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Heterogeneous Lithium-Ion Battery
Introduction
Most lithium-ion battery models make use of a homogenized domain formulation of the porous electrodes, which solve simultaneously for the electrode phase and electrolyte phase potentials in the same domain, defining the electrode reactions by the use of source terms. In these models, the diffusion of lithium into the solid electrode particles is modeled by the use of an extra dimension, representing an average particle for a certain position in the electrode. This modeling approach has great advantages in terms of a relatively small computational load, allowing most models to be formulated in one dimension only, representing the electrode depth (plus the extra dimension for defining the particle diffusion dimension).
However, certain phenomena cannot be captured using the above approach. For instance, the above particle diffusion model inherently assumes either Cartesian, cylindrical, or spherical symmetry, thus not allowing modeling the impact of irregular particle shapes, nor the impact of micro- and macropore distributions. Instead of homogenizing the porous electrode, you can include the structural details of the porous electrodes in the model geometry. Such models are referred to as heterogeneous models. For an introduction to heterogeneous electrode modeling and homogenization, please check out the Heterogeneous NMC Electrode and Homogenizing a Heterogeneous Electrode Model examples.
This tutorial focuses on deformation and changed porosity effects when the strain induced by the lithium concentration distribution in the particles is coupled to a corresponding volumetric expansion in the Solid Mechanics interface.
In the first part of the tutorial, a unit cell model of a lithium ion-battery is set up, excluding deformation effects. The model includes electrolyte ion transport, electrode kinetics and diffusion of lithium atoms in the solid electrode particles, where the negative electrode consists of a blend of graphite and silicon. The second part of the tutorial explores the impact on the charging voltage characteristics when introducing structural mechanics to the model, including fully-coupled geometry deformation due to lithium intercalation strain. Also the local electrolyte volume fraction in the porous conductive binder and separator are affected by the local strain computed by the solid mechanics model.
Due to the usage of the Transport of Solids interface and Hyperelastic material model, the tutorial requires a Structural Mechanics Module license to run.
Model Definition
The model geometry is shown in Figure 1. The geometry consists of a rectangular block, forming a representative unit cell of the model geometry. The two electrodes are defined using a number of quarter spheres. In the negative electrode, the smaller spheres represent silicon, and the larger spheres graphite. On the positive side, all spheres are assumed to be made of a Nickel–Manganese–Cobalt (NMC) alloy as active material. A porous conductive binder material, containing both an electron and electrolyte conducting phase is assumed to surround the active particles in the electrodes.
Figure 1: Model geometry. The negative graphite electrode is located toward the lower left, wherein the smaller quarter spheres define the silicon particles, and the larger quarter spheres the graphite particles, respectively. The positive electrode, containing NMC particles, is located toward the upper right.
The battery current distribution is modeled using a Lithium-Ion Battery interface using the Separator and Porous Conductive Binder nodes to define the concentrated battery electrolyte charge and ion transport as well as the electron conduction in the carbon filler material. Lithium diffusion in the solid electrode particles is defined using a Transport in Solids interface.
The charge transfer reactions occurring at the particle surfaces are defined using an Internal Electrode Surface node in the Lithium-Ion Battery interface. The concentration of solid lithium, solved for by the Transport in Solids interface, is coupled to the Lithium Insertion Reaction electrode Butler–Volmer kinetics, defined in the Electrode Reaction subnodes to the Internal Electrode Surface nodes.
As the lithium concentration increases in the negative graphite electrode material, the material expands. This is modeled in the second part of the tutorial using a Solid Mechanics interface, where the expansion is defined using an Intercalation Strain node (subnode to the Linear-Elastic Material node). The strain is defined as a function of the local solid lithium concentration using an interpolation function, based on experimental data (Ref. 1).
Silicon expands up to 300% upon full lithiation. As a result of these large deforrmations, a linear material model has limited validity in the surrounding binder around the silicon particles. For this reason, a Hyperelastic material model for the binder in the negative electrode is used.
Frame of Reference and the solid lithium concentration
In COMSOL Multiphysics, the partial differential equations of physics interfaces are usually formulated either in a Spatial frame (coordinate system), with coordinate axes fixed in space, or in a Material frame, fixed to the material in its reference configuration, following the material as it deforms. (The former is often referred to as an Eulerian formulation, while the latter is a Lagrangian formulation.)
Most equations and variables in the battery interfaces are defined with reference to the spatial frame (except the particle intercalation concentration in porous electrode nodes). In most battery models however, the spatial and material frames are identical, and no consideration with regards to frame handling needs to be taken.
In this tutorial however, due to the large deformation of the silicon particles, we induce a split between the material and spatial frame (this is performed by having Include Geometric Nonlinearity enabled in the time-dependent solver, which deforms the mesh and spatial frame according to the deformations computed by the solid mechanics interface). As a result of this, certain care needs to be taken when considering the concentration values computed by the mass balance equations in the solid particles.
The equilibrium potential functions used for defining the electrode reactions in the Battery Design material library are defined as functions of concentrations in the reference Material frame, that is, for a control volume excluding material strains induced by lithium intercalation. This means that any concentration value we use for evaluation of the local electrode equilibrium potential needs to be defined with reference to the material frame.
One option to do this is to use the Solid node in the Transport of Diluted Species interface (which solves for the material balance on the spatial frame), and then use one of the built-in transformation variables (such as spatial.detInvF) for converting the spatial concentration variables to the corresponding values for the material frame. In this tutorial we however choose to use the Transport of Solids interface, since this physics interface defines its material balance directly on the material frame, resulting in a reduced number of numerical operations.
To read more about frame handling, check out the Deformed Mesh Fundamentals section in the COMSOL Multiphysics Reference Manual.
Changed Electrolyte Volume Fractions due to Intercalation Strain
In part two of the tutorial, the local volumetric strain ε is computed by the solid mechanics model. In the porous conductive binder and separator domains, which are blended materials, comprising both a solid (conducting or nonconducting) and an electrolyte phase, it is assumed that all volumetric strain is accommodated by a changed porosity, treating the solid part of the blend as incompressible.
The electrolyte volume fraction (porosity) is hence defined as
(1)
where εl,ref and εs,ref are the nondeformed electrolyte and solid volume fractions, respectively, fulfilling the relation εs,ref + εl,ref = 1.
In continuation, effective transport parameters for electrolyte conductivity and diffusivity of the electrolyte salt are defined as
(2)
and
(3)
where σl and Dl are the bulk conductivity and diffusivity values, respectively.
Studies
Two separate studies, both simulating a 1 h charge from 0 to 100% state of charge, are used in the tutorial. The first study excludes structural mechanic effects, whereas the second includes fully two-way solid mechanics-electrochemistry couplings. Automatic remeshing is enabled in the second study in order avoid inverted mesh elements due to deformation of the computational mesh.
Results and Discussion
Figure 2 shows the relative lithiation levels in the particles after 6 min for the fully coupled model. All particles exhibit solid concentration gradients in the radial direction. This early into the charge cycle, the silicon particles have already been lithiated to almost 60%, whereas the graphite has only been lithiated to levels below 10%. This indicates that the silicon material in the electrode will mainly be utilized when operating the battery at low state-of-charge levels. (For a deeper discussion on the behavior of blended graphite-silicon electrodes, see the Silicon–Graphite-Blended Electrode with Thermodynamic Voltage Hysteresis example).
Figure 2: Relative lithiation levels in the electrode particles at t = 6 min.
Figure 3 shows the same plot at the end of the 1 h charge, with added black edges to indicate the original, nondeformed, geometry. Due to the intercalation strain, the negative electrode has expanded somewhat.
Figure 3: Relative lithiation levels in the electrode particles at t = 1 h. The black edges indicate the shape of the original (nondeformed) geometry.
Figure 4 compares the charging voltage curves for the two models. The curves are nearly identical, indicating that including the two-way coupling to solid mechanics has limited impact on the electrochemical model.
Figure 4: Charging voltage for the two models.
Finally, the average outward normal stress on the positive current collector boundary is plotted in Figure 5. The dominating features of the plot are governed by the intercalation strain function of graphite.
Figure 5: Average outward normal stress on the positive current collector boundary.
Reference
1. J.B. Siegel, A.G. Stefanopoulou, P. Hagans, Y. Ding, and D. Gorsich, “Expansion of Lithium Ion Pouch Cell Batteries: Observations from Neutron Imaging,” J. Electrochemical Soc., vol. 160, p. A1031, 2013.
Application Library path: Battery_Design_Module/Heterogeneous_Models/heterogeneous_lib
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  3D.
2
In the Select Physics tree, select Chemical Species Transport > Transport in Solids (ts).
3
Click Add.
4
In the Select Physics tree, select Electrochemistry > Batteries > Lithium-Ion Battery (liion).
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.
Geometry 1
1
In the Geometry toolbar, click Insert Sequence and choose Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file heterogeneous_lib_geom_sequence.mph.
3
In the Insert Sequence dialog, select Geometry 1 in the Select geometry sequence to insert list.
4
Click OK.
5
In the Geometry toolbar, click  Build All.
6
Click the  Transparency button in the Graphics toolbar.
7
Click the  Show Axis Orientation button in the Graphics toolbar.
8
Click the  Zoom Extents button in the Graphics toolbar.
9
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
10
In the Model Builder window, collapse the Geometry 1 node.
Global Definitions
Geometry Parameters
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Geometry Parameters in the Label text field.
Physics Parameters
1
In the Home toolbar, click  Parameters and choose Add > Parameters.
2
In the Settings window for Parameters, type Physics Parameters in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file heterogeneous_lib_physics_parameters.txt.
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 > Electrodes > Graphite, LixC6 MCMB (Negative, Li-ion Battery).
4
Right-click and choose Add to Component 1 (comp1).
5
In the tree, select Battery > Electrodes > Silicon, LixSi (Negative, Li-ion Battery).
6
Right-click and choose Add to Component 1 (comp1).
7
In the tree, select Battery > Electrodes > NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery).
8
Right-click and choose Add to Component 1 (comp1).
9
In the tree, select Battery > Electrolytes > LiPF6 in 3:7 EC:EMC (Liquid, Li-ion Battery).
10
Right-click and choose Add to Component 1 (comp1).
11
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat1)
Set the correct domain selection of the graphite material that you just added.
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Graphite Particles.
3
In the Model Builder window, expand the Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat1) node.
The electrode material nodes define the material strain as a function of the lithium intercalation level. You can view the strain function for graphite as follows:
Interpolation 1 (dVOLdSOL)
1
In the Model Builder window, expand the Component 1 (comp1) > Materials > Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat1) > Intercalation strain (is) node, then click Interpolation 1 (dVOLdSOL).
2
In the Settings window for Interpolation, click  Plot.
Materials
Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat1)
In the Model Builder window, collapse the Component 1 (comp1) > Materials > Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat1) node.
Silicon, LixSi (Negative, Li-ion Battery) (mat2)
1
In the Model Builder window, click Silicon, LixSi (Negative, Li-ion Battery) (mat2).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Silicon Particles.
NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery) (mat3)
1
In the Model Builder window, click NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery) (mat3).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose NMC Particles.
LiPF6 in 3:7 EC:EMC (Liquid, Li-ion Battery) (mat4)
1
In the Model Builder window, click LiPF6 in 3:7 EC:EMC (Liquid, Li-ion Battery) (mat4).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Electrolyte Domains.
Duplicate the electrode materials and change the selections of the duplicates to make the material properties available also at the particle surfaces. These properties will be used when defining the electrode reactions.
Graphite, LixC6 MCMB (Negative, Li-ion Battery) 1 (mat5)
1
In the Model Builder window, under Component 1 (comp1) > Materials right-click Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat1) and choose Duplicate.
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Graphite Electrode Particle Surfaces.
Silicon, LixSi (Negative, Li-ion Battery) 1 (mat6)
1
In the Model Builder window, under Component 1 (comp1) > Materials right-click Silicon, LixSi (Negative, Li-ion Battery) (mat2) and choose Duplicate.
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Si Particle Surfaces.
NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery) 1 (mat7)
1
In the Model Builder window, under Component 1 (comp1) > Materials right-click NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery) (mat3) and choose Duplicate.
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose NMC Electrode Particle Surfaces.
Lithium-Ion Battery (liion)
1
In the Model Builder window, under Component 1 (comp1) click Lithium-Ion Battery (liion).
2
In the Settings window for Lithium-Ion Battery, locate the Domain Selection section.
3
From the Selection list, choose Electrolyte Domains.
Lower the discretization order of the lithium-ion battery interface. This will reduce the computational load.
4
Click to expand the Discretization section. From the Electrolyte potential list, choose Linear.
5
From the Electrolyte salt concentration list, choose Linear.
6
From the Electric potential list, choose Linear.
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_ref_sep.
Porous Conductive Binder 1
1
In the Physics toolbar, click  Domains and choose Porous Conductive Binder.
2
In the Settings window for Porous Conductive Binder, locate the Domain Selection section.
3
From the Selection list, choose Porous Conductive Binder.
4
Locate the Conductive Binder Properties section. In the σs text field, type sigmas_pb.
5
Locate the Porous Matrix Properties section. In the εs text field, type 1-epsl_ref_pb.
6
In the εl text field, type epsl_ref_pb.
7
Locate the Effective Transport Parameter Correction section. From the Electric conductivity list, choose No correction.
Internal Electrode Surface 1
1
In the Physics toolbar, click  Boundaries and choose Internal Electrode Surface.
2
In the Settings window for Internal Electrode Surface, locate the Boundary Selection section.
3
From the Selection list, choose Particle Surfaces.
Electrode Reaction 1
1
In the Model Builder window, click Electrode Reaction 1.
2
In the Settings window for Electrode Reaction, locate the Model Input section.
3
In the c text field, type c.
4
Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Lithium insertion.
5
In the i0,ref(T) text field, type i0_ref.
Electric Ground 1
1
In the Physics toolbar, click  Boundaries and choose Electric Ground.
2
In the Settings window for Electric Ground, locate the Boundary Selection section.
3
From the Selection list, choose Negative Current Collector.
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 Positive Current Collector.
4
Locate the Electrode Current section. From the list, choose Average current density.
5
In the is,average text field, type i_app.
Transport in Solids (ts)
1
In the Model Builder window, under Component 1 (comp1) click Transport in Solids (ts).
2
In the Settings window for Transport in Solids, locate the Domain Selection section.
3
From the Selection list, choose Particles.
Solid - Graphite
1
In the Model Builder window, under Component 1 (comp1) > Transport in Solids (ts) click Solid 1.
2
In the Settings window for Solid, type Solid - Graphite in the Label text field.
3
Locate the Diffusion section. From the Material list, choose Graphite, LixC6 MCMB (Negative, Li-ion Battery) (mat1).
4
From the Dc list, choose Basic (def).
Solid - Silicon
1
In the Physics toolbar, click  Domains and choose Solid.
2
In the Settings window for Solid, type Solid - Silicon in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Silicon Particles.
4
Locate the Diffusion section. From the Material list, choose Silicon, LixSi (Negative, Li-ion Battery) (mat2).
5
From the Dc list, choose Basic (def).
Solid - NMC
1
In the Physics toolbar, click  Domains and choose Solid.
2
In the Settings window for Solid, type Solid - NMC in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose NMC Particles.
4
Locate the Diffusion section. From the Material list, choose NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery) (mat3).
5
From the Dc list, choose Basic (def).
Electrode Surface Coupling 1
1
In the Physics toolbar, click  Boundaries and choose Electrode Surface Coupling.
2
In the Settings window for Electrode Surface Coupling, locate the Boundary Selection section.
3
From the Selection list, choose Particle Surfaces.
4
Locate the Reaction section. From the iloc list, choose Local current density, Electrode Reaction 1 (liion/bei1/er1).
5
In the n text field, type 1.
6
In the νc text field, type 1.
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 Graphite Particles.
4
Locate the Initial Values section. In the c text field, type cs0_Gr.
Initial Values 3
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 Silicon Particles.
4
Locate the Initial Values section. In the c text field, type cs0_Si.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the c text field, type cs0_pos.
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.
4
In the Mesh toolbar, click  Clear Sequence.
Free Tetrahedral 1
In the Mesh toolbar, click  Free Tetrahedral.
Size 1
1
Right-click Free Tetrahedral 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Silicon Particles.
5
Locate the Element Size section. From the Predefined list, choose Extra fine.
Size
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type s_unit_cell/3.
5
In the Minimum element size text field, type s_gap/4.
6
In the Maximum element growth rate text field, type 1.3.
7
In the Resolution of narrow regions text field, type 2.5.
8
In the Model Builder window, right-click Mesh 1 and choose Build All.
9
Click the  Transparency button in the Graphics toolbar.
10
Click the  Show Grid button in the Graphics toolbar.
Definitions
Global Variable Probe 1 (var1)
1
In the Definitions toolbar, click  Probes and choose Global Variable Probe.
2
In the Settings window for Global Variable Probe, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > liion.phis0_ec1 - Electric potential on boundary - V.
3
Locate the Expression section.
4
Select the Description checkbox. In the associated text field, type Cell voltage.
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
From the Time unit list, choose h.
4
In the Model Builder window, click Study 1.
5
In the Settings window for Study, locate the Study Settings section.
6
Clear the Generate default plots checkbox.
7
In the Study toolbar, click  Compute.
Results
Cell Voltage vs. Time
1
In the Model Builder window, under Results click Probe Plot Group 1.
2
In the Settings window for 1D Plot Group, type Cell Voltage vs. Time in the Label text field.
3
Locate the Legend section. Clear the Show legends checkbox.
Lithiation Levels and Lithium Flux
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Lithiation Levels and Lithium Flux in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
5
Locate the Color Legend section. Select the Show titles checkbox.
6
From the Position list, choose Bottom.
Volume 1
1
Right-click Lithiation Levels and Lithium Flux and choose Volume.
2
In the Settings window for Volume, locate the Expression section.
3
In the Expression text field, type c/csmax_Gr.
4
Locate the Coloring and Style section. In the Color legend title text field, type Gr.
Selection 1
1
Right-click Volume 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
From the Selection list, choose Graphite Particles.
Volume 2
1
In the Model Builder window, under Results > Lithiation Levels and Lithium Flux right-click Volume 1 and choose Duplicate.
2
In the Settings window for Volume, locate the Expression section.
3
In the Expression text field, type c/csmax_Si.
4
Locate the Coloring and Style section. From the Color table list, choose Prism.
5
In the Color legend title text field, type Si.
Selection 1
1
In the Model Builder window, expand the Volume 2 node, then click Selection 1.
2
In the Settings window for Selection, locate the Selection section.
3
From the Selection list, choose Silicon Particles.
Volume 3
1
In the Model Builder window, under Results > Lithiation Levels and Lithium Flux right-click Volume 2 and choose Duplicate.
2
In the Model Builder window, click Volume 3.
3
In the Settings window for Volume, locate the Expression section.
4
In the Expression text field, type c/csmax_pos.
5
Locate the Coloring and Style section. From the Color table list, choose AuroraBorealis.
6
In the Color legend title text field, type NMC.
Selection 1
1
In the Model Builder window, click Selection 1.
2
In the Settings window for Selection, locate the Selection section.
3
From the Selection list, choose NMC Particles.
4
In the Lithiation Levels and Lithium Flux toolbar, click  Plot.
Streamline 1
1
In the Model Builder window, right-click Lithiation Levels and Lithium Flux and choose Streamline.
2
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Lithium-Ion Battery > liion.Nposx,...,liion.Nposz - Positive ion flux.
3
Locate the Streamline Positioning section. In the Number text field, type 4.
4
Select Boundary 59 only.
5
Locate the Coloring and Style section. Find the Point style subsection. From the Type list, choose Arrow.
6
From the Arrow distribution list, choose Equal inverse time.
7
From the Color list, choose White.
Surface 1
1
Right-click Lithiation Levels and Lithium Flux and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type 1.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose Gray.
Selection 1
1
Right-click Surface 1 and choose Selection.
2
Select Boundaries 1, 3, 5, 56, 58, 61, 63, and 84 only.
3
In the Lithiation Levels and Lithium Flux toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.
Lithiation Levels and Lithium Flux
1
In the Model Builder window, under Results click Lithiation Levels and Lithium Flux.
2
In the Lithiation Levels and Lithium Flux toolbar, click  Plot.
3
In the Settings window for 3D Plot Group, locate the Data section.
4
From the Time (h) list, choose 0.1.
Animation 1
1
In the Results toolbar, click  Animation and choose Player.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Lithiation Levels and Lithium Flux.
4
Locate the Frames section. From the Frame selection list, choose All.
5
Locate the Playing section. In the Display each frame for text field, type 0.5.
6
Click the  Play button in the Graphics toolbar.
Component 1 (comp1)
Now extend the model to include structural-mechanics effects due to expansion/contraction of the electrode particles upon lithium intercalation/deintercalation.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Structural Mechanics > Solid Mechanics (solid).
4
Click the Add to Component 1 button in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Solid Mechanics (solid)
Due to the slow intercalation processes, mechanical wave propagation can be ignored in the model; set the transient behavior to Quasistatic.
1
In the Settings window for Solid Mechanics, locate the Structural Transient Behavior section.
2
From the list, choose Quasistatic.
Linear Elastic Material 1
1
In the Model Builder window, under Component 1 (comp1) > Solid Mechanics (solid) click Linear Elastic Material 1.
2
In the Settings window for Linear Elastic Material, locate the Linear Elastic Material section.
3
From the E list, choose User defined. In the associated text field, type E_pb.
4
From the ν list, choose User defined. In the associated text field, type nu_pb.
5
From the ρ list, choose User defined. In the associated text field, type rho.
Linear Elastic Material 2
1
In the Physics toolbar, click  Domains and choose Linear Elastic Material.
2
In the Settings window for Linear Elastic Material, locate the Domain Selection section.
3
From the Selection list, choose Particles.
4
Locate the Model Input section. In the c text field, type c.
Intercalation Strain 1
1
In the Physics toolbar, click  Attributes and choose Intercalation Strain.
2
In the Settings window for Intercalation Strain, locate the Model Input section.
3
In the c text field, type c.
Linear Elastic Material 3
1
In the Physics toolbar, click  Domains and choose Linear Elastic Material.
2
In the Settings window for Linear Elastic Material, locate the Domain Selection section.
3
From the Selection list, choose Separator.
4
Locate the Linear Elastic Material section. From the E list, choose User defined. In the associated text field, type E_sep.
5
From the ν list, choose User defined. In the associated text field, type nu_sep.
6
From the ρ list, choose User defined. In the associated text field, type rho.
Hyperelastic Material 1
Due to the presence of silicon in the negative electrode, very large deformations (up to 300%) are expected, which, in turn, are not well described by a linear material model. Therefore, change the model to Hyperelastic in the negative electrode as follows:
1
In the Physics toolbar, click  Domains and choose Hyperelastic Material.
2
Select Domain 1 only.
3
In the Settings window for Hyperelastic Material, locate the Hyperelastic Material section.
4
From the Specify list, choose Young’s modulus and Poisson’s ratio.
5
From the E list, choose User defined. In the associated text field, type E_pb.
6
From the ν list, choose User defined. In the associated text field, type nu_pb.
7
From the ρ list, choose User defined. In the associated text field, type rho.
Roller 1
1
In the Physics toolbar, click  Boundaries and choose Roller.
2
In the Settings window for Roller, locate the Boundary Selection section.
3
From the Selection list, choose Symmetry boundaries.
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
In the Settings window for Fixed Constraint, locate the Boundary Selection section.
3
From the Selection list, choose Current Collectors.
Materials
Silicon, LixSi (Negative, Li-ion Battery) (mat2)
1
In the Model Builder window, under Component 1 (comp1) > Materials click Silicon, LixSi (Negative, Li-ion Battery) (mat2).
2
In the Settings window for Material, locate the Material Contents section.
3
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Young’s modulus
E
150[GPa]
Pa
Young’s modulus and Poisson’s ratio
Poisson’s ratio
nu
0.3
1
Young’s modulus and Poisson’s ratio
NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery) (mat3)
1
In the Model Builder window, click NMC 111, LiNi0.33Mn0.33Co0.33O2 (Positive, Li-ion Battery) (mat3).
2
In the Settings window for Material, locate the Material Contents section.
3
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Density
rho
1000
kg/m³
Basic
Definitions
As a result of the material strain, the porosity (that is, the electrolyte volume fraction) will no longer be constant in the separator and porous conductive binder domains. Add variables for the electrolyte volume fractions as functions of the strain as follows:
Variables - Separator
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, type Variables - Separator in the Label text field.
3
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Separator.
5
Locate the Variables section. In the table, enter the following settings:
Name
Expression
Unit
Description
eps_solid
(1-epsl_ref_sep)/(1+solid.evol)
 
Solid (nonelectrolyte) volume fraction in deformed geometry
epsl
max(1-eps_solid,0.01)
 
Electrolyte volume fraction in deformed geometry
Variables - Porous Conductive Binder
1
Right-click Variables - Separator and choose Duplicate.
2
In the Settings window for Variables, type Variables - Porous Conductive Binder in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Porous Conductive Binder.
4
Locate the Variables section. In the table, enter the following settings:
Name
Expression
Unit
Description
eps_solid
(1-epsl_ref_pb)/(1+solid.evol)
 
Solid (nonelectrolyte) volume fraction in deformed geometry
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.
Porous Conductive Binder 1
1
In the Model Builder window, click Porous Conductive Binder 1.
2
In the Settings window for Porous Conductive Binder, locate the Porous Matrix Properties section.
3
In the εs text field, type eps_solid.
4
In the εl text field, type epsl.
Add Study
The modified model is now ready for solving. Add a second study to solve for the new model, and modify the settings of the first study to solve for the nonstructural mechanics model as follows:
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 > Lithium-Ion Battery > Time Dependent with Initialization.
4
Right-click and choose Add Study.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Definitions
Global Variable Probe 2 (var2)
Create a new cell voltage probe for the second study. In this way, the probe table values of Study 1 will not be overwritten by Study 2.
1
In the Model Builder window, under Component 1 (comp1) > Definitions right-click Global Variable Probe 1 (var1) and choose Duplicate.
2
In the Settings window for Global Variable Probe, click to expand the Table and Window Settings section.
3
Click  Add Table.
Boundary Probe 1 (bnd1)
Also create a probe for the normal stress at the positive current collector to be used in the second study.
1
In the Definitions toolbar, click  Probes and choose Boundary Probe.
2
In the Settings window for Boundary Probe, locate the Source Selection section.
3
From the Selection list, choose Positive Current Collector.
4
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Solid Mechanics > Stress > solid.stn - Normal stress - N/m².
5
Locate the Expression section. In the Expression text field, type -solid.stn.
6
From the Table and plot unit list, choose MPa.
7
Click to expand the Table and Window Settings section. Click  Add Table.
8
Click  Add Plot Window.
Study 1 - Excluding Solid Mechanics
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1 - Excluding Solid Mechanics in the Label text field.
Step 2: Time Dependent
1
In the Model Builder window, under Study 1 - Excluding Solid Mechanics click Step 2: Time Dependent.
2
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Solid Mechanics (solid).
4
Click to expand the Results While Solving section. From the Probes list, choose Manual.
5
In the Probes list box, select Global Variable Probe 2 (var2).
6
Under Probes, click  Delete.
7
In the Probes list box, select Boundary Probe 1 (bnd1).
8
Under Probes, click  Delete.
Study 2 - Full Model
1
In the Model Builder window, click Study 2.
2
In the Settings window for Study, type Study 2 - Full Model in the Label text field.
1
In the Model Builder window, under Study 2 - Full Model click Step 2: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
From the Time unit list, choose h.
4
Locate the Results While Solving section. From the Probes list, choose Manual.
5
In the Probes list box, select Global Variable Probe 1 (var1).
6
Under Probes, click  Delete.
Note that by adding the Hyperelastic Material node under Solid Mechanics, the Include geometric nonlinearity checkbox is enabled by default, and cannot be cleared. Include geometric nonlinearity will displace the computational mesh according to the strain computed by the Solid Mechanics interface.
7
Click to expand the Study Extensions section. Enable automatic remeshing to make sure that the computational mesh does not get compromised as a result of the displacement.
8
Select the Automatic remeshing checkbox.
Solution 3 (sol3)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 3 (sol3) node.
Add a user-defined remeshing condition that performs a remeshing operation if the relative volume of an element shrinks by more than 50%.
3
In the Model Builder window, expand the Study 2 - Full Model > Solver Configurations > Solution 3 (sol3) > Time-Dependent Solver 1 node, then click Automatic Remeshing.
4
In the Settings window for Automatic Remeshing, locate the Condition for Remeshing section.
5
In the Mesh quality expression text field, type comp1.spatial.relVolMin.
6
In the Stop when mesh quality is below text field, type 0.5.
7
Locate the Remesh section. From the Consistent initialization list, choose On.
To increase the robustness of the solver when using Automatic remeshing, change the Jacobian update to Once per timestep. This improves convergence.
8
In the Model Builder window, expand the Study 2 - Full Model > Solver Configurations > Solution 3 (sol3) > Time-Dependent Solver 1 > Segregated 1 node, then click Solid Mechanics.
9
In the Settings window for Segregated Step, click to expand the Method and Termination section.
10
From the Jacobian update list, choose Once per time step.
11
In the Model Builder window, under Study 2 - Full Model > Solver Configurations > Solution 3 (sol3) > Time-Dependent Solver 1 > Segregated 1 click Transport in Solids.
12
In the Settings window for Segregated Step, locate the Method and Termination section.
13
From the Jacobian update list, choose Once per time step.
14
In the Model Builder window, click Study 2 - Full Model.
15
In the Settings window for Study, locate the Study Settings section.
16
Clear the Generate default plots checkbox.
17
In the Study toolbar, click  Compute.
Results
Cell Voltage vs. Time
1
In the Model Builder window, expand the Results > Cell Voltage vs. Time node, then click Cell Voltage vs. Time.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
Select the Show legends checkbox.
Probe Table Graph 1
1
In the Model Builder window, click Probe Table Graph 1.
2
In the Settings window for Table Graph, click to expand the Legends section.
3
From the Legends list, choose Manual.
4
In the table, enter the following settings:
Legends
Without deformation
Probe Table Graph 2
1
In the Model Builder window, click Probe Table Graph 2.
2
In the Settings window for Table Graph, locate the Legends section.
3
From the Legends list, choose Manual.
4
In the table, enter the following settings:
Legends
With deformation
Cell Voltage vs. Time
1
In the Model Builder window, click Cell Voltage vs. Time.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Lower right.
4
In the Cell Voltage vs. Time toolbar, click  Plot.
Study 2 - Full Model/Remeshed Solution 1 (sol5)
1
In the Model Builder window, expand the Results > Datasets node, then click Study 2 - Full Model/Remeshed Solution 1 (sol5).
2
In the Settings window for Solution, locate the Solution section.
3
From the Frame list, choose Spatial  (x, y, z).
Lithiation Levels and Lithium Flux
1
In the Model Builder window, under Results click Lithiation Levels and Lithium Flux.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 2 - Full Model/Remeshed Solution 1 (sol5).
4
From the Time (h) list, choose Interpolation.
5
In the Lithiation Levels and Lithium Flux toolbar, click  Plot.
6
From the Time (h) list, choose Last (1).
7
In the Lithiation Levels and Lithium Flux toolbar, click  Plot.
Enable the data set edges to compare the deformed geometry with the nondeformed geometry.
8
Locate the Plot Settings section. Select the Plot dataset edges checkbox.
9
In the Lithiation Levels and Lithium Flux toolbar, click  Plot.
10
Clear the Plot dataset edges checkbox.
Current Collector Normal Stress vs. Time
1
In the Model Builder window, under Results click Probe Plot Group 3.
2
In the Settings window for 1D Plot Group, type Current Collector Normal Stress vs. Time in the Label text field.
3
Locate the Plot Settings section.
4
Select the y-axis label checkbox. In the associated text field, type Average Normal stress (MPa).
5
Locate the Legend section. Clear the Show legends checkbox.
6
In the Current Collector Normal Stress vs. Time toolbar, click  Plot.
Animation 1
1
In the Model Builder window, under Results > Export click Animation 1.
2
In the Settings window for Animation, locate the Animation Editing section.
3
From the Time selection list, choose Interpolated.
4
In the Times (h) text field, type range(0,0.1,1).
5
Click the  Play button in the Graphics toolbar.