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Lithium-Ion Battery with Multiple Intercalating Electrode Materials
Introduction
Lithium-ion batteries can have multiple intercalating materials in both the positive and negative electrodes. For example, the negative electrode can have a mix of different forms of carbon. Similarly, the positive electrode can have a mix of active materials such as transition metal oxides, layered metal oxides, olivines, and so forth. These materials can have different design properties (such as volume fractions, particle sizes), thermodynamic properties (such as equilibrium potentials, maximum lithium concentrations), transport properties (such as solid diffusivities) and kinetic properties (such as intercalation reaction rate constants).
This model example demonstrates the Additional Porous Electrode Material feature in the Lithium-Ion Battery interface. The model describes a lithium-ion battery with two different intercalating materials in the positive electrode, whereas the negative electrode consists of one intercalating material only. The battery performance during discharge for different mix fractions of the two intercalating materials in the positive electrode is studied. The geometry is in one dimension and the model is isothermal.
Model Definition
This example models the battery cross section in 1D, which implies that edge effects in the length and height of the battery are neglected. The example uses the following domains:
Two active intercalating materials are considered in the positive electrode and the negative electrode consists of a single intercalating material. The model includes the following processes (Ref. 1).
For the porous electrodes, the effective electrolyte properties are calculated using the Bruggeman relation. Transport in the spherical particles is described using the Baker-Verbrugge diffusion model. This diffusion model considers the gradient of the chemical potential of the intercalate lithium as the driving force for diffusion, as opposed to considering the gradient of lithium concentration for dilute solution treatment (Fick’s Law) of lithium transport in the active material particles. The diffusion equation is expressed in spherical coordinates for the material balance of lithium in the particles. Butler-Volmer electrode kinetics describes the local charge transfer current density in the electrodes. The Butler-Volmer expressions are introduced as source or sink terms in the charge balances and material balances.
Material Properties
The electrolyte consists of 1 M LiPF6 salt in 1:1 EC:DEC (by weight) solvent. The two active materials in the positive electrode are NCA (LiyNi0.80Co0.15Al0.15O2) and LMO (LiyMn2O4 spinel). For the negative electrode, MCMB graphite (LixC6) is used in the model. The material properties of the electrolyte and active materials are taken from the Material Library.
The equilibrium potentials of the positive electrode materials are shown in Figure 1
Figure 1: The equilibrium potentials of NCA (top) and LMO (bottom).
The x-axis data in Figure 1 is the state of charge (SOC) in the active material, which is calculated by dividing the surface concentration of lithium with the maximum concentration of lithium in the material.
Results and Discussion
Figure 2 shows the voltage profile for a 1:2 volume ratio of the two positive electrode materials at a constant current discharge of 1C (11.72 A/m2).
Figure 2: Discharge voltage profile at 1C.
Figure 3 shows the lithium concentration at the surface of the active material particles in the positive electrode (at the positive electrode current collector end) during 1C discharge. The variation of the surface concentration with time is different in the two active materials. This is because of the different electrochemical properties of the two active materials.
Figure 3: Surface concentration in the active material particles in the positive electrode during 1C discharge.
Figure 4 shows the lithium concentration inside a particle at a particular position in the negative graphite electrode (at the center of the negative electrode) during 1C discharge. The concentration profiles show characteristic ridges, thereby capturing the staging phenomenon seen in multiphase electrodes like graphite. The Baker-Verbrugge diffusion model accounts for the interactions of the lithium-ions within the solid phase through an activity correction term, and hence provides a realistic representation of the intercalation process.
Figure 4: Concentration profiles inside a particle at a particular position in the negative electrode, at various times during 1C discharge.
The voltage profiles during 1C discharge for different volume mix fractions of the active materials in the positive electrode are shown in Figure 5. The shape of the discharge profile has a pronounced dependence on the mix fraction of the active materials in the electrode.
Figure 5: Voltage profiles during 1C discharge for different volume mix fractions of the active materials in the positive electrode.
Reference
1. P. Albertus, J. Christensen, and J. Newman, “Experiments on and Modeling of Multiple Active Materials in Positive Electrodes for Lithium-Ion Batteries,” J. Electrochem. Soc., vol. 156, no. 7, pp. A606–A618, 2009.
Application Library path: Battery_Design_Module/Batteries,_Lithium-Ion/li_battery_multiple_materials_1d
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  1D.
2
In the Select Physics tree, select Electrochemistry>Batteries>Lithium-Ion Battery (liion).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Time Dependent with Initialization.
6
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Geometry 1
The geometry contains three domains. Create the geometry by specifying the coordinates of the boundaries.
Interval 1 (i1)
1
In the Model Builder window, under Component 1 (comp1) right-click Geometry 1 and choose Interval.
2
In the Settings window for Interval, locate the Interval section.
3
From the Specify list, choose Interval lengths.
4
5
Click  Build Selected.
Materials
All materials are available in the Material Library. Note: In the Materials node, cEeqref denotes the maximum lithium concentration in the active material.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Battery>Electrolytes>LiPF6 in 1:1 EC:DEC (Liquid, Li-ion Battery).
4
Click Add to Component in the window toolbar.
5
In the tree, select Battery>Electrodes>Graphite, LixC6 MCMB (Negative, Li-ion Battery).
6
Click Add to Component in the window toolbar.
7
In the tree, select Battery>Electrodes>NCA, LiNi0.8Co0.15Al0.05O2 (Positive, Li-ion Battery).
8
Click Add to Component in the window toolbar.
9
In the tree, select Battery>Electrodes>LMO, LiMn2O4 Spinel (Positive, Li-ion Battery).
10
Click Add to Component in the window toolbar.
11
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
NCA, LiNi0.8Co0.15Al0.05O2 (Positive, Li-ion Battery) (mat3)
In the Model Builder window, expand the NCA, LiNi0.8Co0.15Al0.05O2 (Positive, Li-ion Battery) (mat3) node.
Interpolation 1 (Eeq_int1)
1
In the Model Builder window, expand the Component 1 (comp1)>Materials>NCA, LiNi0.8Co0.15Al0.05O2 (Positive, Li-ion Battery) (mat3)>Equilibrium potential (elpot) node, then click Interpolation 1 (Eeq_int1).
2
In the Settings window for Interpolation, click  Plot.
LMO, LiMn2O4 Spinel (Positive, Li-ion Battery) (mat4)
In the Model Builder window, expand the LMO, LiMn2O4 Spinel (Positive, Li-ion Battery) (mat4) node.
Interpolation 1 (Eeq_int1)
1
In the Model Builder window, expand the Component 1 (comp1)>Materials>LMO, LiMn2O4 Spinel (Positive, Li-ion Battery) (mat4)>Equilibrium potential (elpot) node, then click Interpolation 1 (Eeq_int1).
2
In the Settings window for Interpolation, click  Plot.
Definitions
Load the variables for this model from a text file.
Variables 1
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Lithium-Ion Battery (liion)
Set up the physics in the insertion electrodes. In this model, the Baker-Verbrugge diffusion model is used in the Particle Intercalation nodes. Note that the Baker-Verbrugge diffusivities are typically different from the Fickian diffusivity values. In this model, the intercalation diffusivites are simply set to the values available in the Material Library.
Porous Electrode 1
1
In the Model Builder window, under Component 1 (comp1) right-click Lithium-Ion Battery (liion) and choose Porous Electrode.
2
3
In the Settings window for Porous Electrode, locate the Electrode Properties section.
4
From the Electrode material list, choose Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat2).
5
Locate the Porous Matrix Properties section. In the εs text field, type liion.epss_neg.
The liion.epss_neg variable is calculated automatically by the Initial Cell Charge Distribution node in order to balance the capacity of the negative electrode to that of the positive.
6
In the εl text field, type epsl_neg.
Particle Intercalation 1
1
In the Model Builder window, click Particle Intercalation 1.
2
In the Settings window for Particle Intercalation, locate the Material section.
3
From the Particle material list, choose Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat2).
4
Locate the Particle Transport Properties section. From the Species concentration transport model list, choose Baker-Verbrugge.
5
In the rp text field, type rp_neg.
The concentration profiles inside a graphite electrode particle will be analyzed during postprocessing. Hence, it is useful to have a finer resolution along the particle dimension, by setting a linear distribution with 20 elements.
6
Click to expand the Particle Discretization section. From the Distribution list, choose Linear.
7
In the Nel text field, type 20.
Porous Electrode Reaction 1
1
In the Model Builder window, click Porous Electrode Reaction 1.
2
In the Settings window for Porous Electrode Reaction, locate the Material section.
3
From the Material list, choose Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat2).
4
Locate the Electrode Kinetics section. In the i0,ref(T) text field, type i0ref_neg.
Porous Electrode 2
1
In the Physics toolbar, click  Domains and choose Porous Electrode.
2
3
In the Settings window for Porous Electrode, locate the Electrode Properties section.
4
From the Electrode material list, choose NCA, LiNi0.8Co0.15Al0.05O2 (Positive, Li-ion Battery) (mat3).
5
Locate the Porous Matrix Properties section. In the εs text field, type epss_pos_NCA.
6
In the εl text field, type epsl_pos.
7
Locate the Effective Transport Parameter Correction section. From the Electrolyte conductivity list, choose User defined. In the fl text field, type epsl_pos^brugl_pos.
8
From the Diffusion list, choose User defined. In the fDl text field, type epsl_pos^brugl_pos.
Particle Intercalation 1
1
In the Model Builder window, click Particle Intercalation 1.
2
In the Settings window for Particle Intercalation, locate the Material section.
3
From the Particle material list, choose NCA, LiNi0.8Co0.15Al0.05O2 (Positive, Li-ion Battery) (mat3).
4
Locate the Particle Transport Properties section. In the rp text field, type rp_pos_NCA.
5
From the Species concentration transport model list, choose Baker-Verbrugge.
Porous Electrode Reaction 1
1
In the Model Builder window, click Porous Electrode Reaction 1.
2
In the Settings window for Porous Electrode Reaction, locate the Material section.
3
From the Material list, choose NCA, LiNi0.8Co0.15Al0.05O2 (Positive, Li-ion Battery) (mat3).
4
Locate the Electrode Kinetics section. In the i0,ref(T) text field, type i0ref_pos_NCA.
Additional Porous Electrode Material 1
1
In the Physics toolbar, click  Domains and choose Additional Porous Electrode Material.
2
3
In the Settings window for Additional Porous Electrode Material, locate the Volume Fraction section.
4
In the εs text field, type epss_pos_LMO.
Particle Intercalation 1
1
In the Model Builder window, click Particle Intercalation 1.
2
In the Settings window for Particle Intercalation, locate the Material section.
3
From the Particle material list, choose LMO, LiMn2O4 Spinel (Positive, Li-ion Battery) (mat4).
4
Locate the Particle Transport Properties section. In the rp text field, type rp_pos_LMO.
5
From the Species concentration transport model list, choose Baker-Verbrugge.
6
Locate the Model Input section. From the c list, choose Solid phase concentration, Additional Porous Electrode Material 1 (liion/addm1/pin1).
Porous Electrode Reaction 1
1
In the Model Builder window, click Porous Electrode Reaction 1.
2
In the Settings window for Porous Electrode Reaction, locate the Material section.
3
From the Material list, choose LMO, LiMn2O4 Spinel (Positive, Li-ion Battery) (mat4).
4
Locate the Electrode Kinetics section. In the i0,ref(T) text field, type i0ref_pos_LMO.
5
Locate the Model Input section. From the c list, choose Insertion particle surface concentration, Additional Porous Electrode Material 1 (liion).
Separator 1
1
In the Physics toolbar, click  Domains and choose Separator.
2
3
In the Settings window for Separator, locate the Porous Matrix Properties section.
4
In the εl text field, type epsl_sep.
5
Locate the Effective Transport Parameter Correction section. From the Electrolyte conductivity list, choose User defined. In the fl text field, type epsl_sep^brugl_sep.
6
From the Diffusion list, choose User defined. In the fDl text field, type epsl_sep^brugl_sep.
Initial Cell Charge Distribution 1
1
In the Physics toolbar, click  Global and choose Initial Cell Charge Distribution.
2
In the Settings window for Initial Cell Charge Distribution, locate the Battery Cell Parameters section.
3
From the Initial battery cell setting list, choose Initial cell state-of-charge.
4
In the SOCcell,0 text field, type SOC_cell0.
5
In the Qcell,0 text field, type Q_batt.
Negative Electrode Selection 1
1
In the Model Builder window, expand the Initial Cell Charge Distribution 1 node, then click Negative Electrode Selection 1.
2
Positive Electrode Selection 1
1
In the Model Builder window, click Positive Electrode Selection 1.
2
Electric Ground 1
1
In the Physics toolbar, click  Boundaries and choose Electric Ground.
2
Electrode Current 1
1
In the Physics toolbar, click  Boundaries and choose Electrode Current.
2
3
In the Settings window for Electrode Current, locate the Electrode Current section.
4
In the Is,total text field, type -liion.I_1C.
Global Definitions
Default Model Inputs
Set up the temperature value used in the entire model.
1
In the Model Builder window, under Global Definitions click Default Model Inputs.
2
In the Settings window for Default Model Inputs, locate the Browse Model Inputs section.
3
In the tree, select General>Temperature (K) - minput.T.
4
Find the Expression for remaining selection subsection. In the Temperature text field, type T.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Fine.
4
Click  Build All.
Definitions (comp1)
Using a boundary integration variable you can access the cell voltage at the end terminal during computation.
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, type EndTerminal in the Operator name text field.
3
Locate the Source Selection section. From the Geometric entity level list, choose Boundary.
4
Study 1
Step 2: Time Dependent
1
In the Model Builder window, under Study 1 click Step 2: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type 0 3600.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
Store the actual steps taken by the solver to make sure to capture any sudden steep voltage changes.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, locate the General section.
4
From the Times to store list, choose Steps taken by solver.
5
Right-click Study 1>Solver Configurations>Solution 1 (sol1)>Time-Dependent Solver 1 and choose Stop Condition.
6
In the Settings window for Stop Condition, locate the Stop Expressions section.
7
8
9
Locate the Output at Stop section. Clear the Add warning check box.
10
In the Model Builder window, click Study 1.
11
In the Settings window for Study, locate the Study Settings section.
12
Clear the Generate default plots check box.
13
In the Study toolbar, click  Compute.
Results
Reproduce the plots in the model documentation for 1C discharge, starting with the voltage profile (Figure 2).
Constant current 1C discharge
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
Right-click 1D Plot Group 1 and choose Rename.
3
In the Rename 1D Plot Group dialog box, type Constant current 1C discharge in the New label text field.
4
Point Graph 1
1
Right-click Constant current 1C discharge and choose Point Graph.
2
3
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Lithium-Ion Battery>phis - Electric potential - V.
Constant current 1C discharge
1
In the Model Builder window, click Constant current 1C discharge.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the y-axis label check box.
4
5
Click to expand the Title section. From the Title type list, choose Label.
6
Locate the Axis section. Select the Manual axis limits check box.
7
In the x minimum text field, type 0.
8
In the x maximum text field, type 3600.
9
In the y minimum text field, type 3.0.
10
In the y maximum text field, type 4.2.
11
In the Constant current 1C discharge toolbar, click  Plot.
Surface concentration Positive
The following steps are for plotting the surface concentration in each active material in the positive electrode during 1C discharge (Figure 3).
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
Right-click 1D Plot Group 2 and choose Rename.
3
In the Rename 1D Plot Group dialog box, type Surface concentration Positive in the New label text field.
4
Point Graph 1
1
Right-click Surface concentration Positive and choose Point Graph.
2
3
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Lithium-Ion Battery>Particle intercalation>liion.cs_surface - Insertion particle concentration, surface - mol/m³.
4
Click to expand the Legends section. Select the Show legends check box.
5
From the Legends list, choose Manual.
6
Point Graph 2
1
Right-click Point Graph 1 and choose Duplicate.
2
In the Settings window for Point Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Lithium-Ion Battery>Particle intercalation>liion.cs_surface_addm1 - Insertion particle surface concentration, Additional Porous Electrode Material 1 - mol/m³.
3
Locate the Legends section. In the table, enter the following settings:
Surface concentration Positive
1
In the Model Builder window, click Surface concentration Positive.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the y-axis label check box.
4
In the associated text field, type Surface concentration (mol/m<sup>3</sup>).
5
Locate the Title section. From the Title type list, choose Label.
6
Locate the Axis section. Select the Manual axis limits check box.
7
In the x minimum text field, type 0.
8
In the x maximum text field, type 3600.
9
In the y minimum text field, type 4000.
10
In the Surface concentration Positive toolbar, click  Plot.
Study 1/Solution 1 (sol1)
The following steps are for plotting the concentration profiles inside a particle at a particular position in the negative electrode, at various times during 1C discharge (Figure 4). To do this, create a Solution dataset that refers to the extra dimension that is set up by the Porous Electrode node corresponding to the negative electrode.
Study 1/Solution 1: xdim Negative
1
In the Model Builder window, expand the Results>Datasets node.
2
Right-click Results>Datasets>Study 1/Solution 1 (sol1) and choose Duplicate.
3
In the Settings window for Solution, type Study 1/Solution 1: xdim Negative in the Label text field.
4
Locate the Solution section. From the Component list, choose Extra Dimension from Particle Intercalation 1 (liion_pce1_pin1_xdim).
Particle concentration Negative
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
Right-click 1D Plot Group 3 and choose Rename.
3
In the Rename 1D Plot Group dialog box, type Particle concentration Negative in the New label text field.
4
5
In the Settings window for 1D Plot Group, locate the Data section.
6
From the Dataset list, choose Study 1/Solution 1: xdim Negative (sol1).
7
From the Time selection list, choose Interpolated.
8
In the Times (s) text field, type range(0,200,2000) range(2030,30,2200) range(2300,200,3300).
Line Graph 1
1
Right-click Particle concentration Negative and choose Line Graph.
2
The atxd1() operator is used to specify the x coordinate in the negative electrode of the battery geometry.
3
In the Settings window for Line Graph, locate the y-Axis Data section.
4
In the Expression text field, type atxd1(25e-6,liion.cs_pce1).
Particle concentration Negative
1
In the Model Builder window, click Particle concentration Negative.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the x-axis label check box.
4
In the associated text field, type Normalized Particle Dimension.
5
Select the y-axis label check box.
6
In the associated text field, type Particle concentration (mol/m<sup>3</sup>).
7
Locate the Title section. From the Title type list, choose Label.
8
In the Particle concentration Negative toolbar, click  Plot.
Root
Now set up a parametric study for different mix fractions of active materials in the positive electrode.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Time Dependent with Initialization.
4
Click Add Study in the window toolbar.
5
In the Model Builder window, click the root node.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 2: Time Dependent
1
In the Settings window for Time Dependent, locate the Study Settings section.
2
In the Output times text field, type 0 3600.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
4
Solution 3 (sol3)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 3 (sol3) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, locate the General section.
4
From the Times to store list, choose Steps taken by solver.
Store only every 3rd time step. This reduces the size of the stored solution and the size of model file.
5
In the Store every Nth step text field, type 3.
6
Right-click Study 2>Solver Configurations>Solution 3 (sol3)>Time-Dependent Solver 1 and choose Stop Condition.
7
In the Settings window for Stop Condition, locate the Stop Expressions section.
8
9
10
Locate the Output at Stop section. Clear the Add warning check box.
11
In the Model Builder window, click Study 2.
12
In the Settings window for Study, locate the Study Settings section.
13
Clear the Generate default plots check box.
14
In the Study toolbar, click  Compute.
Results
You can plot the voltage profiles from the parametric study (Figure 5) by performing the following steps:
Voltage profiles (parametric)
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
Right-click 1D Plot Group 4 and choose Rename.
3
In the Rename 1D Plot Group dialog box, type Voltage profiles (parametric) in the New label text field.
4
Point Graph 1
1
Right-click Voltage profiles (parametric) and choose Point Graph.
2
In the Settings window for Point Graph, locate the Data section.
3
From the Dataset list, choose Study 2/Parametric Solutions 1 (sol5).
4
5
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose phis - Electric potential - V.
6
Locate the Legends section. Select the Show legends check box.
7
From the Legends list, choose Evaluated.
8
In the Legend text field, type eval(fr_pos_NCA*100)/eval((1-fr_pos_NCA)*100) volume mix of NCA and LMO.
Voltage profiles (parametric)
1
In the Model Builder window, click Voltage profiles (parametric).
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the y-axis label check box.
4
5
Locate the Title section. From the Title type list, choose Manual.
6
In the Title text area, type Voltage profiles for different mix fractions of NCA and LMO.
7
Locate the Axis section. Select the Manual axis limits check box.
8
In the x minimum text field, type 0.
9
In the x maximum text field, type 3600.
10
In the y minimum text field, type 3.0.
11
In the y maximum text field, type 4.2.
12
Locate the Legend section. From the Position list, choose Lower left.
13
In the Voltage profiles (parametric) toolbar, click  Plot.