NUMERICALMODELINGOFENERGYSTORAGEMATERIALS ...nj008cy9126/...inspiring people. Without their endless guidance, encouragement, and support, this thesis would not be possible. First and

NUMERICAL MODELING OF ENERGY STORAGE MATERIALS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVIROMENTAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Xiaoxuan Zhang

November 2017

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/nj008cy9126

© 2017 by Xiaoxuan Zhang. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/nj008cy9126

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Christian Linder, Primary Adviser


Ronaldo Borja


Wei Cai

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

iv

Abstract

Lithium-ion batteries (LIBs) are important energy storage devices with applications ranging

from portable electronics to electric vehicles. Their widespread applications persistently

drive the advancement of LIB technologies. To improve energy density of LIBs, one approach

is to use novel electrode materials, such as Silicon (Si), Germanium (Ge), Tin (Sn), or

Sulfur (S), which have been investigated both experimentally and numerically. This work

mainly focuses on the numerical modeling of Si anodes to understand the important role of

mechanics in electrochemical systems.

Si is considered to be a promising anode material for LIBs, characterized by a theoretical

specific capacity as high as 4200 mAh/g, in comparison to 372 mAh/g for graphite, which

currently is the most common commercial anode material. The charging/discharging pro-

cess of Si anodes is highly complex, involving mass diffusion, electrochemical reaction, phase

transformation, and large mechanical deformation (∼300% volume change). The large vol-

ume change can cause fracture of both Si anodes and the solid electrolyte interphase (SEI)

layer, which is a thin layer formed on the anode to prevent unwanted side reactions between

the anode and the electrolyte, resulting in poor cyclic performance of Si-based LIBs. To

better understand the diffusion and mechanical behavior of Si anodes, sophisticated com-

putational models are needed to describe diffusion-induced elastoplastic deformation and

fracture of Si anodes.

In this work, a variational based electro-chemo-mechanical coupled computational frame-

work is formulated to study diffusion-induced mechanical deformation of Si anodes, where

a diffusive phase field fracture model is used to describe the crack formation and propa-

gation. To account for the underlying phase transformation of Si anodes (crystalline Si to

v

amorphous LixSi and amorphous Si to amorphous LixSi) during the initial charging process,

a physically motivated reaction-controlled diffusion (RCD) model is proposed. The RCD

model is incorporated into the newly proposed coupled variational framework to model

diffusion-induced anisotropic deformation for crystalline Si. With this fully coupled compu-

tational framework, we investigate the stress state, Lithium (Li) concentration distribution,

and phase boundary evolution during the lithiation process of Si anodes. We further inves-

tigate how electrode geometry and geometrical constraints affect the fracture behavior of

Si anodes. This work can be extended to study the formation and fracture of the SEI layer

on Si anodes and provides necessary computational tools for optimizing and designing high

energy density Si anodes for LIBs with outstanding cyclic performance in the future.

vi

Acknowledgments

Throughout my journey towards a PhD, I was fortunate to meet and interact with many

inspiring people. Without their endless guidance, encouragement, and support, this thesis

would not be possible.

First and foremost, I would like to thank my advisor, Prof. Christian Linder, for guiding

me to become intellectually independent. Christian taught me how to conduct research at a

high standard, write technically sound scientific reports, and deliver effective presentations.

He is a great mentor, giving me the flexibility to work on topics that personally appeal to

me and being endlessly supportive. His encouragement, guidance, and critiques made me

into a capable researcher today. It was my great honor to have Christian as my supervisor.

I am forever indebted to him.

I would like thank my other reading committee members, Prof. Ronaldo I. Borja and

Prof. Wei Cai, who have provided constructive and valuable feedbacks to improve this work.

I also want to express my gratitude to Prof. Michael Lepech and Prof. William Nix for

serving on my oral examination committee. I greatly appreciate all their time and support.

I also want to acknowledge the financial support provided by the Professor James M.

Gere Graduate Fellowship from Stanford University, the Samsung Global Research Partner-

ship, and the Bosch Energy Research Network (BERN) program. Without these generous

supports, I would never have received my PhD study at Stanford.

In addition, I want to thank Prof. Yi Cui for the insightful comments during our research

collaboration and Prof. Tong-Seok Han for the great research discussion during and after

his visit to the Linder group. I owe many thanks to the current and former members in

the Linder group, Andreas Krischok, Berkin Dortdivanlioglu, Emma Lejeune, Reza Rastak,

vii

Ren Gibbons, Ali Javili, Lihua Jin, Arun Raina, Mykola Tkachuk, Volker Schauer, who

have helped me greatly in the past in different capacities. I also want to give my thanks to

HyeRyoung Lee, Jinhyun Choo, Xiaoyu Song, Martin Tjioe, who have helped me in both

research and personal life.

Finally, I would like to thank my family. My beloved parents always unconditionally

support my studies. Though I often feel remorse that I am unable to help them on their

farm, I hope they will be proud of my PhD degree from “The Farm”. I also want to thank

my older brother, who has been a continuous role model since my childhood. My deep

thanks also go to my parents-in-law, who are willing to provide support whenever needed.

Last and most importantly, I want to give my deepest gratitude to my wife, Yue Zhao. I am

the luckiest man in the world to spend everyday with her. She is always very considerate

and supportive of my research. She sacrificed many weekends to stay in the library with

me when I was coding, writing manuscripts, or preparing this work. Without the countless

support, encouragement, and trust from her in the past several years, I would not have been

able to finish this work. To her, I dedicate this thesis.

viii

To my wife, Yue Zhao

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x

Contents

Abstract v

Acknowledgments vii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Lithium-ion battery . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Half-reaction and half-cell . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Electrochemical terminologies . . . . . . . . . . . . . . . . . . . . . . 3

1.1.4 Battery capacity and cyclic performance . . . . . . . . . . . . . . . . 5

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Experimental investigation . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Numerical investigation . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 An electro-chemo-mechanical coupled variational framework 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Continuous variational framework . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Primary fields and boundary conditions . . . . . . . . . . . . . . . . 16

2.2.2 Mixed variational principles and rate-type potentials . . . . . . . . . 20

2.3 Discrete variational framework . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Time discretization scheme . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Condensation of local fields . . . . . . . . . . . . . . . . . . . . . . . 30

xi

2.3.3 Reduced global variational principle . . . . . . . . . . . . . . . . . . 31

2.3.4 Staggered spatial discretization scheme . . . . . . . . . . . . . . . . . 32

3 Diffusion-induced deformation of Si anodes 35

3.1 Two-phase lithiation of Si anodes . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 Reaction-controlled diffusion model . . . . . . . . . . . . . . . . . . . 37

3.2 Numerical implementation procedure . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Update of the reaction front location . . . . . . . . . . . . . . . . . . 41

3.2.2 Flow chart for modeling diffusion-induced swelling of Si anodes . . . 43

4 Representative numerical examples 47

4.1 Two-phase lithiation with the reaction-controlled diffusion model . . . . . . 47

4.1.1 Pressure effects on the lithiation process . . . . . . . . . . . . . . . . 49

4.1.2 Lithiation of crystalline Si nanoparticles . . . . . . . . . . . . . . . . 56

4.1.3 Lithiation of amorphous Si nanoparticles . . . . . . . . . . . . . . . 56

4.1.4 Lithiation of post amorphous Si nanoparticles . . . . . . . . . . . . . 59

4.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Diffusion-induced deformation of Si anodes . . . . . . . . . . . . . . . . . . 65

4.2.1 Diffusion-induced deformation of 〈100〉 crystalline Si nanopillars . . 65

4.2.2 Diffusion-induced deformation of 〈110〉 crystalline Si nanopillars . . 71

4.2.3 Diffusion-induced fracture of irregular Si nanoparticles . . . . . . . . 75

4.2.4 〈110〉 crystalline Si nanopillars under geometric constraints . . . . . 76

5 Summary and future directions 81

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xii

List of Tables

1.1 Summary of frequently used electrochemical terminologies related to LIBs. . 4

3.1 Summary box of the variational framework to model diffusion-induced large

plastic deformation and phase field fracture during initial two-phase lithiation

of Si electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 Summary of the parameters used in the simulations performed in Section 4.1 49

4.2 Summary of the parameters used in the simulations performed in Section 4.2 66

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xiv

List of Figures

1.1 Illustration of different applications of LIBs: personal portable electronics,

electric vehicles, and energy storage systems. . . . . . . . . . . . . . . . . . 1

1.2 Illustration of a correlative light and electron microscopy image of the cross

section of a cylindrical LIB (Type 18650) (from [5]), which shows a sandwich-

type of internal structure (left) and a schematic illustration of the typical

internal structure of LIBs with arrows indicating the moving directions of

electrons (gray particles) and Li ions (yellow particles) during a charging

process (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Illustration of the evolution and prediction of the specific energy of 18650-

type LIBs in 30 years (from [8]), which shows an approximately 5% cumula-

tive annual growth rate of the cell specific energy. . . . . . . . . . . . . . . 5

1.4 Illustration of the poor cyclic performance of Si nanowire anodes [10]. After

less than 150 charging/discharging cycles, Si nanowire anodes retain only

about 20% of their original capacity. . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Illustration of SEI formation, fracture, and reformation [10]. A thin SEI layer

forms on the surface of the anode during the initial charging process. The

large mechanical deformation of Si anode can cause the SEI layer to fracture,

leading to the formation of additional SEI materials. . . . . . . . . . . . . . 7

xv

1.6 Experimental observation of the two-phase lithiation process in c-Si (a,b) and

a-Si (c-f), and diffusion-induced anisotropic deformation in c-Si (g-h). The

a-LixSi and c-Si (a-Si) form a very sharp phase boundary. (a) Lithiation of

c-Si wafer [17], (b) lithiation of c-Si nanoparticle [18], (c) lithiation of a-Si

coated on a carbon nanofiber (CNF) [19], (d-f) lithiation of a-Si nanoparticle

with the selected area electron diffraction pattern [20]. The diffusion-induced

anisotropic deformation for crystalline nanopillars [21] is shown in (g-h). . 8

1.7 Experimental illustration of diffusion-induced fracture where cracks are formed

in a brittle manner [39]. The scanning electron microscope images show mul-

tiple cracks formed after a lithiation and delithiation cycle of a-Si thin film

electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Illustration of undeformed (left) and deformed (right) configuration of the

electro-chemo-mechanical coupled initial boundary value problem, formu-

lated in terms of the three primary fields ϕ, µ, d and loaded through a

body force γ in B and external loading applied at the boundary ∂B. The lat-

ter is divided into ∂B = ∂Bϕ ∪ ∂Bt, ∂B = ∂Bµ ∪ ∂Bh, and ∂B = ∂Bd ∪ ∂B∇d

for the mechanical, the diffusive, and the fracture problem, respectively. The

outer shell represents the amorphous LixSi alloy and the shaded center repre-

sents the unlithiated Si core, indicating a two-phase lithiation process. The

red line in the deformed configuration represents the diffusion-induced crack

which is approximated by the phase field d with a diffusive crack topology

controlled by a regularized length parameter l. . . . . . . . . . . . . . . . . 17

xvi

3.1 Illustration of the two-phase lithiation mechanism and the numerical simu-

lation setup. (a) The diffusion coefficient D depends on the reaction front

location, where D = D0 for the obtained a-LixSi region behind the reaction

front and D = Ds for the remaining Si region ahead of the reaction front.

(b) The 1D simulations in Section 4.1 represent the lithiation process of Si

nanoparticles with an initial radius of R0. The finite difference method is used

with a second-order central difference approximation of the space derivative

and a backward difference approximation of the time derivative. Along the

radius, a zero-flux boundary condition is applied at x1 and a c = 1 essential

boundary condition is applied at xn+1. The radius is discretized into n ele-

ments. rk and rk+1 represent the reaction front location at time t = k and

t = k+1, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Illustration of the value of f(c) varying with concentration c in (3.5). . . . . 39

3.3 Illustration of the reaction front updating scheme. . . . . . . . . . . . . . . 42

4.1 Illustration of the different reaction front pressure values versus radial lo-

cation used in the simulation, where the left figure represents the pressure

reported in [45] and the right one represents the pressure computed from (4.1). 49

4.2 c-Si simulation – Illustration of the concentration profile versus radial location

at different time steps for E0 = 0.69 eV (a-c), and the reaction front location

versus time (d) corresponding to (a-c). (a-c) An evident reaction front is

observed for each simulation. The Li concentration profile is high in the

lithiated region. (d) The reaction front velocity is constant when the pressure

is neglected and slows down when the pressure is included. . . . . . . . . . 51

4.3 a-Si simulation – Illustration of the concentration profile versus radial loca-

tion at different time steps for E0 = 0.58 eV (a-c), and the reaction front

location versus time (d) corresponding to (a-c). (a-c) An evident reaction

front is observed for each simulation. (d) The reaction front velocity slows

down significantly due to the pressure [45]. . . . . . . . . . . . . . . . . . . 52

xvii

4.4 post-a-Si simulation – Illustration of the concentration profile versus radial

location at different time steps for E0 = 0.50 eV (a-c), and the reaction

front location versus time (d) corresponding to (a-c). (a,c) A one-phase

lithiation process is observed for both simulations. (b) the high pressure

effect alternates the lithiation process from one-phase to two-phase. (d) The

reaction front velocity moves as fast as the diffusion process for (a) and (c)

and slows down for (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


at different time steps for different E0 with pmax = 2.5 GPa. The bond-

breaking energy barrier significantly affects the lithiation speed. For smaller

value of E0, more Si is lithiated. . . . . . . . . . . . . . . . . . . . . . . . . 54


at different time steps for different pmax with the same E0 = 0.65 eV. Like the

bond-breaking energy barrier, the reaction front pressure significantly affects

the lithiation speed. For smaller value of pmax, more Si is lithiated. . . . . . 55


at different time steps for E0 = 0.65 eV and pmax = 2.0 GPa (a,c,e), and

the reaction front location versus time (b,d,f) with the comparison of the

experimental data for three different initial radii. (a,c,e) An evident reaction

front is observed. In the lithiated region, a high Li concentration level is

obtained from the simulation. (b,d,f) The reaction front location curves

match the experimental observation and the gradient of those curves indicate

a slow-down tendency for the reaction front velocity. . . . . . . . . . . . . . 57

xviii


at different time steps for E0 = 0.65 eV and pmax = 2.0 GPa (a,c,e), and

the reaction front location versus time (b,d,f) with the comparison of the

experimental data for three different initial radii. (a,c,e) An evident reaction

front is observed. In the lithiated region, a high Li concentration level is

obtained from the simulation. (b,d,f) The reaction front location curves

match the experimental observation and the gradient of those curves indicate

a slow-down tendency for the reaction front velocity (cont.). . . . . . . . . . 58


tion at different time steps for different E0 with pmax = 2.5 GPa andR0 = 300

nm. The bond-breaking energy barrier significantly affects the concentration

profiles and the lithiation speed. The Li concentration level is higher in the

lithiated region for larger E0, but is much lower than the concentration level

for c-Si in Fig. 4.5. Similar as for c-Si, the lithiation speed is slower for larger

E0 as shown in (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


tion at different time steps for different pmax with E0 = 0.56 eV and R0 = 300

nm. The reaction front pressure significantly affects the concentration pro-

files and the lithiation speed. The Li concentration level is higher in the

lithiated region for higher pressure. Similar as for c-Si, the lithiation speed

is slower for larger pmax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


tion at different time steps for E0 = 0.55 eV (left), and the reaction front

location versus time (right) with comparison of the experimental data. . . . 61


location at different time steps for different E0 with pmax = 2.5 GPa for

R0 = 300 nm. Since the bond-breaking energy barriers used here are very

small, the one-phase lithiation process is recovered for all the different energy

barriers under the given pressure effect. The lithiation process proceeds as

fast as the diffusion process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xix


location at different time steps for different pmax with the same E0 = 0.48

eV for R0 = 300 nm. The reaction front hydrostatic pressure has less effect

on the concentration profiles and the lithiation speed, when compared with

Fig. 4.6 and 4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.13 〈100〉 c-Si nanopillars. (a) Illustration of the diffusion-induced anisotropic

deformation and crystalline direction dependent fracture behavior for 〈100〉

c-Si nanopillars. Fracture locations are indicated by red arrows [21]. (b)

Illustration of the crystalline directions of a 〈100〉 c-Si nanopillar as well as

the finite element mesh and mechanical boundary conditions used in Section

4.2.1. Locations A and B are used to report the evolution of hoop stresses in

Fig. 4.14(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.14 〈100〉 c-Si nanopillars. (a) Evolution of hoop stress at points labelled as A and

B in Fig. 4.13(b) for four different simulation setups: one-phase lithiation

with an elastic model; one-phase lithiation with an elasto-plastic model; two-

phase lithiation for an isotropic elasto-plastic deformation with a uniform E0;

and two-phase lithiation for an anisotropic elasto-plastic deformation with

different values of E0 in different crystalline directions. (b) concentration

profile along the radius of the Si electrode undergoing two-phase lithiation for

different diffusion coefficients at t = 20 s. The larger the diffusion coefficient

the larger is the concentration gradient at the reaction front. . . . . . . . . 68

4.15 〈100〉 c-Si nanopillars. Plots of the concentration field c and the damage

field d for the two-phase lithiation process with gc = 12.5 J/m2 at SOC =

0.20, 0.25, and 0.35. (a) The c-Si nanopillar deforms anisotropically. The

concentration profiles show a high concentration gradient at the reaction

front. The black lines indicate a contour for the damage field d = 0.95. (b)

The diffusion-induced anisotropic deformation causes the fracture to initi-

ate close to the 〈100〉 direction and to propagate inward, restricted by the

reaction front moving speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xx

4.16 〈100〉 c-Si nanopillars. (a) Relation between SOC and crack length of a c-Si

nanopillar with R = 300 nm for three different gc. (b) Plot of SOC at the

onset of crack nucleation in Si nanopillars of different radii and gc. Points

surrounded with a square marker indicate results without fracture onset. The

simulation results are compared with statistical results from experiments,

which are plotted as bar graphs in (b), where nanopillars with radius of 100

nm, 145 nm and 195 nm, 245 nm show 100% non-fracture ratio and 0%

non-fracture ratio, respectively [58]. . . . . . . . . . . . . . . . . . . . . . . 71

4.17 〈110〉 c-Si nanopillars. (a) Illustration of the diffusion-induced anisotropic

deformation and crystalline direction dependent fracture behavior for 〈110〉

c-Si nanopillars. Fracture locations are indicated by red arrows [21]. (b)

Illustration of the crystalline directions of a 〈110〉 c-Si nanopillar as well as

the finite element mesh and mechanical boundary conditions used in Section

4.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.18 〈110〉 c-Si nanopillars. Plots of the concentration field c and the damage

field d for the two-phase lithiation process with gc = 12.5 J/m2 at SOC =

0.28, 0.35, and 0.45. (a) The c-Si nanopillar deforms anisotropically. The

concentration profiles show a high concentration gradient at the reaction

front. The black lines indicate a contour for the damage field d = 0.95. (b)

The diffusion-induced anisotropic deformation causes the fracture to initi-

ate close to the 〈100〉 direction and to propagate inward, restricted by the

reaction front moving speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.19 〈110〉 c-Si nanopillars. (a) Relation between SOC and crack length of a c-Si

nanopillar with R = 300 nm for three different gc. (b) Plot of SOC at the

onset of crack nucleation in Si nanopillars of different radii and gc. Points

surrounded with a square marker indicate results without fracture onset. The

simulation results are compared with statistical results from experiments,

which are plotted as bar graphs in (b), where nanopillars with radius of 130

nm and 145 nm show 100% non-fracture ratio but those with radius of 180

nm and 195 nm show a 95% and 0% non-fracture ratio, respectively [58]. . 74

xxi

4.20 Irregular Si nanoparticles. (a) Experiments show that lithiated irregular Si

nanoparticles may form multiple cracks [159]. (b) Illustration of the finite

element mesh used in Section 4.2.3. . . . . . . . . . . . . . . . . . . . . . . . 74

4.21 Irregular Si nanoparticles. (a-c) Profile of the damage field d for a Si nanopar-

ticle with three different fracture energy release rates at a SOC of 0.35 (ele-

ments with d ≥ 0.96 are deleted in a postprocessing step for plotting purposes

only). The numbers in (a-c) indicate the crack initiation sequence. (d) Values

for the SOC at the onset of crack initiation for each crack. . . . . . . . . . . 75

4.22 〈110〉 c-Si nanopillars under geometric constraints. (a) Experiments show

that geometric constraints affect the fracture behavior of 〈110〉 c-Si nanopil-

lars [160]. The ellipse schematically shows the geometry of a lithiated nanopil-

lar and the potential crack locations at the top and bottom when there is

no geometric constraint. The red arrows show the altered crack locations

when a geometric constraint is present. (b) Illustration of the mesh used in

Section 4.2.4, where the initial gap between the nanopillar and the rigid wall

is denoted as ∆. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.23 〈110〉 c-Si nanopillars under geometric constraints. (a-c) The damage field d

for a 〈110〉 c-Si nanopillar with R = 500 nm under geometrical constraints

with different ∆/R ratios for gc = 12.5 J/m2 at SOC = 0.27 is shown. (d-f)

SOC at fracture onset (maximum damage value d = 0.8) for different radii

R = 400 nm and R = 500 nm, different ratios ∆/R = 0.15 (strong geometric

constraint) to ∆/R = 0.4 (no geometric constraint), and different fracture

energy release rates gc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.23 〈110〉 c-Si nanopillars under geometric constraints. (a-c) The damage field d

for a 〈110〉 c-Si nanopillar with R = 500 nm under geometrical constraints

with different ∆/R ratios for gc = 12.5 J/m2 at SOC = 0.27 is shown. (d-f)

SOC at fracture onset (maximum damage value d = 0.8) for different radii

R = 400 nm and R = 500 nm, different ratios ∆/R = 0.15 (strong geometric

constraint) to ∆/R = 0.4 (no geometric constraint), and different fracture

energy release rates gc (cont.). . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xxii

Chapter 1

Introduction

1.1 Motivation

1.1.1 Lithium-ion battery

The first commercialized lithium-ion battery (LIB) was introduced by Sony Corporation in

1991 [1]. Due to their high energy density, LIBs have become important energy storage

devices for portable electronics and electric vehicles, as shown in Fig. 1.1, with billions of

units being produced each year [2]. In addition, LIBs have also been used in energy storage

systems, as shown in Fig. 1.1, to regulate abundant green energy, such as solar energy and

wind energy.

LIBs are electrochemical systems where chemical reaction, mass diffusion, thermal dis-

sipation, and mechanical deformation are involved. The size and shape of LIBs vary for

Figure 1.1: Illustration of different applications of LIBs: personal portable electronics,electric vehicles, and energy storage systems.

1

2 CHAPTER 1. INTRODUCTION

Cathode Anode

Separator

Al current

collector

Cu current

collector

Anode:Graphite, Si

Cathode:LixCoO2

Separator

ElectrolyteElectrolyte

Current

collector

Figure 1.2: Illustration of a correlative light and electron microscopy image of the crosssection of a cylindrical LIB (Type 18650) (from [5]), which shows a sandwich-type of internalstructure (left) and a schematic illustration of the typical internal structure of LIBs witharrows indicating the moving directions of electrons (gray particles) and Li ions (yellowparticles) during a charging process (right).

different applications. However, the internal structure of LIBs remains similar across differ-

ent geometries. LIBs typically have a sandwich-type of structure, consisting of five major

parts: the cathode (where the reduction reaction occurs during discharging), the anode

(where the oxidization reaction occurs during discharging), the electrolyte (which has good

ionic conductivity, but poor electronic conductivity), the separator (a porous material that

allows ions to pass but blocks electrons), and current collectors, as shown in Fig. 1.2. In

general, conductive binder materials are added to improve mechanical integrity and internal

electronic conductivity of LIBs [3].

During the charging process, the anode is connected to a negative potential, where

the reduction reaction occurs, and the cathode is connected to a positive potential, where

the oxidation reaction occurs [4]. In this process, Li ions move from the cathode to the

anode under an externally applied potential, as shown in Fig. 1.2. The anode is lithiated

and energy is stored in electrodes. During the discharging process, Li ions spontaneously

move from the anode to the cathode. The anode is delithiated and energy is released from

electrodes.

1.1. MOTIVATION 3

For LIBs, their safety, capacity, and cyclic performance are the three most critical fea-

tures that researchers try to understand and improve. Battery safety, where studies focus

on various factors, such as geometry design, thermal management, external mechanical

loading, gassing, or thermal runway, is a challenging research topic beyond the scope of this

research. In this work, we will focus on battery capacity and cyclic performance, which will

be discussed in Section 1.1.4.

1.1.2 Half-reaction and half-cell

For an electrochemical reaction, reduction and oxidation occur on opposite electrodes [6].

For example, considering a LIB with a Si anode and a LiCoO2 cathode, the overall electro-

chemical reaction has the following form

LixSi + Li1−xCoO2 Si + LiCoO2 (1.1)

where we have the cathodic half-reaction as

Li1−xCoO2 + xLi + xe− LiCoO2 (1.2)

and the anodic half-reaction as

LixSi Si + xLi + xe−. (1.3)

The spatially separated reduction and oxidation for an electrochemical reaction divides

an electrochemical cell into two half-cells, where each half-cell contains an electrode and

the electrolyte. When developing novel electrodes, e.g. Si anodes, researchers simplify the

investigation by focusing on only a half-cell rather than a full cell, with the half-reaction

typically being controlled by an externally applied electrical potential [7].

1.1.3 Electrochemical terminologies

For an electrochemical system, the flow of electric charge is called current with a unit of

ampere (A). The electric potential difference between two terminals is called voltage with


Table 1.1: Summary of frequently used electrochemical terminologies related to LIBs.

Name Units Description

Capacity A · hCoulometric capacity, the total Amp-hoursthat a battery can discharge at a particularcurrent.

Specific power W/kgThe maximum available power per unit vol-ume for a battery.

Specific capacity (A · h)/kgThe maximum capacity per unit mass of abattery.

Specific energy (W · h)/kgThe maximum energy per unit mass of a bat-tery.

Energy density (W · h)/m3 The maximum energy per unit volume (vol-umetric energy density) of a battery.

Open circuit potential VVoltage difference between two battery ter-minals without loading.

Overpotential VPotential difference between thermodynami-cally equilibrium potential and measured po-tential of a half-reaction.

Cut-off voltage VA voltage that indicates the empty state of abattery.

C-rates −

Current that indicates battery discharge rate.1C means discharge a entire battery at a con-stant current in 1 hour. And 2C, C/2 meansdeplete a battery at a constant current in 0.5h and 2 h, respectively.

State of Charge (SOC) −The ratio between battery current capacityand its maximum capacity.

a unit of volts (V) or joule (J) per coulomb (C). Power and energy can be calculated as

W = V · A and E = W · h, respectively, where h represents hour. Several frequently

used terminologies related to LIBs are summarized in Table 1.1. Though specific capacity,

specific energy, and energy density have different physical meanings, they can be converted

interchangeably for a given LIB. In this work, we only focus on Si-based half-cells without

specifying cathode materials, packaging materials, current collectors, binder materials, and

separators. Thus, we will characterize half-cells with their capacity, but not specific energy

or energy density, where the latter two explicitly depend on open circuit potential, total

mass, or total volume of battery cells.

1.1. MOTIVATION 5

1990 1995 2000 2005 2010 2015 2020

50

100

150

200

250

300

350

400

Speci

c E

nerg

y [

Wh/k

g]

Panasonic

Freedonia [2012]Linear t

Year

Figure 1.3: Illustration of the evolution and prediction of the specific energy of 18650-typeLIBs in 30 years (from [8]), which shows an approximately 5% cumulative annual growthrate of the cell specific energy.

1.1.4 Battery capacity and cyclic performance

LIB research has led to continuous improvement of energy density and specific energy since

1991, as given in Fig. 1.3, which indicates that 18650-type LIBs have an average specific

energy increase of 11.2 Wh/kg per year or roughly 5% cumulative annual growth rate. Such

improvements can attribute to aspects including but not limited to optimized electrode

structures, improved electrode/electrolyte interface design, better cell internal structures,

or the use of advanced electrode materials [9].

For a LIB cell, its maximum theoretical specific capacity can be estimated as

1

Scell=

1

Scathode+

1

Sanode. (1.4)

This calculation does not consider current collectors, separator, electrolyte, and binder

materials. Here, Scell, Scathode, and Sanode represent the specific capacity for the cell, the

cathode, and the anode, respectively. To illuminate the possible benefits of Si materials,

we consider two LIB cells with the same cathode material LiCoO2 but different anode

materials. Based on Eq (1.4), the one with a Si anode, which has a maximum theoretical


Si nanowire anodes

2000

1500

1000

500

100

80

60

40

20

0

0 200 400

Cycle number

Capacit

y (

mA

h g

-1

Capacit

y r

em

ain

ed

Figure 1.4: Illustration of the poor cyclic performance of Si nanowire anodes [10]. After lessthan 150 charging/discharging cycles, Si nanowire anodes retain only about 20% of theiroriginal capacity.

specific capacity of 4200 mAh/g, can achieve a theoretical cell specific capacity increase

as high as 35% compared with the other one using a graphite anode, which has a specific

capacity of 372 mAh/g. This is a significant capacity improvement compared with the

approximated 5% cumulative annual growth rate since 1991, as shown in Fig. 1.3.

The high theoretical specific capacity makes Si one of the most promising next-generation

anode materials for LIBs. However, the large amount of Li ions diffused into Si anodes causes

large volume change of the electrodes (as high as 300%), resulting in poor cyclic perfor-

mance, as shown in Fig. 1.4, where Si nanowire anodes retain only about 20% capacity

after fewer than 150 cycles, compared with almost 75% original capacity retained after 500

cycles for commercial Panasonic NCR18650 LIBs [11].

Battery cyclic performance is tremendously important since battery lifespan has both

high economical and high environmental impact. Cyclic performance can be impacted by

many factors, such as binding material aging, electrolyte decomposition, formation of solid-

electrolyte interphase (SEI) layer, side reactions, internal resistance increase, etc., [12–15].

Among all the factors, SEI formation is a key factor affecting battery lifetime and safety

[12].

1.1. MOTIVATION 7

Si

Eolyte

LxSi

SEI

chargingcharging discharging many cycles

Figure 1.5: Illustration of SEI formation, fracture, and reformation [10]. A thin SEI layerforms on the surface of the anode during the initial charging process. The large mechanicaldeformation of Si anode can cause the SEI layer to fracture, leading to the formation ofadditional SEI materials.

The SEI layer is a thin layer formed at the interface between the anode and the elec-

trolyte in LIBs due to the irreversible electrochemical decomposition of the electrolyte. For

a LIB, this layer should be thin enough to transport Li ions, but thick enough to prevent

side reactions between the electrode and the electrolyte. The SEI formation consumes active

materials and reduces battery capacity. For Si-based LIBs, their poor cyclic performance is

mainly attributed to the mechanically unstable SEI layer, as illustrated in Fig. 1.5, where

a thick SEI layer forms after many charging/discharging cycles due to the fracture and

reformation of the SEI layer [10], consuming significant amount of active materials and

significantly reducing battery capacity, as shown in Fig. 1.4.

A good understanding of SEI formation, fracture, and reformation processes is extremely

important for designing electrodes with improved cyclic performance. However, the intrinsi-

cally involved complex chemical reactions and its small thickness makes the characterization

of the SEI layer experimentally challenging [16]. For Si anodes, diffusion-induced large me-

chanical deformation poses additional challenges to understand the SEI layer. To improve

our understanding of this layer, sophisticated computational models are needed to describe

this electro-chemo-mechanical coupled problem. In this work, we focus on the development

of coupled computational models to study diffusion-induced deformation and fracture of Si

anodes, which lays the foundation to study SEI formation and fracture in the future.

8 CHAPTER 1. INTRODUCTIONts

(a)

(b)

(c)

(d) (e)

(f)

a-Si

a-LixSi

a-LixSi

a-LixSi

c-Si

c-Si

5nm

Phase Boundary

CNF

a-Si19nm

a-LixSi

7nm

PhaseBoundary

(g) (h)

Figure 1.6: Experimental observation of the two-phase lithiation process in c-Si (a,b) anda-Si (c-f), and diffusion-induced anisotropic deformation in c-Si (g-h). The a-LixSi and c-Si(a-Si) form a very sharp phase boundary. (a) Lithiation of c-Si wafer [17], (b) lithiation ofc-Si nanoparticle [18], (c) lithiation of a-Si coated on a carbon nanofiber (CNF) [19], (d-f)lithiation of a-Si nanoparticle with the selected area electron diffraction pattern [20]. Thediffusion-induced anisotropic deformation for crystalline nanopillars [21] is shown in (g-h).

1.2 Background

1.2.1 Experimental investigation

The concept to use Si as battery electrode material is not new [22–24]. The (de)lithiation

process of Si is highly complex, involving mass diffusion, electrochemical reaction, and

mechanical deformation. Due to the associated diffusion-induced large deformation, Si elec-

trodes fracture and eventually pulverize, resulting in very poor cyclic performance [25, 26].

Recent advancements in nanotechnologies allow the fabrication of nano-sized Si electrodes

with significantly improved cyclic performance [7, 10, 27, 28, 18, 29]. However, nano-sized

Si electrodes face new challenges such as low packing density and high cost [30]. To ad-

dress these new challenges, micrometer-sized electrodes are needed, which requires good

understanding of the diffusion process and mechanical deformation process of Si electrodes.

1.2. BACKGROUND 9

With the development of in situ transmission electron microscopy (TEM) electrochemi-

cal cells technology [31] in recent years, many interesting features of diffusion processes of Li

in Si anodes have been observed. For crystallize Si (c-Si), (i) a two-phase diffusion process

is reported in [18, 32, 33] for the initial lithiation process, as shown in Fig. 1.6(a,b). As the

lithiation proceeds, a layer of amorphous Li-Si alloy (a-LixSi), the transparent region in Fig.

1.6(a,b), is produced. The newly formed a-LixSi and the remaining c-Si forms an evident

phase boundary (or reaction front). The lithiated a-LixSi shell may even crystallize into

the Li15Si4 phase during lithiation with the presence of a c-Li core [32]. (ii) The reaction

front is observed to have a nanoscale thickness with a high Li concentration gradient [17].

(iii) During the initial lithiation of c-Si nanoparticle electrodes, the reaction front is found

to slow down as it progresses towards the core [32]. (iv) The initial lithiation of c-Si is

anisotropic, resulting in an anisotropic deformation, with a preferred crystal direction 〈110〉

[34, 21]. (v) Diffusion-induced large plastic deformation and fracture is observed in experi-

ments [35, 21]. (vi) Fracture of c-Si electrodes during the initial lithiation is size dependent,

with a critical diameter of ∼ 150 nm for nanoparticles [18, 32].

For amorphous Si (a-Si), (i) recent experiments show a similar two-phase diffusion pro-

cess during initial lithiation [19, 20], as illustrated in Fig. 1.6(c-f), where a distinct reaction

front can be observed between a-LixSi and a-Si. However, the Li concentration gradient

around the reaction front is not reported in those experiments. (ii) The phase boundary

is observed to move at a constant speed [19, 20]. (iii) A two-stage lithiation process is

postulated in [19] for a-Si where the Li concentration first reaches x ∼ 2.5 in a-LixSi, fol-

lowed by a second lithiation stage when x ∼ 3.75. (iv) A similar size-dependent fracture

phenomenon is observed with a critical diameter of ∼ 870 nm [20]. As for amorphous Si

obtained after the delithiation process (named as “post-a-Si” in this work), a one-phase

lithiation is observed in experiments [20].

Mechanical properties of Si anodes are measured in different experimental works. Bulk

modulus of crystalline LixSi is studied in [36]. In [17, 35, 37], the yield stress of Si thin-film

electrodes is measured during the (de)lithiation process. Young’s modulus and the hardness

of lithiated Si is measured in [38]. Fracture energy of lithiated Si thin-film electrodes is

measured for various Li concentrations in [39].


1.2.2 Numerical investigation

Even though nanotechnology [7] has shed lights on the practical application of Si as the

anode material, improved understanding of the reaction and diffusion behavior and mechan-

ical behavior of Si anodes is still required, especially when designing micrometer-sized Si

electrodes. To capture the aforementioned two-phase diffusion mechanism, a purely empiri-

cal concentration dependent diffusion coefficient is commonly used to describe the sharp Li

concentration drop in the lithiation of c-Si electrodes [18, 40], where the interfacial reaction

is ignored and simply replaced by a non-linear diffusion process. A flexible sigmoid function

is also used to create Li profiles with either a sharp phase boundary for two-phase lithiation

or a gradually varying concentration for one-phase lithiation in [41] without though mod-

eling the diffusion process as done in this work. The Cahn-Hilliard theory [42] for phase

separation also captures the phase transformation from c-Si to a-LixSi in [43, 44], where a

concentration gradient dependent interfacial contribution in the free energy formulation is

used to describe the interfacial region and to control the interfacial size, but the chemical

reaction rate is not taken into account. In [45], a concurrent reaction model for the lithia-

tion of c-Si is proposed, where both the stress and reaction rate are accounted for, but the

fact that the stress can stall the reaction in the model does not agree with the fact that the

reaction front can be extremely slow but never stops [32]. Also, the model is not able to

capture concentration gradients at the sharp interface region, which might be a key factor

affecting the reaction-limited diffusion process. Similar as in [45], the competition between

chemical reaction and species diffusion is treated by a dimensionless parameter in [46], but

accounting for large deformations with the consideration of the reaction front velocity, which

is assumed to be constant and is thereby not in line with experimental observations for c-Si

[32].

To better understand the experimentally observed complex mechanical behavior of Si

electrodes, numerous numerical investigations are carried out in the literature. Chemo-

mechanical coupled models for elasto-plastic deformation at large strain are developed in

[47–50, 46, 51] to study the diffusion-induced swelling. One can refer to a recent review

paper [52] for a more detailed discussion. Theoretical investigations are carried out to study

1.2. BACKGROUND 11

a−LixSi

Cu

Glass

2µm1µm

Figure 1.7: Experimental illustration of diffusion-induced fracture where cracks are formedin a brittle manner [39]. The scanning electron microscope images show multiple cracksformed after a lithiation and delithiation cycle of a-Si thin film electrodes.

crack nucleation [53, 54], the effect of charging rate on the fracture behavior [55, 56], the size-

dependency of fracture [57], and the resulting anisotropic deformation in c-Si nanopillars [58]

during the diffusion process in Si electrodes. To determine the onset of fracture, different

crack driving forces are proposed based on a chemo-mechanical coupled J-integral [49], the

maximum tensile strength theory [53], or the strain energy release rate [57].

To model the propagation of a crack in a failing material, several numerical tools exist.

Successful approaches are the embedded finite element method [59–74] or the extended

finite element method [75–78], both describing the crack as a discrete entity. Alternatively,

diffusive phase field approaches to fracture [79–83] are currently experiencing a dramatic

upsurge as those do not require the geometric information of a possible failure onset and

perform well when complicated failure surfaces are expected for cases when multiple cracks

are present or when cracks coalesce and branch. The phase field approach to fracture can

be traced back to the seminal work of [79, 80], where the sharp crack topology is replaced

with a regularized crack zone governed by a scalar damage variable and where an elegant

variational description of the resulting energy minimization problem is proposed. Extensions

are made in [84, 82] to account for tension-only induced fracture through a decomposition

of the free energy into a tensile and compressive part, and to prevent crack reversal through

the introduction of a history field for the crack driving force. A staggered update scheme

is proposed in [83, 85] to efficiently solve the resulting system of equations. Phase field

approaches to brittle fracture are further extended to account for dynamic fracture [86, 87],


higher order approximations of the damage field [88, 89], shells [90–93], different types

of materials [94–98], multiphysics problems [85, 99–108], and ductile fracture [109–114].

Additional works on the recent development of phase field approaches to fracture can be

found in the special issue [115].

1.3 Outline

The research goal of this work is to study the diffusion and mechanical behavior of Si

anodes to understand the important role of mechanics in electrochemical systems. The

main content of this work is adapted from previously published journal articles [116, 104].

The rest of this work is organized into Chapters 2-5, which are summarized below.

In Chapter 2, we present a general electro-chemo-mechanical coupled variational frame-

work for diffusion-induced elastoplastic deformation and fracture. Here, fracture is mod-

eled via a diffusive phase field approach, and the electrical information is obtained by a

phenomenological Butler-Volmer kinetic equation. We first summarize the continuous vari-

ational framework in Section 2.2. The discretized framework is then derived by adopting a

temporal and staggered spatial discretization scheme in Section 2.3.

In Chapter 3, a physics-based reaction-controlled diffusion (RCD) model is first proposed

to understand the underlying physics of the two-phase lithiation of Si electrodes during

the initial charging process in Section 3.1. The RCD model is further combined with

the variational framework from Chapter 2 to model the crystalline-direction dependent

diffusion-induced anisotropic deformation of Si anodes, where the numerical implementation

procedure is given in Section 3.2.

In Chapter 4, we present numerical simulations to show the performance of our com-

putational models developed in Chapters 2 and 3. In section 4.1, several 1D numerical

simulations are performed to show the versatility of the proposed RCD model, following by

a discussion on the importance of considering bond-breaking energy barrier and hydrostatic

pressure in the RCD model. We then study diffusion-induced deformation of Si anodes with

various geometries and parameters in Section 4.2. We show that both the electrode sizes

and the geometric constraints can affect the fracture behavior of Si anodes.

1.3. OUTLINE 13

In Chapter 5, we summarize the main contribution of this work and discuss possible

directions for future research of this fascinating problem.


Chapter 2

An electro-chemo-mechanical

coupled variational framework

This chapter is adapted from: X. Zhang, A. Krischok, C. Linder, “A variational framework

to model diffusion induced large plastic deformation and phase field fracture during initial

two-phase lithiation of silicon electrodes”, Comput. Meth. Appl. Mech. Eng., 312:51-77

(2016).

In this chapter, we present a robust electro-chemo-mechanical coupled variational frame-

work for diffusion-induced large elasto-plastic deformation and phase field fracture.

2.1 Introduction

To better understand the complicated interplay of all the complex physical processes arising

during the initial lithiation of Si electrodes, we extend existing phase field approaches for

modeling fracture in electrode materials [117, 118, 107] and propose in this work a varia-

tional based electro-chemo-mechanical coupled computational framework that accounts for

diffusion-induced plastic deformation and diffusion-induced fracture. Supported by exper-

imental observations shown in Fig. 1.7, we treat the diffusion-induced fracture as being

brittle. Due to the variational based origin of the phase field approach to fracture we for-

mulate the diffusion process, elasto-plastic deformation, and diffusion-induced fracture in

15

16 CHAPTER 2. A COUPLED VARIATIONAL FRAMEWORK

terms of mixed variational principles and rate-type potentials. Following [119–122, 44, 123–

125], after properly defining the rate-type potential functional of the problem, the primary

unknowns are solved through an incremental variational formulation for dissipative solids,

resulting in an efficient finite element approximation of this complex coupled problem due

to the inherent symmetry of the incremental variational principle. The notion of algorith-

mic functionals further motivates a staggered scheme in line with [83, 85, 94] to solve the

update of the fracture phase field and the coupled deformation-diffusion problem as two

successive updates to avoid a computationally more expensive monolithic scheme. In this

work, rather than using the concentration field as the primary unknown of the diffusion pro-

cess [126, 51, 127], we derive a variational framework that makes use of a global chemical

potential, similar to works such as [128–130].

2.2 Continuous variational framework

In this section, we summarize the continuous variational framework for diffusion-induced

elasto-plastic deformation and phase field fracture in Si electrodes of LIBs. After defining

the primary fields and boundary conditions of the coupled problem in Section 2.2.1, we

follow [123, 44, 94, 103] and set up the corresponding variational framework in terms of

rate-type potentials in Section 2.2.2 and specify those rate-type potentials in Section 2.2.2.1

with corresponding energy storage and dissipation functions in Section 2.2.2.2 as well as

constitutive relations in Section 2.2.2.3. Finally, we arrive at the Euler equations for the

underlying problem in Section 2.2.2.4.

2.2.1 Primary fields and boundary conditions

The diffusion-induced elasto-plastic deformation and fracture problem is formulated as a

three-field problem characterized by the deformation field ϕ, the chemical potential µ, and

the damage phase field d as global primary fields. The deformation field ϕ : B0 × T → B ∈

R

ndim is a non-linear map of points X in the reference configuration B0 to points x in the

current configuration B at time t ∈ T. The chemical potential µ : B0 × T → R drives the

evolution of the Li concentration c during the (de)lithiation processes and thereby causes

2.2. CONTINUOUS VARIATIONAL FRAMEWORK 17

BB0

γ

∂B∇d

∂B∇d0

∇d · n = 0

n

n

∂Bϕ

∂Bϕ

0

∂Bt

∂Bt0

t

∂Bµ∂Bµ0

∂Bh∂Bh

0

h

X x Γ

ϕ(X, t)

lSiSi

LixSiLixSi

Figure 2.1: Illustration of undeformed (left) and deformed (right) configuration of theelectro-chemo-mechanical coupled initial boundary value problem, formulated in terms ofthe three primary fields ϕ, µ, d and loaded through a body force γ in B and exter-nal loading applied at the boundary ∂B. The latter is divided into ∂B = ∂Bϕ ∪ ∂Bt,∂B = ∂Bµ ∪ ∂Bh, and ∂B = ∂Bd ∪ ∂B∇d for the mechanical, the diffusive, and the fractureproblem, respectively. The outer shell represents the amorphous LixSi alloy and the shadedcenter represents the unlithiated Si core, indicating a two-phase lithiation process. The redline in the deformed configuration represents the diffusion-induced crack which is approxi-mated by the phase field d with a diffusive crack topology controlled by a regularized lengthparameter l.

the large swelling deformation damaging the material. In the presence of fracture, the

crack phase field d : B0 × T → [0, 1] describes the diffusive crack topology with d(X , t) = 0

defining the virgin and d(X , t) = 1 the fully damaged state of the material at X ∈ B0 and

time t ∈ T. The material gradients of the deformation field and the chemical potential are

defined as

F = ∇ϕ(X, t) and M = −∇µ(X, t). (2.1)

We note that the concentration field c : B0 × T → [0, 1] is a normalized quantity, which

represents the actual concentration of the species Li in terms of moles per unit reference

volume with respect to the maximum concentration cmax and is treated as a local quantity


that can be recovered from the chemical potential µ. Following [131, 43], we consider a

multiplicative decomposition of the deformation gradient into

F = F eF cF p with F c = J1/3c 1 and Jc = 1 + Ωc. (2.2)

Here, F e, F c, and F p are the elastic, swelling, and plastic components of the deformation

gradient with Jacobians J = detF > 0, Jc = detF c > 0, and detF p = 1. In (2.2), we

assume that the diffusion-induced swelling is isotropic with Ω being the expansion coefficient.

Consequently, the Lagrangian and Eulerian strain tensors used to describe the elasto-plastic

deformation of the material can be computed as

C = F TF , Cp = F pTF p and b = FF T , be = J−2/3c FCp−1F T . (2.3)

The first Piola-Kirchhoff stress tensor in the bulk is denoted as P , which is related to the

Cauchy stress σ and the Kirchhoff stress τ by σ = J−1PF T and τ = PF T , respectively.

Introducing N as the outward normal of a reference area element dA and n as the outward

normal of a spatial area element da, we have

T = PN and t = σn (2.4)

where T and t are the corresponding traction vectors. Similarly, the material species flux

vector H in B0 and the spatial species flux vector h in B are related to the species outward

fluxes H and h by

H = H ·N and h = h · n. (2.5)

The introduction of the traction vector t and the species outward flux h allows us to specify

the boundary conditions for this coupled problem. As illustrated in Fig. 2.1, the overall

boundary ∂B of the body is divided into ∂B = ∂Bϕ ∪ ∂Bt with ∂Bϕ ∩ ∂Bt = ∅ for the

mechanical problem, ∂B = ∂Bµ ∪ ∂Bh with ∂Bµ ∩ ∂Bh = ∅ for the diffusive problem, and

∂B = ∂Bd ∪ ∂B∇d with ∂Bd = ∅ and ∂B∇d = ∂B for the fracture problem. The essential


and natural boundary conditions for those three fields are then given as

ϕ = ϕ(x, t) on ∂Bϕ and σn = t(x, t) on ∂Bt

µ = µ(x, t) on ∂Bµ and h · n = h(x, t) on ∂Bh

∇d · n = 0 on ∂B∇d

(2.6)

where ϕ, t, µ, and h are the prescribed values for the deformation, traction, chemical

potential, and species outward flux, respectively. In addition to the boundary conditions,

we have to specify the initial values of the chemical potential µ at time t = t0 as

µ(X , t = t0) = µ0(X) in B0 (2.7)

or equivalently written in terms of the Li concentration as c(X , t = t0) = c0(X) in B0.

2.2.1.1 Electrochemical boundary conditions

The natural boundary condition on ∂Bh in Eq (2.6) for the Li-ion diffusion is controlled by

the surface electrochemical reaction. The surface flux h on ∂Bh is related to the current

density i via

h =i

F0cmax(2.8)

where F0 is the Faraday constant. In Eq (2.8), the current density can be approximated by

a phenomenological Butler-Volmer kinetic equation [51, 132, 6]

i = i0

(

exp

[

(1− β)F0ηsRT

]

− exp

[

−βF0ηsRT

])

(2.9)

where β is a symmetry factor, R is the gas constant, T is the room temperature, ηs is the

overpotential, and i0 is the exchange current density. The overpotential ηs in Eq (2.9) is

computed as

ηs = V − V0 (2.10)

where V is the externally applied potential and V0 is the equilibrium potential. The ex-

change current density i0 in Eq (2.9) depends on the Li concentration c at the surface of


the electrodes and the Li concentration camb in the electrolyte with the form

i0 = F0kβa c

βk1−βc (1− c)1−βc1−β

amb (2.11)

where camb is the Li concentration in the electrolyte, ka and kc are the anodic and cathodic

reaction rates, which need to be measured experimentally or calculated with density func-

tional theory (DFT). The equilibrium potential V0 in Eq (2.10) is related to the chemical

potential of Li at the surface of electrodes [132] by

V0 = −µ

F0. (2.12)

During the charging/discharging process of a LIB cell, one can either control the applied

potential and measure the current output, or control the applied current and measure the

potential output. Eqs (2.8) to (2.12) provide the relationships to convert the applied current

or applied potential to a flux-type boundary condition for this IBVP, and to recover the

electrical potential information or current density information for the output.

2.2.2 Mixed variational principles and rate-type potentials

To establish a holistic modeling framework for the complex mechanisms arising during the

(de)lithiation process in Si electrodes of Li-ion batteries, we construct a mixed variational

framework based on a rate-type potential functional whose associated Euler equations yield

the governing equations for the underlying problem.

2.2.2.1 Multi-field rate-type potential

Inspired by similar frameworks for dissipative solids such as [119–122, 44, 123–125], we

construct a variational principle for which the evolution of the deformation field ϕ, damage

field d, and a set of local constitutive variables ξC as well as the state of the chemical

potential µ and non-evolving local fields ξF are governed by the stationarity of a proper

rate-type potential functional Π, which, for the coupled problem, is given as

Π(ϕ, µ, d, ξC, ξF) =d

dtE (ϕ, d, ξC) + D(µ, d, ξC, ξF)− P

ϕ,µext (ϕ, µ) (2.13)


at a given state ϕ, d, ξC. Here, ξF denotes a general array of local dissipative driving forces

and associated Lagrange multipliers. The general form of the energy storage functional E

is given as

E (ϕ, d, ξC) =

∫

B0

ψ(∇ϕ, d, ξC) dV (2.14)

with ψ as the free energy density. The dissipation potential functional D arises from dis-

sipative processes during plastic deformation, diffusion, and fracture. Its general form is

given as

D(µ, d, ξC, ξF) =

∫

B0

χ(µ,M, d, ξC, ξF;∇ϕ, d, ξC) dV (2.15)

with χ as the dissipation potential density. The external power functional Pϕ,µext (ϕ, µ)

consists of a mechanical contribution Pϕext(ϕ; t) and a diffusive contribution P

µext(µ; t),

which are both given as

Pϕext(ϕ; t) =

∫

B0

ρ0γ · ϕ dV +

∫

∂Bt0

T · ϕ dA and Pµext(µ; t) = −

∫

∂Bh0

µHdA (2.16)

where ρ0 is the reference density of the body, γ is the body force, and T and H are

the prescribed traction and outward species flux, respectively. The general forms of the

energy storage functional E and the dissipation potential functional D in (2.14)-(2.15) allow

us to recast the rate-type potential functional in (2.13) into an internal and an external

contribution as

Π(ϕ, µ, d, ξC, ξF) =

∫

B0

π(∇ϕ, µ,M, d, ξC, ξF) dV − Pϕ,µext (ϕ, µ) (2.17)

with π as the mixed internal rate potential density given as

π(∇ϕ, µ,M, d, ξC, ξF) =d

dtψ(∇ϕ, d, ξC) + χ(µ,M, d, ξC, ξF;∇ϕ, d, ξC) (2.18)

depending on the evolution of the free energy density ψ in time and the dissipation potential

density χ. To avoid overcomplicating this coupled problem, we neglect the local mass density

difference induced by the diffusion of Li.


2.2.2.2 Specification of energy storage and dissipation potential functionals

Next, the generalized forms of the energy storage functional E in (2.14) and the dissipation

potential functional D in (2.15) are specified in more detail for the coupled problem at

hand. The free energy density ψ in (2.14) can be written as

ψ(∇ϕ, d, ξC) = ψelas(F , c, d,Cp) + ψplas(Cp) + ψchem(c). (2.19)

Here, Cp ∈ ξC denotes in a generalized way the internal constitutive state array for the

plastic response, which is energetically conjugate to the general array of dissipative internal

forces Fp ∈ ξF. The explicit forms for ψelas, ψplas, and ψchem are given in Section 2.2.2.3

where we specify the chosen constitutive relations. The requirement that the evolution of

the stored energy must be less than the external power due to irreversible dissipative effects

is expressed by the global dissipation postulate

d

dtE (ϕ, d, ξC) ≤ P

ϕ,µext (ϕ, µ) (2.20)

which can be recast to a local dissipation postulate consisting of a part due to local action

Dloc = τ : d+ µc− ψ ≥ 0 (2.21)

and a part due to diffusion

Ddiff = H ·M ≥ 0 (2.22)

where we have introduced d = sym[l] with the spatial velocity gradient l = F F−1. The

definition of (2.19) and the demand (2.21) yields the constitutive relations, by adopting

the postulate that a set of constitutive equations is admissible only if these constitutive

equations are compatible with a non-negative entropy production [133]. Hence, we have

P = ∂Fψelas, µ = ∂cψ, Fp = −∂Cpψelas, βf = −∂dψelas (2.23)


along with the reduced local dissipation inequality

Dlocred = Fp · Cp + βf d ≥ 0 (2.24)

where we introduced the driving force βf ∈ ξF for fracture processes being conjugate to d.

Next, we specify in more detail the dissipation potential density χ in (2.15) for the subse-

quent construction of the variational framework for the dissipative material considered in

this work. For the problem at hand we divide it into plastic, diffusive, and fracture parts as

χ = χplas+χdiff+χfrac. The plastic dissipation potential density for rate-independent prob-

lems is defined by the principle of maximum dissipation [134], that defines the constrained

problem

χplas(Cp;Cp) = supFp∈Ep

Fp · Cp

(2.25)

where Ep = Fp | fplas(Fp;Cp) ≤ 0 models the elastic space of the dissipative forces

in terms of a yield function fplas(Fp;Cp). This constrained problem is solved by Lagrange

multiplier method after introducing the Lagrange multiplier λp ∈ ξF based on

χplas(Cp;Cp) = supFp,λp≥0

Fp · Cp − λpfplas(Fp;Cp)

(2.26)

inducing the flow rule Cp = λp∂Fpfplas along with the Kuhn-Tucker loading-unloading-

conditions

λp ≥ 0, fplas(Fp;Cp) ≤ 0, λpfplas(Fp;Cp) = 0. (2.27)

Second, following [44], we specify the diffusive part of the dissipation potential density as

χdiff(c, µ,M;F , c, d) = −µc− φdiff(M;F , c, d). (2.28)

Here we treat the concentration c ∈ ξC as a local constitutive field. The diffusion dissipation

potential density φdiff whose specific form is given in Section 2.2.2.3 is convex in M and

satisfies condition (2.22) via the constitutive relation

H = ∂M

φdiff. (2.29)


Finally, following [82, 83, 94] the dissipation potential density χfrac for rate-independent

fracture processes at a given state d is constructed based on thermodynamic arguments

∫

B0

[ ∂d χfrac d ] dV ≥ 0 (2.30)

and defined by the principle of maximum damage dissipation as

χfrac(d; d,∆d) = supβf∈Ef

βf d

(2.31)

in terms of the driving force βf , conjugate to d. This constrained problem can be solved by

Lagrange multiplier method

χfrac(d; d,∆d) = supβf ,λf≥0

βf d− λf tc(βf ; d,∆d)

(2.32)

along with the crack irreversibility conditions

tc(βf ; d,∆d) ≤ 0, tc(β

f ; d,∆d)d = 0, d ≥ 0 (2.33)

where λf ∈ ξF is a Lagrange multiplier enforcing the local threshold function tc to define

the reversible domain

Ef =

βf | tc(βf ; d,∆d) ≤ 0

(2.34)

in which tc is defined as

tc(βf ; d,∆d) = βf − gcδdγ(d,∇d) ≤ 0 with γ(d,∇d) =

1

2ld2 +

l

2|∇d|2. (2.35)

Here, γ(d,∇d) is the crack surface density, and gc is the critical energy release rate. The

crack surface density function γ approximates a sharp crack by a diffusive crack topology

controlled by a regularized length parameter l, as illustrated in Fig. 2.1.


2.2.2.3 Constitutive model

In this section we further specify the free energy density ψ in (2.19), the plastic yield function

fplas in (2.26), and the convex diffusion dissipation potential density φdiff in (2.28). Starting

with the elastic free energy density, we assume a perfect von Mises plasticity model without

hardening such that ψplas = 0. To prevent the material from failing under compression, the

elastic free energy ψelas in (2.19) is decomposed into tensile and compressive parts as

ψelas(F , d, c,Cp) = [(1− d)2 + k]ψ+elas(F , c,Cp) + ψ−

elas(F , c,Cp) (2.36)

where only the tensile part of the free energy ψ+elas drives the damage and crack propagation

in the material, and k is a very small positive value to provide numerical stability when d=1

[82]. The objective elastic free energy contribution ψelas in (2.36) in an Eulerian setting is

expressed by the elastic left Cauchy-Green tensor defined in (2.3). Time-differentiating the

elastic part of the free energy density yields

ψelas = ∂beψelas : be + ∂dψelasd = 2∂beψelasb

e :[

l+ 12(Lvb

e)be−1]

+ ∂dψelasd (2.37)

where Lvbe = F d

dt

[

F−1beF−T]

F T with be given in (2.3). By comparison with (2.21) and

(2.24) we obtain

τ = 2∂beψelasbe. (2.38)

The restriction of the considered material response to isotropy motivates the spectral de-

composition of the elastic left Cauchy-Green tensor defined in (2.3) and the Kirchhoff stress

defined in (2.38) as

be =

ndim∑

A=1

(λA)2nA ⊗ nA and τ =

ndim∑

A=1

τAnA ⊗nA (2.39)

where (λA)2 and τA denote the principal values and nA the corresponding principal direc-

tions for A = 1, . . . , ndim.


In the case of isotropy one can show that the Kirchhoff stress tensor can be directly

obtained from

τ = ∂heψelas with he = 12 ln be =

ndim∑

A=1

εeAnA ⊗ nA (2.40)

where he, with the principal elastic logarithmic stretches εeA = ln(λeA), is introduced as

a logarithmic Hencky-type strain tensor for which the time derivative of the free energy

density can be written as

ψ = (∂heψelasF−T ) : ∇ϕ + ∂heψelas :

12(Lvb

e)be−1 + ∂cψc + ∂dψelasd. (2.41)

The elastic free energy density ψ±elas in (2.36) is given in quadratic form as

ψ±elas(h

e) = 12 λ〈ln(J

e)〉2± + µ

ndim∑

A=1

〈εeA〉2± (2.42)

with λ and µ as the Lame constants. The bracket operators 〈•〉± are defined as 〈•〉± =

(•±|•|)/2. From the free energy density in (2.42), the principal Kirchhoff stress components

τA can be computed as

τA =∂ψelas

∂εeA= [(1− d)2 + k]

[

λ〈ln(Je)〉+ + 2µ〈εeA〉+

]

+[

λ〈ln(Je)〉− + 2µ〈εeA〉−

]

. (2.43)

The reduced local dissipation inequality reads

Dlocred = τ : lp + βf d ≥ 0 (2.44)

where we introduced the Eulerian evolution operator lp = −12J

−2/3c FC

p−1F−1be−1. This

motivates the subsequent identification Cp = Cp and Fp = τ. Note in this context

that although lp is energetically conjugate to τ in the subsequent variational framework,

variations with respect to lp and Cp can be used interchangeably and Cp is updated locally.

For perfect von Mises plasticity, the yield function fplas is given in terms of the Kirchhoff


stress τ and the yield stress σ0 as

fplas(τ ) = ||devτ || −√

23σ0. (2.45)

The chemical contribution ψchem(c) to the free energy density in (2.19) is assumed to have

the form [51]

ψchem(c) = cmaxRT [c ln c+ (1− c) ln(1− c)] (2.46)

where cmax is the maximum Li concentration, R is the gas constant, and T is the temper-

ature. Following [44], the convex diffusion dissipation potential density φdiff in (2.28) is

chosen as

φdiff(M;F , c, d) = 12M(c, d)C−1 : (M⊗M) (2.47)

where

M(c, d) =Ddiff(c, d)

cmaxRT[c(1− c)] and Ddiff(c, d) = D0

diff(c)+ d(Dmaxdiff −D0

diff(c)) (2.48)

from which the flux term H can be computed based on (2.29). Motivated by density

functional theory (DFT) calculations [135], the initial diffusion coefficient D0diff for the

undamaged body is considered to be concentration dependent in the numerical simulations

presented in Section 4.2, and in future works even a damage dependent diffusion coefficient

Ddiff could be introduced to allow the diffusion coefficient to reach a maximum value Dmaxdiff

once the body is damaged.

2.2.2.4 Euler equations of the rate-type variational formulation

Based on the previous specifications, the continuous mixed internal rate potential density

introduced in (2.18) reads

π(∇ϕ, µ,M, d, ξC, ξF) = (∂heψF−T ) : ∇ϕ+ (−∂heψ + τ ) : lp + (∂cψ − µ) c+ (βf + ∂dψ) d

− λpfplas(τ )− λf (βf − gcδdγ)− φdiff.

(2.49)


The Euler equations are obtained by taking variations of (2.17) at a given state ϕ, d, ξC

as

(i) δϕΠ ≡ Div[∂heψF−T ] + ρ0γ = 0

(ii) δcΠ ≡ µ− ∂cψ = 0

(iii) δµΠ ≡ c+Div[∂M

φdiff] = 0

(iv) δdΠ ≡ βf + ∂dψ = 0

(v) δβfΠ ≡ d− λf = 0

(vi) δλfΠ ≡ (βf − gcδdγ) ≤ 0, λfδλf

Π ≡ λf (βf − gcδdγ) = 0, λf ≥ 0

(vii) δC

pΠ ≡ (τ − ∂heψ) : ∂C

plp = 0 or δlpΠ ≡ τ − ∂heψ = 0

(viii) δτΠ ≡ lp − λp∂τfplas = 0

(ix) δλpΠ ≡ fplas ≤ 0, λpδλpΠ ≡ λpfplas = 0, λp ≥ 0

(2.50)

along with the boundary conditions defined in (2.6). Here, (2.50)(i) and (2.50)(iii) are the

balance of linear momentum and the balance of species conservation equations. (2.50)(ii)

denotes an implicit form for the chemical potential µ. The evolution equation for the damage

field is given in (2.50)(v) satisfying the crack irreversibility condition (2.50)(vi) along with

the constitutive relation for the driving force (2.50)(iv). (2.50)(viii) describes the Eulerian

flow rule for the plastic internal variable Cp together with the loading/unloading conditions

(2.50)(ix) and the constitutive stress relation (2.50)(vii).

2.3 Discrete variational framework

In this section, we derive the discrete version of the continuous variational framework de-

fined in Section 2.2. Section 2.3.1 defines a time-discrete counterpart of the previously

introduced rate-type functional. The condensation of local fields in Section 2.3.2 motivates

the definition of a reduced functional in Section 2.3.3 that forms a basis for the set of

equations that are subsequently discretized by finite elements in the context of a staggered

scheme in Section 2.3.4.

2.3. DISCRETE VARIATIONAL FRAMEWORK 29

2.3.1 Time discretization scheme

For the mixed variational framework and rate-type potentials described in Section 2.2.2, we

consider a discrete time interval [tn, t] with a time increment ∆t = t− tn > 0. Constitutive

state variables are denoted as

ϕn, dn, ξC,n

at time step tn, which are assumed to be

known, and the primary fields ϕ, d, µ, ξ at time step t without the subscripts, which will

be determined based on the discrete variational framework where we introduced ξ = ξC∪ξF.

The potential functional Π of Section 2.2 is now replaced with an incremental potential

functional

Π∆(ϕ, µ, d, ξ) = E∆(ϕ, d, ξC) + D

∆(µ, d, ξ)− Wϕ,µext (ϕ, µ; t) (2.51)

where E ∆ denotes the incremental energy storage functional, D∆ the dissipative work func-

tional, and Wϕ,µext the external work functional. Equation (2.51) can be written in the form

Π∆(ϕ, µ, d, ξ) =

∫

B0

π∆(∇ϕ, µ,M, d, ξ) dV − Wϕ,µext (ϕ, µ) (2.52)

with the incremental mixed internal potential density per unit volume

π∆(∇ϕ, µ,M, d, ξ) = Algo

∫ t

tn

π(∇ϕ, µ,M, d, ξC, ξF)dt

. (2.53)

In the context of an algorithmic treatment of isotropic finite strain elasto-plasticity an

exact update of the Eulerian flow rule (2.50)(viii) may be derived in line with the update

formulation outlined in [136, 137] that is identical to the return mapping of the infinitesimal

theory. It can be written in the material description as

Cp−1

= [−2λp C−1Np]Cp−1 (2.54)

where Np = F T∂τfplas(τ )F is the pull-back of the yield function normal ∂τ fplas(τ ) to the

reference configuration. The exponential approximation to (2.54) within the interval [tn, t]

can be recast as

Cp−1 = exp[−2∆γp C−1Np]Cp−1n (2.55)


with the incremental plastic parameter ∆γp = (t− tn)λp. The push-forward to the current

configuration and the identification of the trial value be∗ = J−2/3c FCp−1

n F T yields the

update

be = exp[−2∆γp ∂τ fplas(τ )]be∗. (2.56)

The unique spectral decomposition of the Kirchhoff stress tensor τ , the elastic left Cauchy-

Green tensor be and its trial value be∗ =∑ndim

A=1(λ∗A)

2nA ⊗ nA due to their coaxiality for a

restriction to isotropy motivates a return mapping algorithm at fixed principal axis, that is,

(λeA)2 = exp[−2∆γp ∂τAfplas(τ )](λ

e∗A )2. (2.57)

Exploitation of the coaxiality allows further to rewrite the logarithmic elastic strains defined

in (2.40) in additive form as

he = he∗ −∆γp∂τ fplas with he∗ = 12 ln[b

e∗] (2.58)

with the representations he =∑ndim

A=1(lnλA)nA⊗nA and he∗ =∑ndim

A=1(lnλ∗A)nA⊗nA. The

incremental mixed internal potential density defined in (2.53) reads hence in algorithmic

form

π∆(∇ϕ, µ,M, d, ξ) = ψ(he, d, c) + τ : (he∗ − he) + βf (d− dn)− µ(c− cn)

−∆γpfplas(τ )−∆tφdiff(M;F n, cn, dn)−∆γf (βf − gcδdγ(d,∇d))

(2.59)

with the incremental fracture parameter ∆γf = (t− tn)λf .

2.3.2 Condensation of local fields

Due to the local structure of the incremental problem, in an algorithmic treatment ξ can

be obtained locally for given global fields ϕ, µ, d by means of a local stationary problem

ξ = arg

statξ

π∆(∇ϕ, µ,M, d, ξ)

(2.60)


for admissible variations δλp ≥ 0 ∈ L2, δλf ≥ 0 ∈ L2, δc ∈ L2, δβf ∈ L2, δCp ∈ L2 and

δτ ∈ L2, resulting in the local equations:

(i) δcπ∆ ≡ µ− ∂cψ = 0

(ii) δβfπ∆ ≡ d− dn −∆γf = 0

(iii) δλfπ∆ ≡ (βf − gcδdγ) ≤ 0, λf δλf

π∆ ≡ λf (βf − gcδdγ) = 0, λf ≥ 0

(iv) δCpπ∆ ≡ [τ − ∂heψ] : ∂Cphe = 0

(v) δτπ∆ ≡ he∗ − he −∆γp∂τ fplas = 0

(vi) δλpπ∆ ≡ fplas ≤ 0, λpδλpπ

∆ ≡ λpfplas = 0, λp ≥ 0

(2.61)

Note that all the equations in (2.61) are linear with the exception of (2.61)(i), which algo-

rithmically is solved by a local Newton update scheme (see Table 3.1).

2.3.3 Reduced global variational principle

The local stationary principle (2.60) allows us to define a reduced incremental mixed internal

potential density

π∆red(∇ϕ, µ,M, d) = statξ

π∆(∇ϕ, µ,M, d, ξ) (2.62)

associated with a reduced three field potential functional

Π∆red(ϕ, µ, d) =

∫

B0

π∆red(∇ϕ, µ,M, d) dV − Wϕ,µext (ϕ, µ; t). (2.63)

The solution of the global fields is hence obtained from the saddle point problem

ϕ, µ, d = arg

infϕ

supµ

infd

Π∆red(ϕ, µ, d

. (2.64)


Taking variations of (2.64), we obtain the weak form of the relevant set of governing equa-

tions as

DΠ∆red[δϕ] =

∫

B0

Dψ[δϕ] dV −DWϕext[δϕ] = 0

DΠ∆red[δµ] =

∫

B0

− (c− cn)δµ −∆tDφdiff[δµ]

dV −DWµext[δµ] = 0

DΠ∆red[δd] =

∫

B0

Dψ[δd] + gcδdγ δd

dV = 0

(2.65)

where for a generic field η and a functional Θ,

DΘ[δη] ≡d

dǫΘ(η + ǫδη) |ǫ=0 (2.66)

denotes the Gateaux derivative in the direction δη. Hence,

DΠ∆red[δϕ] =

∫

B0

(∂heψF−T ) : δF − ρ0γ · δϕ

dV −

∫

∂Bt0

T · δϕ dA = 0

DΠ∆red[δµ] =

∫

B0

− (c− cn)δµ +∆t ∂M

φdiff · ∇δµ

dV +

∫

∂Bh0

∆t Hδµ dA = 0

DΠ∆red[δd] =

∫

B0

∂dψ δd + gcδdγ δd

dV = 0

(2.67)

with the variations satisfying the requirements δϕ ∈ Wϕ0 =

δϕ|δϕ = 0 on ∂Bϕ0

and

δµ ∈ Wµ0 = δµ|δµ = 0 on ∂Bµ

0 as well as δd = 0 on the entire boundary. The weak

equations in (2.67) provide the starting point for setting up the discretized formulation for

ϕ, µ, d. The set of equations is supplemented by the local Euler equations (2.61) that

define the local fields ξ in terms of the global fields ϕ, µ, d.

2.3.4 Staggered spatial discretization scheme

During the loading process d > 0 we have 2(1−d)ψ+elas = gcδdγ, and during unloading d = 0

we have 2(1− d)ψ+elas ≤ gcδdγ. Following [83], a local history field of the crack driving force

is introduced as

H(X , t) = maxs∈[0,t]

(

ψ+elas(∇ϕ, ξ)

)

(2.68)


where H is the maximal attained value of ψ+elas(∇ϕ, ξ). Replacing the crack driving force

ψ+elas with the newly introduced driving force H ensures d ≥ 0, and allows us to recast the

crack irreversibility condition (2.33) into

2(1− d)H = gcδdγ (2.69)

serving as the evolution equation of the damage field d. The introduction of the history field

H to recast the crack irreversibility condition allows us to solve the problem in a staggered

manner, where the damage field d is solved first, followed by the computation of the other

global fields ϕ, µ. Following [83], at the current time step t, we first update the history

field H as

H =

ψ+elas(∇ϕn, ξn) for ψ+

elas(∇ϕn, ξn) > Hn

Hn otherwise

(2.70)

with H0 = H(X , t = t0) = 0 as the initial condition. Since the history variable H is

determined based on the state variables at time tn, it is constant in the time interval [tn, t].

Thus, to obtain d, we introduce the algorithmic potential

Π∆red(d) =

∫

B0

[

gcγ(d,∇d) + (1− d)2H]

dV (2.71)

with γ as the crack surface density function given in (2.35). The damage field d at time

t is obtained from the incremental stationary principle

d = arg

infd

Π∆red(d)

. (2.72)

Next, we discretize the body B0 into nelem finite elements Be0 with nnode nodal points.

The field variables ϕ, µ, d are approximated by ϕh = Nϕdϕ, µh = Nµdµ, d

h = Nddd in

terms of standard shape functions and their nodal values. The material gradients ∇µ,∇d

are approximated as∇µh = Bµdµ, ∇dh = Bddd and the spatial gradient of the deformation

field is approximated as ∇xϕh = bϕdϕ, all given in terms of standard B-operator matrices.

Employing a Bubnov-Galerkin scheme, all their variations are approximated the same way.

The discretized residual function for the phase field dh is then being determined from


the weak form (2.67)3 as

Rd =nelem

Ae

∫

Be0

NTd

[gcld− 2(1− d)H

]

+ gclBTd∇d

dV = 0 (2.73)

from which the result for dh at time t is obtained through a standard Newton procedure.

From the solution of

dh

computed from (2.73), we continue to solve for the remaining

global fields. For a given chemical potential µ, the concentration variable c is recovered

at the integrating points based on a local Newton procedure [138, 44]. The internal vari-

able Cp is updated based on the von-Mises plasticity return mapping algorithm at finite

deformations [136]. Those parameters are obtained on the element level as soon as the

convergence criterion is met. Knowing c,Cp, the discretized residual functions for ϕ, µ

are determined from the weak equations (2.67)1 and (2.67)2 as

Rϕ =nelem

Ae

∫

Be0

(

bTϕτ − ρ0NTuγ

)

dV −nelem

Ae

∫

∂Be,t0

NTu T dA = 0

Rµ =nelem

Ae

∫

Be0

(

−NTµ (c− cn) + ∆tBT

µH)

dV +nelem

Ae

∫

∂Be,h0

∆tNTµ H dA = 0

(2.74)

from which the result for ϕ, µ at time t is obtained through a standard Newton procedure.

Chapter 3

Diffusion-induced deformation of

Si anodes

This chapter is adapted from: X. Zhang, S. Lee, H. Lee, Y. Cui, C. Linder, “A reaction-

controlled diffusion model for the lithiation of silicon in lithium-ion batteries”, Extreme

Mechanics Letters, 4:61-75 (2015) and X. Zhang, A. Krischok, C. Linder, “A variational

framework to model diffusion induced large plastic deformation and phase field fracture

during initial two-phase lithiation of silicon electrodes”, Comput. Meth. Appl. Mech.

Eng., 312:51-77 (2016).

In this chapter, we first present a RCD model to describe the two-phase lithiation of Si

anodes. We then present the numerical procedure to model diffusion-induced anisotropic

deformation of Si anodes by combining this model with the variational framework presented

in Chapter 2.

3.1 Two-phase lithiation of Si anodes

3.1.1 Introduction

For an electro-chemo-mechanical coupled model, one important aspect is accurately cap-

turing the Li distribution in Si electrodes resulting from the diffusion and reaction process,

which is challenging for existing experiments. More importantly, it is necessary to develop

35

36 CHAPTER 3. DIFFUSION-INDUCED DEFORMATION OF SI ANODES

reaction front LixSi (D=D0)

c-Si & a-Si (D=Ds)

(a) two-phase lithiation for both c-Si and a-Si

x1 x2 xixi+1 xn xn+1

R0

c=1rkrk+1

(b) numerical simluation setup

Figure 3.1: Illustration of the two-phase lithiation mechanism and the numerical simulationsetup. (a) The diffusion coefficient D depends on the reaction front location, where D = D0

for the obtained a-LixSi region behind the reaction front and D = Ds for the remainingSi region ahead of the reaction front. (b) The 1D simulations in Section 4.1 represent thelithiation process of Si nanoparticles with an initial radius of R0. The finite differencemethod is used with a second-order central difference approximation of the space derivativeand a backward difference approximation of the time derivative. Along the radius, a zero-flux boundary condition is applied at x1 and a c = 1 essential boundary condition is appliedat xn+1. The radius is discretized into n elements. rk and rk+1 represent the reaction frontlocation at time t = k and t = k+1, respectively.

a proper model to describe the Li diffusion during the initial lithiation of c-Si, a-Si, and

the subsequent lithiation process, which have different diffusion and reaction behaviors, in

a consistent manner. For electrodes considered at the nanoscale, the chemical reaction rate

rather than the diffusion itself is the limiting factor for the initial lithiation of c-Si and

a-Si [45, 18, 32, 46], whereas the diffusion might be the limiting factor for the one-phase

diffusion of post-a-Si [20]. To capture the lithiation process for all three types of Si in a

uniform way, a one-way coupling model between the reaction and diffusion is proposed in

this work. A physical parameter, the bond-breaking energy barrier, is introduced in this

new RCD model to investigate the similarities and differences between the initial lithiation

process of a-Si and c-Si and the lithiation of post-a-Si. The proposed model will take the

effect of Li concentration and hydrostatic pressure at the reaction front on the reaction

front velocity into account.

3.1. TWO-PHASE LITHIATION OF SI ANODES 37

3.1.2 Reaction-controlled diffusion model

In diffusion-reaction systems, the concentration of species can be described by a reaction

term coupled with the diffusion process through a second order partial differential equation

[139]. During the initial lithiation process of Si discussed here, the reaction rate at the phase

boundary is a limiting factor of the diffusion process [18, 32, 33, 19, 20]. DFT simulations

show that a Li-concentration threshold value of c0 at the reaction front is needed to break

the Si-Si bond to form dangling bonds [140]. Once the reaction front continues to propagate,

Li can diffuse rapidly into LixSi [46], and the concentration c is governed by the diffusion

process in the lithiated region. Therefore, in this case, the instant reaction front velocity

under the current Li concentration profile is more important than the reaction rate at

different locations and time instances. Thus, instead of using the full coupling as in [139],

the reaction can be decoupled from the diffusion process, where the latter will be used to

determine the Li concentration profile. For simplicity, Fick’s second law

∂c

∂t= D

∂2c

∂x2(3.1)

is used, with D as the diffusion coefficient and c as the Li concentration to determine the Li

concentration c. The value of D will depend on the location of the reaction front, as shown

in Fig. 3.1(a). For the a-LixSi region behind the reaction front, D = D0. Although Li could

diffuse into c-Si before breaking Si-Si bonds due to a lower diffusion energy barrier [141], the

sharp reaction fronts observed in [17, 19] indicate that very little amount of Li could diffuse

across the phase boundary and deposit in the Si core for both c-Si and a-Si. The diffused

Li in unreacted c-Si (or a-Si) might only exist at the dopant level, which is negligible for

the results reported here. For post-a-Si, the Si core represents the region to where Li has

not yet diffused. Therefore, we assume D = Ds = 0 in the unreacted Si core, as shown in

Fig. 3.1a. Alternative diffusion models with a concentration-dependent diffusion coefficient

[18] or a pressure-gradient dependent flux term [142] can be used to replace (3.1).


Having obtained the Li concentration from (3.1), the reaction rate between Li and Si at

the reaction front can be approximated by a solid-state reaction function as

dα

dt= A exp

(

−E

kBT

)

f(α) (3.2)

where α, A, E, kB , T , f(α) are the reaction conversion fraction, frequency-related factor,

activation energy, Boltzmann constant, absolute temperature, and reaction model, respec-

tively [143]. During the lithiation process of Si, since Si is the host material, the molar

fraction of Si and Li is 1 in front of the phase boundary. When the reaction front prop-

agates, Li will mix with Si and convert to LixSi. In the newly lithiated region, the LixSi

conversion fraction α in (3.2) is equivalent to the Li concentration c, and the activation

energy E in (3.2) will be the bond breaking energy needed to break the Si-Si covalent bond

to form LixSi alloy under the applied electric potential. Experiments [32], which show a

slow-down tendency as the reaction front progresses towards the core during the initial lithi-

ation of c-Si, suggest a stress-dependent reaction front velocity. A more recent experiment

conducted with Germanium (Ge), which has a similar crystalline structure to Si, indicates a

more evident stress-dependent lithiation process, where a bent Ge nanowire shows a faster

lithiation rate under tensile stresses and a slower lithiation speed under compressive stresses.

Motivated by those experiments, we propose a hydrostatic pressure dependent reaction rate

at the reaction front to replace (3.2) by

vr = A exp

(

−E0 +Bp

kBT

)

f(c) (3.3)

where p is the hydrostatic pressure resulting from a diffusion-induced deformation and E0 is

the initial activation energy at a stress-free state already accounting for a possibly applied

electric potential [18, 32, 33, 19, 20, 144]. The reaction rate (3.3) will be used to determine

the reaction front location. In (3.3), B is a volume related parameter, which is computed

as

B =φSiVSi

2 · 1.60 · 10−19(3.4)

where φSi is the Si-Si bond fraction in the Si electrode, VSi is the atomistic volume of a Si

3.1. TWO-PHASE LITHIATION OF SI ANODES 39

0 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c0

concentration c

f(c)

Figure 3.2: Illustration of the value of f(c) varying with concentration c in (3.5).

atom and 2 represents the two Si-Si bonds that each Si atom has. The coefficient 1.60·10−19

in the denominator will ensure that Bp represents the bond energy change of a single Si-Si

bond with a unit of electronvolt (eV). In (3.3), compressive stresses (positive hydrostatic

pressure) will increase the energy barrier and slow down the reaction rate, and tensile

stresses (negative hydrostatic pressure) will decrease the energy barrier and accelerate the

reaction rate.

A threshold value of c0 at the reaction front is needed to break the Si-Si covalent bond.

Motivated by [145], where the reaction between near-noble metal and Si at low tempera-

ture arises due to the accumulation of the near-noble metal at the interface, and by DFT

calculations in [140], which show a specific threshold of Li concentration above which the

Si bond will be broken permanently, a reaction kinetic model f(c) is proposed for c > 0 as

f(c) = (1 + exp(C(c0 − c)))−1 (3.5)

in a sigmoidal shape, as shown in Fig. (3.2). Here, c0 is the threshold value and C is an

empirical parameter to describe the shape of f(c). In this model, f(c) will increase rapidly

to a maximum value 1 when the Li concentration c at the reaction front is larger than


the threshold value c0, and decrease to 0 when c < c0. In a given time increment ∆t, the

reaction front will propagate a distance ∆r = vf ∆t. The reacted Li at the reaction front

will break Si-Si bonds and create the path to allow extra Li to diffuse into the Si core. Based

on the conservation of species, we approximate the reaction front propagating distance ∆r

as

vr j da dt = ∆r da c0 (3.6)

with j (number of Li particles/(area·time)) as the Li flux at the reaction front. Here, j da dt

represents the total number of particles that could diffuse across the reaction front. The

left hand side of Eq (3.6) gives the total number of reacted Li particles. The reacted Li has

to reach the minimum concentration level c0, as given at the right hand side of Eq (3.6), in

order to create diffusion paths for Li. The new reaction front location will then be updated

as

rnew = rold −∆r (3.7)

for a cylindrical or spherical geometry [143], where the minus sign indicates that the radius

of remaining Si is decreasing due to the reaction.

In our model, the bond-breaking energy barrier E0 is the physical quantity which distin-

guishes different types of Si. It represents the energy barrier to break Si-Si bonds without

considering other dynamic processes at the reaction front, such as bond switching or re-

bonding between Si dangling chains. For the three types of Si discussed in this work, c-Si

has the highest bond-breaking energy barrier, due to the well arranged crystalline structure.

The energy barrier can be different for different crystalline directions [146], which explains

the anisotropic deformation observed in experiments [34, 21]. The energy barrier in a-Si is

expected to be lower than in c-Si due to numerous weak dangling bonds and defects formed

during the preparation process [147]. Post-a-Si has the lowest bond-breaking energy bar-

rier, since it contains not only the largest amount of weak dangling bonds, but also voids or

trapped Li, as indicated by the ∼ 25% remaining volume expansion after delithiation [20].

3.2. NUMERICAL IMPLEMENTATION PROCEDURE 41

3.2 Numerical implementation procedure

In this work, instead of using Cahn-Hilliard type diffusion [131, 43, 44, 148] to account

for the arising two-phase lithiation, we use the RCD model [116] to describe the initial

lithiation of c-Si and a-Si. Doing so allows us to predict the diffusion-induced anisotropic

deformation depending on the crystalline structure of Si. Here, we present a reaction front

location updating scheme based on the solution obtained from the variational framework in

Chapter 2 and the RCD model discussed in section 3.1.2.

3.2.1 Update of the reaction front location

Consider a simple discrete geometry with 13 nodes, 19 edges, and an initial reaction front

(indicated by the red bold line). Each node is assigned with a flag to indicate its lithiated

state. A value of (0) (not shown here) means that the node is located in the unlithiated

region. For the lithiated region, we use (-1) for nodes on front edges (marked with black

bold lines) and (1) otherwise. Front edges are those that intersect the reaction front and

have nodes with a flag of (0) and (-1). A local coordinate s ∈ [0, 1] for each front edge

is used to determine the exact reaction front location. The reaction front is assumed to

propagate along the direction of ∇c (assumed vertically downwards here) with distances

predicted by the reaction-controlled diffusion model. To update the new reaction front

locations, we create a list of front edges and associate with each a forwarding distance

∆r, as shown in (a). No update is performed for edges with a zero forwarding distance.

For those being updated, the new reaction front locations either stay on the existing front

edges (edges e4, e5), or move to different edges (edges e6, e7). (b) For edge e5, project its

associated distance ∆r5 onto edge e5 and update the local coordinate s5 to indicate the new

reaction front location. The forwarding distance associated with edge e5 is then updated

to zero. For edge e6, we obtain the new reaction front location in two steps. First, see (b),

project ∆r6 onto edge e6 and update the local coordinate to s6 = 1 to indicate a temporary

reaction front location at node 7. Update nodal flags for nodes 3 and 7 to (1) and (-1),

respectively, remove edge e6 from the front edge list but add edges e9−11. Second, see (c), a

forwarding distance of ∆r9 = ∆r10 = ∆r11 = ∆r9−11 in the direction of ∇c is assigned and


∆r4 ∆r5 ∆r6

∆r7

1 2 3 4

56

7

8

9 10

11 12 13

e1 e2 e3

e4 e5

e6 e7

e8

e9

e10

e13 e14

e10e12

e16

e15 e17

e18 e19

(a)

∆r5 ∆r6

(-1) (-1)(-1)(-1)

s5

s6

(b)

replacemen

∆r9−11

(1) (-1)

(-1)

(-1)(-1)

s9 s10

(c)

(1)(1)

(-1)(-1)

(-1)(-1)

(d)

Figure 3.3: Illustration of the reaction front updating scheme.

projected onto edges e9−11. The local coordinates s9 and s10 are updated to indicate the

new reaction front location and the forwarding distances associated with edges e9, e10 are

set to zero. The local coordinate s11 = 0 is not updated since the projection ∆r11 does not

intersect edge e11. The local coordinate s11 is updated when updating edge e7. (d) Once

all front edges are updated the new reaction front location is obtained.

Having computed the solution ϕ, µ, d, c at time t, we can now update the reaction front

location based on the reaction-controlled diffusion model summarized in Section 3.1.2. Even

though the procedure is explained for two-dimensional problems, in which the reaction front

is a line, the overall procedure conceptually can be extended to three-dimensional problems.


To compute the reaction rate in (3.3), we create three arrays in which we store the results

obtained from Section 2.3.4 for the hydrostatic pressure p, the Li concentration c, and the

concentration gradient ∇c at the reaction front. Based on the solution of the concentration

c at t, a trial diffusion process is carried out in which the presence of the reaction front

is initially neglected. The result is then used to approximate the Li flow j at the reaction

front. The value of j is determined based on the total amount of Li atoms that diffuse into

the unlithiated Si core in this trial process. Next, the reaction-controlled diffusion model

is used to approximate the new reaction front location based on the bond-breaking energy

barrier E0. For c-Si, the value of E0 is only available for specific crystalline directions, but

the concentration gradient is not always aligned with those. Thus, we must approximate

the reaction front moving velocity based on the available values for E0 in the 〈100〉 and

〈110〉 directions. In the numerical simulation performed in Section 4.2, for each reaction

front location, based on p and c at that front location, we compute two velocities, v〈100〉

for the 〈100〉 direction and v〈110〉 for the 〈110〉 direction. We then approximate the actual

velocity for that location in the direction of the stored value of the concentration gradient

∇c based on a linear interpolation between v〈100〉 and v〈110〉. Doing so allows us to predict

the expected anisotropic deformation during the (de)lithiation process. With the computed

reaction front forwarding distance ∆r in the direction of ∇c, the reaction front locations

are updated. The updating procedure, which is mesh-insensitive, is illustrated in Fig. 3.3

for a simple reaction front geometry.

3.2.2 Flow chart for modeling diffusion-induced swelling of Si anodes

In this section, a summary of the overall solution procedure to model diffusion-induced

deformation and fracture of Si anodes, combining the two-phase lithiation treatment of

Section 3.2.1 with the discrete variational framework for diffusion-induced elasto-plastic

deformation and phase field fracture is provided in Table 3.1.


Table 3.1: Summary box of the variational framework to model diffusion-induced large plas-tic deformation and phase field fracture during initial two-phase lithiation of Si electrodes.

A. Initialize the deformation field ϕn, the phase field dn, chemical potential µn, his-tory field Hn, and the reaction front location rn at time tn. Update the boundaryconditions to the current time step t.

B. Determine the history variable H at t in B0:

H =

ψ+elas(∇ϕn, cn, ξn) for ψ+

elas(∇ϕn, cn, ξn) > Hn

Hn otherwise

C. Compute the phase field d at time t through a standard Newton procedure:

Rd =nelem

Ae

∫

Be0

NTd

[gcld− 2(1− d)H

]

+ gclBTd∇d

dV = 0 .

D. Compute the deformation field ϕ and the chemical potential µ for the fixed phasefield d computed from the previous step:

• Compute the local concentration field c at time t for a given chemical potentialµ as

c⇐ c− k−1c rc until |rc| < tolc

whererc = π∆,c = ∂cψ − µ = 0 and kc = π∆,cc = ∂2ccψ

• The plastic internal variables are updated based on the return mapping algorithm[136] as

εeA = εe∗A − γp∂fplas∂τA

with εeA = ln(λeA), εe∗A = ln(λe∗A ), A = 1, . . . , ndim

• Compute the deformation field ϕ and the chemical potential µ through a stan-dard Newton procedure from

Rϕ =nelem

Ae

∫

Be0

(

bTϕτ − ρ0NTuγ

)

dV −nelem

Ae

∫

∂Be,t0

NTu T dA = 0

Rµ =nelem

Ae

∫

Be0

(

−NTµ (c− cn) + ∆tBT

µH)

dV +nelem

Ae

∫

∂Be,h0

∆tNTµ H dA = 0 .


Continuation of Table 3.1.

E. Update the new reaction front location based on the reaction-controlled diffusionmodel discussed in Section 3.1.2:

• Store the concentration value c, the concentration gradient ∇c, and the hydro-static pressure p from step D at the reaction front location.

• Run a trial diffusion process with the solution µ and c from step D as inputs.Solve Rµ without the presence of the reaction front to approximate the Li flowj.

• From the reaction-controlled diffusion model, compute the reaction rate at thereaction front location as

vr = A exp

(

−E0 +Bp

kBT

)

f(c) with B =φSiVSi

2 · 1.60 · 10−19

and

f(c) =1

1 + exp(C(c0 − c))(for c > 0).

For c-Si, a linear interpolation of the reaction rate is performed between v〈110〉and v〈100〉 based on the orientation difference between the concentration gradientdirection and the 〈100〉 and 〈110〉 crystalline directions.

• Update the new reaction front location r at time t as

r = rn −∆r with vr j da dt = ∆r da c0 cmax.

• Perform the geometric tracking algorithm of the reaction front based on theupdate scheme summarized in Fig. 3.3.


Chapter 4

Representative numerical examples

This chapter is adapted from: X. Zhang, S. Lee, H. Lee, Y. Cui, C. Linder, “A reaction-

controlled diffusion model for the lithiation of silicon in lithium-ion batteries”, Extreme

Mechanics Letters, 4:61-75 (2015) and X. Zhang, A. Krischok, C. Linder, “A variational

framework to model diffusion induced large plastic deformation and phase field fracture

during initial two-phase lithiation of silicon electrodes”, Comput. Meth. Appl. Mech.

Eng., 312:51-77 (2016).

In this chapter, various numerical simulations are conducted to demonstrate the per-

formance of our numerical models proposed in Chapter 2 to Chapter 3. First, several 1D

simulations are carried out to reveal the importance of the consideration of bond-breaking

energy barrier and hydrostatic pressure in the RCD model proposed in Section 4.1. We then

study diffusion-induced deformation of Si anodes with various geometries and parameters

in Section 4.2 by using the proposed variational computational framework and the RCD

model.

4.1 Two-phase lithiation with the reaction-controlled diffu-

sion model

In this section, several 1D numerical simulations are performed to show the versatility of the

proposed RCD model. Those simulations represent the lithiation process of Si nanoparticles

with different initial radii of R0. We first show the necessity of considering the pressure

47

48 CHAPTER 4. REPRESENTATIVE NUMERICAL EXAMPLES

effects in the proposed model. Then, we carry out a parameter sensitivity study for c-

Si, a-Si, and post-a-Si, respectively, and compare the numerical simulation results with

available experimental results. All the simulations are assumed to undergo isotropic defor-

mation without considering different bond-breaking energy barriers for different crystalline

directions [141].

In those simulations, the finite difference method is used with a second-order central

difference approximation for the space derivative and a backward difference approximation

for the time derivative. The chosen parameters used in the simulations are summarized

in Table 4.1. The simulation setup is shown in Fig. 3.1(b). The radius is discretized into

n = 600 elements and a time step of 1·10−2 s is used. Along the radius, a zero-flux boundary

condition is applied at x1 and a c = 1 essential boundary condition is applied at xn+1. At

time t = 0, initial conditions in the form of c = 0 everywhere except at xn+1 are prescribed.

The initial reaction front r0 is located at xn+1. At time step k, the reaction front is located

at rk. A zero flux boundary condition is applied for x < rk since D = 0 for the unlithiated

region. The Li concentration profile is computed based on (3.1), whereas the reaction front

velocity at rk is computed based on (3.3)-(3.6). The reaction front location rk+1 is then

updated based on (3.7) for the time step k+1. The Li flux j at the reaction front in Eq (3.6)

is computed by updating the Li profile through a trial diffusion process without the presence

of a reaction front to determine the number of Li atoms diffusing across the reaction front

per second. For post-a-Si, the predicted reaction velocity might be larger than the diffusion

speed due to smaller chosen E0. To ensure that the reaction can only happen when Li is

present, a cutoff value of Li concentration ccutoff = c0/10 is used to determine the reaction

front location. The simulation repeats for the new time step k + 1.

To account for the diffusion-induced deformation, we consider a linear relationship be-

tween the concentration and the deformation ∆R/R = Ω ∆c as in [53, 40]. An expansion

coefficient Ω = 0.6 as in [40] is used to obtain the ∼ 300% volume expansion at fully lithiated

state. In the simulation procedure, the averaged concentration level in each discretized seg-

ment is computed first. Then, the corresponding segment undergoes a concentration level

driven deformation. The concentration level at the next time step will be computed based

on the deformed geometry. For Fig. 4.2 to Fig. 4.12, the concentration profile versus radial

4.1. TWO-PHASE LITHIATION WITH THE RCD MODEL 49

Table 4.1: Summary of the parameters used in the simulations performed in Section 4.1

parameter value parameter value

E0 (c-Si) 0.59 – 0.69 (eV) [140, 141] D0 1.0 · 10−12 (cm2/s) [149]

E0 (a-Si) 0.51 – 0.58 (eV) T 300 (K)

E0 (post a-Si) 0.45 – 0.50 (eV) [140, 150] kB 8.6173 · 10−5 (eV/K)

c0 (c-Si) 0.10 (-) [140] A 5.0 · 109(s−1) [151]

c0 (a-Si) 0.06 (-) C 50 (-)

c0 (post a-Si) 0.001 (-) pmax 1.0 – 3.5 (GPa)

φSi(c-Si) 1.0 (-) ω 2 (-)

φSi(a-Si) 0.8 (-) cmax 8.867 · 104 (mol/m3) [57]

φSi(post a-Si) 0.7 (-)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

Pressure reported in [24]

location r/R0

hydrostaticpressure

[GPa]

0 0.2 0.4 0.6 0.8 1

Pressure Eq (8)

location r/R0

hydrostaticpressure

[GPa]

pmax

Figure 4.1: Illustration of the different reaction front pressure values versus radial locationused in the simulation, where the left figure represents the pressure reported in [45] and theright one represents the pressure computed from (4.1).

location curves are plotted for the swelled geometry. The makers on the upper right corner

of each figure indicate the new radius of the swelled Si electrodes at the corresponding time

step.

4.1.1 Pressure effects on the lithiation process

In this subsection, we investigate the effect of hydrostatic pressure values at the reaction

front on the lithiation process, which is taken into account in (3.3). We run simulations

with three bond-breaking energy barriers E0 corresponding to c-Si, a-Si, and post-a-Si. It


is shown that by considering a proper value for the hydrostatic pressure, the two-phase

lithiation process for c-Si, a-S, and the one-phase lithiation process for post-a-Si is captured

by the proposed model.

For a nanoparticle undergoing ideal plastic deformation, the reaction front pressure can

be calculated based on p = 2σY log(r/R0) [45]. As in [45], σY = 1 GPa is chosen and

the pressure versus radial location is plotted on the left of Fig. 4.1. Since the plastic

deformation is assumed to be unaffected by the superimposed hydrostatic stress in [45], the

hydrostatic pressure approaches an infinite value at the core.

Motivated by [152], where the yield stress is related to the hydrostatic pressure and a

finite yield stress should correspond to a finite hydrostatic pressure, a location dependent

empirical reaction front pressure distribution with a saturated value pmax, as an alternative

to the one reported in [45], is proposed as

p = pmax(1− exp(−ω(R0 − r)/R0)) (4.1)

where ω is a scalar parameter approximating the shape of the reaction front pressure dis-

tribution during the diffusion-induced plastic deformation. Since currently no fully chemo-

mechanical coupled formulation accounting for Drucker-Prager plasticity given in the liter-

ature provides pressure distribution at the reaction front, we perform a quantitative study

based on the pressure distribution given in [45] and the empirical reaction front pressure

distribution in (4.1).

To do so, the two pressure distributions illustrated in Fig. 4.1 are now inserted into

(3.3) to study the pressure effect on the lithiation process for the three different values of

the bond-breaking energy barrier E0 with pmax = 1.5 GPa. For c-Si, a nanoparticle with

a radius R0 = 100 nm is studied with E0 = 0.69 eV and c0 = 0.1 [140]. For post-a-Si,

the one-phase lithiation process indicates a diffusion-controlled lithiation process. The Si-Si

bonds break at the same speed as the diffusion process. Therefore, a value of E0 = 0.50 eV,

corresponding to the diffusion energy barrier of a-Si [150], is used to study the lithiation

process of a post-a-Si nanoparticle with a radius R0 = 250 nm. A very small but non-zero

value of c0 (= 0.001) is used for the purpose of determining the reaction front location in


0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration

(a) concentraton - without pressure

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

100 105 110 1150.99

0.995

1

location [nm]

concentration

(b) concentraton - with pressure from [45]

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

100 110 1200.99

0.995

1

location [nm]

concentration

(c) concentraton - with pressure from (4.1)

0 50 100 1500

10

20

30

40

50

60

70

80

90

100

without pressure

with pressure [24]

with pressure Eq. (8)

time [s]

reactionfrontlocation[nm]

(d) reaction front location

Figure 4.2: c-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for E0 = 0.69 eV (a-c), and the reaction front location versus time(d) corresponding to (a-c). (a-c) An evident reaction front is observed for each simulation.The Li concentration profile is high in the lithiated region. (d) The reaction front velocityis constant when the pressure is neglected and slows down when the pressure is included.

the numerical simulation. For a-Si, a nanoparticle with a radius R0 = 250 nm is studied

with E0 = 0.58 eV and c0 = 0.06. Since no universal agreement about the bond-breaking

energy barrier exists for c-Si, a-Si, and post-a-Si, we can only estimate the bond-breaking

energy barrier range based on DFT calculations [140, 150, 141], as given in Table 4.1, and

perform parameter studies in later subsections.

The Li concentration profile versus radial location at different time steps for c-Si is

shown in Fig. 4.2(a-c) with and without the consideration of the pressure effects. For the


0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 1500

50

100

150

200

250

without pressure

with pressure [24]


time [s]



Figure 4.3: a-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for E0 = 0.58 eV (a-c), and the reaction front location versus time(d) corresponding to (a-c). (a-c) An evident reaction front is observed for each simulation.(d) The reaction front velocity slows down significantly due to the pressure [45].

lithiation process of c-Si, a sharp reaction front can be observed in all three simulations

in Fig. 4.2(a-c). A high Li concentration is obtained in the lithiated region, matching

the experimental observation [32]. The reaction front location versus time is plotted in

Fig. 4.2(d). For the simulation with pressure in [45] and (4.1), the reaction front velocity

shows a slow-down behavior, matching the experimental observation where the reaction

front velocity becomes smaller as the reaction front approaches the core. This is not the

case when neglecting the pressure effect for which an almost constant reaction velocity is


0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 1500

50

100

150

200

250

without pressure

with pressure [24]


time [s]



Figure 4.4: post-a-Si simulation – Illustration of the concentration profile versus radiallocation at different time steps for E0 = 0.50 eV (a-c), and the reaction front locationversus time (d) corresponding to (a-c). (a,c) A one-phase lithiation process is observed forboth simulations. (b) the high pressure effect alternates the lithiation process from one-phase to two-phase. (d) The reaction front velocity moves as fast as the diffusion processfor (a) and (c) and slows down for (b).

obtained, contradicting experimental results [32].

For a-Si, the Li concentration profile versus radial location at different time steps is

shown in Fig. 4.3(a-c) with and without the consideration of the pressure effect. The simu-

lations in Fig. 4.3(a-c) recover the two-phase lithiation process as observed in experiments

[20]. The pressure from [45] causes the lithiation process to stop, as shown in Fig. 4.3(b),

contradicting experimental observations [20], due to an infinite pressure in [45] at the core


0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s45 50 55

0.99

0.995

1

location [nm]

concentration

(a) concentration - E0 = 0.67 eV

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s45 50 55 60

0.99

0.995

1

location [nm]

concentration

(b) concentration - E0 = 0.65 eV

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

45 50 55 60 650.99

0.995

1

location [nm]

concentration

(c) concentration - E0 = 0.63 eV

0 50 100 15010

15

20

25

30

35

40

45

E

0=0.67eV

E0=0.65eV

E0=0.63eV

time [s]



Figure 4.5: c-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for different E0 with pmax = 2.5 GPa. The bond-breaking energybarrier significantly affects the lithiation speed. For smaller value of E0, more Si is lithiated.

of nanoparticles.

For post-a-Si, the Li concentration profile versus radial location at different time steps

is shown in Fig. 4.4(a-c) with and without the consideration of the pressure effect. The

simulations in Fig. 4.4(a) and (c) are reasonable and recover the one-phase lithiation process

as observed in experiments [20]. The pressure from [45] plays an exaggerated effect on the

reaction rate and alternates the lithiation process from one-phase to two-phase, as shown

in Fig. 4.4(b), due to the infinite pressure value at the core.

The three simulations in Fig. 4.2 indicate the importance of the consideration of the


0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

50 60 700.99

0.995

1

location [nm]

concentration

(a) concentration - pmax = 1.5 GPa

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s45 50 55 60

0.99

0.995

1

location [nm]

concentration

(b) concentration - pmax = 2.5 GPa

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

45 50 550.99

0.995

1

location [nm]

concentration

(c) concentration - pmax = 3.5 GPa

0 50 100 1500

5

10

15

20

25

30

35

40

45

p

max=1.5GPa

pmax

=2.5GPa

pmax

=3.5GPa

time [s]



Figure 4.6: c-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for different pmax with the same E0 = 0.65 eV. Like the bond-breaking energy barrier, the reaction front pressure significantly affects the lithiation speed.For smaller value of pmax, more Si is lithiated.

reaction front pressure effects in the proposed model. The drastic differences in the two

pressure formulations used in Fig. 4.3(b,c) and 4.4(b,c) further outline the importance of

the hydrostatic pressure effect. In the remaining part of this section, we will only consider

the simulations with the pressure effect in (4.1) but not the pressure formulation with an

infinite value at the core of nanoparticles in [45]. In the future, a chemo-mechanical coupled

model will be used to study the diffusion-induced deformation in order to obtain a more

realistic pressure value to be inserted into (3.3).


4.1.2 Lithiation of crystalline Si nanoparticles

Values for the bond-breaking energy barrier can only be obtained from DFT calculations.

To illustrate the impact of the bond-breaking energy barrier and the pressure distribution

on the lithiation process, we carry out a parameter sensitivity study for c-Si with E0 varying

from 0.59 eV to 0.69 eV and pmax varying from 1.0 GPa to 3.5 GPa. Here, we keep one

parameter constant and vary the other one. Fig. 4.5 shows the concentration profile versus

radial location at different time steps and the reaction front location versus time for a

maximum pressure pmax fixed at 2.5 GPa while varying E0 from 0.69 eV to 0.59 eV, with

R0 = 45 nm. Fig. 4.5 indicates that the bond-breaking energy barrier significantly affects

the lithiation speed. For smaller value of E0, more Si is lithiated. The concentration profile

and reaction front location curves for E0 = 0.65 eV and pmax varying from 1.0 GPa to 3.5

GPa are shown in Fig. 4.6. It can be observed that the pressure has an impact on lithiation

speed as well. More Si is lithiated for a smaller pmax.

The numerical results are compared with the available experimental data for nanopar-

ticles in [32]. For one pair of the parameters, E0 = 0.65 eV and pmax = 2.0 GPa, we show

the concentration profile versus radial location in Fig. 4.7(a-c), and the reaction front loca-

tion versus time with comparison to experimental data in Fig. 4.7(d-f), for three different

initial radii. In the Li concentration profile plot, the sharp reaction front is recovered by

the simulation. A high Li concentration is obtained in the lithiated region with c ≈ 1.0, as

shown in Fig. 4.7(a-c), which matches the experimental observation that Li3.75Si is present

with a remaining c-Si core in the initial lithiation process of c-Si nanoparticles [32]. The

reaction front locations, as shown in Fig. 4.7(d-f), matches the experimental observation

given in [32].

4.1.3 Lithiation of amorphous Si nanoparticles

Similar to c-Si, we perform the parameter sensitivity study for a-Si with E0 varying from 0.51

eV to 0.58 eV and pmax varying from 1.0 GPa to 3.5 GPa. Fig. 4.8 shows the concentration

profile versus radial location at different time steps and the reaction front location versus

time for a maximum pressure pmax = 2.5 GPa and a varying E0, changing from 0.58 eV

to 0.51 eV, with R0 = 300 nm. Fig. 4.8 indicates that the bond-breaking energy barrier


0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

45 50 55 60 650.99

0.995

1

location [nm]

concentration

(a) R0 = 45 nm

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

experimental data

simulation data

time [s]reactionfrontlocation[nm]

(b) R0 = 45 nm

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

70 80 900.99

0.995

1

location [nm]

concentration

(c) R0 = 66.5 nm

0 20 40 60 80 1000

10

20

30

40

50

60

experimental data

simulation data

time [s]


(d) R0 = 66.5 nm

Figure 4.7: c-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for E0 = 0.65 eV and pmax = 2.0 GPa (a,c,e), and the reaction frontlocation versus time (b,d,f) with the comparison of the experimental data for three differentinitial radii. (a,c,e) An evident reaction front is observed. In the lithiated region, a highLi concentration level is obtained from the simulation. (b,d,f) The reaction front locationcurves match the experimental observation and the gradient of those curves indicate aslow-down tendency for the reaction front velocity.


0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s100 110 120

0.99

0.995

1

location [nm]

concentration

(e) R0 = 92.5 nm

0 20 40 60 800

10

20

30

40

50

60

70

80

90

experimental data

simulation data

time [s]


(f) R0 = 92.5 nm

Figure 4.7: c-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for E0 = 0.65 eV and pmax = 2.0 GPa (a,c,e), and the reaction frontlocation versus time (b,d,f) with the comparison of the experimental data for three differentinitial radii. (a,c,e) An evident reaction front is observed. In the lithiated region, a highLi concentration level is obtained from the simulation. (b,d,f) The reaction front locationcurves match the experimental observation and the gradient of those curves indicate aslow-down tendency for the reaction front velocity (cont.).

significantly affects the concentration profiles and the lithiation speed. The Li concentration

level is higher in the lithiated region for larger E0, but is much lower than the concentration

level for c-Si in Fig. 4.5. Similar to c-Si, the lithiation speed is slower for larger E0 as shown

in Fig. 4.8(d). The concentration profile and reaction front location curves for E0 = 0.56

eV and pmax varying from 1.0 GPa to 3.5 GPa are shown in Fig. 4.9. It can be observed

that the pressure has an impact on the concentration profiles and the lithiation speed. The

Li concentration level is higher in the lithiated region for higher pmax. Similar as for c-Si,

the lithiation speed is slower for larger pmax.

The numerical results are compared with the experimental data in the supporting file

of [20]. For one pair of the parameters E0 = 0.55 eV and pmax = 1.9 GPa, we show the

concentration profile versus radial location in Fig. 4.10(a) and the reaction front location

versus time with comparison to experimental data in Fig. 4.10(b) for R0 = 291 nm. The

reaction front location from the simulation is comparable to the experimental observation

as shown in Fig. 4.10(b).


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 1500

50

100

150

200

250

300

E

0=0.58eV

E0=0.56eV

E0=0.54eV

time [s]



Figure 4.8: a-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for different E0 with pmax = 2.5 GPa and R0 = 300 nm. The bond-breaking energy barrier significantly affects the concentration profiles and the lithiationspeed. The Li concentration level is higher in the lithiated region for larger E0, but is muchlower than the concentration level for c-Si in Fig. 4.5. Similar as for c-Si, the lithiationspeed is slower for larger E0 as shown in (d).

4.1.4 Lithiation of post amorphous Si nanoparticles

For post-a-Si nanoparticle with R0 = 300 nm, the parameter sensitivity study is carried

out with E0 varying from 0.45 eV to 0.50 eV and pmax varying from 1.0 GPa to 3.5 GPa.

Similar to c-Si and a-Si, we first run simulations with different E0, while keeping pmax = 2.5

GPa constant. Then, we investigate the pressure effect with E0 = 0.48 eV and varying


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 1500

50

100

150

200

250

300

p

max=1.5GPa

pmax

=2.5GPa

pmax

=3.5GPa

time [s]



Figure 4.9: a-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for different pmax with E0 = 0.56 eV and R0 = 300 nm. The reactionfront pressure significantly affects the concentration profiles and the lithiation speed. TheLi concentration level is higher in the lithiated region for higher pressure. Similar as forc-Si, the lithiation speed is slower for larger pmax.

pmax. The concentration profile versus radial location at different time steps and the reac-

tion front location versus time for those two sets of simulations are plotted in Fig. 4.11 and

4.12, respectively. From Fig. 4.11, it can be observed the one-phase lithiation process is

recovered for all the different bond-breaking energy barriers under the given pressure effect.

The lithiation process proceeds as fast as the diffusion process. The different bond-breaking


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration

(a) R0 = 291 nm

0 50 100 1500

50

100

150

200

250

experimental data

simulation data

time [s]


(b) R0 = 291 nm

Figure 4.10: a-Si simulation – Illustration of the concentration profile versus radial locationat different time steps for E0 = 0.55 eV (left), and the reaction front location versus time(right) with comparison of the experimental data.

energy barriers E0 show only a miner effect on the lithiation process. Fig. 4.12 shows the in-

fluence of the hydrostatic pressure on the lithiation process. Like the bond-breaking energy

barriers, also the hydrostatic pressure does not change the lithiation process significantly.

However, if the hydrostatic pressure at the reaction front in the Si core approaches an infi-

nite value as shown in Fig. 4.4(b), the concentration profile from the numerical simulation

might be misleading. But since also the distribution in Equation (4.1) is of empirical na-

ture, we are currently working on a fully chemo-mechanical coupled model accounting for

a Drucker-Prager type plasticity to automatically provide us with the missing information

of the real pressure distribution at the reaction front.

4.1.5 Discussion

The simulations carried out in Section 4.1 outline the performance of the proposed model

and support the consideration of the reaction and diffusion separately. Also, the numerical

simulation support the account of the pressure effect on the reaction rate. By varying

certain physical material parameters, such as the bond-breaking energy barrier and the

initial threshold of Li concentration, which can be obtained from DFT calculations, the

proposed model is able to describe the lithiation process of all three types of Si discussed


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 1500

50

100

150

200

250

300

E

0=0.50eV

E0=0.48eV

E0=0.46eV

27.1 27.2 27.3 27.40

0.05

0.1

time [s]



Figure 4.11: post-a-Si simulation – Illustration of the concentration profile versus radiallocation at different time steps for different E0 with pmax = 2.5 GPa for R0 = 300 nm.Since the bond-breaking energy barriers used here are very small, the one-phase lithiationprocess is recovered for all the different energy barriers under the given pressure effect. Thelithiation process proceeds as fast as the diffusion process.

here.

The proposed model can be easily implemented within existing diffusion models and

thereby improve the computational efficiency without introducing additional degrees of

freedom. Since the chemical reaction is coupled with the Li concentration in one way, the

reaction rate is determined based on the computed Li concentration profile on a local level.

Consequently, the reaction front location is determined locally, differing from [46], where

the reaction front location is introduced as one additional degree of freedom. Because of the


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0s

t=10s

t=25s

t=50s

t=75s

t=100s

t=125s

t=150s

location [nm]

concentration


0 50 100 1500

50

100

150

200

250

300

p

max=1.5GPa

pmax

=2.5GPa

pmax

=3.5GPa

27.1 27.2 27.3 27.40

0.05

0.1

time [s]



Figure 4.12: post-a-Si simulation – Illustration of the concentration profile versus radiallocation at different time steps for different pmax with the same E0 = 0.48 eV for R0 = 300nm. The reaction front hydrostatic pressure has less effect on the concentration profiles andthe lithiation speed, when compared with Fig. 4.6 and 4.9.

automatically generated two-phase diffusion process with a high Li concentration drop at

the reaction front, the usage of the Cahn-Hilliard theory to capture the phase transformation

in the continuum model [43, 44], which would require the introduction of one additional

degree of freedom when employing the finite element method to solve the 4th order partial

differential equation with a piecewise linear approximation of the Li concentration, can be

avoided.

By considering the reaction and diffusion process separately, rather than considering


the reaction as a non-linear diffusion process, the proposed model is capable of providing

additional insight to understand the competition between diffusion and reaction processes

during (de)lithiation. It also suggests different ways to mitigate high concentration gradients

at c-Si. If particular non-damage technologies like, X-ray, heating, applied electrical field,

magnetic field, could be used to agitate the c-Si, the reaction process might be increased

and the stress state will be altered, leading to an improvement of the performance of c-Si.

Also, the critical size of a-Si might be increased by doping technologies which decrease the

energy barrier and mitigate the concentration at the interface region.

The proposed model is capable to study the lithiation process involving multiple phases.

A two-stage lithiation process is postulated in [19] for a-Si, based on computing the average

Li concentration in the lithiated region, where the Li concentration first reaches x ∼ 2.5

in a-LixSi, followed by a second lithiation stage when x ∼ 3.75. If it is true that there

exists a second phase of lithiation for a-Si, in addition to the initial bond-breaking energy

barrier needed to break the covalent bonds to create the diffusion paths, a second energy

barrier and an additional reaction front will be needed to capture the second lithiation

stage. However, the simulation of such two-stage lithiation with the proposed model is not

attempted in this work, due to the lack of experimental information for the second stage

lithiation. As shown in the first simulation for E0 = 0.58 eV, the concentration gradient is

small at the interfacial region. The obtained x ∼ 2.5 in the experiment [19] might be caused

by incomplete diffusion, since there is no evidence that shows that the lithiated a-LixSi has

a homogeneous Li distribution.

For other materials like Germanium, which involves different phases during the lithia-

tion and delithiation process, such as Li9Ge4, LixGe and Li15Ge4 [144], this model can be

very useful. If the multiple phases are caused by different bond-breaking energy barriers

at different applied electric potential levels, with additional information about the physical

quantities such as different bond-breaking energy barriers, diffusion coefficients, and dif-

ferent concentration threshold levels, the proposed model will be capable to capture very

complicated (de)lithiation processes of Germanium by introducing multiple energy barriers

and reaction fronts.

4.2. DIFFUSION-INDUCED DEFORMATION OF SI ANODES 65

4.2 Diffusion-induced deformation of Si anodes

In this section, we carry out several numerical simulations to demonstrate the performance

of the proposed variational framework and the importance of considering the reaction front

in this chemo-mechanical coupled fracture problem. All the material parameters that are

used in the numerical simulation part for this chemo-mechanical coupled fracture problem

are summarized in Table 4.2. We restrict ourselves to ndim = 2 in this paper, but the

extension to 3D is conceptually straightforward. Experimental results reveal a diffusion-

induced anisotropic deformation and a crystalline direction dependent fracture behavior for

〈100〉 and 〈110〉 c-Si nanopillars [21], as shown in Fig. 4.13(a) and Fig. 4.17(a). To show

that our computational framework is capable of capturing diffusion-induced anisotropic

deformation and the crystalline direction dependent fracture behavior, we investigate the

initial lithiation process for 〈100〉 c-Si nanopillars in Section 4.2.1 and 〈110〉 c-Si nanopillars

in Section 4.2.2. In Section 4.2.3, we show how multiple cracks form and propagate in a

realistic irregular Si electrode. Finally, we investigate the lithiation process of a constrained

〈110〉 c-Si nanopillar and show how geometric constraints affects the fracture behavior of Si

electrodes in Section 4.2.4.

4.2.1 Diffusion-induced deformation of 〈100〉 crystalline Si nanopillars

We start investigating diffusion-induced large plastic deformation under the presence of

the reaction front in 〈110〉 c-Si nanopillars initially without considering damage in Section

4.2.1.1 and thereby calibrate the concentration dependent diffusion coefficient. Damage is

considered in Section 4.2.1.2, where we study diffusion-induced fracture. We model one

quarter of the circular nanopillar geometry with 11638 Q1 elements under plane strain,

where the field variables ϕ, µ, d are approximated with standard linear shape functions,

and the mechanical boundary conditions shown in Fig. 4.13(b). In all simulations, an initial

chemical potential µ0 = −11.5 J/mol (equivalent to c0 = 0.01) is applied throughout the

electrode. The prescribed chemical potential at the outer surface linearly increases from µ0

to µ = 11.5 J/mol (equivalent to c = 0.99) within the first second and is kept constant for

the remainder of the simulation.


Table 4.2: Summary of the parameters used in the simulations performed in Section 4.2

Name Parameter Value Unit Eqn. Ref

Young’s Modulus (c-Si) E 169 GPa (2.42) [153]Poisson’s Ratio (c-Si) ν 0.28 - (2.42) [153]Young’s Modulus (a-Si) E 80 GPa (2.42) [154]Poisson’s Ratio (a-Si) ν 0.22 - (2.42) [154]Young’s Modulus (Li3.75Si) E 41 GPa (2.42) [38]Poisson’s Ratio (Li3.75Si) ν 0.24 - (2.42) [155]Yield stress σ0 0.5∼2.0 GPa (2.45) [35, 156]Fracture energy release rate gc 2.4∼14.9 J/m2 (2.35) [39]Diffusion coefficient D0

diff(1∼106) · 10−13 cm2/s (2.48) [149, 157]

Expansion coefficient Ω 2.8 - (2.2)Bond-breaking energy barrier (c-Si) E0 0.59∼0.69 eV (3.3) [116]Bond-breaking energy barrier (a-Si) E0 0.51∼0.58 eV (3.3) [116]Gas constant R 8.3144 J/(mol·

K)(2.46)

Boltzmann constant kB 8.6173 · 10−5 eV/K (3.3)Room temperature T 300 K (3.3)Maximum concentration cmax 8.867 · 104 mol/m3 (2.46) [57]Frequency parameter A 5.0 · 109 s−1 (3.3) [116]Volumetric parameter (c-Si) φcSi 1.0 - (3.3) [116]Volumetric parameter (a-Si) φaSi 0.8 - (3.3) [116]Scaling parameter C 50 - (3.5) [116]Threshold value (c-Si) c0 0.10 - (3.5) [116]Threshold value (a-Si) c0 0.06 - (3.5) [116]

4.2.1.1 One-phase lithiation versus two-phase lithiation with plastic deforma-

tion

Both, elastic and elasto-plastic models are commonly used in the literature [57, 158, 126]

to predict diffusion-induced swelling of Si electrodes. To investigate their differences and

demonstrate the importance of considering large plastic deformations for modeling diffusion-

induced fracture of Si electrodes, we perform elastic and elasto-plastic simulations for

nanopillars of radius R = 300 nm. We do this initially without considering the presence of

a reaction front, thereby mimicking a one-phase lithiation process. As the diffusion-induced

deformation is isotropic in this case, we only report the evolution of hoop stresses at the

location labeled as A in Fig. 4.13(b). We observe from Fig. 4.14(a) that hoop stresses

at the outer surface of the electrodes remain compressive throughout the simulation when

neglecting plastic effects, whereas otherwise those hoop stresses change from compressive

to tensile during the lithiation process, thereby leading to the fracture of Si electrodes.


(a) experimental results

〈100〉

〈110〉A

〈110〉

〈110〉

〈110〉

〈100〉B

〈100〉

〈100〉

〈100

〉

R

(b) simulation setup

Figure 4.13: 〈100〉 c-Si nanopillars. (a) Illustration of the diffusion-induced anisotropic de-formation and crystalline direction dependent fracture behavior for 〈100〉 c-Si nanopillars.Fracture locations are indicated by red arrows [21]. (b) Illustration of the crystalline direc-tions of a 〈100〉 c-Si nanopillar as well as the finite element mesh and mechanical boundaryconditions used in Section 4.2.1. Locations A and B are used to report the evolution ofhoop stresses in Fig. 4.14(a).

Next, we consider the presence of the reaction front and determine its influence on the

evolution of hoop stresses in Si electrodes, thereby mimicking the two-phase lithiation pro-

cess. The crystalline directions for a 〈100〉 c-Si nanopillar are given in Fig. 4.13(b). First, we

consider a uniform bond-breaking energy barrier E0 = 0.60 eV to model diffusion-induced

isotropic elasto-plastic deformation. Second, we consider two different bond-breaking en-

ergy barriers for the 〈100〉 and the 〈110〉 directions with E0 = 0.60 eV and E0 = 0.66


0 5 10 15 20 25 30-10

-8

-6

-4

-2

0

1

one-phase elas: Aone-phase plas: Atwo-phase (iso): Atwo-phase (aniso): Atwo-phase (aniso): B

20 25 30

0.2

0.3

0.4

0.5

time [s]

hoopstresses

[GPa]

(a) evolution of hoop stresses at points A and B

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

D0

diff= 1 · 10

-12

D0

diff=1~15 · 10

-12

D0

diff=1~25 · 10

-12

D0

diff=1~50 · 10

-12

240 260 280 300

0.5

0.6

0.7

0.8

0.9

1

radius [nm]

concentration[-]

(b) concentration profile along the radius

Figure 4.14: 〈100〉 c-Si nanopillars. (a) Evolution of hoop stress at points labelled as Aand B in Fig. 4.13(b) for four different simulation setups: one-phase lithiation with anelastic model; one-phase lithiation with an elasto-plastic model; two-phase lithiation foran isotropic elasto-plastic deformation with a uniform E0; and two-phase lithiation for ananisotropic elasto-plastic deformation with different values of E0 in different crystallinedirections. (b) concentration profile along the radius of the Si electrode undergoing two-phase lithiation for different diffusion coefficients at t = 20 s. The larger the diffusioncoefficient the larger is the concentration gradient at the reaction front.

eV, respectively, to account for anisotropic effects. In Fig. 4.14(a) we report the hoop

stresses at point A for the isotropic case and at points A and B for the anisotropic case.

Both two-phase lithiation results indicate a rapid transition from compressive to tensile at

point A of the outer electrode surface. Furthermore, for the anisotropic case, our model

predicts a possible fracture location at point B (close to the 〈100〉 crystalline direction)

with even higher tensile hoop stresses, which is consistent with experimental results shown

in Fig. 4.13(a). The ability to model diffusion-induced anisotropic deformation in different

crystalline directions based on the different bond-breaking energy barriers E0 is a unique

feature of our computational framework.

We so far assumed that the diffusion coefficient D0diff = 10−12 cm2/s remains constant

throughout the lithiation process. This simplification ensures that the difference of hoop

stresses reported in Fig. 4.14(a) arises from the mechanical models or the presence of the

reaction front. However, it is reported in the literature that the diffusion coefficient highly

depends on the concentration level, varying orders of magnitude [149, 157]. To investigate


the effect of the diffusion coefficient on the isotropic two-phase lithiation results reported

above, we perform four different simulations, whose results are shown in Fig. 4.14(b).

First, the diffusion coefficient is kept constant at D0diff = 1 · 10−12 cm2/s. For the other

three cases, D0diff increases linearly from 1 · 10−12 to 15 · 10−12, 25 · 10−12, and 50 · 10−12

cm2/s with c increasing from 0 to 1. For all four cases, the concentration profile along

the radial direction at t = 20s is plotted in Fig. 4.14(b). One can observe that a high

concentration gradient is obtained at the reaction front for the concentration dependent

diffusion coefficients, which is consistent with experimental results [32]. Therefore, we

consider a linearly varying concentration dependent diffusion coefficient with D0diff = 1 ∼

15 · 10−12 cm2/s for c varying from 0 to 1 for the remaining simulations in this work.

Contrary to the diffusion coefficient, we though neglect a possible dependency of Young’s

modulus and Poisson’s ratio of LixSi with varying concentration c as reported in [149, 157].

Instead, we choose the corresponding values from Li3.75Si. This assumption is supported by

the fact that the concentration profile shown in Figs. 4.15 and 4.18 remains almost constant

in the lithiated region with c ≈ 1.

4.2.1.2 Diffusion-induced fracture of 〈100〉 crystalline Si nanopillars

In this section we now account for damage and investigate the diffusion-induced size depen-

dent fracture behavior for c-Si 〈100〉 nanopillars. We perform simulations with the same

mesh as in Section 4.2.1.1 but with varying radii and fracture energy release rates. We again

choose bond-breaking energy barriers E0 = 0.60 eV and E0 = 0.66 eV for the 〈110〉 and

the 〈100〉 directions, respectively. Motivated by the results obtained in Section 4.2.1.1, the

diffusion coefficient varies linearly from 10−12 cm2/s to 15 · 10−12 cm2/s with c increasing

from 0 to 1. Its dependency on the damage field shown in (2.48) is neglected here.

For a nanopillar with a radius of R = 300 nm and a fracture energy release rate of

gc = 12.5 J/m2, we follow the criteria given in [82] and choose a length parameter of

l = 8 nm. The evolution of the concentration field c and the damage field d during the

lithiation process are shown in Fig. 4.15 at different states of charge (SOC), defined as the

ratio between the total amount of Li particles in the electrode and the maximum Li the


(a) concentration profiles in c-Si electrode at SOC = 0.20, 0.25, and 0.35

(b) damage field profiles in c-Si electrode at SOC = 0.20, 0.25, and 0.35

Figure 4.15: 〈100〉 c-Si nanopillars. Plots of the concentration field c and the damagefield d for the two-phase lithiation process with gc = 12.5 J/m2 at SOC = 0.20, 0.25, and0.35. (a) The c-Si nanopillar deforms anisotropically. The concentration profiles show ahigh concentration gradient at the reaction front. The black lines indicate a contour forthe damage field d = 0.95. (b) The diffusion-induced anisotropic deformation causes thefracture to initiate close to the 〈100〉 direction and to propagate inward, restricted by thereaction front moving speed.

electrode can hold. Since our model allows us to distinguish between the two different bond-

breaking energy barriers for the two different crystalline directions of the c-Si nanopillar, it

is shown in Fig. 4.15 that an anisotropic deformation is obtained for different SOC while

continuously charging the Si nanopillar, which results in a crystalline direction dependent

fracture behavior, taking place dominantly in 〈100〉 direction. Also, the linearly varying

diffusion coefficient results in a sharp concentration gradient at the reaction front as shown

in Fig. 4.15(a), matching experimental results [32]. As fracture takes place only in the

lithiated region, the crack growth is limited by the reaction front moving speed.

In Fig. 4.16(a) the relation between SOC and the crack length is shown for different

energy release rates gc. A continuous increase of the crack length, defined as∫

B0γ(d,∇d) dA


0 50 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

gc=7.5J/m 2

gc=10J/m 2

gc=12.5J/m 2

crack length [nm]

SOC

[-]

(a) crack length vs SOC for different gc

0

0.2

0.4

0.6

0.8

1

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

gc=7.5J/m 2

gc=10J/m 2

gc=12.5J/m 2

experimental data

radius [nm]

SOC

[-]

non-fra

cture

ratio

(b) SOC vs radii for different gc

Figure 4.16: 〈100〉 c-Si nanopillars. (a) Relation between SOC and crack length of a c-Sinanopillar with R = 300 nm for three different gc. (b) Plot of SOC at the onset of cracknucleation in Si nanopillars of different radii and gc. Points surrounded with a square markerindicate results without fracture onset. The simulation results are compared with statisticalresults from experiments, which are plotted as bar graphs in (b), where nanopillars withradius of 100 nm, 145 nm and 195 nm, 245 nm show 100% non-fracture ratio and 0%non-fracture ratio, respectively [58].

in the 2D setting with γ given in (2.35), is observed during the charging process. In Fig.

4.16(b), the SOC when the maximum value of the damage field reaches d = 0.8 is plotted

for different radii of the Si nanopillar and different gc. The smaller the radius and larger

gc, the larger the attainable SOC, before the crack initiates. The points surrounded by

a square marker for a SOC of approximately 0.4 indicate numerical results for which the

crack has not yet formed for the corresponding radius, but severe mesh distortion did not

allow us to continue the charging process. Fig. 4.16(b) also includes experimental results

taken from [58], which report no fracture for nanopillars of radius 100 nm and 145 nm but

fracture taking place for nanopillars of radius 195 nm and 245 nm.

4.2.2 Diffusion-induced deformation of 〈110〉 crystalline Si nanopillars

In this section, we now investigate diffusion-induced fracture for 〈110〉 c-Si nanopillars.

Similar as for 〈100〉 c-Si nanopillars, experiments again reveal a diffusion-induced anisotropic

deformation and crystalline direction dependent fracture behavior, as shown in Fig. 4.17(a).

We use the same mesh, initial conditions, and loadings as in Section 4.2.1, but change the



〈110〉

〈100〉

〈100〉

〈110〉〈110〉 R


Figure 4.17: 〈110〉 c-Si nanopillars. (a) Illustration of the diffusion-induced anisotropic de-formation and crystalline direction dependent fracture behavior for 〈110〉 c-Si nanopillars.Fracture locations are indicated by red arrows [21]. (b) Illustration of the crystalline direc-tions of a 〈110〉 c-Si nanopillar as well as the finite element mesh and mechanical boundaryconditions used in Section 4.2.2.

crystalline directions to those of a 〈110〉 c-Si nanopillar, as shown in Fig. 4.17(b).

For a nanopillar with a radius of R = 300 nm and a fracture energy release rate gc = 12.5

J/m2, the evolution of the concentration field c and the damage field d during the lithiation

process are shown in Fig. 4.18 at different SOC. Since our model allows us to distinguish

between the two different bond-breaking energy barriers for the two different crystalline

directions of the c-Si nanopillar, Fig. 4.18 shows that also for 〈110〉 c-Si nanopillars an

anisotropic deformation is obtained for different SOC while continuously charging the Si


(a) concentration profiles in c-Si electrode at SOC = 0.28, 0.35, and 0.45

(b) damage field profiles in c-Si electrode at SOC = 0.28, 0.35, and 0.45

Figure 4.18: 〈110〉 c-Si nanopillars. Plots of the concentration field c and the damagefield d for the two-phase lithiation process with gc = 12.5 J/m2 at SOC = 0.28, 0.35, and0.45. (a) The c-Si nanopillar deforms anisotropically. The concentration profiles show ahigh concentration gradient at the reaction front. The black lines indicate a contour forthe damage field d = 0.95. (b) The diffusion-induced anisotropic deformation causes thefracture to initiate close to the 〈100〉 direction and to propagate inward, restricted by thereaction front moving speed.

nanopillar, which again results in a crystalline direction dependent fracture behavior, taking

place dominantly in 〈100〉 direction. As fracture takes place only in the lithiated region,

the crack growth is again limited by the reaction front moving speed.

In Fig. 4.19(a) the relation between SOC and the crack length for different energy

release rates gc is now also shown for 〈110〉 c-Si nanopillars. Again, a continuous increase

of the crack length is observed during the charging process. In Fig. 4.19(b), again the SOC

when the maximum value of the damage field reaches d = 0.8 is plotted for different radii

of the Si nanopillar and different gc. Again, the smaller the radius and larger gc, the larger

the attainable SOC, before the crack initiates. The points surrounded by a square marker

for a SOC of approximately 0.4 indicate again numerical results for which the crack has

not yet formed for the corresponding radius, but severe mesh distortion did not allow us to

continue the charging process. Fig. 4.19(b) also includes experimental results taken from

[58], which report no fracture for nanopillars of radius 130 nm and 145 nm but fracture


0 50 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

gc=7.5J/m 2

gc=10J/m 2

gc=12.5J/m 2

crack length [nm]

SOC

[-]

(a) crack length vs SOC for different gc

0

0.2

0.4

0.6

0.8

1

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

gc=7.5J/m2

gc=10J/m2

gc=12.5J/m2

experimental data

radius [nm]

SOC

[-]

non-fra

cture

ratio

(b) SOC vs radius for different gc

Figure 4.19: 〈110〉 c-Si nanopillars. (a) Relation between SOC and crack length of a c-Sinanopillar with R = 300 nm for three different gc. (b) Plot of SOC at the onset of cracknucleation in Si nanopillars of different radii and gc. Points surrounded with a square markerindicate results without fracture onset. The simulation results are compared with statisticalresults from experiments, which are plotted as bar graphs in (b), where nanopillars withradius of 130 nm and 145 nm show 100% non-fracture ratio but those with radius of 180nm and 195 nm show a 95% and 0% non-fracture ratio, respectively [58].

(a) experimental results (b) mesh

Figure 4.20: Irregular Si nanoparticles. (a) Experiments show that lithiated irregular Sinanoparticles may form multiple cracks [159]. (b) Illustration of the finite element meshused in Section 4.2.3.

taking place for 5% of the nanopillars with a radius of 180 nm whereas all of those with a

radius of 195 nm fracture.


#1

#2

#3

#4

#5

#6

#7

(a) 7.5 J/m2 (SOC = 0.35)

#1

#2

#3

#4

#5

#6#7

(b) 10 J/m2 (SOC = 0.35)

#1

#2

#3

#4(c) 12.5 J/m2 (SOC = 0.35)

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

gc=7.5J/m 2

gc=10J/m 2

gc=12.5J/m 2

crack sequence

SOC

[-]

(d) crack sequence vs SOC

Figure 4.21: Irregular Si nanoparticles. (a-c) Profile of the damage field d for a Si nanopar-ticle with three different fracture energy release rates at a SOC of 0.35 (elements withd ≥ 0.96 are deleted in a postprocessing step for plotting purposes only). The numbers in(a-c) indicate the crack initiation sequence. (d) Values for the SOC at the onset of crackinitiation for each crack.

4.2.3 Diffusion-induced fracture of irregular Si nanoparticles

Next, we investigate the lithiation process of a realistic irregular Si nanoparticle. Experi-

mental results [159] suggest that multiple cracks can form in one Si particle, as shown in

Fig. 4.20(a). To reproduce these results with our computational framework, we consider a

nanoparticle with a diameter of approximately 1.0 µm and a mesh of 11805 Q1 elements

under plane strain conditions, as shown in Fig. 4.20(b). Two nodes in the center of the


particle are fixed to avoid rigid body motions. The same initial and loading conditions as

in Section 4.2.1 are used.

As the crystalline directions for such irregular Si nanoparticles are in general not avail-

able, we treat the Si nanoparticle as amorphous and choose a uniform bond-breaking energy

barrier E0 = 0.58 eV when modeling the two-phase lithiation process here. To investigate

the fracture behavior of irregular Si nanoparticles, we perform a number of simulations

with a length parameter l = 20 nm and different fracture energy release rates gc. We plot

the damage field d in Fig. 4.21(a-c) for a SOC of 0.35 and indicate the crack formation

sequence by numbers for gc = 7.5 J/m2, gc = 10 J/m2, and gc = 12.5 J/m2. Fig. 4.21(d)

shows the SOC when each of those cracks form (maximum damage value reaches d = 0.8).

The smaller the fracture energy release rate, the smaller the SOC at crack initiation and

the more cracks form during the lithiation process. Still, the locations of the major, earlier

forming cracks depend more on the geometry of the nanoparticle rather than the energy

release rate. To conclude, our computational framework has the potential to be beneficial

for designing fracture resistant electrode geometries.

4.2.4 〈110〉 crystalline Si nanopillars under geometric constraints

In our final example, we want to investigate an experimentally observed increase in the

fracture resistance of lithiated crystalline Si nanopillars with radius R under geometric

constraints [160], as shown in Fig. 4.22(a). To do so, we revisit the problem investigated

in Section 4.2.2 but now restrict the deformation of the 〈110〉 nanopillar by adding a rigid

wall at a horizontal distance ∆, as shown in Fig. 4.22(b). We again model one quarter of

the geometry with a mesh consisting of 10924 Q1 elements under plane strain conditions.

The crystalline directions are distributed as shown in Fig. 4.17(b) and the initial and

loading conditions are identical to those in Section 4.2.1. The values of the initial gap

∆ are chosen such that the volume expansion during the lithiation process will result in

frictional contact at the electrode/wall interface, thereby affecting the fracture process. The

constraint condition is applied by using a frictional contact model with Lagrange multiplier

method provided in the finite element analysis program FEAP [161].

In particular, we perform simulations with different radii R and ∆/R ratios for three



R ∆


Figure 4.22: 〈110〉 c-Si nanopillars under geometric constraints. (a) Experiments show thatgeometric constraints affect the fracture behavior of 〈110〉 c-Si nanopillars [160]. The ellipseschematically shows the geometry of a lithiated nanopillar and the potential crack locationsat the top and bottom when there is no geometric constraint. The red arrows show thealtered crack locations when a geometric constraint is present. (b) Illustration of the meshused in Section 4.2.4, where the initial gap between the nanopillar and the rigid wall isdenoted as ∆.

different fracture energy release rates gc and a length scale of l = 8 nm. In Fig. 4.23(a-c),

we plot the damage field d for a nanopillar with radius R = 500 nm and ratios ∆/R = 0.15,

∆/R = 0.3, and ∆/R = 0.4 for gc = 12.5 J/m2 at a SOC of 0.27. The former two ratios

result in contact between the lithiated nanopillar and the rigid wall, which subsequently

results in a delayed crack initiation and propagation compared to the latter case, where there

is no contact, for the SOC considered here. The resulting crack lengths are computed as 0

nm (no fracture), 118 nm, and 123 nm for the three ratios. Thus, the geometric constraint


indeed has an affect on the crack formation and increases the nanopillars fracture resistance.

We argue that the geometric constraint increases the pressure inside the nanoparticle and

thereby slows down the lithiation speed in the 〈100〉 direction, which subsequently reduces

the level of anisotropy and fracture.

To further quantify these results, we plot in Fig. 4.23 (d-f) the SOC at fracture onset

(maximum damage value reaches d = 0.8) for different radii R = 400 nm and R = 500

nm, different ratios ∆/R = 0.15 (strong geometric constraint) to ∆/R = 0.4 (no geometric

constraint), and different fracture energy release rates gc = 7.5 J/m2, gc = 10 J/m2, and

gc = 12.5 J/m2. We can observe that for a radius of R = 400 nm, the ratio ∆/R = 0.15

prevents crack initiation for all three values of gc and for a SOC of up to 0.27 (those points

surrounded with a square marker indicate again that no fracture takes place). On the other

hand, for a weaker constraint ∆/R ≥ 0.2, almost no difference of the SOC at fracture onset

is obtained when compared to the SOC of the fracture onset for the unconstrained case

∆/R = 0.4. Also the resistance increase is diminished for the larger radius of R = 500 nm,

where e.g. for the case of gc = 7.5 J/m2 the SOC = 0.21 at fracture onset for ∆/R = 0.15

is only slightly larger than the SOC = 0.18 of fracture onset in the unconstrained case.

The results for the constrained Si electrodes are consistent with the critical size study in

Section 4.2.1.2 and Section 4.2.2, where electrodes with larger radius or smaller fracture

energy release rate fracture at a lower level of SOC.

The results of SOC versus ∆/R for different radii and gc are plotted in Fig. 4.23 (d-f).

For a smaller radius with R = 400 nm, a constraint with ∆/R = 15% can delay the crack

initiation for all three different fracture energy release rates. For example, at SOC = 0.27,

no crack nucleates in the electrodes for all the three gc. The non-fractured electrode is

indicated by the point surrounded with a square marker. The geometric constraints have

less influence on crack initiation with ∆/R ≥ 20%, where cracks initiate at almost the

same level of SOC. The SOC for ∆/R = 40% can be viewed as a lithiation process of a

free standing electrodes without geometric constraints, because cracks nucleate before the

swelled electrodes contact with the rigid wall, as shown in Fig. 4.23(a). For a larger radius

with R = 500 nm, the geometric constraints delay the crack initiation for different gc as for

R = 400 nm. The delay effect for R = 500 nm is not as effective as for R = 400 nm with


(a) R = 500 nm, ∆R

= 0.15 (SOC = 0.27)

0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

R=400nmR=500nm

∆/R ratio [-]

SOC

[-]

(b) gc = 7.5 J/m2

(c) R = 500 nm, ∆R

= 0.3 (SOC = 0.27)

0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

R=400nmR=500nm

∆/R ratio [-]

SOC

[-]

(d) gc = 10.0 J/m2

Figure 4.23: 〈110〉 c-Si nanopillars under geometric constraints. (a-c) The damage field d fora 〈110〉 c-Si nanopillar with R = 500 nm under geometrical constraints with different ∆/Rratios for gc = 12.5 J/m2 at SOC = 0.27 is shown. (d-f) SOC at fracture onset (maximumdamage value d = 0.8) for different radii R = 400 nm and R = 500 nm, different ratios∆/R = 0.15 (strong geometric constraint) to ∆/R = 0.4 (no geometric constraint), anddifferent fracture energy release rates gc.


(e) R = 500 nm, ∆R

= 0.4 (SOC = 0.27)

0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

R=400nmR=500nm

∆/R ratio [-]SOC

[-]

(f) gc = 12.5 J/m2

Figure 4.23: 〈110〉 c-Si nanopillars under geometric constraints. (a-c) The damage field d fora 〈110〉 c-Si nanopillar with R = 500 nm under geometrical constraints with different ∆/Rratios for gc = 12.5 J/m2 at SOC = 0.27 is shown. (d-f) SOC at fracture onset (maximumdamage value d = 0.8) for different radii R = 400 nm and R = 500 nm, different ratios∆/R = 0.15 (strong geometric constraint) to ∆/R = 0.4 (no geometric constraint), anddifferent fracture energy release rates gc (cont.).

gc = 7.5 J/m2, where the former cracks at SOC = 0.20 and the latter does not at SOC

= 0.27. The results for the constrained Si electrodes are consistent with the critical size

study in Section 4.2.1.2 and Section 4.2.2, where electrodes with larger radius or smaller

fracture energy release rate fracture at a lower level of SOC.

Chapter 5

Summary and future directions

5.1 Summary

The goal of this work is to study the diffusion and mechanical behavior of Si anodes to

understand the important role of mechanics in electrochemical systems. To do so, we first

formulate a variational-based electro-chemo-mechanical coupled computational framework

to model diffusion-induced large deformation and phase field fracture in Chapter 2. The

variational framework suggests treatment of chemical potential, instead of the concentration

field, as an independent unknown. In this framework, we split the elastic free energy

into a tensile contribution and a compressive contribution, where only the tensile part

will drive crack propagation. The electrical field is coupled with the chemical field via a

phenomenological Butler-Volmer kinetic equation, which allows us to compute the electric

potential or current density of the Si-based half-cells.

To study diffusion-induced large deformation of Si anodes, we first need to understand

the two-phase lithiation of Si anodes during the initial charging process. In Chapter 3,

we propose a RCD model to universally describe the lithiation process for three types of

Si materials (c-Si, a-Si, and post-a-Si) based on a key physical material parameter, the

bond-breaking energy barrier E0. In this model, we account for the influence of hydrostatic

pressure on the reaction rate at the reaction front. The RCD model is combined with the

variational framework in Chapter 2 to study diffusion-induced anisotropic deformation and

fracture of Si anodes.

81

82 CHAPTER 5. SUMMARY AND FUTURE DIRECTIONS

Various numerical simulations are presented in Chapter 4 to demonstrate the perfor-

mance of our models. To study the diffusion behavior of Si anodes, we use several 1D

simulations to show how the reaction front moving rate is related to the bond-breaking

energy barrier and the hydrostatic pressure. The numerical simulations demonstrate the

importance of accounting for the hydrostatic pressure at the reaction front on the reaction

rate. By treating diffusion and reaction separately, the RCD model might suggest different

possibilities for advancing the performance of Si by either improving the chemical reaction

speed or diffusion property between Li and Si. This RCD model also has the potential to

explain the complicated diffusion-induced phase transformation in other materials, such as

Ge. We then use several 2D simulations to show that the reaction front plays a significant

role in modeling diffusion-induced fracture of Si anodes. Our fully coupled model allows us

to investigate how the fracture energy release rate, the electrode geometry, and geometrical

constraints affect the fracture behavior of Si anodes.

The novel contributions of this work include:

• a physics-motivated reaction-controlled diffusion model to study the initial two-phase

lithiation of Si anodes;

• an electro-chemo-mechanical coupled computational framework to study the diffusion-

induced large deformation and fracture for Si anodes;

• investigation of the impact of the electrode geometry and the geometrical constraints

on the fracture behavior of Si anodes.

5.2 Future directions

Electrochemical systems generally involve multiple physical phenomena, making the compu-

tational modeling of such systems a rich field of research with many unanswered questions.

This dissertation only scratches the surface of this fascinating topic. In this work, several

assumptions have been made to simplify our model complexity. To improve the fidelity of

this model, one can work in the following areas in the future.

• Identify material parameters. In this work, we assume material parameters either

5.2. FUTURE DIRECTIONS 83

have a linear compositional dependence or to be constant. But material parameters,

such as the diffusion coefficient, chemical reaction rate, or mechanical properties, can

not only depend on Li concentration but also the operating conditions [162, 163]. One

can conduct first principle calculation or molecular dynamics simulations to identify

different material parameters, and investigate how these parameters affect the simu-

lation results from the proposed continuum models.

• Validate simulation results with experimentally measurable quantitative data. As our

simulation object is at the nanoscale level, quantitatively measured experimental data

is limited. One potential area is to work closely with experimental groups to design

feasible experiments, such as in [164, 165], to collect quantitative data and thoroughly

validate the simulation results.

• Improve the treatment of the multiphysics modeling at the failure surface. In the

current study, we simply use a damage variable dependent diffusion coefficient to

account for the influence from the diffusive fracture. Advanced techniques to apply

flux-type boundary conditions in the diffusive fracture region are needed.

• Account for thermal effects. Thermal effects are extremely important for operating

LIBs, which are neglected in this work, as our main focus is the effects of mechanical

deformation. In the future, one can further study how the thermo-mechanical and

thermo-chemical coupling will affect battery performance [166, 167].

• Investigate surface effects on the mechanical and electrochemical behavior of Si anodes.

Continuum bodies show strong size effects when they are getting smaller [168, 169].

Surface effects can potentially influence the overall response of the electrodes [170].

• Investigate the formation and fracture of the SEI layer. The SEI layer is tremen-

dously important to the cyclic performance of Si-based LIBs. One can incorporate a

growth model [171, 172] to the proposed coupled variational framework in Chapter 2

to improve our understanding of the SEI layer.

• Resolve the microstructure of LIBs. As illustrated in Fig. 1.2, electrodes of LIBs

are highly heterogeneous. One potential direction of future research is to reconstruct

84 CHAPTER 5. SUMMARY AND FUTURE DIRECTIONS

large-scale heterogeneous electrode samples from high resolution images with advanced

imaging techniques [173] and conduct large-scale multiphysics simulations [174] to

understand the effects of mechanics on the overall performance of LIBs.

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