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Copper Current-Collector Dissolution
Introduction
The copper current collector on negative graphite electrodes in lithium-ion batteries have been seen to dissolve at over-discharge. This can be a safety concern as the dissolution damages the current collector irreversibly and dissolved copper ions can redeposit and form dendrites. Batteries are usually operated within specified voltage limits, either on cell or pack level, in order to avoid over-discharge. However, over-discharge may still occur, either in a pack due to charge imbalances between the individual cells, or, as we will partly investigate in this tutorial, due to local heterogeneities in the electrodes within a single battery cell.
This tutorial investigates the copper current-collector dissolution during over-discharge from an electrode with an inhomogeneous graphite electrode material coating. The cuprous ion concentration, as well as the redeposited copper distribution in the cell, are predicted.
Model Definition
general
The model is set up in 2D for a graphite/NMC battery cell with a 1.0 M LiPF6 in EC:EMC (3:7 by weight) electrolyte. Properties for these materials are taken from the Battery material library.
The model consists of the following three domains:
•
Negative porous electrode: Graphite (MCMB LixC6) active material (~64 μm).
•
Separator (25 μm).
•
Positive porous electrode: NMC 811 active material (40 μm).
The Lithium-Ion Battery interface is used, accounting for:
•
Electronic conduction in the electrodes,
•
Ionic charge transport in the electrodes and electrolyte/separator,
•
Material transport in the electrolyte, allowing for including the effects of concentration on ionic conductivity and concentration overpotentials,
•
Material transport within the solid spherical particles in the electrodes
•
Butler–Volmer electrode intercalation kinetics using experimentally measured equilibrium potentials.
More information about how to use the Lithium-Ion Battery interface can be found in the 1D Isothermal Lithium-Ion Battery example.
The Transport of Diluted Species interface is used to define the copper (cuprous) ion reaction and transport in the cell. Additional features in the Lithium-Ion Battery interface are used to define the electrochemical copper dissolution/deposition reaction (as described below).
The model is solved for two shapes of dents in the graphite active material coating. Figure 1 displays the two shapes of dents investigated.
Figure 1: The 2D model geometry. Two dent shapes (gray) with the same volume are studied.
Electrolyte phase potential definitions
In this model we assume the cuprous ion concentration to be low in relation to the Li+/PF6- (+ solvent) electrolyte, and assume a dilute approximation of the Cu+ transport and electrochemical reactions to be valid. We will hence assume that the electrolyte phase potential field is governed by the lithium-ion transport, and that the contribution to the electrolyte potential field from the cuprous ion transport may be neglected.
A caveat here is that the electrolyte phase potential variable ϕl solved for by the Lithium-Ion Battery interface is defined by concentrated electrolyte theory, where ϕl is defined with reference to a Li+/Li(s) electrode of the same local concentration cl of lithium ions, also solved for by the Lithium-Ion Battery interface. We need hence to correct for the local lithium concentration before we can use the electrolyte phase potential variable as input to the cuprous reaction and transport terms. With the absence of concentration dependent Li+ activities, an ideal electrolyte concentration dependence for single charged ions is used as an approximation, as follows:
(1)
where cl,ref refers to a chosen (1 M) reference salt-concentration of the battery electrolyte.
A pseudo-potentiostatic electrolyte potential, ϕl,ps (V), to be used for the cuprous ion transport and kinetic definitions can now be defined as:
(2)
Copper Dissolution/Deposition Reaction
On the negative porous electrode, the copper reaction is defined as an additional reaction to the main graphite-lithium intercalation reaction. This is done using an additional Porous Electrode Reaction in the Porous Electrode node for the negative electrode. At the copper current-collector boundary, the reaction is added as an Electrode Reaction in an Internal Electrode Surface node.
The reaction is reversible, purely electrochemical, and considers cuprous ions (Ref. 1):
The electrochemical reaction current, iCu, is defined by concentration-dependent Butler–Volmer kinetics (Ref. 2):
(3)
where the pre-exponential factors for the anodic, Fa, and cathodic, Fc, terms are defined as
(4)
In the expressions above:
•
i0,ref is the reference exchange current density.
•
 cCu is the cuprous ion concentration in the electrolyte.
•
  cCu,ref is the reference cuprous ion concentration, set to 1 M.
•
 cCu(s),cov is the concentration of a monolayer fully covered with deposited copper on the negative active material. The monolayer thickness is set to 0.1 nm.
•
  cCu(s) is the deposited copper concentration (equal to cCu(s),cov at the current-collector boundary).
•
 ηCu is the overpotential of the copper reaction with the equilibrium potential, Eeq,Cu, set to 3.5 V versus Li+/Li(s) in a 1M Li+/PF6- electrolyte.:
(5)
 The deposited copper concentration is computed using the in-built Dissolving–Depositing Species feature in the main Porous Electrode node. The equation that is solved takes the following form for the copper reaction:
(6)
where av is the specific surface area (m2/m3) of the active material particles.
Copper Flux and Mass Transport
In the Lithium-ion Battery interface, the flux to/from the copper reaction site is assumed to be in the form of lithium ions to maintain charge neutrality and binary salt transport conditions in the electrolyte.
The Transport of Diluted Species interface solves for the transport of cuprous ions by diffusion and migration. For the migration term, the electrolyte potential ϕl,ps is used, as defined above. On the copper current collector boundary, the flux due to dissolution of copper is computed using an Electrode Surface Coupling node. At the negative porous electrode, the Porous Electrode Coupling node defines the redeposition of copper. Both these coupling nodes refer to the electrochemical reactions defined in the Lithium-Ion Battery Interface.
With the definitions above, the net electrolyte salt concentration, cl,net, is given by the difference in electrolyte salt concentration, cl, as computed by the Lithium-Ion Battery interface and the cuprous ion concentration:
(7)
Results and Discussion
A galvanostatic cell discharge between 4.1 V and −0.05 V is modeled to capture the overdischarge behavior. In Figure 2, the cell voltage and cuprous ion concentration in the cell is shown near the end of the discharge. The voltage is well below the normal lower cutoff voltage (~2.7 V) when the amount of cuprous ions starts to increase. The results are similar for both dent shapes.
Figure 2: Cell voltage and cuprous ion concentration in the cell near the end of the discharge for the two dent shapes.
In Figure 3, the electrode potential distribution in the porous electrodes are shown at the time when the potential is higher than 3.5 V in the negative electrode at the copper current collector.
Figure 3: Electrode potential distribution once local negative electrode potential is higher than 3.5 V at the copper current collector (~3605 s).
As shown in Figure 4, this marks the initiation of copper dissolution and correlates to the copper reaction equilibrium potential set to 3.5 V in the model. Comparing the two different dent shapes shows that the deeper dent contributes to a more uneven potential distribution and copper dissolution.
Figure 4: Cuprous ion concentration in the cell once local negative electrode potential is higher than 3.5 V at the copper current collector (~3605 s).
The dissolved cuprous ions will redeposit at lower local electrode potentials. This can be a safety concern, for instance, if the deposits take the form of dendrites. The deposited copper concentrations at −0.05 V (soon after battery polarity reversal) are displayed in Figure 5. Higher local concentrations and more uneven distribution are shown for the deeper dent.
Figure 5: The deposited copper concentration at -0.05 V cell voltage.
The dissolution current along the copper current collector is displayed in Figure 6. Faster dissolution of the current collector is shown beneath the tip of the dent. Results point to higher risk of local damage for the deep dent.
Figure 6: Copper dissolution current at the copper current collector boundary once local negative electrode potential is higher than 3.5 V at the copper current collector (~3605 s) and at cell voltage -0.05 V.
References
1. L. Guo, D.B. Thornton, M.A. Kanonfel, I.E.L. Stephens, and M.P. Ryan, “Degradation in lithium ion battery current collectors,” J. Phys. Energy., vol. 3, pp. 2874, 2000.
2. S.E.J. O’Kane, I.D. Campbell, M.W.J. Marzook, G.J. Offer, and M. Marinescu, “Physical Origin of the Differential Voltage Minimum Associated with Lithium Plating in Li-Ion Batteries,” J. Elec. Soc., vol. 167, pp. 090540, 2020.
Application Library path: Battery_Design_Module/Lithium-Ion_Batteries,_Aging_and_Abuse/cu_current_collector_dissolution
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  2D.
2
In the Select Physics tree, select Electrochemistry > Batteries > Lithium-Ion Battery (liion).
3
Click Add.
4
In the Select Physics tree, select Chemical Species Transport > Transport of Diluted Species (tds).
5
Click Add.
6
Click  Study.
7
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Lithium-Ion Battery > Time Dependent with Initialization.
8
Click  Done.
Global Definitions
Parameters 1
Load the parameters from a text file.
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
Browse to the model’s Application Libraries folder and double-click the file cu_current_collector_dissolution_parameters.txt.
Add material data for the electrolyte, the electrodes, and the copper current collector from the material library.
Add Material
1
In the Materials 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:DMC (Liquid, Li-ion Battery).
4
Click the Add to Component button in the window toolbar.
5
In the tree, select Battery > Electrodes > Graphite, LixC6 MCMB (Negative, Li-ion Battery).
6
Click the Add to Component button in the window toolbar.
7
In the tree, select Battery > Electrodes > NMC 811, LiNi0.8Mn0.1Co0.1O2 (Positive, Li-ion Battery).
8
Click the Add to Component button in the window toolbar.
9
In the tree, select Built-in > Copper.
10
Click the Add to Component button in the window toolbar.
11
In the Materials toolbar, click  Add Material to close the Add Material window.
Geometry 1
Draw the model geometry in 2D to investigate different shapes of dents.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type w_battery.
4
In the Height text field, type L_neg_cc.
Rectangle 2 (r2)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type w_battery.
4
In the Height text field, type L_neg.
5
Locate the Position section. In the y text field, type L_neg_cc.
Rectangle 3 (r3)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type w_battery.
4
In the Height text field, type L_sep.
5
Locate the Position section. In the y text field, type L_neg_cc+L_neg.
Rectangle 4 (r4)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type w_battery.
4
In the Height text field, type L_pos.
5
Locate the Position section. In the y text field, type L_neg_cc+L_neg+L_sep.
Rectangle 5 (r5)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type w_dent.
4
In the Height text field, type depth_dent.
5
Locate the Position section. In the x text field, type w_battery/2-w_dent/2.
6
In the y text field, type L_neg+L_neg_cc-depth_dent.
7
In the Home toolbar, click  Build All.
Definitions
In the Model Builder window, expand the Component 1 (comp1) > Definitions node.
View 1
Use the View node to scale the geometry in the Graphics window to better visualize the drawn domains.
Axis
1
In the Model Builder window, expand the Component 1 (comp1) > Definitions > View 1 node, then click Axis.
2
In the Settings window for Axis, locate the Axis section.
3
From the View scale list, choose Manual.
4
In the y scale text field, type 100.
5
Click  Update.
6
Click the  Zoom Extents button in the Graphics toolbar.
Add selections to facilitate building the model.
Negative Electrode
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Negative Electrode in the Label text field.
3
Select Domain 2 only.
Positive Electrode
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Positive Electrode in the Label text field.
3
Select Domain 4 only.
Separator
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Separator in the Label text field.
3
Select Domain 3 only.
Dent
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Dent in the Label text field.
3
Select Domain 5 only.
Copper Current Collector
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Copper Current Collector in the Label text field.
3
Select Domain 1 only.
Copper Current-Collector Boundary
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Copper Current-Collector Boundary in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 4 only.
Aluminum Current-Collector Boundary
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Aluminum Current-Collector Boundary in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 9 only.
Integration - Electrolyte-Filled Domains
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
Create an integration operator for all electrolyte filled domains to compute, for example, the total amount of cuprous ions in the cell.
2
In the Settings window for Integration, type Integration - Electrolyte-Filled Domains in the Label text field.
3
In the Operator name text field, type int_el_doms.
4
Select Domains 2–5 only.
Now assign the previously added materials to the different domains of the geometry.
Materials
LiPF6 in 1:1 EC:DMC (Liquid, Li-ion Battery) (mat1)
1
In the Model Builder window, under Component 1 (comp1) > Materials click LiPF6 in 1:1 EC:DMC (Liquid, Li-ion Battery) (mat1).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
Click  Clear Selection.
4
Select Domains 3 and 5 only.
Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat2)
1
In the Model Builder window, click Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat2).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Negative Electrode.
NMC 811, LiNi0.8Mn0.1Co0.1O2 (Positive, Li-ion Battery) (mat3)
1
In the Model Builder window, click NMC 811, LiNi0.8Mn0.1Co0.1O2 (Positive, Li-ion Battery) (mat3).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Positive Electrode.
Copper (mat4)
1
In the Model Builder window, click Copper (mat4).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Copper Current Collector.
Definitions
Add model variables.
Variables 1
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file cu_current_collector_dissolution_variables.txt.
Now begin defining the physics. Start with the Lithium-Ion Battery interface.
Lithium-Ion Battery (liion)
Separator 1
1
In the Model Builder window, under Component 1 (comp1) > Lithium-Ion Battery (liion) click Separator 1.
2
In the Settings window for Separator, locate the Porous Matrix Properties section.
3
In the εl text field, type epsl_sep.
Electrolyte 1
1
In the Physics toolbar, click  Domains and choose Electrolyte.
2
In the Settings window for Electrolyte, locate the Domain Selection section.
3
From the Selection list, choose Dent.
Current Conductor 1
1
In the Physics toolbar, click  Domains and choose Current Conductor.
2
In the Settings window for Current Conductor, locate the Domain Selection section.
3
From the Selection list, choose Copper Current Collector.
Porous Electrode - Negative
1
In the Physics toolbar, click  Domains and choose Porous Electrode.
2
In the Settings window for Porous Electrode, type Porous Electrode - Negative in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Negative Electrode.
4
Locate the Electrolyte Properties section. From the Electrolyte material list, choose LiPF6 in 1:1 EC:DMC (Liquid, Li-ion Battery) (mat1).
5
Locate the Electrode Properties section. In the σs text field, type sigmas_neg.
6
Locate the Porous Matrix Properties section. In the εs text field, type epss_neg.
7
In the εl text field, type epsl_neg.
8
Locate the Effective Transport Parameter Correction section. From the Electric conductivity list, choose No correction.
In the negative electrode, add copper metal as dissolving–depositing species.
9
Click to expand the Dissolving–Depositing Species section. Click  Add.
10
In the table, enter the following settings:
Species
Density (kg/m^3)
Molar mass (kg/mol)
CuMetal
rho_Cu
M_Cu
11
Clear the Add volume change to electrode volume fraction checkbox.
12
Clear the Subtract volume change from electrolyte volume fraction checkbox.
Particle Intercalation 1
1
In the Model Builder window, click Particle Intercalation 1.
2
In the Settings window for Particle Intercalation, locate the Species Settings section.
3
In the cs,init text field, type cs_init_neg.
4
Locate the Particle Transport Properties section. In the rp text field, type rp_neg.
Porous Electrode Reaction - Intercalation
1
In the Model Builder window, under Component 1 (comp1) > Lithium-Ion Battery (liion) > Porous Electrode - Negative click Porous Electrode Reaction 1.
2
In the Settings window for Porous Electrode Reaction, type Porous Electrode Reaction - Intercalation in the Label text field.
3
Locate the Electrode Kinetics section. In the i0,ref(T) text field, type i0_ref_neg.
Porous Electrode Reaction - Copper
1
Right-click Porous Electrode Reaction - Intercalation and choose Duplicate.
Define the copper dissolution/deposition reaction in an additional Porous Electrode Reaction node. To maintain electrolyte neutrality and binary conditions in the interface, cuprous ions are assumed to behave as lithium ions; that is, the copper reaction introduces a flux of lithium ions.
2
In the Settings window for Porous Electrode Reaction, type Porous Electrode Reaction - Copper in the Label text field.
3
Locate the Equilibrium Potential section. From the Eeq list, choose User defined. In the associated text field, type Eeq_Cu+delta_phil.
4
Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Concentration dependent kinetics.
5
In the i0 text field, type i0_ref_CuMetal.
6
In the CR text field, type min(liion.c_pce1_CuMetal/(cCucov*liion.Av_pce1_per2),1).
7
In the CO text field, type cCu/cCu_ref.
8
Locate the Stoichiometric Coefficients section. In the νLiθ text field, type 0.
9
In the Stoichiometric coefficients for dissolving–depositing species: table, enter the following settings:
Species
Stoichiometric coefficient (1)
CuMetal
1
Porous Electrode - Positive
1
In the Physics toolbar, click  Domains and choose Porous Electrode.
2
In the Settings window for Porous Electrode, type Porous Electrode - Positive in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Positive Electrode.
4
Locate the Electrolyte Properties section. From the Electrolyte material list, choose LiPF6 in 1:1 EC:DMC (Liquid, Li-ion Battery) (mat1).
5
Locate the Electrode Properties section. In the σs text field, type sigmas_pos.
6
Locate the Porous Matrix Properties section. In the εs text field, type epss_pos.
7
In the εl text field, type epsl_pos.
8
Locate the Effective Transport Parameter Correction section. From the Electric conductivity list, choose No correction.
Particle Intercalation 1
1
In the Model Builder window, click Particle Intercalation 1.
2
In the Settings window for Particle Intercalation, locate the Species Settings section.
3
In the cs,init text field, type cs_init_pos.
4
Locate the Particle Transport Properties section. In the rp text field, type rp_pos.
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 Electrode Kinetics section.
3
In the i0,ref(T) text field, type i0_ref_pos.
At the boundary between the copper current-collector and the porous negative electrode, define the copper dissolution/deposition reaction. In contrast to the porous electrode, the metallic copper amount can be set as constant.
Internal Electrode Surface - Copper
1
In the Physics toolbar, click  Boundaries and choose Internal Electrode Surface.
2
In the Settings window for Internal Electrode Surface, type Internal Electrode Surface - Copper in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Copper Current-Collector Boundary.
Electrode Reaction 1
1
In the Model Builder window, click Electrode Reaction 1.
2
In the Settings window for Electrode Reaction, locate the Material section.
3
From the Material list, choose Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat2).
4
Locate the Equilibrium Potential section. From the Eeq list, choose User defined. In the associated text field, type Eeq_Cu+delta_phil.
5
Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Concentration dependent kinetics.
6
In the i0 text field, type i0_ref_CuMetal.
7
In the CO text field, type cCu/cCu_ref.
Electric Ground 1
1
In the Physics toolbar, click  Boundaries and choose Electric Ground.
2
Select Boundary 2 only.
Electrode Current 1
1
In the Physics toolbar, click  Boundaries and choose Electrode Current.
2
In the Settings window for Electrode Current, locate the Boundary Selection section.
3
From the Selection list, choose Aluminum Current-Collector Boundary.
4
Locate the Electrode Current section. In the Is,total text field, type -I_1C_cell.
To improve the initialization of the computations set the approximate potential (~cell voltage) in the positive electrode. This is necessary due to the highly nonlinear copper reaction kinetics.
Initial Values 2
1
In the Physics toolbar, click  Domains and choose Initial Values.
2
Select Domain 4 only.
3
In the Settings window for Initial Values, locate the Initial Values section.
4
In the phis text field, type V0.
Continue by defining the Transport of Diluted Species interface that is used to track the cuprous ion movement in the electrolyte. All fluxes due to copper dissolution/deposition needs to be coupled to the Lithium-Ion Battery interface.
Transport of Diluted Species (tds)
1
In the Model Builder window, under Component 1 (comp1) click Transport of Diluted Species (tds).
2
In the Settings window for Transport of Diluted Species, locate the Domain Selection section.
3
In the list box, select 1.
4
Click  Remove from Selection.
5
Select Domains 2–5 only.
6
Locate the Transport Mechanisms section. Clear the Convection checkbox.
7
Select the Migration in electric field checkbox.
8
Select the Mass transfer in porous media checkbox.
9
Click to expand the Dependent Variables section. In the Concentrations (mol/m³) table, enter the following settings:
cCu
Species Charges
1
In the Model Builder window, under Component 1 (comp1) > Transport of Diluted Species (tds) click Species Charges.
2
In the Settings window for Species Properties, locate the Charge section.
3
In the zcCu text field, type zCu.
Fluid 1
1
In the Model Builder window, click Fluid 1.
2
In the Settings window for Fluid, locate the Diffusion section.
3
In the DcCu text field, type D_Cu.
4
Locate the Migration in Electric Field section. In the V text field, type phil_ps.
Electrode Surface Coupling - Copper
1
In the Physics toolbar, click  Boundaries and choose Electrode Surface Coupling.
2
In the Settings window for Electrode Surface Coupling, type Electrode Surface Coupling - Copper in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Copper Current-Collector Boundary.
Reaction Coefficients 1
1
In the Model Builder window, expand the Electrode Surface Coupling - Copper node, then click Reaction Coefficients 1.
2
In the Settings window for Reaction Coefficients, locate the Reaction Current Density section.
3
From the iloc list, choose Local current density, Electrode Reaction 1 (liion/bei1/er1).
4
Locate the Stoichiometric Coefficients section. In the νcCu text field, type -1.
Porous Electrode - Negative
1
In the Physics toolbar, click  Domains and choose Porous Medium.
2
In the Settings window for Porous Medium, type Porous Electrode - Negative in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Negative Electrode.
Fluid 1
1
In the Model Builder window, click Fluid 1.
2
In the Settings window for Fluid, locate the Diffusion section.
3
In the DF,cCu text field, type D_Cu.
4
From the Effective diffusivity model list, choose Bruggeman model.
5
Locate the Migration in Electric Field section. In the V text field, type phil_ps.
Porous Matrix 1
1
In the Model Builder window, click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the εp list, choose User defined. In the associated text field, type epsl_neg.
Porous Electrode - Positive
1
In the Physics toolbar, click  Domains and choose Porous Medium.
2
In the Settings window for Porous Medium, type Porous Electrode - Positive in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Positive Electrode.
Fluid 1
1
In the Model Builder window, click Fluid 1.
2
In the Settings window for Fluid, locate the Diffusion section.
3
In the DF,cCu text field, type D_Cu.
4
From the Effective diffusivity model list, choose Bruggeman model.
5
Locate the Migration in Electric Field section. In the V text field, type phil_ps.
Porous Matrix 1
1
In the Model Builder window, click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the εp list, choose User defined. In the associated text field, type epsl_pos.
Separator
1
In the Physics toolbar, click  Domains and choose Porous Medium.
2
In the Settings window for Porous Medium, type Separator in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Separator.
Fluid 1
1
In the Model Builder window, click Fluid 1.
2
In the Settings window for Fluid, locate the Diffusion section.
3
In the DF,cCu text field, type D_Cu.
4
From the Effective diffusivity model list, choose Bruggeman model.
5
Locate the Migration in Electric Field section. In the V text field, type phil_ps.
Porous Matrix 1
1
In the Model Builder window, click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the εp list, choose User defined. In the associated text field, type epsl_sep.
Porous Electrode Coupling - Copper
1
In the Physics toolbar, click  Domains and choose Porous Electrode Coupling.
2
In the Settings window for Porous Electrode Coupling, type Porous Electrode Coupling - Copper in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Negative Electrode.
Reaction Coefficients 1
1
In the Model Builder window, click Reaction Coefficients 1.
2
In the Settings window for Reaction Coefficients, locate the Reaction Current Source section.
3
From the iv list, choose Local current source, Porous Electrode Reaction - Copper (liion/pce1/per2).
4
Locate the Stoichiometric Coefficients section. In the νcCu text field, type -1.
The highly nonlinear copper reaction kinetics require the steps in the study to set different initial cuprous ion concentrations to initialize the computations. Use an additional Initial Values node for this purpose.
Initial Values 2
1
In the Physics toolbar, click  Domains and choose Initial Values.
2
In the Settings window for Initial Values, locate the Domain Selection section.
3
From the Selection list, choose All domains.
4
Locate the Initial Values section. In the cCu text field, type cCu0.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Size 1
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size 1.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 4, 6, 10, 11, 13, and 14 only.
5
Locate the Element Size section. From the Predefined list, choose Extremely fine.
Free Triangular 1
1
In the Model Builder window, click Free Triangular 1.
2
In the Settings window for Free Triangular, click to expand the Scale Geometry section.
3
In the y-direction scale text field, type 50.
4
Click  Build All.
Study 1
Use a parametric sweep to study the copper dissolution for two geometrical shapes of dents.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
Click  Add.
5
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
w_dent (Width of dent in negative electrode coating)
0.5[mm] 7.5[mm]
m
depth_dent (Dent depth in negative active material)
50[um] 20[um]/6
m
The cuprous ion concentration needs to be zero in Step 1. The second step runs smoother with nonzero concentration. Use the Modify model configuration support to achieve this.
Step 1: Current Distribution Initialization
1
In the Model Builder window, click Step 1: Current Distribution Initialization.
2
In the Settings window for Current Distribution Initialization, locate the Study Settings section.
3
From the Current distribution type list, choose Secondary.
4
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
5
In the tree, select Component 1 (comp1) > Transport of Diluted Species (tds) > Initial Values 2.
6
Right-click and choose Disable.
Step 2: Time Dependent
1
In the Model Builder window, 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 4000.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, under Study 1 > Solver Configurations > Solution 1 (sol1) click Time-Dependent Solver 1.
4
In the Settings window for Time-Dependent Solver, locate the General section.
5
From the Times to store list, choose Steps taken by solver.
6
In the Store every Nth step text field, type 10.
The following solver settings improve and speed up the computations.
7
Right-click Study 1 > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 and choose Fully Coupled.
8
In the Settings window for Fully Coupled, click to expand the Method and Termination section.
9
From the Nonlinear method list, choose Automatic (Newton).
Define a stop condition that terminates the computations at the lower cutoff cell voltage.
10
Right-click Time-Dependent Solver 1 and choose Stop Condition.
11
In the Settings window for Stop Condition, locate the Stop Expressions section.
12
Click  Add.
13
In the table, enter the following settings:
Stop expression
Stop if
Active
Description
-comp1.liion.phis0_ec1>-V_low
True (>=1)
√
Stop expression 1
14
Locate the Output at Stop section. From the Add solution list, choose Steps before and after stop.
15
Clear the Add information checkbox.
16
In the Home toolbar, click  Compute.
Results
Cell Voltage and Concentration
The following steps reproduce the figures found in the documentation and fix the plot groups under the Results node.
1
In the Settings window for 1D Plot Group, type Cell Voltage and Concentration in the Label text field.
2
Click to expand the Title section. From the Title type list, choose None.
3
Locate the Plot Settings section.
4
Select the x-axis label checkbox. In the associated text field, type Discharge time (s).
5
Select the y-axis label checkbox. In the associated text field, type Cell voltage (V).
6
Select the Two y-axes checkbox.
7
Select the Secondary y-axis label checkbox. In the associated text field, type Cu<sup>+</sup> concentration in cell (mol/m<sup>3</sup>).
8
Locate the Axis section. Select the Manual axis limits checkbox.
9
In the x minimum text field, type 3500.
10
In the x maximum text field, type 3750.
11
In the y minimum text field, type -0.25.
12
In the y maximum text field, type 2.
13
In the Secondary y minimum text field, type -10.
14
In the Secondary y maximum text field, type 70.
15
Locate the Legend section. Select the Show legends checkbox.
16
From the Position list, choose Middle right.
Global 1
1
In the Model Builder window, expand the Cell Voltage and Concentration node, then click Global 1.
2
In the Settings window for Global, click to expand the Legends section.
3
From the Legends list, choose Manual.
4
In the table, enter the following settings:
Legends
Deep
Shallow
Global 2
1
Right-click Results > Cell Voltage and Concentration > Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
cCu_cell
mol/m^3
Cuprous ion concentration in cell
4
Locate the y-Axis section. Select the Plot on secondary y-axis checkbox.
5
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
6
From the Color list, choose Cycle (reset).
7
In the Cell Voltage and Concentration toolbar, click  Plot.
Cuprous Ion Concentration
1
In the Model Builder window, under Results click Concentration (tds).
2
In the Settings window for 2D Plot Group, type Cuprous Ion Concentration in the Label text field.
3
Click to expand the Title section. Locate the Data section. From the Dataset list, choose None.
4
Locate the Title section. From the Title type list, choose Manual.
5
In the Title text area, type Cu<sup>+</sup> concentration (mol/m<sup>3</sup>).
6
Clear the Parameter indicator text field.
7
Locate the Plot Settings section. From the View list, choose View 1.
8
Click to expand the Plot Array section. From the Array type list, choose Linear.
Arrow Surface 1
1
In the Model Builder window, expand the Cuprous Ion Concentration node.
2
Right-click Arrow Surface 1 and choose Delete.
Surface 1
1
In the Settings window for Surface, locate the Data section.
2
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol3).
3
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 1: w_dent=5E-4 m, depth_dent=5E-5 m.
4
Locate the Coloring and Style section. From the Color table list, choose Lichen.
Line 1
1
In the Model Builder window, right-click Cuprous Ion Concentration and choose Line.
2
In the Settings window for Line, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol3).
4
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 1: w_dent=5E-4 m, depth_dent=5E-5 m.
5
Locate the Expression section. In the Expression text field, type 1.
6
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
7
From the Color list, choose Black.
8
Click to expand the Plot Array section. Select the Manual indexing checkbox.
Selection 1
1
Right-click Line 1 and choose Selection.
2
Select Boundaries 3–11, 13, 14, and 16–18 only.
Annotation 1
1
In the Model Builder window, right-click Cuprous Ion Concentration and choose Annotation.
2
In the Settings window for Annotation, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol3).
4
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 1: w_dent=5E-4 m, depth_dent=5E-5 m.
5
Locate the Annotation section. In the Text text field, type Deep.
6
Locate the Coloring and Style section. Clear the Show point checkbox.
7
From the Anchor point list, choose Center.
8
Click to expand the Plot Array section. Select the Manual indexing checkbox.
Annotation 1, Line 1, Surface 1
1
In the Model Builder window, under Results > Cuprous Ion Concentration, Ctrl-click to select Surface 1, Line 1, and Annotation 1.
2
Right-click and choose Duplicate.
Surface 2
1
In the Settings window for Surface, locate the Data section.
2
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 2: w_dent=0.0075 m, depth_dent=3.3333E-6 m.
3
Click to expand the Range section. Locate the Coloring and Style section. Clear the Color legend checkbox.
4
Click to expand the Inherit Style section. From the Plot list, choose Surface 1.
5
Click to expand the Plot Array section. Select the Manual indexing checkbox.
6
In the Index text field, type 1.
Line 2
1
In the Model Builder window, click Line 2.
2
In the Settings window for Line, locate the Data section.
3
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 2: w_dent=0.0075 m, depth_dent=3.3333E-6 m.
4
Locate the Plot Array section. In the Index text field, type 1.
Annotation 2
1
In the Model Builder window, click Annotation 2.
2
In the Settings window for Annotation, locate the Data section.
3
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 2: w_dent=0.0075 m, depth_dent=3.3333E-6 m.
4
Locate the Annotation section. In the Text text field, type Shallow.
5
Locate the Plot Array section. In the Index text field, type 1.
Deposited Copper Concentration
1
In the Model Builder window, right-click Cuprous Ion Concentration and choose Duplicate.
2
In the Model Builder window, click Cuprous Ion Concentration 1.
3
In the Settings window for 2D Plot Group, type Deposited Copper Concentration in the Label text field.
4
Locate the Title section. In the Title text area, type Deposited copper concentration (mol/m<sup>3</sup>).
Surface 1
1
In the Model Builder window, click Surface 1.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > Dissolving–depositing species > liion.c_pce1_CuMetal - Dissolving–depositing species concentration - mol/m³.
3
Locate the Coloring and Style section. From the Color table list, choose Caissara.
Surface 2
1
In the Model Builder window, click Surface 2.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > Dissolving–depositing species > liion.c_pce1_CuMetal - Dissolving–depositing species concentration - mol/m³.
3
In the Deposited Copper Concentration toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.
Electrode Potentials
1
In the Model Builder window, right-click Deposited Copper Concentration and choose Duplicate.
2
In the Settings window for 2D Plot Group, type Electrode Potentials in the Label text field.
3
Locate the Title section. In the Title text area, type Surface: Electrode potential (V) Arrow Surface: Electrode current density vector.
Surface 1
1
In the Model Builder window, expand the Electrode Potentials node, then click Surface 1.
2
In the Settings window for Surface, locate the Data section.
3
From the Time (s) list, choose Interpolation.
4
In the Time text field, type 3604.7.
5
Locate the Expression section. In the Expression text field, type phis-phil.
6
Locate the Coloring and Style section. From the Color table list, choose MetasepiaBlue.
Surface 2
1
In the Model Builder window, click Surface 2.
2
In the Settings window for Surface, locate the Data section.
3
From the Time (s) list, choose Interpolation.
4
In the Time text field, type 3604.7.
5
Locate the Expression section. In the Expression text field, type phis-phil.
Arrow Surface 1
1
In the Model Builder window, right-click Electrode Potentials and choose Arrow Surface.
2
In the Settings window for Arrow Surface, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol3).
4
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 1: w_dent=5E-4 m, depth_dent=5E-5 m.
5
From the Time (s) list, choose Interpolation.
6
In the Time text field, type 3604.7.
7
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > liion.Isx,liion.Isy - Electrode current density vector.
8
Locate the Coloring and Style section.
9
Select the Scale factor checkbox. In the associated text field, type 5e-5.
10
From the Color list, choose Black.
11
Click to expand the Plot Array section. Select the Manual indexing checkbox.
Arrow Surface 2
1
Right-click Arrow Surface 1 and choose Duplicate.
2
In the Settings window for Arrow Surface, locate the Data section.
3
From the Parameter value (w_dent (m),depth_dent (m)) list, choose 2: w_dent=0.0075 m, depth_dent=3.3333E-6 m.
4
Locate the Plot Array section. In the Index text field, type 1.
5
In the Electrode Potentials toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Surface 1
1
In the Model Builder window, under Results > Cuprous Ion Concentration click Surface 1.
2
In the Settings window for Surface, locate the Data section.
3
From the Time (s) list, choose Interpolation.
4
In the Time text field, type 3604.7.
Surface 2
1
In the Model Builder window, click Surface 2.
2
In the Settings window for Surface, locate the Data section.
3
From the Time (s) list, choose Interpolation.
4
In the Time text field, type 3604.7.
5
In the Cuprous Ion Concentration toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Electrolyte Salt Concentration
1
In the Model Builder window, right-click Electrode Potentials and choose Duplicate.
2
In the Model Builder window, click Electrode Potentials 1.
3
In the Settings window for 2D Plot Group, type Electrolyte Salt Concentration in the Label text field.
4
Locate the Title section. In the Title text area, type Surface: c<sub>l</sub> concentration (mol/m<sup>3</sup>) Arrow Surface: Electrolyte current density vector.
Surface 1
1
In the Model Builder window, click Surface 1.
2
In the Settings window for Surface, locate the Data section.
3
From the Time (s) list, choose Last (3749.3).
4
Locate the Expression section. In the Expression text field, type cl-cCu.
5
Locate the Coloring and Style section. From the Color table list, choose Prionace.
Surface 2
1
In the Model Builder window, click Surface 2.
2
In the Settings window for Surface, locate the Data section.
3
From the Time (s) list, choose Last (3726).
4
Locate the Expression section. In the Expression text field, type cl-cCu.
Arrow Surface 1
1
In the Model Builder window, click Arrow Surface 1.
2
In the Settings window for Arrow Surface, locate the Data section.
3
From the Time (s) list, choose Last (3749.3).
4
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > liion.Ilx,liion.Ily - Electrolyte current density vector.
5
Locate the Coloring and Style section. From the Color list, choose Cyan.
Arrow Surface 2
1
In the Model Builder window, click Arrow Surface 2.
2
In the Settings window for Arrow Surface, locate the Data section.
3
From the Time (s) list, choose Last (3726).
4
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > liion.Ilx,liion.Ily - Electrolyte current density vector.
5
Locate the Coloring and Style section. From the Color list, choose Cyan.
Current Collector Dissolution Current
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Current Collector Dissolution Current in the Label text field.
3
Locate the Data section. From the Dataset list, choose None.
4
Locate the Title section. From the Title type list, choose None.
Line Graph 1
1
Right-click Current Collector Dissolution Current and choose Line Graph.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol3).
4
From the Parameter selection (w_dent, depth_dent) list, choose First.
5
From the Time selection list, choose Interpolated.
6
In the Times (s) text field, type 3604.7.
7
Select Boundary 4 only.
8
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > Electrode kinetics > liion.iloc_er1 - Local current density - A/m².
9
Click to expand the Legends section. Select the Show legends checkbox.
10
From the Legends list, choose Manual.
11
In the table, enter the following settings:
Legends
Deep, local E<sub>neg</sub> >3.5 V
Line Graph 2
1
Right-click Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Time selection list, choose Last.
4
Locate the Legends section. In the table, enter the following settings:
Legends
Deep, E<sub>cell</sub>=-0.05 V
Line Graph 1, Line Graph 2
1
In the Model Builder window, under Results > Current Collector Dissolution Current, Ctrl-click to select Line Graph 1 and Line Graph 2.
2
Right-click and choose Duplicate.
Line Graph 3
1
In the Settings window for Line Graph, locate the Data section.
2
From the Parameter selection (w_dent, depth_dent) list, choose Last.
3
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
4
Locate the Legends section. In the table, enter the following settings:
Legends
Shallow, local E<sub>neg</sub> >3.5 V
Line Graph 4
1
In the Model Builder window, click Line Graph 4.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Parameter selection (w_dent, depth_dent) list, choose Last.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
5
Locate the Legends section. In the table, enter the following settings:
Legends
Shallow, E<sub>cell</sub>=-0.05 V
6
In the Current Collector Dissolution Current toolbar, click  Plot.
Finally, remove redundant plots.
Average Electrode State of Charge (liion), Electrode Potential with Respect to Adjacent Reference (liion), Electrode Potential with Respect to Ground (liion), Electrolyte Salt Concentration (liion), Particle Surface State of Charge (liion), Separator Current Density Magnitude (liion)
1
In the Model Builder window, under Results, Ctrl-click to select Average Electrode State of Charge (liion), Electrode Potential with Respect to Ground (liion), Electrode Potential with Respect to Adjacent Reference (liion), Electrolyte Salt Concentration (liion), Separator Current Density Magnitude (liion), and Particle Surface State of Charge (liion).
2
Right-click and choose Delete.
Cuprous Ion Concentration
Click the  Zoom Extents button in the Graphics toolbar.