Cathode Materials Design Lab

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Introduction

  • Example : Lobby of Cathode Materials Design Lab
  • Example : Working Studio of Cathode Materials Design Lab

Status bar

Status-bar.JPG

Both lobby and working-studio pages in the lab have common status bar on the top. It contains the buttons for lobby Lobby-button.JPG and working studio Working-studio-button.JPG. 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 Help.JPG to see this document and the job status button Job-status-button.JPG that shows the present status of on-going jobs in this lab.


How to use the module

User Manual : Lobby

When you first come the the cathode materials design lab, you are in lobby. This is the default setting of the lab.
You can go to working studio anytime by clicking the working studio button Working-studio-button.JPG in the status bar.
You can come to lobby anytime by clicking the lobby button Lobby-button.JPG in the status bar.

Lobby has two sections:

  • Visualization Window to show the atomic structure of Cathode or the final configuration of the job.
  • Results Table to list the Cathode you have worked in the present project.
  • Lobby

Visualization Window

On the left of the lobby is the visualization window that shows the atomic structure or final configuration of the job. Cathode structure (final configuration of the job) is displayed depending on your selection in the Results table (Job table). Color of atom in this visualization is based on the CPK coloring convention. However, slight modification is applied for better distinction. See color of atom page for the color list.

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
  • Visualization window
caption
will align the image from +x, +y, +z directions and two buttons for filtering and brief cell information , respectively.

You can take the snapshot of the image of the visualization window by using the camera button on the right upper corner, Camera.JPG. 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.

  • Snapshots of the LiCoO2 taken by the camera

Results Table

  • Results Table

All jobs you and your colleagues have done are listed on the Results Table. Data is generated when the user executes calculation in the working studio.

  • Owner : Name of the user
  • U/R/A : It shows whether the job is 'unrelaxed' or 'relaxed' status. If you execute advance calculation for analysis, it will be turn into 'advanced' status.
  • Job Name : Name of the job
  • Del.JPG : This button is to delete the cathode job

The results data can be filtered by many factors. To filter the data, click the title of the list that you want to filter with. Then the filtering box will be shown. choose the condition and type the value. Click the Apply to apply. Multiple filters can be applied in once. See the example below.

  • Results Table Filtered by the Volume

To send the works from Results Table to Working Studio, just click the checkbox of the works and move on to Working Studio.

  • Selecting the works as you want to implement

User Manual : Working Studio

  • Working studio

Working studio in the Cathode section consists of 6 sub parts

In the Working studio, you can design and simulate the cathode material as your choice


  • Crystal builder : Generating the crystal structure of cathode material
  • Visualization : Showing the atomic structure of crystal structure made in Crystal builder
  • High Throughput Calculation : Calculating the stoichiometric chemical formula from the atomic structure
  • Results : Showing the calculated results
  • Analysis : Simulating the cathode material through the various method
  • Informatics: Enumerating the information of cathode material


How to create the cathode material

In Cathode section, you can build the crystal structure and simulate the the crystal structure to analyze the cathode material

Build the crystal structure of cathode material

To create the cathode material, first click the caption button on the Lobby page

  • In the Working studio page,you can design cathode material
  • Working studio


  • In the working studio page, you can create crystal structure of cathode
  • crystal_builder
  • On the crystal builder window, you can enter specific crystal information of cathode material you want to simulate
  • LiCoO2 is the one of the typical cathode material. LiCoO2 is R-3m, O3 type structure and has being used as cathode material. you also load the CIF information to build the cathode material. The atomic information of LiCoO2 is entered in crystal_builder window and then click the Generate button
  • crystal_information_LiCoO2
  • And then, Working studio show the atomic information and cell information of LiCoO2
  • crystal_information_LiCoO2
  • In the middle of the working studio page, there is a visualization window that show the atomic structure you built right before
  • Visualization window provide the atomic structure viewer. Color of atom is based on the CPK coloring convention, with slight modification for better distinction. See color of atom page for the color list.
  • On the visualization window, there are caption button
  • If you click the button, you will see the plane of the crystal structure perpendicular to x,y,z axis and brief cell information
  • crystal_information_LiCoO2
  • In the working studio page, there are High Throughput Calculation window
  • High Throughput Calculation
  • If you just use the crystal structure entering the input on the crystal builder window, click the Enumerate button
  • Then, possible stoichiometric composition come out
  • You can save the structure by entering the Job name box and then you also calculate by clicking the Calculate button
  • High Throughput Calculation of LiCoO2
  • After clicking the Calculate button, results come out on the Result window
  • Results from the High Throughput Calculation

Doping from the crystal structure

To dope the initial atom to the other atoms, you should first create the crystal structure as like before Build crystal structure section

  • From the LiCoO2 crystal structure, there are each atom on the Visualization window. You can click the atom(Selected atom) you want to dope to the other atom(Target element) and then the selected atom automatically come out beside Selected atoms(s). You can also select more than one atom
  • Selecting atom from initial structure


  • After that, you can enter doping atom in the Target element(s) box and then click the Enumerate button. (X is for empty space)
  • Then, possible stoichiometric composition come out
  • You can save the structure by entering the Job name box and then you also calculate by clicking the Calculate button
  • High Throughput Calculation of Ni doped - LiCoO2
  • After clicking the Calculation button, results with initial stoichiometric composition and doped stoichiometric composition come out on the Result window
  • Results from the High Throughput Calculation with doping

How to Analyze Cathode Material

To Analyze the Cathode Material, you should first complete Design Cathode Materials process and should first simulatecaption

  • There are Analysis window on the Working studio page
  • Analysis window
  • You can simulate Relaxation,Diffusion,Voltage,Free Energy,Phase and Density of States from the crystal structure already created


Relaxation

To Relax the crystal structure, you should first calculate the structure like before Design Cathode Materials section

  • On the Analysis window, click the caption button and then click the 확 인 . Then Relaxation process will progress
  • Confirm the Relax
  • After the Relaxation, then Results window change the status from Unrelaxed to Relaxed
  • Relaxing the crystal structure


Diffusion by NEB

To Diffuse the crystal structure, you should first calculate the structure like before Design Cathode Materials section

  • On the Analysis window, click the caption button and then click specific information as following steps
  • You can confirm the relaxed cathode information, which are Material name, diffusion path and caculation status
  • Diffusion by NEB
  • Then you can select the atom that you want to move by clicking select and then clicking the atom
  • Selecting the atom
  • Then you can select the atom which is destination by clicking select and then clicking the atom's destination
  • After that, you shoud click the Simulate to caculate Diffusion by NEB
  • Diffusion destination
  • After the calculation end, the energy barrier and diffusion path is shown on the window
  • Energy barrier and diffusion path

Potential

To calculate Potential, you should first relax the structure like before Relaxation section

  • On the Analysis window, click the caption button and then you can confirm the Potential figure on the Delithiation Potential window
  • Delithiation Potential

Free energy

To calculate Free energy, you should first relax the structure like before Relaxation section

  • On the Analysis window, click the caption button and then you can confirm the Free energy figure on the Free Energy window
  • Free Energy

Diffusivity of Li by molecular dynamics (MD) simulation

To compute diffusivity of Li ion in cathode materials, one should first calculate the structure as in the Design Cathode Materials section.

  • Mean square displacement: The mobility of atoms can be described by the mean square displacement (MSD).
         Diffusivity eq1.png
  • Diffusivity: The diffusivity (or diffusion coefficient) D, can be determined from the MSD by the Einstine relation.
         Diffusivity eq2.png
  • On the Analysis window, click the Diffusivity button.png button and then input temperature (K) and time (ps) for molecular dynamics (MD) simulation.
         Diffusivity snapshot1.png
  • The mean square displacement (MSD) is computed for a half MD simulation time (see 4.3). A simulation time greater than 100 ps is recommended for the computation of MSD.
  • Click the simulate button to start the molecular dynamics simulation. The sample is automatically duplicated along the x, y and z axis to reach a box size greater than 24 Å (the minimum size for the present simulation technique). The number of atoms in those samples is 1,000~2,000, and the computing would take 2~3 days for a simulation time of 200 ps.
  • When the MD simulation is completed, the jobs are listed in a table. Click the job to plot the MSD curve in the graph, and Drag your mouse to the region you want to compute diffusivity by the slope.
         Diffusivity snapshot2.png

Phase field

By clicking the phase field button Phase.png, new window for phase field model opens.

User interface
  • User interface of phase field model

User interface of phase field model divided into two window, input status window and result window.

Input status window
caption
  • Material properties obtained from DFT data.

- Process : Selection of discharge and charge process.

                        Dischare-charge.png

- Geometry : The structure modeling for job. Cubic, Cylinder, Sphere, Polyhedron, and Hexagonal plate structures are available. A user could apply porosity for each structure. Structure modeling.png

- Porosity : User could select even and uneven distribution of pores. Selectable porosities are 0, 10, 20, 30, 40, and 50 %. Porosity.png

- C-rate : Charge rate, charge or discharge rate equal to the capacity of a battery in one hour. When high c-rate condition set, discharge process ended in short time.

- Time : The time user wants for the job. When discharge process is finished, job could end even the set time is not reached.

< Geometry model detail >

Cube: Select cube length Cube.jpg

Cylinder : Select radius and length Cylinder.jpg

Sphere : Select radius Sphere.jpg

Polyhedron : Select circumscribed sphere radius Polyhedron.png

Hexagonal plate : Select length Hexagonal.png

Phase field model input example

Process: Charge

Geometry: Sphere with 250 nm radius, 30% porosity, evenly distributed

Charge/discharge rate: 2 c-rate

1. Select charge process

E1.png

2. Select sphere with 250 nm radius

E2.png

3. Select evenly distributed, 30% porosity

E3.png

4. Select 2 c-rate

E4.png

5. Enter job name and click simulation button

E5.png

Result window
Result left top.jpg

By clicking left top button of the result window, user could change result window into [Li fraction], [Phase distribution],[Temperature] and [2d plot]


[Li fraction], [Phase distribution] and [Temperature]

  • Output status window.png
    Phase distribution.jpg
  • On the left of the result window is the visualization window that shows the Li-ion fraction or phase distribution or temperature of the job.
    On the right of the result window, 2 dimensional slice plane visualized. Position and direction of slice plane could be changed by button xy, yz, zx and scroll bar.

    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

    Three buttons on the left lower corner, Xyz-button.JPG will align the image from +x, +y and +z directions, respectively.

    [2d plot]

  • 2D plot.jpg
  • The result window shows graph of Li fraction and phase distribution of simulated job.

    The graph calculated from adding all data from [Li fraction], [Phase distribution], [Temperature] tab and divided by cathode volume.

    Specific capacity showed in the bottom of interface calculated from Li fraction at expected time for designed c-rate.

    For example, in 4-crate condition, shows specific capacity of cathode at 900 sec(3600sec/4).

    <Phase field model output example>

    1. Process: Charge, Geometry: Sphere, 30% porosity, evenly distributed, Charge/discharge rate: 2 c-rate

    [figure][figure][figure][figure]

    2. Process: Charge, Geometry: Sphere, 10% porosity, unevenly distributed, Charge/discharge rate: 4 c-rate

    [figure][figure][figure][figure]

    3. Process: Discharge, Geometry: cube, Charge/discharge rate: 4 c-rate

    [figure][figure][figure][figure]

    Density of States

    To calculate Density of States, you should first relax the structure like before Relaxation section

    • On the Analysis window, click the caption button and then Densty of states window show up
    • Density of States window
    • On the Density of states window, therer are Atoms to draw : with check circles where you can select whether to see density of states of all atoms or specific atom
    • Then, you also can choice specific orbitals among integrated states in the Orbitals to draw : with check boxes
    • Selecting the specific information of Density of States
    • First, if you check the ⊙ All such as caption , Density of States window will show the integrated energy states of all atoms.
    • Next, you can choice integrated states or specific states by clicking the states you wnat to see below the Orbitals to draw :
    • After these step, click the Draw
    • Integrated Density of States of all atoms
    • Integrated Density of States of all atoms


    • Slected orbital's Density of States of all atoms
    • Slected orbital's Density of States of all atoms


    • Combined with all state and each orbital state can be calculted all together about all atoms
    • Combined Density of States of all atoms


    • As mentioned above, Density of States of speicif atom can be simulated by clicking the specific atom on the visualization window and then the name of selected atom come out next to the ⊙ Atom , as like caption
    • Integrated Density of States of specific atoms
    • Integrated Density of States of specific atoms
    • Slected specific Density of States of all atoms


    • Slected orbital's Density of States of specific atoms
    • Slected orbital's Density of States of specific atoms


    • Combined with all state and each orbital state can be calculted all together about specific atoms
    • Combined Density of States of specific atoms

    How to create the surface model of cathode material

    Build the surface model of cathode material

    • Vacuum setting


    In the Crystal builder , you can control vacuum setting to create the surface model of cathode material

    • After design the cathode material in crystal builder, you can set the surface information
    • Surface setting information
    • First, you can select the surface directions ( x, y, z directions )
    • Surface directions
    • And then, you can also select the vacuum layer
    • If you enter the vacuum layer and click the Generate button, surface model of your cathode material will be created
    • surface model as vacuum layer variation

    Analyze the surface model

    • After creating surface model, you can analyze this model as like the bulk mode described in above documentation sections( 3.1 & 3.2 )
    • To analyze the surface model, just follow the same process described in the documentation 3.2 and then you can check the calculated results(NEB, Potential, Free energy ...) for your surface cathode model
    • Example : dos of surface model

    How to build the carbon coating surface model of cathode material

    • After creating surface model, you could coating carbons on the your surface model
    • Coating setting
    • In the Crystal builder , you can control carbon coating(c coating) thickness to create the carbon coated surface model of cathode material
    • Surface setting information

    To coating carbon on your surface model, just enter the coating thickness of your choice and click Apply button

    • Carbon coated surface model as coating thickness variation
    • To simulate the coating simulation, enter the target temperature and simulation time
    • Coating simulation

    Next,click AIMD Relax check box and enter the AIMD name for AIMD simulation

    • AIMD user input

    And also, motion of AIMD model could be check by click the Explore button in the Crystal builder window

    In the AIMD Explorer window, select AIMD name and click the Animate button

    • AIMD Explore

    Infromatics

    To conform the information on the cathode material that you made through before process, you click the caption button And then you can check the information

    Technical Information

    Density Functional Theory Simulation

    Density Functional theory (DFT) Simulation is carried out to predict phase behaviors, structural stability, electrochemical voltage and electronic structures for redox reaction and to design a new cathode. (Click this link for details of DFT scripts)

    Delithiation simulation

    Delithiation simulation of cathodes is the most significant calculations to predict electrochemical voltage behaviors, structural stability associated with phase variations and redox behaviors of transition metals. Most cathode materials in lithium ion battery originally contain lithium ion as the same experiments. Therefore, a user firstly have to model lithiated cathode structures except for specialized cathodes that should electrochemically insert lithium ion into the structures. With generated lithiated cathodes, a user can extract lithium ion with all possible Li-vacancy configuration at the specific content of lithium ion from a fully lithiated structure to a fully delithiated structure.

    Analysis

    Relaxation

    To find optimized structures with thermodynamically stable phases, relaxation should be performed. Initial atomic coordinates and lattice parameters are initial guess, which means that those information would not be ground states. Therefore, relaxation process should be conducted to reach ground states with given calculation conditions by users.

    From the relaxation, you can obtain various parameters such as specific atomic coordinates and configurations, lattice parameters, cell volume.

    Mixing enthalpy

    From the perspective of thermodynamics, the mixing enthalpy calculated by Li/vacancy configurations indicates relatively structural stability of the different configurations at each Li concentrations as follows:

    [math]\Delta H_{mixing}= E_{Li_{1-x}TMO_{2}}-xE_{LiTMO_{2}}-(1-x)E_{TMO_{2}} [/math]

    , where [math]E_{Li_{1-x}TMO_{2}}[/math] is the total energy of a specific Li-vacancy configuration at a mole fraction [math]x[/math], [math]E_{LiTMO_{2}}[/math] is the total energy of fully lithiated structure and [math]E_{TMO_{2}}[/math] is the total energy of fully delithiated structure. The configurational mixing enthalpy is crucially important to analyze the phases (e.g., two or one phase), electrochemical voltage behaviors and cycleability. Using configurational mixing enthalpy, users can obtain thermodynamically stable phase at targeting Li concentrations with specific atomic configurations. See the related reference as below.

    • J.-M. Lim, D. Kim, Y.-G. Lim, M.-S. Park, Y.-J. Kim, M. Cho, K. Cho, "The origins and mechanism of phase transformation in bulk Li2MnO3: first-principles calculations and experimental studies", J. Mater. Chem. A, 2015.
    • D. Kim, J.-M. Lim, M.-S. Park, K. Cho, M. Cho, "Phase Separation and d Electronic Orbitals on Cyclic Degradation in Li-Mn-O Compounds: First-Principles Multiscale Modeling and Experimental Observations", ACS Appl. Mater. Interfaces, 2016.
    Voltage

    In electrochemical system, calculations of voltage are very important process because voltage is directly related to energy density of electrode. The equilibrium voltage, denoted as the open circuit voltage (OCV), between the anode (e.g., negative electrode) and cathode (e.g., positive electrode) is calculated by the difference of Li chemical potential between the anode and cathode as follows:

    [math]V=-\frac{\mu_{Li}-\mu_{Li}}{nF}[/math]

    , where F is the Faraday constant and n is the number of Li mol involving in redox reaction and [math]\mu_{Li}[/math] refers to the Li chemical potential in cathodes and [math]\mu_{Li}[/math] indicates the Li chemical potential as Li metal in anodes. Even though it is difficult to calculate the chemical potential of Li in cathode materials as a function of Li concentration, the average delithiation potential, AverageV.jpg, can be easily obtained from the total system energies calculated using DFT.

    AverageV eq.jpg

    , where Li limit.jpg is the two Li ions intercalation limits, Gibbs.jpg refers the change of Gibbs free energy according to the delithiation process. The variation of the Gibbs free energy, Gibbs eq.jpg,can be approximated by the change of the internal energy(Internal.jpg)at 0 K. A very small change in volume (Volume.jpg) and the thermal energy term (File:Thermal.jpg) almost do not affect the free energy.

    See the related reference as below.

    • M. K. Aydinol, A. F. Kohan, G. Ceder, K. Cho, J. Joannopoulos "Ab initio study of lithium intercalation in metal oxides and metal dichalcogenides", Phys. Rev. B, 1997.
    Diffusion by NEB

    From the perspective of kinetics, the migration barrier of Li ion is significant factor to power density related to C-rate in Li ion battery system. The barrier can be varied as a function of Li concentration and is calculated by NEB (Nudged Elastic Band) presenting a method to find saddle points and minimized energy pathway between reactants and products. For the calculation, a number of predicted intermediate images along the reaction path are required to work. Also, the migration barrier of transition metals directly correlated with phase transformations in cathodes is calculated by NEB method. The transition metal barrier can be different as a function of Li concentration and affected by doping other elements. See the related reference as below.

    • J.-M. Lim, D. Kim, M.-S. Park, M. Cho, K. Cho, "Underlying mechanisms of the synergistic role of Li2MnO3 and LiNi1/3Co1/3Mn1/3O2 in high-Mn, Li-rich oxides", Phys. Chem. Chem. Phys., 2016.
    Density of states

    In solid-state and condensed matter physics, the density of states (DOS) of a system describes the number of states per interval of energy at each energy level that are available to be occupied. In Li-ion battery system, electronic behaviors of general redox from transition metals in oxides are major driving force during the lithiation/delithiation, that is, DOS to investigate electronic structures and behaviors should be understood to design cathode materials. The DOSDOS.jpgis actually derived from the difference of the integrated DOS between two energy states.

    DOS 1.jpg

    , whereDeltaE.jpgis the difference between two energy states, andNstates.pngrefers the integrated DOS expressed as the following. Nstates eq.jpg.

    This method indicates the total number of electrons in the whole energy range or in the specific range of calculated system by users. See the related reference as below.

    • D. Kim, J.-M. Lim, Y.-G. Lim, J.-S. Yu, M.-S. Park, M. Cho, K. Cho, "Design of Nickel-rich Layered Oxides Using d Electronic Donor for Redox Reactions", Chem. Mater., 2015.
    • J.-M. Lim, D. Kim, Y.-G. Lim, M.-S. Park, Y.-J. Kim, M. Cho, K. Cho, "Mechanism of Oxygen Vacancy on Impeded Phase Transformation and Electrochemical Activation in Inactive Li2MnO3", ChemElectroChem, 2016.
    • Y.-G. Lim, D. Kim, J.-M. Lim, J.-S. Kim, J.-S. Yu, Y.-J. Kim, D. Byun, M. Cho, K. Cho, M.-S. Park, J. Mater. Chem. A, 2015.

    Molecular dynamics simulation

    To investigate dynamic properties in atomic scale, the molecular dynamics (MD) simulation technique can be employed. An interatomic potential model is required to perform MD simulations and the potential should be able to cover the metal oxide systems for the application of cathode materials. (Click this link for details of MD script)

    • Interatomic potential model: ‘2NN MEAM + Qeq’ potential: See the references for details.
      • E. Lee, K.-R. Lee, M. I. Baskes, and B.-J. Lee, ‘A modified Embedded-Atom method interatomic potential for ionic systems: 2NNMAM+Qeq’, Physical Review B 93, 144110 (2016)
    • Materials systems that can be covered by the MD simulation
      • Li-Mn-O
      • Li-Co-O
      • Li-Ni-O

    Phase field

    Phase Field Modeling(PFM) simulation could simulate discharge and charge process about microstructures. With PFM simulation, it is possible to enhance efficiency of cathode by comparing capacity of various shape of structures.

    Cahn-hilliard equation is used for PFM simulation. [math]\mu[/math] represents chemical potential and is composed by derivative of chemical energy [math]f(c)[/math] and interface energy [math]0.5h\left\vert \nabla c\ \right\vert^2[/math].

    [math]\frac{\partial c}{\partial t}=\nabla \ \cdot [D \nabla \ \mu], \mu=f'(c)-h \nabla^2 \ c [/math]

    <Discretization>

    Semi-implicit Fourier spectral method applied to increase space resolution & calculation speed.

    [math]\frac{\partial c}{\partial t}=A \nabla^2 \ c + f(c) , f(c)=\nabla \ \cdot [D \nabla \ \mu]- A \nabla^2 \ c [/math]

    Additionally, 2-SBDF (second order semi-implicit backward differentiation) applied to increase time integration stability.

    [math]\frac{3}{2}c^{n+1}-2c^n+\frac{1}{2}c^{n-1}=\Delta t(\nabla \ ^2 c^{n+1})+2Q^n-Q^{n+1}[/math]

    [math]Q^n=\Delta t[\nabla \ \cdot (D\nabla \ \mu^n)-A\nabla^2 \ c^n][/math]

    The equation above are effectively calculated with the Fourier transform.

    [math]\hat c^{n+1}=\frac{4\hat c^n-\hat c^{n-1}+4\hat Q^n-2 \hat Q^{n-1}}{3+2\Delta t k^2}[/math]

    [math]\hat Q^n=\Delta t [i \mathbf{K} \cdot \left \{ D_n (i \mathbf{K} \hat \mu^n)_r \right \}_k +k^2 \hat c ^n] [/math]

    Vector [math] \mathbf{K} [/math] denotes the wave vector in Fourier space, and [math] k^2=k^2_1+k^2_2+k^2_3. [/math]

    The subscript [math] r [/math] denotes the inverse Fourier transform.

    <Normalization>

    The governing equation is normalized with the characteristic time and length are set to be [math] t_c = 0.5 s [/math] and [math] l_c = 5.0 nm [/math], respectively.

    Boundary condition

    The boundary condition for Cahn-Hilliard equation is that the flux([math]\mathbf{J}[/math]) on the particle surface is related to the discharge/charge current density [math]i_{loc}[/math] as

    [math]\mathbf{J}\cdot\mathbf{n}=i_{loc}/F[/math]

    [math]F[/math] is Faraday constant.

    Joule heat

    [math]\dot{q}=i_{loc}\eta[/math]

    1) [math]i_{loc}[/math] (local current density)

    Local current density.jpg

    2) [math]\eta[/math] (overpotential)

    Overpotential.jpg

    [math]F[/math] : Faraday constant (96485 C/mol)

    [math]k_0[/math] : Surface electrochemical reaction-rate constant

    [math]c[/math] : Surface Li-ion concentration

    [math]c_{max}[/math] : Maximum Li-ion concentration

    [math]\beta[/math] : Anodic transfer coefficient (=0.5)

    [math]R[/math] : Ideal gas constant (=8.314 J/mol/K)

    [math]T[/math] : Temperature (K)

    heat equation

    Heat equation.jpg

    [math]\rho[/math] : Density

    [math]c_p[/math] : Specific heat capacity

    [math]\kappa[/math] : Thermal conductivity

    Contact

    Prof. Dongchoul Kim, Sogang University (Tel: +82-2-705-8643)

    Connection to Other Documents