# Reliability and Full Cell Simulation Lab

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## Introduction

Battery reliability module is about the Diffusion induced stress(DIS), Crack propagation and Full cell.

• Example : Lobby of Reliability module
• Example : Working Studio of Reliability module

## Status bar

Both lobby and working-studio pages in the lab have common status bar on the top. It contains the buttons for lobby and working studio . It also shows the present user ID (e-mail address) and the present project name in bold with login time (In this example, krlee@kist.re.kr in project "3rd year demo" (20-15-09-15 10:59:01)). On the right corner of the status bar are the help button to see this document and the job status button which shows the present status of on-going jobs in this lab.

## User Manual

When you first enter the the Full Cell & Reliability module, you are in lobby. This is the default setting of the module.
You can go to working studio anytime by clicking the working studio button in the status bar.
You can come to lobby anytime by clicking the lobby button in the status bar.

### Lobby

Lobby has four sections:

• Visualization Window to show the single or combined atomic structure and geometry of the results.
• Electrode Materials to list the materials you want to work for the present project.
• Electrode Job to list the electrode results you have worked in the present project.
• Full Cell Job to list the Full cell results you have worked in the present project.
• Lobby

#### Visualization Window

On the left side of the lobby, there is the visualization window that shows the single or combined atomic structure and geometry of the results. Structure (final configuration of the job) is displayed depending on your selection in the Electrode Materials table or Results table (Electrode Job). Slight modification is applied for better distinction.

A number of mouse action can be used to change the visualization.

• Scrolling the center wheel to zoom in & zoom out
• Drag with left button to change the viewing angle
• Drag with right button to move the image
• Structure of Electrode Materials
• Geometry of Electrode Job

x, y, and z buttons align the image in +x, +y, +z directions, respectively. The other two buttons are for filtering and brief cell information , respectively.

You can take a snapshot of the image of the visualization window by using the camera button on the right upper corner, . As you click the camera button, new window with the sanpshot opens. Every snapshot appears in separate windows so that the user can compare the images. The snapshot image can be stored as a file that can be used later for the user's purpose. In order to save the image, place the cursor on the snapshot image then right click to invoke the menu of Chrome browser.

#### Electrode Materials

Materials that are used for electrode such as graphite and Si are listed on Electrode Materials table. If you have experimental or simulation data, you can generate electrode materials. To generate materials, you should input the Young's Modulus, Shear Modulus, Poisson's ratio, Diffusion coefficient, Reaction rate coefficient and Volume Expansion.

• Electrode Materials
•  : This button is to add the electrode material
•  : This button is to delete the electrode material

#### Electrode Job

All electrode jobs you and your colleagues have done are listed on the Electrode Job Table. Data is generated when the user executes calculation in the working studio. There is the information about Job name, the number of cycle, C-rate, Maximum stress and Result status.

• Electrode Job
•  : This button is to delete the Electrode Job

#### Full Cell Job

All full cell jobs you and your colleagues have done are listed on the Full Cell Job Table. Data is generated when the user executes calculation in the working studio. There is the information about Job name, Material for Anode, Material for Cathode and result status.

• Full cell Job
•  : This button is to delete the Full Cell job

### Working Studio

#### Electrode Job for the diffusion induced stress analysis

If you select one material in Electrode Materials table at Lobby and click the working studio button in the status bar, you can go to working studio.
You also can go to the working studio by selecting a job in the Electrode Job table and clink the working studio button.

• Working studio(Electrode Job)

In the Working studio, you can design and simulate the stress and crack propagation in the electrode material as you want.

Working studio in the Electrode Job section consists of 6 sub parts.

• Sample : Generating the geometry of electrode structure
• Mesh : Set the mesh density of structure and assign boundary conditions
• Simulation : Set the electrochemical condition(ex. c-rate, time step, etc)
• Analysis : Simulating the crack propagation and stress
• Results : Showing the calculated results
• Visualization : Showing the geometry of the electrode structure made in Sample and mesh made in Mesh

#### Full Cell Job

If you select two materials in Electrode Materials table at Lobby and click the working studio button in the status bar, you can go to working studio.
You also can go to the working studio by selecting a job in Full Cell Job table and clink the working studio button.

• Working studio(Full Cell Job)

In the Working studio, you can do the Full cell simulation for electrode materials as you want.

Working studio in the Full Cell Job section consists of 4 sub parts.

• Simulation : Set the electrochemical condition(ex. cutoff voltage, time step, etc)
• Material property : set the material property of the anode and cathode, which you chose in lobby, and the electrolyte
• Analysis : Showing the results for capacity, voltage, concentration and SEI
• Open Circuit Voltage : Showing the graph about open circuit voltage of each electrodes

## Models

### Stress analysis

#### Introduction

A diffusion induced stress (DIS) model is used to analyze mechanical stress for electrode material for use in Li-ion batteries.
To determine the DIS for various electrode shapes, a full finite element scheme was implemented to simulate the stress induced by lithium insertion and extraction.
As is usually the case with the elasticity and kinetics of materials, the governing equations, the equilibrium equations and the conservation laws, are fully coupled.
During the simulation, this model will show the distribution of lithium concentration and stress in the designed electrode. The interactive results and post-processed results are displayed as 3D images or 2D contour plot.

• Stress analysis model interface

#### Model definition

The conservation law (Fick’s second law) coupled with stress effects is derived from the chemical potential which can be found from:

$\mu = \mu_o + RT\mathcal{ln}X - \Omega\sigma_h$

where $μ_{0}$ is an initial chemical potential, $R$ is the gas constant, $T$ is the absolute temperature, $X$ is the is the molar fraction of lithium ions, $Ω$ is the the molar volume of Li in LixSi, and $σ_{h}$is the the hydrostatic stress, which represents a diagonal average of stress components

The Li flux is expressed as product of the gradient of chemical potential and lithium diffusivities. Then the Fick’s second equation is:

${\partial c \over \partial t} + \nabla \cdot \mathcal{J} = 0$

${\partial c \over \partial t} = D \left (\nabla^2 c - {\Omega \over RT} \nabla c \nabla \sigma_h - {\Omega c \over RT} \nabla^2 \sigma_h \right)$

Where $D$ is the diffusion coefficient and $c$ is the concentration of lithium.

During the insertion and extraction of Li, the change in volume of the host electrode can be assumed to change linearly with the volume of Li inserted. Thus, the stress-strain relation for an elastic body can be written as:

$\varepsilon_\mathcal{ij} = {1 \over Y} \left[(1+\mathcal{v})\sigma_h - \mathcal{v}\sigma_\mathcal{kk}\delta_\mathcal{ij} \right] + {c - c_0 \over 3} \Omega \delta_\mathcal{ij}$

where $Y$ and $v$ are Young’s modulus and Poisson’s ratio as a function of Li concentration. And the components of stress can be written as:

$\sigma_\mathcal{ij} = 2 \mu \varepsilon_\mathcal{ij} + \left( \lambda\varepsilon_\mathcal{kk} - \beta \left( c - c_0 \right) \right) \delta_\mathcal{ij}$

where $\mu = {Y \over 2(1+\nu)}$, $\lambda = {2\nu\mu \over 1-2\nu}$, and $\beta = {\Omega(3\lambda + 2\mu) \over 3}$.

Finally, the diffusion induced stress is obtained from the equilibrium equations:

$\nabla \cdot \sigma = 0$

#### Instruction of model

The example model geometry consists of a 1000 nm tall nanowire with a radius of 100 nm. Electrode materials for the simulation can be selected from material DB in Lobby. The material in this tutorial was given silicon properties. The users are able to insert a new material.

##### Geometry of model

To design the model geometries, you can select the dimension and the shape of sample with size that you want to simulate in a geometry setting tab,.

• Model geometry setting box
• Geometry type

As a example, the type of cylinder was selected in the geometry tab. The radius and height were set to 100 nm and 1000 nm, respectively. Click build icon.

• Model geometry of nanowire
##### Boundary conditions

In the Mesh Builder window, the selected boundary conditions will be listed. For the fixed boundary conditions, the Planes of 1, 2, 3 and 4 were selected using plane selection icon. After click plane selection icon in flux boundary tab, the planes of 9, 10, 11 and 12 were also selected for lithium flux boundary condition.

• Boundary condition setting tab
• Select the boundary condition by the click
##### Meshing

Mesh was generated in Density tab. The number of node on each line controls the element density in the model structure.

• Mesh setting tab
• Different mesh density

The Density is set to be 7 for the simulation for the example.

• Meshing in example model
##### Operating condition and Running

For the charging and discharging simulation, the process parameters have to set in the simulation window.

• Simulation setting tab for process parameters

The time step was set to be 10 sec. Charge Rate was 2.0 C, which 1C is the charge-discharge cycle number during 1 hour. The number of cycle is 1. After writing down the job-name, click Submit button.

#### Result analysis

In the results window, the simulation results of stress and Li concentration were plotted every iteration. The contour plot of middle surface in the 3D geometry was also plotted on the right part in results window.

• Results after running

The bottom plot shows the stress evolution and the Li concentration profiles versus time during simulation.

• Results after running

### Crack propagation

#### Introduction

As the battery is used for a long time, the capacity fading is inevitable. When the battery goes through charge-discharge cycles, the lithium diffuses into the anode, and then volumetric expansion/contraction is induced.
This repeated volume change causes stress concentration and finally initiates the crack growth inside the battery cell. This model performs a crack propagation simulation in a silicon thin film that is commonly used in battery anode.
The diffusion equation is based on Fick's second law with a diffusion induced stress term. Once the diffusion equation is solved, one can get the lithium ion concentration and corresponding stress distribution. XFEM is then applied to simulate crack propagation. A simultaneous diffusion-crack propagation analysis on electrode considering its large deformation and non-linear behavior.

• Crack propagation model interface

#### Model definition

In charge cycle, the lithium ions are diffused into the anode. To simulate this, one must solve a diffusion equation.

${\partial c \over \partial t} - \nabla \cdot \left[ D \left( \nabla c - {\Omega c \over RT} \nabla \sigma_h \right) \right] = 0$

This diffusion equation is based on Fick’s second law with a diffusion induced stress term like a DIS model.

By solving diffusion equation, one can get the lithium ion concentration and corresponding stress distribution. XFEM is then applied to simulate crack propagation. XFEM is the acronym of eXtended Finite Element Method which can simulate crack propagation without remeshing. First, XFEM defines the crack geometry independent of the mesh by level set. Then, special values are assigned to the crack tip element as well as the cracked elements called enriched elements. Finally, the finite element approximation can be written as below.

The modified staggered approach is considered to account the time effect. Diffusion takes much more time than a crack propagation phenomena. So it solve XFEM analysis several times until the crack arrests, and then it solve the Diffusion analysis for the next iteration. The crack arresting criteria is determined by comparing the critical stress intensity factor and effective delta K value.

• Modified staggered approach

#### Instruction of model

##### Boundary conditions

The model geometry is an amorphous silicon thin film of width (200 nm) and height (50 nm) in the finite element method (FEM) assuming plane strain condition The silicon thin film lithiated by prescribing a constant flux (Charge rate) on the top surface. Geometrical constraints are imposed like a below figure and the top free surface of the film is taken to be traction-free.

• Boundary conditions of crack propagation model
##### Model parameters
• Parameters setting tab for crack propagation model

### Full cell

#### Introduction

This model provide the Li ion battery full cell interface for characterizing the discharge and charge behavior for a given set of material properties. This model is used to investigate the influence of various design parameter for battery developers. It is easy to simulate the battery performance under different operating conditions based on a study by J. Newman and Doyle and numerous other researchers.

• Schematic of Li-ion battery full cell

#### Model definition

##### Single particle full cell model

A one-dimensional model is shown in below figure. The negative and positive electrode are assumed an isothermal single particle in one dimension.
The potential and concentration gradient in electrolyte are neglected and assumed as a solution resistance. Additionally the potential gradient of the electrode are neglected and reaction current density is uniform within the electrode.
The single particle formulation accounts for solid diffusion in the electrode particles and the intercalation reaction kinetics. These assumptions are reasonable for highly conductive electrodes and thin structure electrodes.

##### Effective diffusivity by numerical simulation

The effective diffusivity is calculated by the numerical approach.

#### Instruction of model

When you get into the reliability platform, The lobby window will be displayed.

• Full cell analysis model interface
##### Select modeling materials to simulate the full cell model

Then select the two materials to simulate the full cell model on the materials window

• Select materials

After click the process icon, the fullcell simulation window will be displayed.

• Click process icon for simulation
• Full cell analysis window
##### Parameters

On the right parameter setting window, fill in the parameters.

• Full cell analysis model interface

##### Setting Equilibrium open Potential

On the Open circuit voltage window, Click the Equilibrium open voltage icon.

Write the OCV data on the appeared window, then click the OK. It generate the 2D plot of OCV

• Write the OCV data
• 2D plot for OCV

##### Setting operating condition

On the simulation window, fill in the battery operating conditions as boundary conditions.

Write the Job name then click the Submit icon.

• Operating conditions

#### Result analysis

After finishing the simulation, Select the your job on the Lobby window.

On the Analysis window, Click result icon to create the 2D plot and the 2D image for the analysis.

• Full cell analysis model results
• The Capacity Vs. Voltage icon, generate the 2D plot for the cyclic behavior.
• The Cycle capacity icon, generate the 2D plot for the capacity variation.
• The Concentration icon, generate the 2D plot for Li concentration on each electrode.
• The SEI icon, generate the SEI 2D image.
• Full cell analysis model results

## Technical Information

### Finite element method simulation

Finite element method (FEM) simulation is employed to find approximate solutions for the diffusion induced stress, crack propagation, and full cell simulation during lithiation/delithiation for Li-ion battery system. FEM is the most widely used numerical analysis method together with finite difference method (FDM) and boundary element method (BEM) in structural analysis, thermal analysis, and fluid analysis. This method divides the object into finite number of regions (elements) and determine the representative point (node) of this region, and solve the governging differential equation of this point by approximating the solution. Our in house code for FEM simulation includes element partitionining is called a preprocessing, solver that is solve the linear equation, and a post processing that displays analysis results in graphic.

### Diffusion induced stress

To investigate the diffusion induces stress in anode/cathode materials during lithiation and delithiation, the squeezing effect incurred by the hydrostatic stress duirng the Li diffusion is considered. So, the FE formulations involve the coupling effect between stress and diffusion. ${\partial c \over \partial t} = D \left (\nabla^2 c - {\Omega \over RT} \nabla c \nabla \sigma_h - {\Omega c \over RT} \nabla^2 \sigma_h \right)$

• Finite element analysis

#### Model properties

User can select the target material that user want to analyze the the diffusion induces stress from the lobby window. The selected material contains material properties such as Young's modulus, Poisson's ration and diffusion coefficient. This information is automatically utilized for the diffusion-induces stress analysis.

#### Model dimension

The user can choose the dimension of the materials he/she want to analyze. In the 2-dimensional analysis, a circular shape and a rectangular shape can be chosen and in the 3-dimensional analysis, a cylinderical shape and a spherical shape can be selected. Then the size of each shape to be analyzed can be directly input in the nanemeter scale.

#### Meshing

Mesh density can be entered from 1 to 7, the higher number means the higher desity of mesh. The high density of the mesh increases the accuracty fo the solution, but it has a disadvantage that it takes a lot of calculation time. Therefore, user should put the value to suit their own research purpose.

#### Simulation

The time step refers to the time interval during which lithium enters and exits. The larger time step means the faster calculation speed, but the lower accuracy of the solution. The charge and discharge rates are controlled by the "charge rate value". As demonstrated in most of the experimental papers, the higher charing rate induces the greater stress in the active materials absoring or releasing lithium, leading to the breakage of it. The fast charge leads to a larger difference in the Li concentration distribution in the inside, and thus, it causes a large increase in teh chemical strain mismatch, thereby rapidly increasing the internal stress. The number of cycles refers to the number of charge and discharge cycles of the battery. If user wants to charge and discharge once, enter 1.

• Charge-Discharge cycle

#### Simulation results

The calculated results are the Li concentration distribution and stress distribution in the active materials. However, this platform shows the maximum stress value while lithium enters and exits. This value is an important factor in designing the active material.

• Li concentration and hoop stress distribution

### Crack propagation

In the case of the crack propagation model, the above DIS in house code is applied for the calculation of Li concentration distribution and stress analysis in silicon material. The XFEM method is used for crack propagation, and the details are described in "4.2.2 Model definition".

#### Simulation

Only thin film shape is provided for the crack propagation analysis during lithium absorption. In this platform, it is possible to analyze the phenomenon of crack propagation that occurs when lithium enters from the top of a thin film. The material properties needed for crack propagation are automatically read as in the DIS module. The user can perform the analysis by inserting the size information of the thin film or the position of the first crack through the x and y coordinate values.

### Full cell

#### How the battery works

Like a normal battery system, the lithium ion battery also composes of anode, cathode and electrolyte. the electrolyte help the lithium ion move freely inside the battery. When you charge the battery, the lithium in cathode material dissociate into lithium ion and electron. By the voltage difference between two electrodes, these lithium ions move from the cathode to the anode, simulataneously electrons go through the external electric circuit. Finally they recombine within the anode material. The reproduced lithium resettle down between the original atomic structure of anode, which induces the volume change and generates the stress within the anode material. When discharging the battery, the reverse process occurs.

#### Assumption in the full cell model

In the case of the full-cell simulation on the platform, the model takes into account various assumptions and increases the efficiency of the computation time. In the first assumption, the porous effect of the anode and cathode materials was neglected. Instead, to compensate for this assumption, the lithium flux values at the surface of each electrode were taken into account with this assumption. Therefore, the lithium concentration at each electrode is assumed as a single domain. So, Li diffusion in active particle as follows.

The active particles in the electrode are considered as spherical shape particle with same size. Fick’s second law is assumed to simulate the Li diffusion in active particle in the negative and positive electrode.

${\partial C \over \partial t} = D \left[ {\partial^2 C \over \partial r^2} + {2 \over r}{\partial C \over \partial r} \right]$

Boundary condition at active particles There are some of the boundary conditions.

At center of the particle, ${\partial C \over \partial r} = 0|_{r=0}$

At surface of the particle, $D{\partial C \over \partial r} = R_{Li}|_{r=r_p}$
where r is the radius of the active particle and R is the Li flux value.
Lithium flux value reflecting porous effect.
$R_{Li}={i_{loc} \over F}={\pm i_{applied} \over F(3/r_p)\epsilon_sL}$

• Li concentration at each electrode
• The second assumption is that the voltage drop caused by the transfer of lithium ions in the electrolyte is replaced by a small resistance value. Depending on the rate of charge and discharge condition, the over-potential in the electrolyte is different. However, in case of non-fast charging condition, the electrical over-potential is relatively small, so that the model result and the experimental result can show a similar tendency.

#### Electrochemical kinetics

The Butler-Volmer kinetic equation was used for electrochemical reactions. This describes electrical current flow between the electrolyte and the electrode.

$i_{loc} = i_0\left( \exp\left( \left( 0.5F\eta \over RT \right)-\exp\left( -{0.5F\eta \over RT} \right) \right) \right)$

$\eta = \phi_s - \phi_l - E_{eq}$

$i_0 = Fk\sqrt{\left( c_{s}^{max}-c_s \right)c_s\left( c_l/c_{l,ref} \right)}$

The causes of the degradation of the performance of lithium-ion batteries come from various reasons. For example, dissolution of the current collect, debonding and dendrite formation between the current collect and the active particles, breakage and loss of the active particles, and formation of the SEI. In the proposed full cell model, the degradation of the electrochemical performance of the battery is reflected by formation of the undesired SEI in the anode material during the charging and discharging cycles. The amount of SEI is calculated by the Monte Carlo methode, and it results in increasing the internal resistance of battery system.

Solid Electrolyte Interface growth model is decribed as follows. Active and passive SEI cyclic growth model is simulated by kinetic Monte Carlo method. The phenomena such as adsorption, desorption, diffusion, and passive material formation is considered. Active SEI is associated with the desorption and diffusion, however, the passive SEI layer is associated with the adsorption and formation of passive materials.

• Event Catalog
• Reaction at SEI

The nonlinear reaction rates $K_{1}$, $K_{2}$ are considered as function of the surface coverage $θ$ and $η$ ($η=V-U_{n}$) is the overpotential.

$K_{3}$ is a function of the exchange current density

The results show that the active SEI and passive SEI are in the graphite during the charging and the discharging. During the charging, there is no change in the passive SEI, but the active SEI increases and saturates. On the other hand, during the discharge, the active SEI decreases and the passive SEI increases.

• SEI growth image during charging/discharging cycling
• Active SEI and Passive SEI growth rate

The figure below shows how each SEI changes as the charge / discharge cycle progresses. As the cycle repeats, the graphite decreases and the passive SEI increases. This increase in passive SEI increases the internal resistance of the battery, thereby degradating the performance of the battery.

• SEI formation during 50 cycle with KMC simulation

The upper part does not reflect the SEI effect. Even if the charging and discharging occur, the performance degradation is not observed. However, when the SEI effect is considered, the capacity decreases as shown in the figure.

• Electrochemical degradation during 50 cycle

## Reference

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7. Newman J, Tiedemann W. Porous-Electrode Theory with Battery Applications. Aiche J. 1975; 21: 25-41.

8. Doyle M, Newman J, Gozdz AS, Schmutz CN, Tarascon JM. Comparison of modeling predictions with experimental data from plastic lithium ion cells. J Electrochem Soc. 1996; 143: 1890-903.