanical coupling relationship in Chapter 5, as the focus was
on the stress effect on Si lithiation rather than Li-Si mixing theories, as the first step, the ideal
solution assumption was adopted[57,83,150]. Therefore, Eq. 3.3 became
0
𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) = 𝜇𝐿𝑖
+ 𝑅𝑇 ln 𝜒𝐿𝑖 + ∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔)
𝑥 𝑆𝑖

(3.4)

0
where 𝜒𝐿𝑖 is the mole fraction of Li in LixSi, and 𝜇𝐿𝑖
is a constant corresponding to the
𝑥 𝑆𝑖

chemical potential of Li in Si at its standard state. The value of 𝜇0𝐿𝑖𝑥𝑆𝑖 is unknown. Since the
0
interests here are the change of chemical potential 𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) − 𝜇𝐿𝑖
, and the gradient ∇𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔),
0
the exact value of 𝜇𝐿𝑖
is not needed.
𝑥 𝑆𝑖

For ∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔), as the first step, a simple form of chemo-mechanical coupling relationship
was adopted based on[57,83,150]
∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) = 𝛼Ω𝜎ℎ

(3.5)

where 𝜎ℎ is the hydrostatic stress, i.e., the average of principal stresses 𝜎ℎ = (𝜎1 + 𝜎2 + 𝜎3 )/3;
ι is the coefficient for the influence of hydrostatic stress on chemical potential; Ί is the partial
molar volume of Li defining the volume change due to one mole of guest atom insertion into the
host material. For Li in Si, Ω = 9×10-6 m3/mol[151].

32

When the MRM model was used to investigate the cell performance in Chapter 6, 𝜇𝐿𝑖𝑥𝑆𝑖 in
Eq. 3.3 is then mathematically determined by fitting the open-circuit potential (OCP) U using Eq.
3.6[152,153]. Thus, the change of Li composition is related to chemical potential and, in turns, to
the cell potential
𝜇𝐿𝑖𝑥𝑆𝑖 − 𝜇𝐿𝑖 = −𝑛𝐹𝑈

(3.6)

where 𝜇𝐿𝑖 is the chemical potential of the Li metal, which is set as 𝜇𝐿𝑖 =0; 𝑛 is the number of
charge Li+ carries, which equals to 1; 𝐹 is the Faraday constant, and F=96485 C/mol.
To describe the Li+ diffusion and charge balance in the electrolyte, Nernst-Planck equations
based on the concentrated solution theory[139] are adopted. As an initial step, the influence of
convective flow on Li+ transport in the fluid electrolyte is not considered. The fluxes of the
binary ionic species are described by[130]


2k t t RT   ln f  
kt
c  2   2
N  =   D2  2   2 1 
z F
c z  F   ln c2 

2

2k t RT   ln f 
N  =   D2  2  2 1 
 ln c 2
c  z  F 



k t
c   2   2
z F


(3.7)

(3.8)

where N  and N  are the positive and negative ion molar fluxes; t  and t  are the transport
numbers of the positive and negative ions; z  and z  are the charge numbers of the positive and
negative ions; c and c are the positive and negative ion concentrations; c 2 is the Li+
concentration in electrolyte; D2 is the Li+ diffusivity in electrolyte; k 2 is the ionic conductivity;

f  is the average molar activity coefficient;  2 is the electrical potential of the electrolyte.
The electrolyte current density can be expressed as[130]

33

i 2 = k 2 2 
3.1.2

2 RTk2
c2 F

  ln f 
1 
  ln c2


1  t  c2


(3.9)

Electron Transport

The electron transport in the cell is considered in the Si electrode and current collector
through Ohm’s law. In Si electrode, the current i 1 is calculated as

k11  i1

(3.10)

where k1 is the electrical conductivity of Si, 1 is the electrical potential, and i 1 is the current
density in Si.
Similarly, the electron transport inside the current collector is calculated as

k c  c   i c

(3.11)

where kc is the electrical conductivity of the current collector; and ic is the current density inside
the current collector.
The relationship between i 1 and i 2 at the Si-electrolyte interface is

i1  i 2
3.1.3

(3.12)

Electrochemical Kinetics

The electrochemical reactions occurring in the cell are:
Si electrode: x𝐿𝑖 + + 𝑆𝑖 + x𝑒 − = 𝐿𝑖𝑥 𝑆𝑖 , (0 ≤ 𝑥 ≤ 3.75)
Li electrode: 𝐿𝑖 + + 𝑒 − = 𝐿𝑖

(3.13)
(3.14)

The electrochemical reaction is considered though the Butler-Volmer equation [130,138],
which describes the relationship among the reaction kinetics, lithium concentration, and
electrical potential at the electrochemical interface

34

   F 
   c F 
Si side: i1, 2  i Si exp a   exp

 RT 
  RT 

(3.15)

     2 F 
   F 
Li side: i0, 2  i Li exp a
  exp c 2 
RT

 RT 
 

(3.16)

where i1, 2 and i 0 , 2 are the transfer current density at the Si-electrolyte and Li-electrolyte interface,
respectively; Ρ is the surface overpotential; and  a and  c are the negative and positive charge
2
transfer coefficients, respectively. For LIBs, since n=1,  a  1   c [139]. i Li  12 .6 A/m is the

exchange current density at Li-electrolyte interface; i Si is the exchange current density at the Sielectrolyte interface
c
iSi  Fk0 c2a ctheory  csurf  a csurf


(3.17)

In Eq. 3.17, k0 is the reaction rate constant; c 2 is the Li+ concentration in the electrolyte;
c theory is the theoretical maximum Li concentration in Si, and c surf is the Li concentration at Si

surface. Both c surf and c theory are in reference volume V0 to represent the LixSi composition in
Butler-Volmer equation [139].
The overpotential Ρ in Eq. 3.15 is defined as

  1  2  U

(3.18)

where  1 and  2 are the local electrical potential of Si and electrolyte, respectively. The
overpotential in Eq. 3.16 is reduced to   2 because the potential of Li is kept as 0V, and the
OCP of Li is also 0V [138,139].
The Li+ mass flux N  into the Si is related to i1, 2 such that

35

N   i1, 2 / F
3.1.4

(3.19)

Boundary Conditions

Fig. 3.1(a) presents the boundary conditions for the “battery” sub-model, which corresponds
to the conditions experienced by a basic cell. Because the charged mobile species in the
electrolyte cannot diffuse into the current collectors or beyond the upper and lower boundaries,
and there is no Li flux between Si and the current collector. At these boundaries, the flux of Li
and the charged mobile species was set as

N  N  0

(3.20)

A zero potential is assigned for the Li reference electrode and a constant current is applied to
the current collector at Si electrode side. As the electrons do not enter the electrolyte, all
boundaries and interfaces of the electrolyte are treated as electrically insulated. The interfacial
resistance is not considered at the Si-current collector interface at this time.
3.2 “Stress” Sub-Model
3.2.1

Deformation of Si

To model the large deformation of Si electrode, the finite deformation theory[31], also called
finite strain theory, is used. Similar approaches have been used for lithiation of Si in other
works[57,58,63,67,76,77,79]. The deformation of a-Si anode during battery cycling includes the
deformation due to mechanical constraint and those induced by Li concentration and temperature
variations[133,136,154]. At this stage, the heat generation and the thermal effect are not
considered and hence the thermal deformation is neglected.
The deformation gradient 𝐅 can be calculated as[77]
𝐅 = (𝐈 + ∇𝐮𝒔𝒐𝒍𝒊𝒅)
where 𝐈 is the identity matrix, 𝐮𝒔𝒐𝒍𝒊𝒅 is the displacement vector.

36

(3.21)

The deformation gradient is composed of an elastic deformation gradient 𝐅𝑒𝑙 and an
inelastic deformation gradient 𝐅𝑖𝑛𝑒𝑙 . In a Si anode, the inelastic deformation includes the plastic
deformation gradient 𝐅𝑝 and the Li concentration induced deformation gradient 𝐅𝑐 such
that[57,67]
𝐅 = 𝐅𝑒𝑙 𝐅𝑖𝑛𝑒𝑙 = 𝐅𝑒𝑙 𝐅𝑝 𝐅𝑐

(3.22)

Li insertion causes Si anode to increase its volume. The ratio of the current volume
𝑉 to the initial volume 𝑉0 is proportional to the change in Li concentration[79]. In a stress-free Si
network
𝑉
= det(𝐅𝑐 ) = 1 + Ω∆𝑐
𝑉0

(3.23)

where ∆𝑐 is the Li concentration increment calculated by using the initial volume 𝑉0.
To describe the large deformation, the stretch ratio 𝜆 instead of strain is often used. It is
defined as[55,59]
𝜆=

length in current state
𝑙
=
length in reference state 𝐿

(3.24)

For a-Si, the deformation is isotropic. The lithiation induced stretches 𝜆1 , 𝜆2 and 𝜆3 ,
therefore, are assumed to be equal[79]
isotropic

det(𝐅𝑐 ) = 1 + Ω∆𝑐 = 𝜆1 𝜆2 𝜆3  𝜆3

(3.25)

The lithiation induced deformation is determined through the deformation gradient as
3

𝐹𝑐,𝑖𝑗 = √1 + Ω∆𝑐 𝛿𝑖𝑗
where 𝛿𝑖𝑗 is the Kronecker delta.

37

(3.26)

3.2.2

Coupling between Li Transport inside Si and Volume Expansion

The volume expansion of a-Si during lithiation was computed in COMSOL FSI module
through the thermal-mass diffusion analogy. The thermal strain is calculated as
𝜀𝑖𝑗𝑇 = 𝛽∆𝑇𝛿𝑖𝑗

(3.27)

where ∆𝑇 is the temperature increment, which is equivalent to the effective concentration
increment ∆𝑐𝑒𝑓𝑓 ; 𝛽 is the thermal expansion coefficient. Comparing Eq. 3.27 to Eq. 3.26, the
mass expansion coefficient for the analogy is derived as

 c1  

3

1   c  1
c1

(3.28)

where 𝑐1 is the effective Li concentration calculated based on the current Li xSi volume 𝑉. It
should be noted that 𝑐1 differs from the Li concentration 𝑐 calculated with the unlithiated Si
volume 𝑉0, 𝑐1 /𝑐 = 𝑉0 /𝑉, as discussed in Ref. [155]. Therefore, SOC is determined by
SOC =

𝑐𝑉0
𝑐𝑡ℎ𝑒𝑜𝑟𝑦 𝑉0

=

𝑐1 𝑉
𝑐𝑡ℎ𝑒𝑜𝑟𝑦 𝑉0

(3.29)

where 𝑐𝑡ℎ𝑒𝑜𝑟𝑦 is the value corresponding to 𝑉0, which is 3.11×105 mol/m3[138]. The relationship
between 𝑐 and 𝑐1 is shown in Fig. 3.2.

38

Figure 3.2 The ratio of c 1 to c as functions of volume ratio V /V0 .

In this work, the Li diffusion equation was formulated in PDE module in the spatial
coordinate system. The large deformation of LixSi was considered in FSI module in the material
coordinate system. In simulations, the Li transport was bi-directionally coupled with LixSi large
deformation through the Thermal Expansion node within FSI module (i.e. through the time
derivative of the Li concentration that depends on the deformation). However, the mass
conservation was not automatically satisfied[156]. In simulations in spatial coordinates, the Li
flux applied at the Si boundary does not account for its dependency on the deformation. As a
result, a constant flux would result in an increase in the total number of Li entering the Si host if
the surface area increases. To account for this effect, a mass correction coefficient 𝑚𝑐 was
introduced at the boundary with Li source such that 𝑚𝑐 = 𝑉/𝑉0[155].
The mass conservation in the coupled analysis was examined using two electrodes, a sphere
(Fig. 3.3(a)) and a thin film on a substrate (Fig. 3.3(b)). The simulations were carried first using

mc  1 , i.e. without the correction. Fig. 3.3(c) plots the mass rate ratio between the mass

increasing rate of Li inside the Si calculated with

39





dM * d  cdV

to that entered the Si electrode
dt
dt

through the surface. The results showed that mc  V /V0 was valid for the two examined cases
with a variety of model representations. Simulations were then carried out for both thin film and
sphere geometries with correction mc  V /V0 with a Li diffusivity, D  5.0  10 16 m 2 /s . The
error in mass conservation was within 3%, Fig. 3.3(d).

(a)

(b)

(c)

(d)

Figure 3.3 Verification of the mass correction coefficient with (a) a free expanding Si sphere and
(b) a constrained thin film Si on a substrate, expanding only in the through thickness direction. (c)
The mass rate ratio as a function of volume ratio V/V0 . The results showed that the mass
correction coefficient follows the relation of mc  V/V0 . (d) The mass conservation was
maintained for both thin film and sphere geometries.

40

3.2.3

Stress Generation in Si

The Cauchy stress tensor 𝛔 is related to the elastic strains in Si by Eq. 3.30 to 3.32
𝛔 = 𝐽−1 𝐅𝐒𝐅 𝐓

(3.30)

where 𝐅 is the deformation gradient tensor, 𝐽 is the determinant of 𝐅, 𝐒 is the second PiolaKirchhoff stress tensor which can be derived from the relation
𝐒 = 𝐒𝟎 + 2

𝜕𝑊𝑠
𝜕𝐂

(3.31)

𝑇
where 𝐂 is the elastic right Cauchy-Green deformation tensor, 𝐂 = 𝐅𝑒𝑙
𝐅𝑒𝑙 . The Cauchy stresses

refer to the current configuration, whereas the second Piola-Kirchhoff stresses refer to the
reference configuration. 𝑊𝑠 is the strain energy density calculated based on the elastic Lagrange
strain tensor 𝐄 as
𝑊𝑠 = 𝛬tr(𝐄)2 + 𝛭tr(𝐄 𝟐 )

(3.32)

where 𝐄 = (𝐂 − 𝐈)/2 ; The 𝛬 and 𝛭 are the Lamé constants which are giving in terms of
Young’s modulus 𝐸 and Poisson’s ratio 𝜈, 𝛬 = 𝜈𝐸/((1 + 𝜈)(1 − 2𝜈)), 𝛭 = 𝐸/(2(1 + 𝜈)).
The non-homogeneous volume expansion in a solid results in stresses. The high stress may
lead to the yielding of the Si anode. The yield function 𝑓 is defined as
𝑓 = 𝜎𝑉𝑀 − 𝜎𝑌

(3.33)

where 𝜎𝑉𝑀 is the Von Mises stress; 𝜎𝑌 is the yield stress, which is a function of the Li
concentration. The post yielding response was assumed to be perfect plastic.
3.2.4

Fluid Electrolyte

The fluid flow is described by the Navier-Stokes equation

 v fluid
 v fluid  v fluid

t


 fluid 

41


  p   fluid  2 v fluid


(3.34)

where v fluid is the electrolyte fluid velocity;  fluid is the density of electrolyte; p is atmosphere
pressure at 1atm and  fluid is the electrolyte dynamic viscosity.
3.2.5

Current Collector

In literature, both Cu[157] and stainless steel[158] have been used as the current collector
for Si anode. The Cu and stainless steel are modeled by piece-wise linear elastic-plastic models.
Their stress-strain curves are given in Fig. 3.4.

(a)

(b)

Figure 3.4 The stress-strain curve for (a) Cu substrate[157], and (b) stainless steel substrates[158]

3.2.6

Boundary Conditions

The fluid/solid interaction at the Si-electrolyte interface is considered by a force f exerted on
Si boundary by the fluid, which is calculated as






2


T
f  n   pI    fluid v fluid  v fluid    fluid   v fluid I 
3




(3.35)

Both the Si electrode and the current collector can be viewed as a wall for the fluid
electrolyte. Therefore,

42

n  v fluid  0

(3.36)

For fluid phase, the upper and lower boundaries are treated as no-slip boundary condition, so
that
v fluid  0

(3.37)

For solid phase (Si electrode, current collector and Li reference electrode), the upper and
lower boundaries are treated as roller boundaries, so that

n  u solid  0

(3.38)

3.3 Modeling Parameters
3.3.1

Parameters in “Battery” Sub-Model

3.3.1.1 Electrode Properties
Besides the uncertainty in Li diffusivity as mentioned earlier, the electronic conductivity of
LixSi was also reported to vary with lithiation of Si. McDowell et al.[107] measured the
electronic conductivity of Li-Si alloy in amorphous Si thin film. The results from three different
tests were presented in Fig. 3.5. As shown, the conductivity ranges widely from 10 -3 to 102 S/cm.
On the other hand, the Chandrasekaran and Fuller modeled the 1D Li-ion cell with Si
composites[138] using the conductivity value (k1 in Eq. 3.10) of 33 S/m. Comparing the
changing range in experiments to the simulation data, we found that the 33 S/m could be used as
a reasonable electronic conductivity value in the MRM model. The influence of k1 to the cell
performance is also investigated and discussed in Appendix C.

43

Figure 3.5 Measurement of electronic conductivity in amorphous Si thin film held at various
potentials between 350 and 0 mV where the Li-Si alloy phase form [107].

3.3.1.2 Electrolyte Properties
The electrolyte was modeled as an incompressible fluid. In literature, the electrolyte density
and viscosity have been reported as functions of salt concentration, solvent composition, and
temperature, etc.[159,160]. The electrolyte density of LiPF6/EC/DMC system varies around
1100-1200 kg/m3 and the dynamic viscosity varies around 0.5-2 mPa∙s [159,160]. For
simplification, the density and viscosity of the fluid electrolyte were assumed to be those of
water, i.e.  fluid =1000 kg/m3 and  fluid =0.9 mPa∙s.
Table 3.1 presents a summary of the parameter values used in the current model. For Si, the
Li diffusivity in Si, the apparent transfer coefficient (Îąa and Îąc) in Butler-Volmer equation (Eq.
3.15) and the reaction rate constant k0 in Eq. 3.17 and have not been well established. The values
of these parameters are labeled as to be determined (TBD) and they will be determined in later
chapters.

44

Table 3.1 Parameters used in the MRM model.
Symbol
Îąa, Îąc

 fluid

Name/Description
Apparent transfer coefficient
Li diffusivity in LixSi
Diffusion coefficient of LiPF6 in organic solvent
(EC:DMC=2:1 vol.%) [131]
Exchange current density for the electrochemical
reaction at the lithium foil [138]
Cation transport number representing the
percentage of the current in the electrolyte carried
by Li-ion [138]
Electronic conductivity of LixSi [107,138]
Reaction rate constant
Theoretical maximum Li concentration [151]
Gas constant
Temperature
Partial molar volume [151]
Electrolyte density

 fluid

Dynamic viscosity of electrolyte

D1

D2
iLi
t+
k1
k0
𝑐𝑡ℎ𝑒𝑜𝑟𝑦
𝑅
𝑇
Ί

3.3.2

Value
TBD in Chapter 6
TBD in Chapter 4
1.27×10-11 (m2/s)
12.6 (A/m2)
0.38
Averaged at 33 (S/m)
TBD in Chapter 6
3.11×105 (mol/m3)
8.314 J/(K mol)
298 (K)
9×10-6 (m3/mol)
1000 (kg/m3)
0.9 (mPa s)

Mechanical Properties

The mechanical properties of lithiated Si, including the elastic modulus, the Poisson’s ratio
and the yield stress, were described with three piecewise linear relationships vs. SOC as shown
in Fig. 3.6. The elastic modulus was from the first principle calculation of Li xSi subjected to a
uniaxial tension reported by Zhao et al.[105]. The Poisson’s ratio was calculated by Shenoy et
al.[108]. For the yield stress, there was a significant discrepancy between the computed
values[65] and the experimental results[60,63]. Based on the range reported in literature, a
simple linear yielding function verses SOC was used in the current model based on experiments
by Sethuraman et al.[60].

45

(a)

(b)

(c)

Figure 3.6 Mechanical properties of LixSi used: (a) elastic modulus, (b) Poisson’s ratio, and (c)
yield stress.

3.3.3

Source of Parameter and Form of Si Used

As the interest of this work is focus on a-Si, when the parameter was searched in the
literature, only that obtained from a-Si is obtained. Or in some cases where the data for a-Si
phase was not available, the parameters from other form of Si, such as c-Si phase or the Si
composites were used. Parameters are adopted from both experimental measurement and
simulation predictions are adopted. Some may be evaluated from multiple sources. The source of
parameter, type of study and the form of Si used is listed in Table 3.2.

Table 3.2 Source of parameter used for Si
Parameter
Partial molar volume
m3/mol
OCP in Fig. 6.1(a)
Elastic modulus
Poisson’s ratio of LixSi
Estimated yield stress

Source
[151]
[15]
[118]
[105]
[108]
[60,63]
[138]

Si conductivity
[107]

Type of study
Experiment
Experiment
Experiment
Molecular dynamic Simulation
Molecular dynamic Simulation
Experiment
Continuum simulation
assumption
Experiment

46

Form of Si used
Si composite
a-Si thin film
a-Si thin film
a-Si
a-Si
a-Si thin film
Si composites, phase
not specified
Si nanowire, phase not
specified

Chapter 4

Study of Lithium Diffusivity in Amorphous Silicon via Finite Element Analysis

4.1 Introduction
This chapter studies the key material property for a-Si, the Li diffusivity in Si, which is a key
parameter determining the lithiation time and Li concentration gradient, the latter in turns
influences the intercalation stresses.
The reported Li diffusivity values, however, vary a great deal, ranging from 10 13 m 2 /s to
1019 m2 /s [63,85,94–99]. Furthermore, the experimentally measured diffusivity is based on the

initial Si volume. Such values cannot be used directly in analysis with large volume variations.
This chapter investigates the possible range of the Li diffusivity in a-Si through correlations
of numerical simulations with the in-situ lithiation experiments by McDowell et al.[42]. To
simulate the large deformation of Si, a large strain based formulation for the concentration
induced volume expansion is proposed. The problem is solved using COMSOL MultiphysicsÂŽ.
4.2 Model Development and Implementation
In literature, McDowell et al.[42] investigated the lithiation of a-Si nanospheres in-situ in a
transmission electron microscope (TEM). The experimental setup is shown in Fig. 4.1(a). Li was
supplied to a-Si spheres through crystalline silicon (c-Si) pillars. Potentiostatic charge and
discharge under 3V were carried out. The images and video clips showed that during lithiation
and delithiation, the a-Si spheres expanded and contracted radially and remained a spherical
shape. During the first lithiation, a boundary between Li-rich and Li-poor regions (named as
reaction front) was clearly visible, Fig. 4.1(b). This phenomenon is known as the two-phase
lithiation.
The experiment by McDowell et al.[42] provides a good study case. The spherical growth
pattern suggests that the constraint of the Si pillar to the spheres was negligible, the spheres

47

behaved as free-standing structures; and the sphere grew as having a uniform Li supply at its
surface. In addition, since the spheres were not in contact with the electrolyte, there were no
solid-electrolyte interface (SEI) layers on the particle surface.
The a-Si sphere was modeled as 2D axisymmetric. Fig. 4.1(c) presents the schematic of the
model along with the boundary condition. The potentiostatic charge was simulated by a constant
Li concentration boundary condition. The Li concentration corresponding to Li3.75Si was applied
during lithiation. To ensure the stability of the simulations, it was introduced with a ramp of 3s.
The sphere was allowed to expand freely during lithiation.

(a)

(b)

(c)

Figure 4.1 Axisymmetric model for modeling lithiation of a-Si nanoparticle. (a)
Experimental setup by McDowell et al.[30]; (b) two-phase lithiation in the first lithiation
observed in experiment [30]; and (c) simulating two-phase lithiation of a-Si nanospheres using
an axisymmetric 2D model.

In an MRM battery model[161], the Butler-Volmer equation would be used to consider the
electrochemical reaction at the electrode/electrolyte interface. In the experiment[30], however,
the a-Si spheres were not in direct contact with the electrolyte and they were lithiated with excess

48

Li supply. Therefore, a simple Li concentration boundary condition was sufficient to
approximate the experimental setup.
The physical phenomena involved inside the a-Si sphere are Li diffusion (studied by Eq. 3.1),
the diffusion induced elastic-plastic deformation and Li intercalation induced stresses (Section
3.1-3.3). Table. 4.1 summarizes the governing equations for this problem. The problem
formulations and their implementation in the model are presented next.

Table 4.1 Governing equations for Si model in Chapter 4 (Symbols are explained in
Nomenclature).
Physics
Li Diffusion

Governing Equations
𝜕𝑐1
+ ∇ ∙ (−𝐷∇𝑐1 ) = 0
𝜕𝑡

Eq. No.

Deformation

𝐅 = 𝐅𝑒𝑙 𝐅𝑖𝑛𝑒𝑙 = 𝐅𝑒𝑙 𝐅𝑝 𝐅𝑐
𝑉
= det(𝐅𝑐 ) = 1 + Ω∆𝑐
𝑉0

(3.22)

𝛔 = 𝐽 −1 𝐅𝐒𝐅 𝐓
𝜕𝑊𝑠
𝐒 = 𝐒𝟎 + 2
𝜕𝐂
𝑊𝑠 = 𝛬tr(𝐄)2 + 𝛭tr(𝐄 𝟐 )

(3.30)

Stress

(3.1)

(3.23)

(3.31)
(3.32)

4.3 Li Diffusivity in a-Si
The literature data on Li diffusivity in Si varies over a wide range[63,85,94–99]. Table 4.2
provides a summary of reported values for a-Si. For comparison, the representative values for Li
in c-Si are also listed in the first three rows. As shown, the Li diffusivity is about one to two
orders higher in a-Si than that in c-Si. Furthermore, the values estimated by ab initio molecular
dynamics (MD) simulations[85] and potential step chronoamperometry (PSCA)[95], where
chemical diffusion took place, appears to be much higher than the values determined by

49

electrochemical experiments such as the cyclic voltammetry (CV)[99], electrochemical
impedance spectroscopy (EIS)[98], galvanostatic intermittent titration technique (GITT)[22],
and potentiostatic intermittent titration technique (PITT) [63,97].
The large variation in diffusivity measurement is attributed to the complex physics involved
in lithiation of Si. The diffusivity measured with electrochemical experiments may include the
effect of interface reaction, stress-diffusion interaction, the contribution of SEI layer, and etc.
For instance, Bucci et al.[63] considered the stress-diffusion interaction and chemical interaction
of Li with Si host and determined an intrinsic diffusivity of 1019 m2 /s based on PITT experiment.
Li et al.[97] reported that their samples were cycled four times between 1.0 to 0.3V to form a
stable SEI layer before measurement. SEI layer is reported to act as a Li-ion diffusion barrier,
limiting Li-ion transport with a high resistivity[63,162]. Furthermore, the diffusivity determined
by experiments is an averaged value across the electrode over one or multiple cycles, and may be
a function of potential or state-of-charge (SOC)[94,98]. Results may be strongly dependent on Si
geometry and size[96]. Finally, the experimentally determined diffusivity is based on the initial
volume. Such values cannot be used directly in analysis considering volume changes.

50

Table 4.2 Literature data on Li diffusivity in Si.
Method
ab initio MD [85]
CV, EIS, GITT
[94]
PSCA [95]
ab initio MD [85]
PSCA [95]

Li Diffusivity in Si
(m2/s)
14
1.67 10 ~ 4.88 1013

Material

Current density

c-Si slab

chemical diffusion

11016

nano c-Si

0.2 mA/cm2

2 1015
1.25 1013 ~ 3.69 1012

single crystal wafer

chemical diffusion
chemical diffusion
chemical diffusion

PITT [97]

7.7 1018 ~ 1.0 1017

EIS [98]

3 1017 ~ 3 1016

a-Si slab
a-Si thin film
100nm and 1000nm
a-Si film
a-Si thin film

CV [99]

1.9 1017

120nm a-Si thin film

PITT [63]

11019

104nm a-Si thin film

110

13

0.07mA/cm2
0.4V, 0.0025 mA/cm2
0.1 mA/cm2
discharge, 0.395 to
0.39V, Intrinsic value

To estimate the order of Li diffusivity in Si with volume change, a numerical parameter study
was conducted for the experiment in Fig. 4.1. Both the two-phase lithiation observed in the first
lithiation and the single-phase lithiation observed in the subsequent cycles were investigated.
The two-phase phenomenon in the first lithiation is related to a lower Li diffusivity in
pristine Si. In general, the diffusivity in Li-poor phase is about an order lower than that in Lirich phase. In the current investigation, for the first lithiation, the diffusivity values were
assigned separately for the Li-poor core and the Li-rich shell with a rapid increase of Li
diffusivity at reaction front, as shown in Fig. 4.2. The exact composition of the metastable aLixSi at the reaction front was difficult to determine[163,164]. Based on literature
discussions[48,165,166], the composition was assumed as Li2Si in the current study. Within each
phase, the diffusivity was assigned as a constant and the Li composition dependency was not
considered. The diffusivity determined for Li-rich phase was then used in simulations of single-

51

phase lithiation. Considering the uncertainty in Li diffusivity, the stress effect to the diffusivity
was ignored in this work.

Figure 4.2 Constant diffusivity was assigned separately for Li-rich and Li-poor phases with a
small transition at around Li2Si.

Table 4.3 presents the initial six test cases for diffusivity study. The Li diffusivity in Li-poor
and Li-rich phases were denoted as D poor and Drich , respectively. The tests were divided into two
sets. The first set (Tests 1 to 3) was used to investigate Drich , where D poor was kept constant
while Drich varied from 1.0  1015 m2 /s to 1.0  1014 m2 /s . The second set (Tests 3 to 5) was used to
investigate the D poor , where Drich was kept constant while D poor varied from 1.0  1018 m2 /s to
1.0 1016 m 2 /s . Test 6 was added based on the results of Tests 1-5. Results will be discussed in

later session.

52

Table 4.3 Diffusivity study test cases.
Test label

Li-poor Diffusivity D poor (m2/s)

Li-rich Diffusivity Drich (m2/s)

Test 1
Test 2
Test 3
Test 4
Test 5
Test 6

1.0 1017
1.0 1017
1.0 1017
1.0 1018
1.0 1016
2.0  1017

1.0 1015
1.0  1014
2.5  1015
2.5  1015
2.5  1015
2.0  1015

4.4 Results and Discussion
4.4.1

Two-Phase Lithiation

TEM images from McDowell et al.[42] showed the lithiation process of an a-Si sphere with
diameter about 570 nm at several time frames, as presented in the first row in Table 4.4. An a-Si
sphere of the same size was simulated according to the six tests in Table 4.3. The simulation
results were compiled in Table 4.4, which compares the Li concentration distributions and the
reaction front in the sphere with the experimental result. The scaled bar was normalized to the
maximum achieved c 1 (81884 mol/m3).
In the first set simulations (Tests 1 to 3), the results showed that when Drich  1.0  10 15 m 2 /s ,
the two-phase lithiation was still in progress at 129s, which was much longer than the time for
Li-poor phase to disappear observed in the experiment. On the other hand, when
D rich  1.0  10 14 m 2 /s , the Li-poor phase disappeared at about 60s, which was much shorter than

the time in the experiment. Test 3, with Drich  2.5  10 15 m 2 /s , matched the experiment reasonably
well.

53

Table 4.4 Comparison of simulations with experimental results of two-phase lithiation.
Test label

48s

61s

129s

Comments

Experiment –
first lithiation of
a-Si[42]
Scale

bar

for

Li

concentration in simulatons.
Drich  1.0  10 15 m 2 /s

,

the

time for two-phase lithiation

Test 1

was much longer than that
in the experiment.
Drich  1.0  10 14 m 2 /s , the Li-

poor
Test 2

phase

disappeared

before 60s, which was much
shorter than the time in the
experiment.

Drich  2.5  10 15 m 2 /s

Test 3

matched

the

,

experiment

reasonably well.

Dpoor  1.0 1018 m2/s
Test 4

lithiation
greatly.

`

54

time

,

increased

Table 4.4 (cont’d)

Dpoor  1.0 1016 m2/s resulted

Test 5

in a diffused reaction front.

Best

Test 6

correlation

with

experimental results.

The results in the second set of simulations (Tests 3 to 5) showed that D poor , the diffusivity
in Li-poor phase, had a significant influence on the lithiation time. When D poor was small (Test
4), the lithiation time greatly increased. When D poor was large (Test 5), the lithiation time
reduced but the reaction front was diffused.
The effect of D poor was further examined by comparing the concentration profiles obtained
from simulation (Fig. 4.3(b)) with the experimental results (Fig. 4.3(a)). The experimental
profiles were established according to the light intensity in TEM images[42]. It should be noted
that the light intensity may not have a linear relationship with the Li concentration. The profiles
generated this way, therefore, were only qualitative indications. As shown in Fig. 4.3(b), Test 3
and 4 resulted in Li concentration oscillations in the Li-poor phase near the reaction front. This
was caused by the relatively steep diffusivity gradient at the reaction front. The smaller D poor in
Test 4 caused a larger scale of concentration oscillation. Diffusivity values in both tests were

55

rejected because they did not match well with the experimental profile in Fig. 4.3(a). Test 5
resulted in a much diffused reaction. Therefore, a new test case, Test 6, was added. The
concentration profiles in Test 6 had a distinguishable reaction front at 48s and 61s, resembling
those depicted in experiments. Among all test cases, Test 6 appeared to provide the best
correlation between simulation and experiment. This yielded the diffusivity values of

Dpoor  2.0 1017 m2/s and Drich  2.0  10 15 m 2 /s .

(a)

(b)

Figure 4.3 Concentration profile comparison: (a) The image intensity during lithiation of the
570 nm diameter sphere during lithiation in experiments[42]. Red circles label the concentration
Figure 4.3 (cont’d) drop at the reaction front. (b) Lithiation of Si in Tests 3 to 6 in simulations:
concentration profile vs. distance to particle center.

Comparing with the diffusivity values in Table 4.4, the value of Drich  2.0  10 15 m 2 /s is one to
two orders higher than that measured by electrochemical experiments[97–99] but two orders
lower than the value determined by chemical diffusion[85,95]. The electrochemical experiments

56

include the effect of SEI layer, and the Li diffusivity is determined with the dimension of
undeformed Si electrode. Taking these factors into consideration, Drich  2.0  10 15 m 2 /s is in a
reasonable range.
McDowell et al.[42] also plotted the thickness of the Li-rich phase (lithiated-thickness)
versus time, as showed in Fig. 4.4(a). The results showed the thickness increased linearly with
time. However, the slopes of these curves varied over a wide range, which may be caused by
differences in experimental conditions, such as the particle sizes and quality of electrical/Li +
pathway connection. The thickness plot was used to examine the Li diffusivity values established
in Test 6. The specific information on the range of sphere sizes was not provided in Ref. [30]. To
compare the results, simulations were carried out for spheres with diameter of 570 nm and 400
nm, because these two sizes were mentioned in Ref. [30]. As shown in Fig. 4.4(a), the lithiated
thickness growth curves of both spheres obtained from simulations fell within the experimental
range. The lithiated thickness increased almost linearly during the two-phase lithiation, as seen in
the experiment. These results further confirmed that the Li diffusivity values in Test 6 were
reasonable estimations. Finally, in simulations, the slope of the curves reduced to nearly zero
when the Li-poor phase was consumed. Further lithiation proceeded in Li-rich phase as the
sphere continued to grow.
4.4.2

Single-Phase Lithiation

As reported in Ref. [42], Si undergoes two-phase lithiation only in the first lithiation. The
subsequent delithiation and lithiation display as a single-phase process without a visible reaction
front. Furthermore, the time for lithiation is greatly reduced. For a-Si sphere of 400 nm, the
lithiation time was 122 seconds and 31 seconds for the first and second lithiation, respectively.
The disappearance of the two-phase lithiation is likely due to the further amorphization of a-Si.

57

a-Si may have some short range order in local areas. Li insertion disturbed the ordered structure
as in the amorphization of c-Si[165]. Diffusion proceeds fast in amorphous host. Therefore the
subsequent lithiation is similar to that in Li-rich phase.
The single-phase lithiation was simulated for a-Si sphere of 400 nm with D  2.0 10 15 m 2 /s ,
the value for the Li-rich phase in two-phase lithiation. In single-phase lithiation, the reaction
front was assumed at Li2Si, the same as in two-phase lithiation. The location of the reaction front
was determined from concentration profile plots at different time intervals. Fig. 4.4(b) showed
that the single-phase lithiation of the sphere took about 30 seconds while the two-phase lithiation
took more than 70 seconds. The simulation results agreed with the experimental observations
quite well, confirming that D  2.0 10 15 m 2 /s is a reasonable estimation for a-Si single-phase
lithiation.

(a)

(b)

Figure 4.4 Lithiated thickness as a function of time during lithiation of a-Si spheres, including: (a)
Two-phase lithiation comparison between results from experiments and simulations.
Experimental curves (black) #1-#4 are from McDowell et al.[42] out of 26 nanospheres with
unspecified diameter range. Simulation curves (red) are a-Si spheres of 400 nm and 570 nm. (b)
Comparing lithiation speed between single-phase lithiation and two-phase lithiation.
58

4.4.3

Concentration Dependency of Diffusivity

Li diffusivity may be a function of Li concentration. Metastable a-LixSi phases form during
lithiation before a-Si becomes fully lithiated. Therefore, the Li diffusivity may vary when LixSi
composition and the network structural change. The concentration dependency of the diffusivity,
however, is unclear.
To investigate the effect of the concentration dependency of the diffusivity, additional
simulations were performed with a relationship proposed by Berla et al.[167]

D  1  c / cmax  D poor  c / cmax  Drich
3

3

(4.1)

Two new tests were performed using a diffusivity described by Eq. 4.1. In Test 7, the values
used in Ref. [51], Dpoor  1.0 1017 m2/s and Drich  1.0  10 16 m 2 /s , were adopted. In Test 8, the
values established in the current work, Dpoor  2.0 1017 m2/s and Drich  2.0  10 15 m 2 /s , were used.
As shown in Fig. 4.5(a), Eq. 4.1 described a smooth curve instead of a sharp change. As a result,
the reaction front between Li-poor and Li-rich was not maintained in Test 7 and 8. As shown in
Fig. 4.5(b), the lithiated thickness growth curves predicted by Test 7 and 8 deviated from the
nearly linear growth tend observed in experiments. In comparison, the simulation with a simple
approximation of constant diffusivity values in Test 6 matched the experimental trend closer.

59

(a)

(b)

Figure 4.5 Simulations of lithiation of Si sphere of 570 nm diameter with a concentration
dependent diffusivity. (a) Diffusivity functions; (b) Lithiated-thickness versus time, comparing to
experimental data reported in Ref. [30] as shown in Fig. 4.4(a).

4.5 Conclusion
The Li diffusivity in the two-phase lithiation and the subsequent single-phase lithiation of aSi was investigated by simulating the in-situ TEM experiments of a-Si nanospheres[42]. To
simulate the large deformation of Si, a large strain based formulation for the concentration
induced volume expansion was used. The coupled diffusion and deformation problem was
solved in COMSOL MultiphysicsÂŽ. The diffusivity of Li in a-LixSi was approximated with two
distinct diffusivity values for the Li-poor phase and Li-rich phase, respectively. Simulations with
Li diffusivity values ranging from 1.0 10 18 m 2 /s to 1.0 10 14 m 2 /s were carried out and the
results were compared with experiments for the movement of the reaction front, the lithiatedthickness growth, and the lithiation time. The Li diffusivity was found at the order of
2.0  1015 m2 /s for Li-rich phase and 2.0  1017 m 2 /s for Li-poor phase in the two-phase lithiation.

The single-phase lithiation of a-Si after the first cycle was simulated with a Li diffusivity of
2.0  1015 m2 /s . The computed lithiation time agreed well with the experimental results.

60

The Li diffusivity for single-phase lithiation determined in this work is two orders lower than
that determined by chemical diffusion process but one to two orders higher than that established
in electrochemical experiments. The value reported here is in a reasonable range taking into
account that the Li diffusivity is calculated for deformed Li xSi volume and the analysis is for a
case without the limiting effect on Li-ion transport from SEI layer.

61

Chapter 5

Investigation of the Chemo-Mechanical Coupling in Lithiation of Amorphous Si
through Simulations of Si Thin Films and Si Nanospheres

5.1 Introduction
Si anode experiences large volume expansion/contraction (up to 400 %) of Si during
lithiation cycles. The large volume variation of Si during lithiation/delithiation cycles can induce
high stresses. This presents a great challenge to the structural integrity of LIBs. The high stresses
and large volume variation have been cited as a primary cause for rapid capacity fading of LIBs
with Si anode. Therefore, understanding the lithiation process, stress evolution and its effect on
Si lithiation is of great importance for Si electrode design.
The stress evolution has been measured in situ during lithiation/delithiation for amorphous Si
(a-Si) thin films[60,63,65,168]. The measured stress can reach as high as 1.5 GPa in both
compression and tension. The stress also influences the lithiation process[61,62]. It was reported
that, after the crystalline Si (c-Si) nanosphere had been lithiated for some time, the growth rate of
lithiated thickness (thickness of LixSi shell in the first lithiation) reduced significantly[61]. The
phenomenon was primarily attributed to the high compressive stress at the reaction front that
eventually prohibited Li from further insertion. However, the mechanism of how the stress
interacts with Si lithiation is not clear.
In literature, two categories of chemical-mechanical coupling theories have been proposed
for the stress effect on Li diffusion into Si: (I) by influencing Li diffusivity, and (II) by
influencing the chemical potential of Li in Si.
In category (I), it assumes that the Li diffusivity is modulated by the stress induced activation
free energy barrier shift[69,74,75]. The rate of reaction depends on the activation energy (i.e.
free energy between the ground state and transition state). So far, the relationship is qualitative

62

only. The activation energy is adjusted by either hydrostatic stress[69,169] or biaxial stress (in
thin film) [170] with a coefficient which varies in a wide range.
In category (II), it postulates that the chemical potential of Li in Si changes because the free
energy difference between Si and LixSi is altered by stress[63,64,76–79]. This effect is still quite
argumentative given different treatments in literature[63,74,77,82,83,149,171,172]. For example,
some proposed that only the hydrostatic stress influenced the chemical potential for ideal
solution[83,124] or non-ideal solution[74] whereas others cast the chemical potential as a
function of both hydrostatic stress and deviatoric stress components[63,78].
The chemical-mechanical coupling has two facets: 1) the diffusion induced deformation and
stress; 2) the stress effect on diffusion process. Because the co-existence of these two facets for
any given geometry, it is difficult to determine the coupling relationship with a single geometry.
In this work, two geometries will be investigated: a-Si thin film and a-Si nanosphere. In
experiments with a-Si thin film, Li et al.[173] has observed a strong asymmetric rate behavior
such that the lithiation capacity reduced much faster than that of delithiation with increasing the
charging/discharging rates, i.e. the C rates. So far, the asymmetric rate behavior has only been
reported for a-Si anodes in the form of thin films, but not for other geometries such as anodes
with nanoparticles[174]. Since the stress state developed in a substrate-constrained thin film
would be very different from that of a weakly constrained sphere during lithiation/delithiation,
these two cases provide a unique opportunity for the investigation of chemical-mechanical
coupling. A model with a correct coupling relationship would predict the rate behaviors of both
geometries.
It should be noted that, although the abovementioned chemical-mechanical coupling theories
have been formulated using both the small strain[63,171,172] and large strain[77,124]

63

frameworks, the Li transport is commonly considered with the normalized Li concentration
calculated using the dimension of unlithiated Si. As a result, the analytical/simulation results
could not be compared directly to the dimension/shape changes observed in experiments. In our
previous work, a model that considers the lithiation/delithiation of Si with its accompanied
volume change has been developed and used to determine the Li diffusivity in a-Si[115]. The
current work extends the model to include chemo-mechanical coupling by considering the stress
effect on chemical potential[142].
5.2 Model Setup
5.3 FE Models and Simulation Conditions
The physical phenomena involved inside a-Si electrodes are the diffusion of Li, diffusion
induced deformations and stresses, and the coupling among them. Table 5.1 summarized the
governing equations. This section presents the problem formulations and their implementation in
the model.

Table 5.1 Governing equations for Si model in Chapter 5 (Symbols are given in Nomenclature).
Physics
Li Diffusion

Deformation

Stress

Governing Equations
𝜕𝑐1
𝐷1
(𝛔)) = 0
+ ∇ ∙ (−
𝑐 ∇𝜇
𝜕𝑡
𝑅𝑇 1 𝐿𝑖𝑥𝑆𝑖
0
𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) = 𝜇𝐿𝑖
+ 𝑅𝑇 ln 𝜒𝐿𝑖 + ∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔)
𝑥 𝑆𝑖
∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) = 𝛼Ω𝜎ℎ

Eq. No.

𝐅 = 𝐅𝑒𝑙 𝐅𝑖𝑛𝑒𝑙 = 𝐅𝑒𝑙 𝐅𝑝 𝐅𝑐
𝑉
= det(𝐅𝑐 ) = 1 + Ω∆𝑐
𝑉0

(3.22)

𝛔 = 𝐽 −1 𝐅𝐒𝐅 𝐓
𝜕𝑊𝑠
𝐒 = 𝐒𝟎 + 2
𝜕𝐂
𝑊𝑠 = 𝛬tr(𝐄)2 + 𝛭tr(𝐄 𝟐 )

(3.30)

64

(3.1)
(3.4)
(3.5)

(3.23)

(3.31)
(3.32)

Two experiments with distinct geometries were selected to study the stress effect on Si
lithiation: the first was the cycling experiment of a-Si thin film conducted by Li et al.[173]; the
second was that of Si nanospheres by Liu et al.[174].
These two experiments were carried out galvanostatically but with two different conditions.
In Ref. [173], the thin film was evaluated independently for single-phase lithiation and
delithiation, i.e., a slow lithiation/delithiation at C/10 was performed to reach the same SOC
before each delithiation/lithiation measurement at the desired rate. In Ref. [174], the anode with
nanospheres was lithiated and delithiated under the same rate for at least ten cycles before
moving to the next rate. These are simulated as Condition 1 and Condition 2 as shown in Table
5.2. Simulations for the two geometries were performed with both conditions.

Table 5.2 Two conditions used in simulations.
Condition 1
Condition 2

5.3.1

A slow lithiation/delithiation at C/10 is conducted before each
delithiation/ lithiation measurement at the desired rate
The lithiation/delithiation measurement is conducted following a
delithiation/ lithiation at the same rate

Amorphous Si Thin Film Model

Li et al.[173] galvanostatically cycled the a-Si thin film of 70 nm thickness in a Li-ion cell
over a wide range of rates between 0.05 V and 2.0 V. The rate performance was evaluated
following Condition 1. The cells were cycled a few times in advance to minimize the effect of
solid-electrolyte interface (SEI)[173]. Their results are reproduced in Fig. 5.1, which shows that
the reduction of capacity with C rate was much faster in lithiation than in delithiation. This was
referred as the asymmetric rate behavior. It was attributed to the voltage shift introduced by the
electrode Ohmic resistance at higher C rates, as the capacity is highly dependent on the voltage

65

at the end of lithiation but not so much at the end of delithiation. Although the authors suspected
that the stress may be a contributing factor, it effect was not investigated in Ref. [173].

Figure 5.1 Asymmetric rate performance of 70nm a-Si thin film observed in experiment[173].

To simulate this experiment, a two-dimensional (2D) plane strain FE model was built, as
shown schematically in Fig. 5.2. Since the experimental setup was not elaborated in detail in Ref.
[173], the boundary conditions for the model were defined based on other Si thin film cycling
experiments[60,175]. The Si thin film is usually deposited on a rigid substrate which restricts the
in-plane expansion of the thin film. This boundary condition is often referred as substrateconstrained in literature. In the FE model, this boundary condition was simulated by constraining
all degrees of freedom of the nodes at the bottom of the film. The thickness of the film was 70nm,
the same as in Ref. [173]. In experiments, the width of the thin film is usually in the range of
micrometers, which results in a large aspect ratio (width/thickness) of several hundred. In the FE
model, the large aspect ratio was simulated with a film with a width/thickness ratio of 10
constrained by a symmetric (roller) boundary condition at the left and right hand sides. The
upper surface of the film was allowed to expand freely. The Li flux was applied to the upper

66

surface according to the C rate used in Ref. [173]. The thin film was meshed with 5400
quadrilateral elements with an average size about 3 nm. The mesh is shown in Fig. 5.2.

Figure 5.2 Schematic illustration of the 2D model for a-Si thin film with boundary conditions
(BCs), and the FE mesh. The schematic is not drawn to scale.

The model for the thin film was first verified with the experiment of Sethuraman et al.[60],
which measured the curvature change of substrate-constrained Si thin film during lithiation and
delithiation. The in-situ stress variation was then determined using the Stoney equation[176,177].
The results were reported in terms of biaxial stress 𝜎𝑏 , which equals to the averaged in-plane
𝑎𝑣𝑒
𝑎𝑣𝑒
stress in isotropic LixSi such that 𝜎𝑏 = 𝜎𝑥𝑥
= 𝜎𝑦𝑦
. Fig. 5.3(a) compares the model prediction

with the experimental result for biaxial stress variation during cycling of a film of 250 nm
thickness under C/4 rate. The chemo-mechanical coupling effect was not considered in the model
because of the low rate. As seen, the film undergoes compression (up to -1.5 GPa) during
lithiation, and reverses to tension (up to 1.5 GPa) upon delithiation. The predicted stress was in
good agreement with the experiment both in the trend and the range. The stress profiles in thin
film at various SOC under lithiation were showed in Fig. 5.3(b). This showed that, as the cycling
rate was small, the Li concentration variation through the thickness was small and the stress
distribution was almost uniform in the film.
67

(a)

(b)

Figure 5.3 Biaxial stress in thin film during cycling of a-Si thin film of 250 nm thickness under
C/4 rate were plotted, (a) the stress history vs. SOC was compared with the experimental result
was from Sethuraman et al.[60]. In simulations, the lithiation started with a residual stress from
delithiation at the same rate; (b) the stress profiles inside the thin film showed that the stress
distribution was almost uniform in the film during lithiation.

5.3.2

Si Nanosphere Model

Besides a-Si thin films, the asymmetric rate behavior has not been reported in Si anodes of
other geometries, like Si nanosphere[174], hollowed sphere[36], and nanotube[178].
The experiment with Si nanospheres by Liu et al.[174] was investigated here. In Ref. [174],
the spheres were encapsulated inside graphite shells. The graphite shells were then bonded with
alginate binder to form the anode, as shown in Fig. 5.4(a). Since Si was not in direct contact with
an electrolyte, there was no SEI formation on Si surface. The cell with this anode was
galvanostatically lithiated and delithiated under the same rate ranging from C/10 to 4C, i.e.
following Condition 2. The measured lithiation and delithiation capacities were almost the same,

68

i.e. symmetric (Fig. 5.4(b)), although the cycling capacity decreased about 80% from C/10 to 4C
due to the poor ionic conductivity of the alginate binder[174].

(a)

(b)

(c)

Figure 5.4 (a) The anode with Si nanospheres encapsulated inside graphite shells[174] and (b) its
capacity measured under various cycling rates, reproduced from Fig. S8 in Ref. [174]. (c) The
FE model and its boundary conditions used in simulations.

In Ref. [174], the average diameter of the spheres was around 100 nm. The void between Si
sphere and graphite shell allowed enough room for Si expansion. The in-situ transmission
electron microscopy (TEM) characterization[174] suggested that the constraint of the graphite
shell to the spheres was negligible. The spheres behaved as freestanding structures and grew as
having a uniform Li supply at their surface.
Liu’s experiment can be simplified as a free-standing single Si nanosphere. The 100 nm
sphere was simulated using a 3D model. The sphere was meshed with nearly 20000 elements (for
half sphere), with a mix of both hexahedral and tetrahedral elements. The average element size
was about 3nm. Fig. 5.4(c) presents a 2D cross section of the 3D model with the boundary
conditions and local meshes. A constant current for the corresponding rate was applied to the
69

particle surface. The sphere was allowed to freely expand upon lithiation/delithiation. In Ref.
[174], the initial nanosphere was c-Si but the measurement was performed over a wide range of
rates from 0.1C to 4C with more than 10 cycles at each rate. Since c-Si will turn into amorphous
phase after the first few cycles[42], the responses in Fig. 5.4(b) can be considered as that of a-Si.
Therefore, the Si sphere was modeled as the same material as for the a-Si thin film.
5.4 Results and Discussion
5.4.1

The Rate Behaviors

The simulations for Si thin film and nanosphere were carried out for charging and
discharging at C rate between C/10 to 100C under the two conditions. The chemo-mechanical
coupling was investigated by varying the coefficient in Eq. 3.5: =0 for the case of without the
stress effect; and =0.01 for with the stress effect. To mimic the cut-off voltage in galvanostatic
cycling experiment, the lithiation was terminated when the SOC at the geometry surface reached
to about 0.83 in lithiation and 0 in delithiation.
To compare with the experimental trends, the predicted specific capacities were plotted vs. C
rate for the thin film and nanosphere in Fig. 5.5. In these plots, the solid lines are the results with

=0, and the dashed lines are that with =0.01. The lithiation and delithiation are represented by
triangle and round symbols, respectively. Table 5.3 provides a summary of the remaining
capacities at 100C under various conditions in respect to the capacity at 0.1C. In this
investigation, 100% capacity is equivalent to SOC=0.83.

70

(a)

(b)

(c)

(d)

Figure 5.5 The predicted specific capacities vs. C rate for both 70nm a-Si thin film and a single
a-Si nanosphere with (=0.01) and without (=0) the stress effect (a) thin film under Condition 1;
(b) thin film under Condition 2; (c) nanosphere under Condition 1 and (d) nanosphere under
Condition 2.

71

Table 5.3 The specific lithiation/delithiation capacity at 100C in respect of the capacity at 0.1C
for the thin film and nanosphere under Conditions 1 and 2.


Thin film
Sphere

0
0.01
0
0.01

100C Lithiation
Condition 1
Condition 2
76.4%
74.6%
49.5%
47.8%
99.7%
99.1%
90.7%
90.2%

100C Delithiation
Condition 1
Condition 2
96.1%
74.6%
94.4%
47.3%
99.7%
99.1%
99.0%
89.9%

Fig. 5.5 reveals that the asymmetric rate behavior occurs only under Condition 1. For the thin
film, the asymmetric rate behavior was pronounced. Over the range of 0.1C to 100C, when =0,
the decrease in capacity was about 23.6% for lithiation and 3.9% for delithiation. With =0.01,
the decrease was 50.5% for lithiation and 5.6% for delithiation. Under Condition 2, both the
lithiation and delithiation capacities reduced at a similar pace which resulted in a symmetric rate
behavior.
The Si nanosphere displayed a symmetric rate behavior under Condition 1 and under
Condition 2 when =0. However, it showed a small asymmetric rate behavior under Condition 1
with =0.01. The capacity decreased about 9.3% for lithiation and 1.0% for delithiation.
For Si nanosphere, the predicted decrease in capacity is not as dramatic as that in the
experiment as shown in Fig. 5.4(b). This is due to the fact that only one Si nanosphere was
simulated and the effect of the binder was not included.
The simulations with chemical-mechanical coupling correctly captured the trends of the rate
behavior of these two geometries. To understand why the two geometries displayed distinctly
different rate behaviors, the simulation results were analyzed further. Three aspects were
investigated: the electrode geometry, the chemo-mechanical coupling and the effect of the prior
process.
72

5.4.2

Electrode Geometry

Under Condition 1, with =0, the asymmetric rate behavior was observed in the thin film but
not in the nanosphere. This result is not related to the chemo-mechanical coupling. A close
inspection indicates that the surface limitation associated with the electrode geometry and its
boundary constraint is a likely cause, as explained below and Appendix A.
Let us compare a substrate-constrained thin film with a free-standing sphere. In lithiation, the
surface area would remain the same for the thin film but increase to (𝑟/𝑟0 )2 times for the sphere
(𝑟 and 𝑟0 correspond to the radius of the deformed sphere and the original sphere). Under
galvanostatic charging, the rate of Li insertion into the electrode remains constant. Furthermore,
a sphere has the highest surface/volume ratio among all geometries. As a result, the Li
concentration at the surface layer of the nanosphere is much lower than that in the thin film. The
lithiation terminates when Li reaching the maximum concentration at the surface layer. This will
happen at a much lower charging rate in a thin film than that in a nanosphere if the thickness and
the radius of the two are comparable.
5.4.3

Chemo-Mechanical Coupling

Under Condition 1, with =0.01, the asymmetric rate behavior was much pronounced in the
thin film. A small asymmetry was also observed in nanosphere. The results indicate that the
chemo-mechanical coupling played a role in the rate behavior.
To understand this process, the profiles of Li concentration (𝑐1 ), hydrostatic stress and the Li
flux obtained with =0 and =0.01 are compared. Fig. 5.6 and 5.7 present the results at 100C
during lithiation/delithiation for the thin film and nanosphere, respectively. For the thin film, the
profiles were plotted against the instantaneous thickness measured from the substrate to the top
surface of the film at selected time intervals. For the sphere, they were plotted against the

73

instantaneous radial distance from the center to the surface. The solid lines are for =0 and the
dash lines for =0.01. The Li flux was calculated according to Fick’s first law as 𝐍 =
−𝐷1 𝑐1 ∇𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔)/𝑅𝑇.
5.4.3.1 Thin film
During lithiation, the hydrostatic stress in the thin film changed quickly from a tensile
residual stress resulted from the prior slow delithiation to a large compressive stress, as in Fig.
5.6(c). The Li concentration profiles at different time had similar shapes and the curves moved
up steadily with time (Fig. 5.6(a)). The compressive stress suppressed the Li diffusion. With the
stress effect, the driving force for Li diffusion and the flux reduced, the Li concentration gradient
increased. With the progress of lithiation, the difference between =0 and =0.01 increased. The
process terminated when the maximum SOC was reached at the surface of the film. The
predicted termination time was 27.5s and 17.8s, corresponding to a capacity retention of 76.4%
and 49.5% for =0 and =0.01, respectively.
In delithiation, the hydrostatic stress changed from compression to tension, in Fig. 5.6(d).
Unlike in lithiation where the stress transition took less than 2s, the transition in delithiation took
about 10s and 15s to propagate through the thickness of the film for =0 and  =0.01,
respectively. With  =0.01, the stress profiles had a steeper slope, and the Li flux had a steep
step over the transition region. A stress profile that was tensile at the surface and compressive at
interior resulted in a large Li concentration gradient in the film. When the stress in the film fully
converted to tensile, the Li concentration gradient reduced, as seen in Fig. 5.6(b). Towards the
end of delithiation, the Li concentration profiles for the two cases were very close. The process
terminated at 34.6s and 34.0s, corresponding to a capacity retention of 96.1% and 94.4% for =0

74

and  =0.01, respectively. The results show that although the profiles differed significantly, the
chemo-mechanical coupling had a relatively small effect on the overall delithiation capacity.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.6 Profiles of Li concentration, hydrostatic stress and Li flux vs. the distance to the
bottom of the thin film at 100C under Condition 1. Solid line for without ( =0) and dash line for

75

Figure 5.6 (cont’d): with the stress effect (=0.01). (a) and (b) plot the Li concentration; (c) and
(d) plot the hydrostatic stress; (e) and (f) plot the Li flux 𝑵𝒛 . (a)(c)(e) are under 100C lithiation
and (b)(d)(f) are under 100C delithiation.

5.4.3.2 Nanosphere
The Li concentration, hydrostatic stress and Li flux profiles in the nanosphere were quite
different from those in the thin film. The tensile and compressive hydrostatic stress always coexisted in the sphere. During lithiation, it was compression at the surface and gradually changed
to tensile at the core (Fig. 5.7(c)). It was vice versa for delithiation (Fig. 5.7(d)). The stress
profile in lithiation suppresses the Li transport at the surface layer but enhances it in the core area.
It is vice versa in delithiation. As a result, the Li distribution was much more uniform across the
sphere as compared to that in thin film, as in Fig. 5.7(a) and (b). Furthermore, the Li flux at the
surface of the sphere was not a constant. It reduced with lithiation and increased with delithiation
as the surface area varied, as discussed in section 4.2 and presented in Fig. 5.7(e) and (f).
Comparing the cases of =0 and =0.01, the stress and Li flux profiles differed but the Li
concentration profiles were almost identical with the exception towards the end of lithiation and
at the beginning of delithiation where the profiles for =0.01 were steeper. As shown in the case
of thin film, a steeper Li distribution towards the end of a process reduced the capacity. The
lithiation capacity was affected. The value was 99.7% and 90.7% for =0 and =0.01,
respectively. The delithiation capacity was almost unaffected. The values were 99.7% and 99%,
respectively.

76

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5.7 Profiles of Li concentration, hydrostatic stress and Li flux vs. the radial distance for
the sphere at 100C under Condition 1. Solid line for without ( =0) and dash line for with the

77

Figure 5.7 (cont’d): stress effect (=0.01). (a) and (b) plot the Li concentration; (c) and (d) plot
the hydrostatic stress; (e) and (f) plot the Li flux 𝑁𝑧 . (a)(c)(e) are under 100C lithiation and
(b)(d)(f) are under 100C delithiation.

The above results show that the stress profiles in the two geometries play an important role in
their rate behavior. During lithiation, a compressive hydrostatic stress exists throughout the thin
film. In a sphere, it is only in the surface layer. Although Li transport was suppressed at the
surface for both cases, the stress profile in the sphere promoted a more uniform Li distribution.
The stress profile together with the electrode geometry discussed in section 5.4.2 provided a
better lithiation capacity for a Si nanosphere than that for a Si thin film.
During delithiation, the stress profiles were vice versa to that of lithiation. A tensile
hydrostatic stress promotes the Li transport. As shown in Fig. 5.6 and 5.7, under Condition 1, the
chemo-mechanical coupling had a very small effect on the delithiation capacity of the two
geometries. Again, the stress profile in the sphere favored a more uniform Li distribution. The
condition that resulted in a larger nonuniform Li distribution towards the end of the process
reduced the overall capacity.
5.4.4

Effect of Prior Process

The asymmetric rate behavior occurred only under Condition 1. The results indicate that the
prior process has an effect on the capacity of the electrode in the next step. It was suspected that
the residual stress from the prior process might have played a role. To investigate this effect, the
Li concentration and the hydrostatic stress profiles for the thin film at 100C with =0.01 under
the two conditions were compared in Fig.5.8.

78

As seen from the profiles at t=0s, Condition 1, a slow process, resulted in a nearly uniform Li
distribution, whereas Condition 2, a fast process, resulted in a nonuniform Li distribution through
the thickness of the thin film.
For lithiation, the stress profiles at t=0s by the two conditions were very close, both were
tensile and had a uniform distribution. During lithiation, although the stress gradient for
Condition 2 was higher initially, the stress profiles of the two conditions became
indistinguishable after 5s. As the lithiation progressed, the Li profiles of the two conditions
became closer. As discussed above, the Li profile toward the end of a process appears to
determine the lithiation capacity. The retaining capacity was 49.5% and 47.8% for Conditions 1
and 2, respectively.
For delithiation, the stress profiles of the two conditions differed significantly. At t=0s,
although both were compressive, the one for Condition 1 was uniform whereas the other had a
gradient. At later time intervals, the shapes of the profiles between the two conditions had some
similarity but the curves remained apart. Nevertheless, a closer inspection reveals that the
difference in the delithiation capacity between the two conditions was originated from their
initial SOC, rather than the residual stress. The Condition 1 resulted in a complete charged state
(SOC=0.83) whereas Condition 2 resulted in a partially charged electrode (SOC=0.41),
corresponding to 100% and 47.8% of the capacity, respectively. The delithiation capacities for
the two conditions were 94.4% and 47.3%. These values correlated well with the amount of Li
initially in the electrodes. The same phenomenon was observed for the nanosphere. The
delithiation capacity 89.9% at 100C under Condition 2 can be traced back to its initial SOC
which was at 90.2% of the capacity.

79

Based on these results, we conclude that the reduction in delithiation capacity under cyclic
charge/discharge is related to the lower SOC in the electrode.

(a)

(b)

(c)

(d)

Figure 5.8 Profiles of Li concentration and hydrostatic stress vs. the distance to the bottom of the
thin film at 100C with =0.01 under Condition 1 (dashed line) and Condition 2 (solid line). (a)
and (b) are Li concentration; (c) and (d) are hydrostatic stress. (a) and (c) are under 100C
lithiation; (b) and (d) under 100C delithiation.

Besides the voltage shift introduced by the Ohmic resistance, Li et al.[173] pointed out that
the stress may also contribute to the asymmetric rate behavior of the thin film by affecting the Li
diffusivity. The current study confirms that the stress is indeed an important factor contributing
80

to the asymmetric rate behavior. In future work, the chemo-mechanical coupling will be
incorporated into a multiphysics battery model to consider other effects such as electrochemistry,
kinetics, temperature, charge transport in other battery components, and etc.[133,154,161].
It is worth pointing out that, although we only model the amorphous Si structures as the
crystalline Si is usually amorphized in the first lithiation and remains as amorphous phase in later
cycles[42], this model could be developed to consider the two-phase lithiation of crystalline Si in
the first lithiation, and also could be revised to study other high capacity electrodes which
inherent high deformations under lithiation, such as sulfur cathode and tin anode, etc.
5.5 Conclusion
The chemo-mechanical coupling of Si lithiation was investigated using finite element
analysis with a model that incorporates the stress effect into the diffusion equation through
influencing the chemical potential of Li in Si. The coupled diffusion and deformation problem
was solved in COMSOL MultiphysicsÂŽ.
The model was examined in the simulations of two geometries: a-Si thin film and a-Si
nanosphere. In experiments, a strong asymmetric rate behavior has been observed in a-Si thin
film such that the decrease in capacity with increasing the C rate was much faster in lithiation
than that in delithiation. However, such a behavior has not been reported for other geometries.
FEA simulations captured the asymmetric rate behavior of a-Si thin film and the symmetric rate
behavior of Si nanospheres. The simulation results revealed that both the electrode geometry and
the chemo-mechanical coupling played important roles in the asymmetric rate behavior.
The lithiation of a substrate-constrained Si film is more likely to become surface limited than
other geometries due to its low surface/volume ratio and constant surface area. Furthermore,
such a thin film is mostly under compression during lithiation. The compressive hydrostatic

81

stress suppresses the Li diffusion. The two factors result in a significantly lower lithiation
capacity. In delithiation, the stress reverses to tension which promotes Li diffusion. The
simulations show that the delithiation capacity was almost unaffected by the discharging rate.
The large differential rate dependence in charge and discharge contributes to the strong
asymmetric rate behavior of the thin film.
A free standing nanosphere has an increasing surface area with volume expansion and hence
is less likely to become surface limited in lithiation. In a sphere, the tensile and compression
stresses always co-existed. The stress suppressed and enhanced diffusion happened
simultaneously at different radial distances across a sphere. This profile promotes a uniform Li
distribution. The nanosphere is less likely to suffer capacity reduction with increasing the rate.
The delithiation capacity of the two geometries is affected very little by the electrode
geometry and the chemo-mechanical coupling. However, the delithiation capacity is limited by
the amount of Li available in the electrode. Therefore, under cyclic charging/discharging
condition, the delithiation capacity decreased at the same pace as that of lithiation. The
asymmetric rate behavior was absent. On the other hand, a slow lithiation process will lead to a
better retained delithiation capacity at higher rates.

82

Chapter 6

A Microstructure-Resolved Multiphysics Model for Li-ion Battery Si Anode

6.1 Introduction
Although with an ultra-high theoretical capacity, the capacity of the a-Si anode that can be
actually realized in a LIB is related to its shape, packing, electrical pathway, and Li transport in
the electrolyte and in the active materials. The design options are numerous, as discussed in 1.1.
A composite Si anode will bring in additional flexibility and another level of complexity. The
ability to find the best combinations through computation-guided optimization can drastically
accelerate the process of bringing a promising novel nanostructure into products. Such
optimizations require a multiphysics microstructure-resolved model (MRM) for LIB cell. The
model should include the electrochemical reaction, Li transport in electrolyte and electrodes,
electrical connectivity, dimensional changes, stresses, and the coupling relationships.
Multiphysics numerical models have been used in LIB research, such as coupled thermalelectrochemical model for the investigation of thermal response[133–135], coupled
electrochemical-mechanical model for the study of stress and deformation of the electrode
particles

and

battery

models[132,137,140,179–181]

components[136],
and

microstructure

resolved

multiphysics

multiscale coupled electrical/electrochemical-thermal-

mechanical models for the prediction of mechanical crush introduced short circuit and thermal
responses of LIBs[182,183]. All these models were for LIBs with graphite anodes.
The objective of the current work is to incorporate Si anodes in these models. Si anode
presents new challenges for numerical models, including the numerical issues in considering
mass balance with finite deformation[23],

the uncertainty in some key properties for Si

anodes[23,24], and the modeling of electrolyte and its interaction with other cell components. In
previous Chapters, we have investigated the modeling of Li diffusivity in amorphous Si (a83

Si)[115], chemo-mechanical coupling[155] of LixSi, and the mass balance with finite
deformation[115,155] separately using models that only consist of a-Si anode. The current work
is focused on a LIB cell.
This chapter presents a model for a half cell configuration consisting of a-Si anode and Li
metal reference electrode. The model has been implemented in COMSOL, a Multiphysics Finite
Element (FE) package. This paper is organized as follows. In section 6.2-6.3, the model setup
and model parameters are presented. Key properties (reaction rate constant and apparent transfer
coefficient) are investigated by correlating with experiments with Si nanowire anode at various C
rates. In section 6.4, the model is used to examine a-Si thin film anode for validation. In section
6.5, a parameter study is performed for half cell with Si nanowalls anode.
6.2 Model Description
The model development is described in detail in Chapter 3 previously. The governing
equation was categorized in Table 6.1 for MRM model for both “battery” and “stress” submodels. For computational efficiency, a two-dimensional (2D) MRM model has been developed
first. The methodology developed for 2D model can be extended to a three-dimensional (3D)
model. The model described below is for implementation in COMSOL. The model used the
following COMSOL modules: General Form Partial Differential Equation (PDE) module,
Nernst-Planck (CHNP) module, Electric Currents (DC) module and Fluid-Structure Interaction
(FSI) module.

84

Table 6.1 Governing equations for cell model in Chapter 6 (Symbols are in Nomenclature).
Physics
Governing Equations
“Battery” sub-model
Mass Balance
(1) Li Diffusion 𝜕𝑐1
𝐷1
(𝛔)) = 0
+ ∇ ∙ (−
𝑐 ∇𝜇
in Si
𝜕𝑡
𝑅𝑇 1 𝐿𝑖𝑥𝑆𝑖
𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) = 𝜇𝐿𝑖𝑥𝑆𝑖 + ∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔)
𝜇𝐿𝑖𝑥𝑆𝑖 − 𝜇𝐿𝑖 = −𝑛𝐹𝑈
∆𝜇𝐿𝑖𝑥𝑆𝑖 (𝛔) = 𝛼Ω𝜎ℎ
(2) Ionic

2k 2 t  t  RT   ln f 
1 
Transport in N  =   D2 
 ln c 2
c  z  F 2 

Electrolyte
2

2k 2 t  RT   ln f 
1 
N  =   D2 
2 
c
z
F
 

  ln c 2

Charge Balance
(1) Fluid
Electrolyte
(2) Si anode
(3) Current
Collector
Electrochemical
Interface

i 2 = k 2  2 

2 RTk2
c2 F

  ln f 
1 
 ln c 2


Eq. No.

(3.1)
(3.3)
(3.6)
(3.5)


k t
c   2   2
z F


(3.7)


k t
c   2   2
z F


(3.8)


1  t  c 2


k11  i 1
k c  c   i c

(3.9)
(3.10)
(3.11)

   F 
   c F 
i1, 2  i Si exp a
  exp

 RT 
  RT 

iSi  Fk0 c2a ctheory  csurf



a

c
csurf

(3.15)
(3.17)

  1   2  U

(3.18)

N   i1, 2 / F

(3.19)

“Stress” Sub-model
Si Deformation

𝐅 = 𝐅𝑒𝑙 𝐅𝑖𝑛𝑒𝑙 = 𝐅𝑒𝑙 𝐅𝑝 𝐅𝑐
𝑉
= det(𝐅𝑐 ) = 1 + Ω∆𝑐
𝑉0
Stress in Si
𝛔 = 𝐽 −1 𝐅𝐒𝐅 𝐓
𝜕𝑊𝑠
𝐒 = 𝐒𝟎 + 2
𝜕𝐂
𝑊𝑠 = 𝛬tr(𝐄)2 + 𝛭tr(𝐄 𝟐 )
Fluid
Electrolyte
 v fluid
 fluid 
 v fluid  v fluid
Motion
 t

85

(3.22)
(3.23)
(3.30)
(3.31)
(3.32)

  p   fluid  2 v fluid


(3.34)

6.3 Model Parameters
In addition to the model parameters listed in section 3.3, there are more parameters that
should be determined based on the target system and theories adopted. Therefore, the following
parameters needed in the MRM model are described below.
6.3.1

Chemical Potential

The OCP has been measured by Sethuraman et al. through a series of ten-hour relaxation
experiments performed on a 140 nm porous a-Si thin film[118]. The results are replotted in Fig.
6.2(a), where the red dots were from lithiation and blue dots were from delithiation of Si with the
state of charge (SOC) changing from 0.03 to 0.8. The OCP value at SOC=0 equals to the
theoretical potential difference between Li and Si, and the value reduces to 0V when
SOC=1[184]. Based on these observations, the OCP as a function of the SOC is established, as
shown by the black curve in the Fig. 6.1(a). Based on Eq. 3.6, the chemical potential 𝜇𝐿𝑖𝑥𝑆𝑖 as a
function of SOC is calculated, as presented in Fig. 6.1(b).

(a)

(b)

Figure 6.1 (a) The relaxation potential of lithiated Si during lithiation and delithiation from
Sethuraman et al.[118] and the OCP curve used in the current work. (b) The chemical potential
derived from extrapolated OCP curve using Eq. 3.6.
86

6.3.2

Li Diffusivity in a-Si

The Li diffusivity in Si was determined through correlating numerical simulations with the
in-situ lithiation experiments of Si nanospheres by McDowell et al.[42], as presented in Ref.
[115]. The Li diffusivity in single phase lithiation of Si is found to be in the range of 3×10 -16
m2/s with the chemical potential as the driving force for lithiation (Eq. 3.2 and 3.6) instead of the
concentration gradient in Ref. [115] (Eq. 3.1). It is an averaged value without the consideration
of Li concentration-dependency and stress effect. In this study, 𝐷1 = 3×10-16 m2/s is used in Eq.
3.2 for single phase lithiation.
6.3.3

Electrolyte Properties

The experiments to be simulated in this work used two electrolyte systems, as listed in Table
6.1. Literature data showed that the ionic conductivity of different electrolyte system varies in a
similar range[131,133,185–187], and the solvent ratio had a negligible impact on the
performance of the cell[188,189]. Since the ionic conductivity k2 for the two systems in Table
6.2 were not available, k2 was assumed to follow the relationship for 1 M LiPF 6 in EC/DMC (2:1
vol.%)[137], as shown in Fig. 6.2.

Table 6.2 Electrolyte systems used in chosen experiments
Experiments
Nanowire [30]
Thin film [60]

Electrolyte system
1 M LiPF6 in 1:1 (vol%) EC:DEC
1.2 M LiPF6 in 1:2 (vol%) EC:DEC with 10% FEC additive

87

Figure 6.2 The ionic conductivity k2 as a function of the salt concentration of LiPF 6 in EC:DMC
(2:1 vol.%)[131].

Table 3.1 presents a summary of the parameter values used in the current model. For Si, the
apparent transfer coefficient (Îąa and Îąc) in Butler-Volmer equation (Eq. 3.15) and reaction rate
constant k0 in Eq. 3.17 and have not been well established. The values of these parameters are
labeled as to be determined (TBD) in Table 3.1, and they will be determined by simulating a Liion cell with Si nanowires experiment of Zhang et al.[30] to be presented next.
6.3.4

Determine the Reaction Rate Constant and Apparent Transfer Coefficient

The current density at the Si-electrolyte interface is determined by the Butler-Volmer
equation, Eq. 3.15-3.17, where the key parameters are k 0 ,  a and  c . For systems with a charge
transfer number n=1,  a +  c =1[139]. Table 6.3 provides a summary of the values of these
parameters reported in literature, where they were determined by fitting the experimental data
with Si continuum models[63,138,190] or analytical solution[118,191] for various Si anodes,
like a-Si thin film[63,118,192], polycrystalline Si thin film[191,193], and Si composite[138,190].
As shown in Table 6.3, these values varied a great deal. This may be due to the wide range of

88

experimental techniques and conditions used in these investigations. There is a need for
standardization in these measurements.

89

Table 6.3 Comparison of reaction rate constant, exchange current density and apparent transfer coefficient between literature and
current work.

Reference

Reaction
rate constant
(m/s)(mol/m
3 -Îąa
)

Exchange
current
density
(A/m2)

Apparent transfer
coefficient

Baggetto et al.
[191,193]

-

10

 a =0.26;  c =0.74

Li et al. [192]

-

1.3

-

Swamy et
al.[194]

-

1

-

Sethuraman et
al. [118]

1×10−14
(estimated)

10-9 (in OCP
relaxation)

Bucci et al.
[63]

2.5-10×10−8

105

Chandrasekara
n et al. [190]

1×10−102.5×10−9

Chandrasekara
n et al. [138]

System used

Experiment technique

70 nm polycrystalline Si thin film;
LiPON solid-state electrolyte
100 nm a-Si thin film;
1M LiPF6 in EC/DMC (1:1)
500 Îźm c-Si wafer;
1M LiPF6 in EC/DMC (1/1 wt%) +
FEC (10 wt %) + VC (2 wt%)

Butler-Volmer fitted
GITT
Analytical solution fitted
PITT

 a  1.77SOC  1.65
Porous 300 nm c-Si thin film;
 c  0.63SOC  1.52 1.2M LiPF6 in 1:2 (vol.%) EC/DEC
0<  <1, values were
100 nm a-Si thin film;

EIS
A Tafel equation fitted
OCP relaxation

not provided
 a =0.5-0.8

1M LiPF6 in EC/DEC/DMC (1:1:1)

1D electrochemical
model fitted cell voltage

10-2

 c =0.5-0.2
 c +  a =1

silicon composite electrode;
1M LiPF6 in EC/DEC/DMC (1:1:1)

Single Si particle model
fitted GITT

10-13
(assumed)

10-2

 c =  a =0.5

Pal et al. [128]

1.5-60×10-13

-

 c =  a =0.5

This work

8×10−15

10-3

 a =0.7;  c =0.3

Si composite electrode;
no certain electrolyte system used
250 nm a-Si thin film; No
electrolyte given
Si nanowires; LiPF6 in EC:DMC
(2:1 vol.%)

1D cell continuum
model fitted cell voltage
2D electrochemical
model fitted cell voltage
2D MRM model fitted
cell voltage

90

In the current work, k 0 ,

 a and  c

were determined by correlating simulations with the

experiment by Zhang et al. [30]. In Ref. [30], half cells with Si nanowires were galvanostatically
cycled under C rates ranging from C/20 to 1C and the cell voltage against cell capacity was
measured for cycles after the first cycle. Each C-rate was measured with a different cell. To
compare with simulations, the measured cell capacity was converted to SOC and plotted in Fig.
6.3(b). The SOC decreased about 40% when the C rate increased from C/20 to 1C. The SEM
observation showed that individual nanowire is a standalone structure with minimal contact with
other cell components, in good contact with the stainless steel current collector, and remain intact
without fracture during cycling. Therefore, the decrease of capacity with C rate was not due to
mechanical failure but the characteristic response of the cell. This experiment serves as a good
study case.
To simulate this experiment, a 2D plane strain FE model was built, as shown schematically
in Fig. 6.3(a). In experiments, the nominal diameter of the nanowire is 90 nm and the length is in
the range of micrometers, which results in a large aspect ratio (length/diameter). In the FE model,
the nanowire was simulated with an aspect ratio of 30. Besides the constraint by the current
collector at the bottom, the nanowire was surrounded by a large amount of fluid electrolyte
which provides sufficient supply of Li. The model geometry is set as in Table 6.4.

Table 6.4 Model dimensions used in the Si nanowire model
Quantity
Si Nanowire
Current Collector
Li Electrode Electrolyte
2.7
2
1
5.3a
Height (m)
Width (nm)
90
1000
1000
1000
a. Height of electrolyte is measured from top of Si nanowire to bottom of Li electrode

91

By adjusting the value of the parameters and correlating the cell voltage at various C rates
with the experimental curves, the parameter set that produced the best fit to the experimental
curves was determined as k0 = 8×10-15 (m/s) (mol/m3)-0.7, αa=0.7 and αc=0.3. Fig. 6.3(b)
compares the cell voltage vs. SOC plots obtained in the experiment with the simulation results
using this set of parameters. As seen, the predicted curves (solid lines) were in reasonable
agreement with the experimental data (markers) under C rates from C/5 to 1C. This set of
parameters is presented in the last row in Table 6.3. Compared to literature data, the value of k0
is at the same order as that reported by Sethuraman et al.[118]. The trend of  a >  c and the
range of the values agreed with that reported by Chandrasekaran et al.[190]. Trial runs with
symmetric apparent transfer coefficients  a =  c =0.5 have also been tested and the resulted cell
voltage locus is always symmetric to the OCP, which differ from the experimental trend.

(a)

(b)

(c)

Figure 6.3 (a) The schematic of cell model with Si nanowire and (b) the mesh. (c) Comparisons
of the cell voltage vs. SOC curves obtained by simulations with k0 = 8×10-15 (m/s) (mol/m3)-0.7,
Îąa=0.7 and Îąc=0.3 (solid lines) and the experimental results (markers) by Zhang et al.[30].

92

6.4 Model Validation
For model validation, the experiment reported by Sethuraman et al. [60] was simulated. In
Ref.[60], the a-Si thin film of 250 nm thick was cycled galvanostatically at C/4 rate. The
available results include the cell voltage and the in-situ biaxial stress variation with SOC for
cycles after the first cycle. The biaxial stress was determined through the Stoney equation based
on the substrate curvature measurement[176,177].
The FE model for the thin film experiment is shown schematically in Fig. 6.4(a). The
dimensions of model is listed in Table 6.5. The simulations were performed with parameter
values reported in Table 3.1 and k 0 ,  a and  c values determined in this work. The predicted
cell voltage and biaxial stress are compared with the experimental results in Fig. 6.4(b) and
6.6(c), respectively. As seen, the predicted stress variation with SOC was in good agreement
with the experiment. The cell voltage prediction, although less accurate, agrees with the
experimental trend. The simulation was also performed with  a =  c =0.5 and k 0  8 10 15 , and
the results are plotted in Fig. 6.4(b). As shown, the predicted cell voltage locus during
delithiation deviated further from the experimental curve with  a =  c =0.5.

Table 6.5 Model dimensions used in the Si thin film model
Quantity
Height (nm)
Width (m)

Si Nanowire
250
1

Current Collector
200
1

93

Li Electrode
100
1

Electrolyte
250
1

(a)

(b)

(c)

(d)

Figure 6.4 Simulation of the experiment of the thin film a-Si electrode by Sethuraman et al. [60].
The FE model used was depicted in (a) and the mesh used in (b). The comparison between
simulations and experiments were given for both (c) cell voltage and (d) biaxial stress vs. SOC
during cycling at C/4 rate.

6.5 Cell Design with Si Nanowalls
In order to achieve a high volumetric/specific cell capacity, the Si should be packed as
densely as possible in the electrode. On the other hand, the electrode should have sufficient Li
supply within the designed charging range. The overall efficiency of the cell depends on a
number of design parameters such as the shape and the size of Si nanostructures, the packing
density, and the Li+ salt concentration in the electrolyte. The MRM model is a useful tool in the
94

investigation of design options and in design optimization. The use of MRM model is
demonstrated by a parameter study for a half cell with Si nanowalls (SiNWs) anode.
6.5.1

Design Parameter Study

Fig. 6.5 shows the FE model for this investigation. Considering the symmetry, the model is
simplified by including two half SiNWs with fluid electrolyte filling the space between them.
The model also contains a current collector at the Si side and a Li counter electrode.

Figure 6.5 Schematic of the model used for the parameter study for Si half cell. Line ① and
Point A are for the data analysis in later discussion.

The parameters studied include the wall thickness; the aspect ratio (AR) - the ratio of the
length to thickness of the wall; the spacing ratio (SR) - the ratio of the space between the wall to
the wall thickness; and the initial Li+ concentration in the electrolyte. The baseline was for 200
nm SiNWs, with AR=45, SR=1, and in 1M LiPF6 in EC/DMC. The dimensions used for baseline
test was listed in Table 6.6. The design parameter was changed one at a time in respect to the
baseline. Table 6.7 provides a summary of the cases studied and the maximum SOC achieved
under 1C charging rate. Since the amount of Si varies in cells of different designs, to keep the

95

same rate, the applied current has to vary accordingly. As a result, the current density at the Sielectrolyte interface remained the same for the cases with 200 nm SiNWs but almost doubled for
400 nm SiNWs.

Table 6.6 Model dimensions used in the baseline model
Quantity
Height (m)
Width (nm)
Initial concentration
(mol m-3)
Maximum concentration
(mol m-3)

Si Nanowire
9
200

Current Collector
1
400

Li Electrode
1
400

Electrolyte
2
400

0

/

/

1000

311307.4/ V

/

/

/

Table 6.7 Half cells with SiNWs of different design parameters and the maximum SOC at 1C
Aspect
Spacing
Ratio
Ratio
Baseline Test
45
1
AR=15
15
AR=90
90
SR=0.5
0.5
SR=2
2
2M LiPF6
3M LiPF6
Thickness=400 nm
“-” = the same as the baseline

LiPF6 in
electrolyte (Mole)
1
2
3
-

Nanowall
thickness (nm)
200
400

Maximum
SOC
0.17
0.42
0.050
0.089
0.31
0.34
0.43
0.032

Fig. 6.6 examines the effect of electrode design parameters. Fig. 6.6(a) plots the cell voltage
vs. SOC. In simulations, the cell voltage was measured at the bottom of the current collector with
respect to the Li electrode. The lithiation terminates when the cell voltage reaches 0V and hence
the SOC at 0V represents the maximum SOC for the cell. These values are also listed in Table
6.7. As shown, the maximum SOC increases with increasing the SR and Li + concentration in the
electrolyte, and decreases with increasing the AR and nanowall thickness.
96

6.5.2

Concentration Polarization

The maximum SOC that a cell can achieve appears to relate to the level of Li+ and the
concentration polarization in the cell. Fig. 6.6(b) examines the Li+ concentration in the
electrolyte along the Si-electrolyte interface (line ① in Fig. 6.5) at the end of lithiation. As
shown, the cell with a lower Li+ and a steeper Li+ profile corresponds to a lower maximum SOC.
For the cases of AR=90 and Thickness=400 nm, the concentration polarization has led to Li +
depletion near the current collector, and the lowest SOC. For all other cases, ample supply of Li+
was still available at the Si-electrolyte interface when the lithiation stopped.

(a)

(b)

Figure 6.6 (a) The cell voltage changes with SOC at various test cases, and (b) the Li+
concentration along the Si-electrolyte interface (line ① in Fig. 6.5) at the end of lithiation in
each test case.

6.5.3

Potential Evolution

Since the lithiation process is driven by the potential difference between the two electrodes,
the evolution of the potentials was examined. Four potentials are involved in this process: the Si

97

potential  1 , OCP U , electrolyte potential  2 , and overpotential  . These potentials are related
by Eq. 3.18 as 1     2  U . While U is determined by the local SOC as shown in Fig. 6.2(a),

 1 ,  2 and  are governed by the local currents.
Fig. 6.7 examines the evolution of four potentials for the baseline case at point A (in Fig. 6.5),
which is on the Si-electrolyte interface and at the base of NW. As the voltage drop in the current
collector is negligible,  1 at Point A approximately equals to the cell voltage. Fig. 6.7 shows that

 and  2 display a sudden change at the beginning of the lithiation process and then become
relatively stable whereas U reduces steadily with the local SOC over time. After the initial stage,

 1 reduces at a similar rate as U. The stabilized  and  2 values determine the range of  1 .
Further away from zero are these potentials, smaller is the range of  1 , which in turns leads to a
lower maximum SOC.

98

(a)

(b)

(c)
Figure 6.7 The evolution of potentials at point A for (a) the baseline test; (b) Thickness = 400 nm,
and (c) 3M LiPF6 electrolyte.

As shown by Eq. 3.18,  2 is influenced by c 2 , the Li+ concentration in the electrolyte, and

k 2 , the ionic conductivity, and the latter is also a function of c 2 . The local  2 , therefore, is
influenced by the concentration polarization. The factor enhancing the concentration polarization
would lead to a lower stabilized  2 value. The stabilized  value follows the same trend. Table
6.8 summarizes the trends how various design parameters influence these potentials. Fig. 6.8
compares the profiles of  2 and  at the Si-electrolyte interface at SOC=0.03 for different cases.

99

Increasing the SR and electrolyte concentration, and reducing the AR and nanowall thickness
lead to reduced concentration polarization, a lower magnitude of  2 and  , and higher SOC.

Table 6.8 The influences of design parameters on c 2 polarization,  2 ,  ,  1 and SOC

Test Conditions

Concentration
polarization

Electrolyte
potential (  2 )

Overpotential
( )

Overall Si
potential
( 1 )

Final SOC

AR 











SR 
Li+ c 2 





















Wall thickness 
 =increases,  =decrease,  = same level

(a)

(b)

Figure 6.8 The profiles of (a) electrolyte potential and (b) overpotential along line ① in Fig. 6.5
at SOC=0.03 for various test cases.

100

6.5.4

Cell Capacity

The capacity of a cell is compared by the specific capacity (mAh/g) and the volumetric
capacity (mAh/cm3). Table 6.9 presents a comparison of the cell capacity for SiNWs half cells
with different AR and SR values under 0.25C, 1C and 4C charging rates. The capacity was
calculated based on the total weight or volume of Si, 10 Îźm thick Cu current collector[114], and
the fluid electrolyte contained inside the cell. The capacities as a function of AR and SR are also
shown in Fig. 6.9. Table 6.10 presents the cell capacity retention in reference to 0.25C.
The results show that the cell with the highest SOC is not necessarily the one with the highest
capacity, such as the case of AR=15 and SR=2. Furthermore, the capacity is not a linear function
of AR and SR. A high level of specific capacity is found in cells with AR=90 and SR=2 at low C
rate (0.25C), and with AR=45 and SR=2 at higher C rates (1C-4C). On the other hand, the
volumetric capacity is relatively high when SR in the range of 1-2, AR is in range of 45-90 at
low C rate (0.25C-1C) and is in the range of 15-45 at high C rate (1C-4C). The capacity retention
is judged by the SOC ratio with reference of 0.25C. As shown, the AR increases resulted in
decrease in capacity retention.

101

Table 6.9 Comparison of the specific and volumetric capacity of half cell with SiNWs under
0.25C, 1C and 4C
AR

SR

15
15
15
45
45
45
90
90
90

0.5
1
2
0.5
1
2
0.5
1
2

Specific capacity (mAh/g)
0.25C
1C
4C
78.27
54.16
8.54
96.59
55.42
7.92
67.59
40.75
6.35
148.82
42.25
2.14
216.66
60.88
6.66
178.64
75.45
8.76
131.80
10.51
0.81
262.56
31.33
1.57
280.90
51.86
3.43

Volumetric capacity (mAh/cm3)
0.25C
1C
4C
566.05
391.68
61.78
693.66
398.00
56.86
482.02
290.60
45.26
834.78
237.01
12.01
1182.45
332.26
36.35
956.39
403.92
46.91
578.41
46.13
3.55
1115.13
133.07
6.65
1153.28
212.91
14.09

Table 6.10 Comparison of the SOC and the capacity retention of half cell with SiNWs under
0.25C, 1C and 4C
AR
15
15
15
45
45
45
90
90
90

SR
0.5
1
2
0.5
1
2
0.5
1
2

0.25C
0.45
0.73
0.76
0.32
0.61
0.73
0.16
0.42
0.65

SOC
1C
0.31
0.42
0.46
0.09
0.17
0.31
0.013
0.050
0.12

4C
0.049
0.060
0.072
0.0046
0.019
0.036
0.0010
0.0025
0.0079

102

Capacity Retention
1C/0.25C
4C/0.25C
0.69
0.11
0.58
0.082
0.61
0.095
0.28
0.014
0.28
0.031
0.42
0.049
0.081
0.006
0.12
0.006
0.18
0.012

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6.9 Si nanowalls of 200 nm thickness with various AR and SR in 1 M LiPF 6. (a) The
specific capacity and (b) the volumetric capacity at 0.25C rate, (c)(d) at 1C, and (e)(f) at 4C.

103

6.6 Conclusion
A multiphysics microstructure-resolved model (MRM) has been developed for Li-ion
battery (LIB) cell with a-Si anode. The model couples the electrochemical kinetics, species
transport, and the structural/stress evolutions. The coupled diffusion and deformation problem
was solved in COMSOL MultiphysicsÂŽ. Besides the parameters determined in our previous
studies, the reaction rate constant (k0 = 8×10-15 (m/s) (mol/m3)-0.7) and apparent transfer
coefficient (  a =0.7;  c =0.3) were determined in this work. The model was validated by
simulating a-Si experiments from literature.
The model was used to investigate the parameter design of Si nanowall anodes with a half
cell configuration. The results showed that the maximum SOC of a cell can achieve under a
given charging rate depends on the electrolyte potential and overpotential. The two potentials in
turns depend on concentration polarization. The factors reducing concentration polarization will
enhance the maximum achievable SOC of the cell. Such measures include increasing the space
ratio and Li+ concentration in the electrolyte, and reducing the aspect ratio and nanowall
thickness. On the other hand, the cell with the highest SOC does not necessarily lead to the
highest capacity. The specific/volumetric capacities display a nonlinear relationship with the
aspect ratio and spacing ratio. The MRM model can be used for design optimization.

104

Chapter 7

Conclusion and Outlook

7.1 Conclusion and Contributions
In this work, an MRM model has been developed for a basic Li-ion battery cell with Si anode
in Li-ion. This model couples the electrochemical kinetics, species transport, and the
structural/stress evolutions. The model can be used for the design optimization of Si anode and
the battery performance of the cell. It can also be used to investigate the stress and deformation
in different cell components. The specific contributions are
7.1.1

Identify the Range of Li Diffusion for a-Si Anode

The lack of key material properties for Si/LixSi, especially the Li diffusivity in Si, has
prevented the efficient use of such numerical tools in the design of lithium-ion batteries with Si
anodes. Therefore, parameter investigations were firstly performed using FEA on the single Si
particles outside the MRM model. The Li diffusion and volume expansion were coupled and
solved in COMSOL MultiphysicsÂŽ. By comparing with the lithiation experimental observations
for the a-Si nanospheres, the range of Li diffusivity for a-Si anode was identified. This value is
one of the key input parameter for MRM model for Si.
7.1.2

Identify the Chemical-Mechanical Two-Way Coupling Mechanism for Si under Large
Deformation Framework

A strong asymmetric rate behavior between lithiation and delithiation has been observed in
amorphous Si (a-Si) anode in the form of thin films, but not in other geometries, such as Si
nanospheres. The chemo-mechanical coupling of Si under large deformation was investigated by
simulating the rate behavior of the two geometries. The results reveal that the rate behavior is
affected by the geometry and the constraint of the electrode, the chemo-mechanical coupling,
and the prior process. A substrate-constrained film has a relatively low surface/volume ratio and

105

a constant surface area. Its lithiation has a great tendency to be hindered by surface limitation.
The chemo-mechanical coupling also plays an important role. The stress profiles differ in the
two geometries but both affect the lithiation process more than the delithiation process. The
overall delithiation capacity is affected very little by the chemo-mechanical coupling. In cyclic
loading, the delithiation capacity is reduced at the same rate as the lithiation capacity because of
the low initial state of charge in the electrode. The asymmetric rate behavior was absent under
cyclic loading. On the other hand, a slow charging process resulted in a better retained
delithiation capacity and an asymmetric rate behavior at higher rates.
7.1.3

Identify the Range of the Electrochemical Kinetic Parameters

In order to correctly predict the cell performance, the electrochemical reaction of Si in cell
under the large deformation framework was studied first. Through correlating the simulation
results to the cell cycling experiments, the electrochemical reaction related parameters were
characterized, including the reaction rate constant and apparent transfer coefficient. The model
was validated with Li-ion cell experimental results in literature, obtained with both Si nanowires
and thin films at different cycling rates
7.1.4

A Design Tool for the Optimal Design of Battery Cells with a-Si Anode

The use of MRM model was demonstrated in a parameter study of Si nanowall|Li cells. The
specific and volumetric capacities of the cell as a function of the size, length/size ratio, spacing
of the nanostructure, and the Li+ concentration in the electrolyte were investigated. The results
show that the cell with the highest SOC is not necessarily the one with the highest capacity.
Furthermore, the capacity is not a linear function of design parameters, such as aspect ratio or
spacing ratio. For example, a larger aspect ratio of Si nanowall is beneficial to specific and

106

volumetric capacities at low cycling rates, and vice versa. Study also showed that the spacing
ratio between Si nanowall is best for volumetric capacity when ranges from 1-2.
7.2 Outlook of Future Work
The study of high capacity Li-ion battery is a field full of opportunities. The MRM model
developed in this work can be used not only in improving battery design but also in advancing
our understanding of the coupled electrochemical and mechanical processes.
7.2.1

Interfacial Interplay between Si and Other Components

Interfacial problems are present from a number of places in Li-ion battery, including
1) When Si is surrounded by electrolyte, the solid electrolyte interface (SEI) layer is
generated at the Si surface during cell cycling. Interface evolves between Si and SEI.
2) To prevent SEI, coatings were deposit on Si surface using material such as Cu and
graphite. The interplay between Si and Cu or graphite raises another interaction problem.
3) Most electrode matrices consist of not only active Si particles but also binder. The
interface between Si and binder present an interfacial problem.
4) When Si is deposited directly onto the current collector, there is an interface between Si
and current collector. A preliminary study on this problem is showed in Appendix D.
By adjusting the MRM model with the inclusion of interface and the possibility of interfacial
delamination, the unidentified interaction mechanisms may be revealed.
7.2.2

Thermal Management of Li-Ion Cell

The thermal management of commercialized Li-ion cell has been one of the design criteria in
industries such as small electronics and automobile. With a new type of electrode material (Si),
the heat generated during battery cycling may be greatly altered. So the design of the thermal
management system strongly relies on the prediction of heat generation and temperature variance.

107

The MRM model can be further developed to consider the temperature influence on the
electrochemical reactions, mass transport and structural and stress evolutions under various
cycling conditions.
7.2.3

Extension of Model to Other High Energy Density Electrode Material Li-Ion Cell

Besides Si, other high capacity electrode materials such as Sn, Ge and Al, etc., also exhibit
large volume change with lithiation/delithiation. The current model can be adopted to investigate
these electrode materials as well.
The model can also be used in a multi-scale framework, or in conjunction with other software,
to investigate the performance of battery packs.

108

APPENDICES

109

Appendix A

Effect of Li Transport Distance Change and Geometrical Constraint on Different
Si Geometries

Under galvanostatic cycling, a constant Li flux 𝑵 is maintained at the Si electrode surface.
The Fick’s first law corresponding to this boundary condition is
−

𝑫𝟏
𝒄 𝜵𝝁 = 𝑵
𝑹𝑻 𝟏 𝑳𝒊

Eq. A1

The differences between Si thin film and sphere geometries under the influence of volume
expansion and geometrical constraint will be discussed in A.1 and A.2.

(a)

(b)

Figure A.1 Control elements at electrode surface in Si geometry (a) thin film and (b) sphere

A.1 Si Thin Film
For a substrate-constrained Si thin film, as it is confined to expand only along the z direction,
the 𝛻𝜇𝐿𝑖 in Eq. A1 is reduced to 1D with 𝜕𝜇𝐿𝑖 /𝜕𝑧. Fig. A.1(a) shows a rectangular control
element at the surface. It has a surface area of 𝐴0 and an initial thickness of ∆𝑧0. The Li flux
applied at 𝐴0 is 𝑵𝟎 , Eq. A1 becomes
−

D1 ∂μLi
c
= 𝐍𝟎
RT 1 ∂z0

110

Eq. A2

The control volume thickness changes to ∆z during lithiation and the surface area remains as
A0 . Eq. A2 becomes
−

D1 ∂μLi
c
= 𝐍𝟎
RT 1 ∂z

Eq. A3

The relationship between ∆z and ∆z0 is
∆z = det(𝐅)∆z0 = 𝐽∆z0

Eq. A4

where 𝐅 is the deformation gradient. 𝐽 is the determinant of 𝐅, which reveals the volume
change of the element, i.e., 𝑱 = 𝑽/𝑽𝟎 [195]. Therefore
∂z
=𝑱
∂z0

Eq. A5

Apply the chain rule to Eq. A3 and substitute Eq. A5 in, we have
−

D1 𝜕𝜇𝐿𝑖 𝜕𝑧0
D1 ∂μLi
c1
= 𝐽−1 (−
c
) = 𝐍𝟎
RT 𝜕𝑧0 𝜕𝑧
RT 1 ∂z0

Eq. A6

The lithium induced volume expansion increases the Li transport distance and results in a 𝑱
times slower diffusion. Therefore, lithiation stopped at lower capacity when maximum
concentration reached at the thin film surface. Vice versa, during delithiation, the delithiation
accompanied volume shrinking decreases the Li transport distance, and resulted in a 𝑱 times
faster delithiation. Thus, the end-state of delithiation was close to full discharge. Thus, the
asymmetric rate behavior would be observed even without considering the stress effect on Li
diffusion in thin film.
A.2 Si Nanosphere
In contrary to thin film geometry, Si nanosphere is considered to expand freely without
constraint, as explained in the main text. According to the spherical symmetry, the 𝛻𝜇𝐿𝑖 in Eq.
A1 can be reduced to 1D with 𝜕𝜇𝐿𝑖 /𝜕𝑟. Fig. A.1(b) shows the control element at the particle

111

surface with an initial thickness of ∆𝑟0 and initial surface area 𝐴0 . Li flux applied at 𝐴0 is 𝑁0 . Eq.
A1 becomes
−

D1 ∂μLi
c
= N0
RT 1 ∂r0

Eq. A7

As lithiation proceeds, the radius of the sphere increased from r0 to r and the surface area
increased to 𝐴 = 𝐴0 𝑟 2 /𝑟02 . Therefore, the surface Li flux decreases from N0 to N
r02
N = N0 2
r

Eq. A8

Substitute Eq. A8 to Eq. A7, then
−

D1 ∂μLi
r02
c1
= N0 2
RT
∂r
r

Eq. A9

Apply chain rule to Eq. A9

−

D1 𝜕𝜇𝐿𝑖 ∂r0
r02
c1
= N0 2
RT 𝜕r0 ∂r
r

Eq.
A10

The relationship between ∆r and ∆r0 is
Eq.
1/3

∆𝑟 = 𝑑𝑒𝑡(𝐹)∆𝑟0 = 𝐽

∆𝑟0
A11

where F and J share the same definiations as in Eq. A4.
The relationship between r and r0 is
V 1/3
r = r0 ( )
V0 sphere

Eq.
A12

where (V/V0 )sphere is the volume ratio (current volume/initial volume) of the sphere.
Substitute Eq. A11 and 12 to Eq. A10, we have
V 2/3
D1 ∂μLi
J −1/3 ( )
(−
c
) = N0
V0 sphere
RT 1 ∂r0

112

Eq.

A13
In spherical geometry, although the lithium induced volume expansion increases the Li
transport distance to J −1/3 times, the surface area also increases along with the expansion to
(V/V0 )2/3
sphere times. If the Li concentration is uniform inside the Si sphere when lithiating at low
C rate, then J = (V/V0 )sphere . Eq. A13 reduces to
J

1/3

Eq.

D1 ∂μLi
(−
c
) = N0
RT 1 ∂r0

A14

Overall, the Li diffusion became J1/3 times faster. Therefore, the asymmetric rate behavior
was not seen in Si nanosphere with Îą=0. At high C rate, the J > (V/V0 )sphere due to the higher
2/3

Li concentration at the particle surface. Yet, J −1/3 (V/V0 )sphere may still be larger than one
unless cycling rate is ultrahigh. As a result, no asymmetric rate behavior was observed for Si
nanosphere even at 100C.

113

Appendix B

Transport Phenomena in Si-Li Cells

The use of MRM model is demonstrated by a parameter study for a half cell with Si
nanowalls (SiNWs) anode. Fig. B.1 shows the FE model for this investigation. It includes two
half SiNWs with fluid electrolyte filling the space between them, a current collector at the anode
side and a Li counter electrode Fig. B.1(b)-(e) present selected results at the end of lithiation for
the case of 200nm SiNWs with SR=1, AR=15, and in 1M LiPF 6 in EC/DMC. The MRM model
can provide not only the overall cell information, like the cell voltage, but also details inside the
cell, such as the Li concertation in Si (Fig. B.1(b)) and in the electrolyte (Fig. B.1(c)), individual
stress component in solid components (Fig. B.1(d)) and the velocity of the electrolyte (Fig.
B.1(e)), etc.

114

(a)

(b)

(c)

(d)

(e)

Figure B.1 (a) Schematic of the model used for the parameter study for Si half cell. Line ① and
Points A-C are for the data analysis in later discussion; (b)-(f) are profiles at the end of charging
at 1C for the case of 200nm SiNWs, with SR=1, AR=15, in 1M LiPF 6 in EC/DMC. The original
sizes and shapes are shown in dark lines. (b) Li concentration in Si; (c) Li+ in electrolyte; (d) the
hydrostatic stress in Si and in current collector; (e) the magnitude of fluid velocity in the
electrolyte.

Therefore, by using the MRM model, the cell transport phenomena and other parameters of
research interest could be investigated locally. As shown in Table B1, the baseline test in
Chapter 6 were plotted along line ①, including the Si potential

115

 1 , electrolyte potential  2 , OCP

U , Overpotential at Si surface (bottom to top)  , Li concentration in Si c1 , Li concentration in
electrolyte c 2 , Current at Si surface (bottom to top) i1, 2 , Exchange current density at Si surface
(bottom to top) iSi .

Table B.1 Results in baseline test at t=130s (SOC=0.03)

1

Electrolyte potential

2

Si Potential (V)

Electrolyte Potential (V)

Si potential

Distance to bottom (Îźm)

Distance to bottom (Îźm)
Li concentration in Si

c1

c1 (mol/m3)

c1 (mol/m3)

OCP U

Distance to bottom (Îźm)

Distance to bottom (Îźm)

116

Table B.1 (cont’d)
Overpotential at Si surface (bottom to top)

c2



c2 (mol/m3)

Overpotential (V)

Li concentration in electrolyte

Distance to bottom (Îźm)

Current at Si surface (bottom to top) i1, 2

Exchange current density at Si surface
(bottom to top) iSi

i1,2 (A/m2)

iSi (A/m2)

Distance to bottom (Îźm)

Distance to bottom (Îźm)

Distance to bottom (Îźm)

117

Appendix C

Electrical Conductivity

The order of Si conductivity is selected based on references [107,138]. The reported range
varies between 0.1~100 S/m. The conductivity influence on the cell performance is investigated
in the model presented in Fig. C.1(a). The tests were conducted for conductivity equals to 33 S/m,
0.33 S/m and 0.033 S/m, respectively. The potential profiles along the Si nanowall were plotted
in Fig. C.1(b)-(e). The potential drop across line ① proportionally increases as the
conductivity decreases. For example, k1=33 S/m, the potential drop across line ① is about 10-5
V/Îźm; k1=0.033 S/m, the potential drop is about 10 -2 V/Îźm. Overall, the results showed that,
even conductivity drops to a very small value, the potential of Si nanowall is still very small and
would not present much influence on cell SOC. Therefore, the k1 is kept at 33 S/m in the study in
Chapter 6.

118

(a)

(b)

(c)

(d)

Figure C.1 The investigation of electronic conductivity in Li-Si alloy by plotting the Si potential
along ① in (a) at same time (t=150s) when (b) k1=33 S/m; (c) k1=0.33 S/m; (d) k1=0.033 S/m.

119

Appendix D

Si-Current Collector Interaction

The interface in Li-ion battery with Si anode includes the Si/substrate interface, Si/coating
interface and Si/binder interface, etc. The interface between different materials often induces
complexities such as compound formation and interdiffusion[196]. Furthermore, during battery
operation, the movement of Li at this region makes the problem more complex. The interface is
extremely important for battery performance prediction in MRM model. Here, we take the Sicurrent collector interface as an example, to show the importance of studying the interface
interactions.
Only a few investigations have focused on the interface between Si and current collector. Cu
is usually used as current collector for Si. Experimental observations showed that the interface
sliding happened during lithiation[50] accompanied by large volume change at Si/Cu interface.
A first principle simulation predicted that there was Li segregation at interface, which allowed an
almost frictionless interface sliding by changes from Si-Cu to Li-Cu and Li-Si bond, forming
interdiffused Li-Si-Cu phase[197,198]. Therefore, the lithiation of interface region and its
property evolution may be different from bulk Si and bulk Cu. On the other hand, the adhesion
strength at Si/Cu interface is weaker than the bulk materials[199]. The first principle simulation
predicted the interface adhesion strength is further reduced by Li segregation[197] that is prone
to delamination failure as seen in experiments[199]. Therefore, the material properties at
interface are very important input in the modeling Si anode in Li-ion battery. However,
quantitative studies for interface properties at Si/Cu or for other kinds, such as Si/Ni, during
lithiation are not available in literature. This presents another obstacle in simulating Si anode in
Li-ion battery.

120

The experiment of lithiation of a-Si pillars by Bucci et al.[167] provide a good study case. As
shown in Fig. D.1 (a), the a-Si pillar was on a Ni pillar of same diameter formed by deposition
and etching process. After lithiation, the a-Si pillar almost doubled its size while the Ni portion
showed little deformation. Pillars were sectioned with focused ion beam (FIB) after lithiation or
one cycle of lithiation and delithiation. Cracks were observed only in samples after delithiation,
as shown in Fig. D.1(b-c). The crack was initiated from the surface of the Si pillar near the
interface and propagated into the bulk Si.

(a)

(b)

(c)

Figure D.1Lithiation and delithiation of a-Si pillars on Ni substrate: (a) lithiation of pillars
with diameter of 2100 nm and height of 2300nm; (b) delithiation of pillars with diameter of 2600
nm and height of 2300nm, in which crack is observed at Si surface near Si-Ni interface; (c)
delithiation of pillar with diameter of 3150 nm and height of 2100nm, in which the crack
propagates toward center of Si pillar[167].

The behavior of a-Si pillar during lithiation/delithiation has been investigated without the
consideration of chemical-mechanical coupling and using the Li diffusivity reported in Bucci et
al.[167]. The results are showed in Table D.1. For comparison, the FEA results for this

121

experiment by Bucci et al.[167] were also presented as Simulation I for comparison, where the
Ni pillar was treated as rigid material and Si-Ni was treated as perfect bonding. Apparently, this
treatment produced better results in Si. However, the change in Ni substrate was not considered.
Bucci et al. did not show the interaction between Si and Ni substrate. Therefore, two more
studies conducted in this work are presented in Table D.1. In Simulation II, the a-Si and Ni pillar
were modeled as continuous solid with different properties. Ni was modeled with an elasticplastic material. As shown, the Ni pillar deformed with Si pillar and a high stress was generated
in Ni close to the interface. This deformation mode was different from the experimental
observation. In Simulation III, a soft interface with properties as shown in Fig. D.2 was
introduced between Si and Ni. This resulted in a large deformation in the interface layer while
the Ni pillar remained almost undeformed. The shape changes of the Si and Ni pillars after
lithiation and delithiation closely agreed with the experimental observation.

(a)

(b)

Figure D.2Mechanical properties for the bulk Si and the interface region are modeled separately.
(a) Young’s Modulus as a function of concentration; (b) yield stress as a function of
concentration.
122

Table D.1 Preliminary study on Si-Ni interface

Process

Substrate and
interface
properties

Experiment
[167]

-

Simulation
I [167]

Simulation
II

Lithiation

Delithiation

Ni substrate as
rigid substrate
Perfect bonding
between Si and Ni

Ni substrate with
realistic
mechanical
property

-

Perfect bonding
between Si and Ni

Simulation
III

Ni substrate with
realistic
mechanical
property
Thin film region
for Li segregated
interface
Start of
delithiation

123

End of
delithiation

This preliminary FEA tests in Table D.1 showed that the interface between Si and current
collector may influence the prediction of both shape change and the stress state in Si electrode
and substrate. Given Si interface with different current collector/binder/coating materials, this
kind of study should be conducted to correctly predict the mechanical response of them and their
influence to cell performance.

124

Appendix E

Stress State in Si Nanosphere

E.1 Si Nanosphere with Two Phase Lithiation
The mean stress evolution along particle radius in test 6 in Chapter 4 is shown in Fig. E.1(a).
The two-phase lithiation presents a huge effect on stress generated at reaction front because of
the high concentration gradient. Homogeneous mean stress in Li-poor core and inhomogeneous
in Li-rich shell are accumulated along with lithiation. A stress jump is seen at the reaction front
over a small shell thickness, showing the newly lithiated element is further compressed. The
similar analytical prediction has seen in literature [64]. Effect of the reaction front extends
further into the bulk phases. The Li-poor phase experiences tensile stress at first and gradually
reduces to compressive stress as more Si got lithiated. In the meantime, the lithiated silicon
located just outside reaction front suffers an elastic reloading due to the push from newly
lithiated silicon. As the lithiation is a dynamic process, reaction front effect is repeated and
added on to lithiated silicon. A stress fluctuation is seen in Li-rich phase. Overtime, stress
becomes smooth in early lithiated silicon. Eventually the stress contour in Fig. E.1(a) is seen
along particle radius as the particle got lithiated. The highest compressive stress stays at particle
center after two-phase lithiation finishes. Under the high compressive stress along the particle
lithiation, the particle tends not to fracture during the first lithiation.

125

(a)

(b)

Figure E.1 Hydrostatic stress along particle radius at selected times during (a) two phase
lithiation, and (b) single phase lithiation.

E.2 Si Nanosphere with Single Phase Lithiation
The single phase lithiation of Si nanosphere with diameter of 500 nm was also studied, with
the hydrostatic stress plotted in Fig. E.1(b). Distinct stress state was observed in single-phase
lithiation from the two-phase lithiation process. The stress changes smoothly without the
existence of the reaction front. The stress changes from compressive near the particle surface to
tensile inside. Especially at early stage of lithiation, the particle center experiences high tensile
hydrostatic stress, which would easily cause particle to fracture from inside and form hollow
spaces. Therefore, by designing hollow structured Si structure would be beneficial to the
structural stability and increase cyclability.

126

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