Protocol for modeling and simulating lithiation-induced stress in largely deformed spherical nanoparticles using COMSOL

Summary

Here, we present a finite element method-based scheme for solving coupled partial differential equations (PDEs) for the analysis of lithiation-induced stress in largely deformed spherical nanoparticles via the PDE module in COMSOL. We describe steps for software installation and setting PDEs, initial/boundary conditions, and mesh parameters. We then detail procedures for dividing the mesh and analyzing lithium trapping during electrochemical cycling. This protocol can also be extended to analyze a wide range of problems involving diffusion-induced stress.

For complete details on the use and execution of this protocol, please refer to Li et al.¹

Graphical abstract

Highlights

• •Protocol for analyzing lithiation-induced stress with COMSOL
• •Guidance on analyzing largely deformed electrode with non-linear continuum mechanics
• •Steps for numerically investigating lithium-trapping effects with COMSOL

Publisher’s note: Undertaking any experimental protocol requires adherence to local institutional guidelines for laboratory safety and ethics.

Here, we present a finite element method-based scheme for solving coupled partial differential equations (PDEs) for the analysis of lithiation-induced stress in largely deformed spherical nanoparticles via the PDE module in COMSOL. We describe steps for software installation and setting PDEs, initial/boundary conditions, and mesh parameters. We then detail procedures for dividing the mesh and lithium trapping during electrochemical cycling. This protocol can also be extended to analyze a wide range of problems involving diffusion-induced stress.

Before you begin

Using the PDE module in commercial software COMSOL makes it possible to solve the coupled partial differential equations involving chemo-mechanical interactions and likely helps researchers understand the stress evolution and mass transport during electrochemical cycling of metal-ion battery. Using such a methodology, we have analyzed the chemical stress in a largely deformed electrode² taking into account the contribution of defect evolution,³ free-volume evolution,⁴ anisotropic deformation,⁵ lithium-trapping effects,¹ and so forth. Among these analyses, it requires more computational skills to investigate the lithium-trapping effects with COMSOL.

This protocol illustrates the computational details involved in using the PDE module of COMSOL by taking the analysis of the lithium-trapping effects as a typical case. It provides detailed information needed for the settings of physical parameters, initial and boundary conditions in modeling one-dimensional-spherical problem, and the solver parameterization.

This section outlines the necessary preparations, including hardware setup, software installation, and the correspondence between physical parameters and COMSOL parameters.

Hardware condition and software installation

Timing: 10 min

• 1.We deploy the protocol on a laptop with OS of Windows 10 Professional Edition. The CPU of the laptop is 12^th Gen Intel Core i9-12900H 2.50 GHz.
• 2.We solve the coupled partial differential equations using the commercial software COMSOL Multiphysics, which we have installed on the laptop. You can download the software from the official website https://www.comsol.com/. This protocol uses version 5.5.

Correspondence between the variables in software and physical variables

Timing: 5 min

Before solving the coupled partial differential equations with the PDE module of COMSOL, we need to define the variables in COMSOL. Table 1 lists all the parameters needed in the numerical calculation, and Table 2 lists the correspondence between the variables used in the numerical modeling and the formulas presented in the work by Li et al.¹

Table 1.

Parameters used in numerical calculation

Property/Parameter	Symbol	Variables’s name in COMSOL	Value
Young’s Modulus of pristine Si	Eh	E	90 GPa6
Poisson’s ratio	ν	v	0.286
Volumetric expansion coefficient per mole of diffusive species	Ω1	Omega1	8.18 × 10−6 m3·mol−16
Volumetric change per mole for immobile atoms	$\mathrm{\varpi }$	w_Omega	Assumed to be the same as Ω1
Partial molar volume of the host material	Ω2	Omega2	Assumed to be the same as Ω1
Diffusion coefficient	D	D	1 × 10−16 m2·s−16
Maximum concentration	Cmax	rhomax	3.67 × 105 mol·m−36
Backward reaction rate constant	λ	lampda	0.05
Forward reaction rate constant	$k{C}_2^b$	fk	0.03
Initial radius	R0	R0	250 nm
Gas constant	Rg	Rg	8.3145 J⋅mol−1⋅K−1
Absolute temperature	T	T	298 K

Table 2.

Correspondence between the variables used in numerical modeling and the formulas presented in the work by Li et al.¹

Parameter	Symbol	Variables’s name in COMSOL
Charge parameter (1 for lithiation and 0 for de-lithiation)	—	charge_parameter
Radial component of total deformation gradient tensor	$F_R$	F11
Tangential component of total deformation gradient tensor	$F_{\mathrm{\Theta }}$	F22
Tangential component of total deformation gradient tensor	$F_{\mathrm{\Phi }}$	F33
Radial component of elastic deformation gradient tensor	$F_R^e$	F11e
Tangential component of elastic deformation gradient tensor	$F_{\mathrm{\Theta }}^e$	F22e
Tangential component of elastic deformation gradient tensor	$F_{\mathrm{\Phi }}^e$	F33e
Radial component of elastic Green Lagrangian strain tensor	$E_R^e$	E11e
Tangential component of elastic Green Lagrangian strain tensor	$E_{\mathrm{\Theta }}^e$	E22e
Tangential component of elastic Green Lagrangian strain tensor	$E_{\mathrm{\Phi }}^e$	E33e
Radial and tangential components of eigen-transformation deformation gradient tensor	$F_R^c$	Fc
Inelastic volumetric expansion ratio	$\det \left({\mathbf{F}}^c\right)$	Jc
Total flux	J	J
Strain energy density in current configuration	w	w
Radial component of the first Piola-Kirchhoff stress	$P{k}_R$	Pk11
Tangential component of the first Piola-Kirchhoff stress	$P{k}_{\mathrm{\Theta }}$	Pk22
Tangential component of the first Piola-Kirchhoff stress	$P{k}_{\mathrm{\Phi }}$	Pk33
Radial component of Cauchy stress tensor	${\sigma }_R$	sigma11
Tangential component of Cauchy stress tensor	${\sigma }_{\mathrm{\Theta }}$	sigma22
Tangential component of Cauchy stress tensor	${\sigma }_{\mathrm{\Phi }}$	sigma33
Cauchy hydrostatic stress	${\sigma }_h$	Sigmam
Radial component of displacement vector	u	u
Radial coordinate in Lagrangian configuration	R	x
Concentration of diffusive atoms (mobile atoms) in Lagrangian configuration	C	rhoc
Concentration of immobile solute atoms	S	rhos

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFER
Software and algorithms
COMSOL 5.5	COMSOL	https://www.comsol.com/
Windows 10	Microsoft	https://www.microsoft.com/en-us/software-download/windows10
Other
CPU laptop	Lenovo	12th Gen Intel Core i9-12900H 2.50 GHz

Materials and equipment

• •You can install COMSOL Multiphysics on computers with different operating systems, including Linux, MacOS, and Windows. Simply install the software of version 5.5 designed for the specific operating system. The mentioned hardware configuration determines the time spent in the following steps. However, the actual time required may vary, depending on the hardware conditions.
• •The software version used in Li et al.¹ is 3.5. Here, we extend the modeling analysis to the newer software version 5.5. Note that the PDE module in the 5.5 version is nearly the same as that in the 3.5 version. However, the software of 5.5 and newer versions has made some optimizations in the solver, and the method presented in this protocol is also applicable to COMSOL of newer versions. Note that the numerical results in this work are identical to the results given by Li et al.¹ As a typical example, we only present the numerical results for the concentration of immobile lithium.

Step-by-step method details

This section outlines the procedures for configuring the model and resolving the issues in COMSOL. Commencing with the establishment of Partial Differential Equations (PDEs), initial/boundary conditions, and mesh parameters for the mechanical equations, the subsequent steps involve configuring the PDEs, initial/boundary conditions, and mesh parameters for the diffusion equations. Subsequently, the electrochemical cycling is implemented by incorporating the de-lithiation process. Ultimately, a post-processing step is introduced to analyze the constant and nonuniform concentration of residual trapped lithium.

Major step one: Variable definitions

Timing: 60 min

• 1.
  Create a spherically symmetric model in the COMSOL PDE module.

  • a.
    Create a mechanical equation: Opening COMSOL Multiphysics→Model Wizard→1D→Mathematics→PDE Interfaces→Coefficient Form PDE (c) →Add. The default, “dependent variable” is u, which represents the Radial component of displacement vector, as listed in Table 2.
  • b.
    Create a mass transport equation: Coefficient Form PDE (c)→Add→First Dependent variables change to “rhoc”→Add Dependent Variable→Second Dependent variables change to “rhos”.

    Note: The dependent variables rhoc and rhos represent the concentrations of mobile atoms and immobile atoms in Lagrangian configuration, respectively, as listed in Table 2.

  • c.
    Click “Study” button→Time Dependent→done. The spherically symmetric model with a blank mechanical equation and mass transport in the COMSOL PDE module is established, as shown in Figure 1.
• 2.Create parameters used in the numerical calculation.
  In the spherically symmetric model with a blank mechanical equation and mass transport in the COMSOL PDE module→Global Definitions→Parameters table→create all the parameters used in the numerical calculation as follows:

  • a.
    Create charge parameter:

    • i.
      Name→charge_parameter;
    • ii.
      Expression: 1;
    • iii.
      Description: Charge parameter, 1 for charge and 0 for discharge.
  • b.
    Create diffusion coefficient:

    • i.
      Name→D;
    • ii.
      Expression:1e-16;
    • iii.
      Description: Diffusion Coefficient.
  • c.
    Create elastic modulus:

    • i.
      Name→E;
    • ii.
      Expression: 90e9;
    • iii.
      Description: Elastic Modulus.
  • d.
    Create forward reaction rate constant:

    • i.
      Name→fk;
    • ii.
      Expression: 0.03;
    • iii.
      Description: kC2ˆb, where k represents the forward reaction rate constant.

      Note: There is a typo in Li et al.’s work¹ – the parameter kC₂ˆb should be 0.03 rather than 0.01.

  • e.
    Create backward reaction rate constant:

    • i.
      Name→flampda;
    • ii.
      Expression: 0.05;
    • iii.
      Description: backward reaction rate constant.
  • f.
    Create Volumetric expansion coefficient per mole of the diffusive species:

    • i.
      Name→Omega1;
    • ii.
      Expression: 8.18e-6;
    • iii.
      Description: Volumetric expansion coefficient per mole of the diffusive species.
  • g.
    Create partial molar volume of the host material:

    • i.
      Name→Omega2;
    • ii.
      Expression: 8.18e-6;
    • iii.
      Description: Partial molar volume of the host material.
  • h.
    Create volumetric change per mole for immobile atoms:

    • i.
      Name→w_Omega;
    • ii.
      Expression: 8.18e-6;
    • iii.
      Description: Volumetric change per mole for immobile atoms.
  • i.
    Create initial radius:

    • i.
      Name→R0;
    • ii.
      Expression: 250e-9;
    • iii.
      Description: Initial radius.
  • j.
    Create Gas Constant:

    • i.
      Name→Rg;
    • ii.
      Expression: 8.3145;
    • iii.
      Description: Gas Constant.
  • k.
    Create max concentration of solute atoms:

    • i.
      Name→rhomax;
    • ii.
      Expression: 3.67e5;
    • iii.
      Description: max concentration.
  • l.
    Create room temperature:

    • i.
      Name→T;
    • ii.
      Expression: 298;
    • iii.
      Description: room temperature.
  • m.
    Create Poisson’s ratio:

    • i.
      Name→v;
    • ii.
      Expression: 0.28;
    • iii.
      Description: Poisson’s ratio.
      Finally, we create and show the parameters used in the numerical calculation in Figure 2.
• 3.Create the variables part 1 used in the numerical calculation.
  Component 1→right click Definitions→create Variables 1→create first part of the variables used in the numerical calculation as follows:

  • a.
    Create Radial component of Green Lagrangian strain tensor:

    • i.
      Name→E11e;
    • ii.
      Expression: ((F11e∗F11e)-1)/2;
    • iii.
      Description: Radial component of Green Lagrangian strain tensor.
  • b.
    Create two tangential components of Green Lagrangian strain tensor:

    • i.
      Name→E22e and E33e;
    • ii.
      Expression: ((F22e∗F22e)-1)/2 and ((F33e∗F33e)-1)/2, respectively;
    • iii.
      Description: Tangential components of Green Lagrangian strain tensor.
  • c.
    Create radial component of total deformation gradient tensor:

    • i.
      Name→F11;
    • ii.
      Expression: 1 + ux;
    • iii.
      Description: Radial component of total deformation gradient tensor.
  • d.
    Create two tangential components of total deformation gradient tensor:

    • i.
      Name→F22 and F33.
    • ii.
      Expression: 1 + u/x.
    • iii.
      Description: Tangential components of total deformation gradient tensor.
  • e.
    Create radial component of elastic deformation gradient tensor:

    • i.
      Name→F11e;
    • ii.
      Expression: F11/Fc;
    • iii.
      Description: Radial component of elastic deformation gradient tensor.
  • f.
    Create two tangential components of elastic deformation gradient tensor:

    • i.
      Name→F22e and F33e;
    • ii.
      Expression: F22/Fc and F33/Fc, respectively;
    • iii.
      Description: Tangential components of elastic deformation gradient tensor.
  • g.
    Create inelastic volumetric expansion ratio:

    • i.
      Name→Jc;
    • ii.
      Expression: 1+Omega1∗(rhoc)+w_Omega∗(rhos);
    • iii.
      Description: Inelastic volumetric expansion ratio.
  • h.
    Create eigen-transformation deformation gradient tensor:

    • i.
      Name→Fc;
    • ii.
      Expression: (abs(Jc)+0.01)ˆ(1/3);
    • iii.
      Description: Eigen-transformation deformation gradient tensor.

      Note: The original expression for the eigen-transformation deformation gradient tensor is (Jc)ˆ(1/3). Here, we modify it to (abs(Jc)+0.01)ˆ(1/3) to avoid zero initial conditions and negative non-integer powers, which helps the convergence of numerical computations.

      Finally, we create and show the variable 1 used in the numerical calculation in Figure 3.
• 4.Create the variables part 2 used in the numerical calculation.
  Component 1→right click Definition→create Variables 2→create second part of the variables used in the numerical calculation as follows:

  • a.
    Create radial component of the first Piola-Kirchhoff stress:

    • i.
      Name→Pk11;
    • ii.
      Expression: Fc∗E/((1 + v)∗(1–2∗v))∗((1-v)∗E11e+2∗v∗E22e)∗F11;
    • iii.
      Description: Radial component of the first Piola-Kirchhoff stress.
  • b.
    Create two tangential components of the first Piola-Kirchhoff stress:

    • i.
      Name→Pk22 and Pk33;
    • ii.
      Expression: Fc∗E/((1 + v)∗(1–2∗v))∗(v∗E11e+E22e)∗F22 and Fc∗E/((1 + v)∗(1–2∗v))∗(v∗E11e+E22e)∗F33, respectively;
    • iii.
      Description: Tangential components of the first Piola-Kirchhoff stress.
  • c.
    Create Radial component of Cauchy stress:

    • i.
      Name→sigma11;
    • ii.
      Expression: Pk11/(F22∗F22);
    • iii.
      Description: Radial component of Cauchy stress.
  • d.
    Create two tangential components of Cauchy stress:

    • i.
      Name→sigma 22 and sigma 33;
    • ii.
      Expression: Pk22/(F11∗F33) and Pk33/(F11∗F22), respectively;
    • iii.
      Description: Tangential components of Cauchy stress.
  • e.
    Create hydrostatic stress:

    • i.
      Name→sigmam;
    • ii.
      Expression: (sigma11+sigma22+sigma33)/3;
    • iii.
      Description: Hydrostatic stress.
  • f.
    Create strain energy density:

    • i.
      Name→w;
    • ii.
      Expression: E∗(v∗(E11e+E22e+E33e)ˆ2/(1–2∗v)+(E11eˆ2 + E22eˆ2 + E33eˆ2))/(2 + 2∗v);
    • iii.
      Description: Strain energy density.
  • g.
    Create flux:

    • i.
      Name→J;
    • ii.
      Expression: -d(rhoc/(F11∗F22∗F33),x)∗D∗F22∗F33/F11.
    • iii.
      +D∗rhoc∗Omega1/(F11ˆ2∗Rg∗T)∗d(sigmam,x).
    • iv.
      Description: Flux.

      Note: The diffusive flux can be typically calculated from the chemical potential of diffusive species as $J=-\frac{Dc}{R_{g}T}\nabla \mu =-\frac{Dc}{R_{g}T}\nabla \left({\mu }_0+R_{g}T\ln c-{\mathrm{\Omega }}_1{\sigma }_h\right)$, giving the expression in COMSOL as, d(rhoc/(F11∗F22∗F33),x)∗D∗F22∗F33/F11 + D∗rhoc∗Omega1/(F11ˆ2∗Rg∗T) ∗d(sigmam,x). If we want to consider the effects of strain energy density⁷ and local deformation rate⁸ on the migration of diffusive species, the expression should be changed to -d(rhoc/(F11∗F22∗F33),x)∗D∗F22∗F33/F11

       +D∗rhoc∗Omega1/(F11ˆ2∗Rg∗T)∗d(sigmam,x)
       -D∗rhoc∗Omega2/(F11ˆ2∗Rg∗T)∗d(w,x)+ut∗rhoc/(F11).
      Finally, we create and show the variable 2 used in the numerical calculation in Figure 4.
• 5.Create spherically symmetric geometry of a spherical nanoparticle.

Figure 1.

The spherically symmetric model with a blank mechanical equation and mass transport in the COMSOL PDE module

Figure 2.

Parameter used in numerical calculation

Figure 3.

Variable 1 used in numerical calculation

Figure 4.

Variable 2 used in numerical calculation

Geometry→Interval→modify the value on the second line of Coordinates(m) to R0→Build Selected. The final 1-D model is found in Figure 5.

Note: Since a spherical nanoparticle with a given solute-atom flux or concentration at the surface of the spherical particle exhibits spherically symmetric, we simply create a one-dimensional model with the internal coordinate of 0 and the external coordinate of R0.

Figure 5.

1-D spherically symmetric model of a spherical nanoparticle with the internal coordinate of 0 and the external coordinate of R0

Major step two: Equation, initial and boundary conditions

Timing: 20 min

• 6.
  Create mechanical equilibrium equation and corresponding initial/boundary conditions through Coefficient Form PDE (c). (As shown in Figure 6).

  • a.
    Establishment of mechanical equations.
    Coefficient Form PDE (c)→Coefficient Form PDE 1→modify the Source Term into -2∗(Pk11-Pk22)/x and Conservative Flux Source into Pk11; set all other terms as 0.

    Note: The general form of a partial differential equation is

    $$e_a\frac{{\partial }^{2}u}{\partial t^2}+d_a\frac{\partial u}{\partial t}+\nabla ·\left(-c\nabla u-\alpha u+\gamma \right)+\beta ·\nabla u+au=f\text{.}$$ (Equation 1)
    Substituting e_a = 0, d_a = 0, c = 0, α = 0, γ = Pk11, β = 0, a = 0, f = -2∗(Pk11-Pk22)/x in Equation 1 yields.

    $$\frac{\partial Pk11}{\partial x}+2\frac{Pk11-Pk22}{x}=0\text{,}$$ (Equation 2)
    which is the same as the mechanical equilibrium equation in Li et al.¹
  • b.
    Initial and boundary conditions.

    • i.
      Coefficient Form PDE (c)→Initial Values1→Initial Values→Initial value for u→set u as 1e-9. Here, we set a small initial displacement as 1 nm, which helps the convergence of the numerical calculation.
    • ii.
      Coefficient Form PDE (c)→right click and create a Dirichlet Boundary Condition 1→Activate Selection and select the left point of the geometry→Dirichlet Boundary Condition→Prescribed value of u and set r as 0.

      Note: Since a spherical nanoparticle exhibits spherical symmetry, we set a zero-displacement boundary condition at the center of the spherical nanoparticle. The current model automatically satisfies the traction-free outer surface of the spherical nanoparticle, so there is no need to add additional boundary conditions.

• 7.
  Create mass transport equation and corresponding initial/boundary conditions through Coefficient Form PDE (c2). (As shown in Figure 7).

  • a.
    Establishment of mass transport equation.
    Coefficient Form PDE (c2)→Coefficient Form PDE 1→We need to set the coefficients of the partial differential equation in the coefficient form, so that the partial differential equation in a general form degenerates to the diffusion equation with a term representing the chemical reaction. The coefficients are set as follows:

    • i.
      Diffusion Coefficient: c11 = 0, c12 = 0, c21 = 0, c22 = 0,
    • ii.
      Absorption Coefficient: a11 = 0, a12 = 0, a21 = -fk∗charge_parameter, a22 = flampda∗(1∗(rhos>=0.2∗rhomax)+0∗(rhos<0.2∗rhomax))∗(1-charge_parameter).
    • iii.
      Source Term: f11 = -2∗J/x, f12 = 0,
    • iv.
      Mass Coefficient: e_a11 = 0, e_a12 = 0, e_a21 = 0, e_a22 = 0,
    • v.
      Damping or Mass Coefficient: d_a11 = 1, d_a12 = 1, d_a21 = 0, d_a22 = 1,
    • vi.
      Conservative Flux Convection Coefficient: α11 = 0, α12 = 0, α21 = 0, α22 = 0,
    • vii.
      Convection Coefficient: β11 = 0, β12 = 0, β21 = 0, β22 = 0,
    • viii.
      Conservation Flux Source: γ11 = J, γ12 = 0.
      The general form of the coefficient form of a partial differential equation is.

      $$e_a\frac{{\partial }^{2}u}{\partial t^2}+d_a\frac{\partial u}{\partial t}+\nabla ·\left(-c\nabla u-\alpha u+\gamma \right)+\beta ·\nabla u+au=f\text{.}$$ (Equation 3)
      Substituting e_a, d_a, c, α, γ, β, a, f in Equation 3, one obtains the mass transport equation during lithiation by setting charge_parameter as 1,

      $$\frac{\partial \text{rhoc}}{\partial t}+\frac{\partial \text{rhos}}{\partial t}+\frac{\partial J}{\partial x}+\frac{2J}{x}=0\text{,}$$ (Equation 4)

      $$\frac{\partial \text{rhos}}{\partial t}=fk·rhoc\text{.}$$ (Equation 5)
      which is the same as the mass transport equation given by Li et al.¹

      Note: Different from the coefficient form of the PDE with “dependent variable” of displacement u, the coefficients in the PDE with “dependent variables” of the concentrations of mobile solute atoms C(rhoc) and immobile solute atoms S(rhos) are represented in a matrix [2 × 2] or vector [1 × 2].

  • b.
    Initial and boundary conditions.

    • i.
      Coefficient Form PDE (c2)→Initial Values1→Initial Values→keep all the default values of the initial values.
    • ii.
      Coefficient Form PDE (c2)→right click and create a Dirichlet Boundary Condition 1→Activate Selection and select the right point of the geometry→Dirichlet Boundary Condition→Prescribed value of rhoc and set r1 as rhomax∗(t/(0.001∗R0ˆ2/D)∗(t<=0.001∗R0ˆ2/D)+1∗(t > 0.001∗R0ˆ2/D))→Untick the checkbox in front of r2.

      Note: Since it is difficult to calculate the numerical results under potentiostatic operation due to a large concentration gradient at the onset of lithiation, we introduce a linear transition function when the dimensionless time is smaller than 0.001∗R0ˆ2/D. To avoid the computational error caused by this linear transition function, the domain size of the linear transition function is limited. Note that the smaller the domain size, the smaller the error. However, this can lead to a convergence issue of the computation.

    • iii.
      Coefficient Form PDE (c)→right click and create a Flux/Source boundary condition→Select the left point of the geometry→keep all the default values of the Boundary Flux/Source and boundary Absorption/Impedance Term.

Figure 6.

The setting of the Coefficient Form PDE for mechanical equilibrium equation (left), and corresponding initial conditions (middle) and boundary conditions (right)

Figure 7.

The settings of the Coefficient Form PDE of mass transport equation, and corresponding initial and boundary conditions

Major step three: Mesh generation and solver parameterization

Timing: 10 min

• 8.Create 1000 mesh in the 1D geometry.

Component 1 (comp 1)→Mesh 1→Size→Element Size→Change the element size method from “Predefined” to “Custom”→Set Maximum element size as R0/1000→keep all the other values as default→press Build Selected to generate the mesh.

• 9.
  Solver parameterization.

  • a.
    Rename the “Study 1” as “1cycle-lithiation”→Step 1: Time Dependent→Study Settings→Set the Output times as 0, 0.001∗R0ˆ2/D, 0.005∗R0ˆ2/D, 0.01∗R0ˆ2/D, 0.015∗R0ˆ2/D, 0.025∗R0ˆ2/D.
  • b.
    1cycle-lithiation→Solver Configurations→Solution 1→Time-Dependent Solver 1→Fully Coupled 1→Method and Termination→set Nonlinear method as “Automatic (Newton)”→set initial damping factor as 1e-1 and minimum damping factor 1e-4. The final solver configuration can be found in Figure 8.
  • c.
    1cycle-lithiation→press Compute to start analyzing the diffusion induced stress in a spherical nanoparticle during lithiation.

Figure 8.

The solver parameterization of lithiation

Major step four: Adding the de-lithiation process

Timing: 10 min

• 10.Change the charge_parameter from lithiation to de-lithiation.

  Global Definitions→Parameters table→Parameters→change the “charge_parameter” from 1 to 0, which indicates that the simulation process changes from lithiation to de-lithiation.

  In this case, all the parameter in the Coefficient Form of PDE 2 are as follows,

  • a.
    Diffusion Coefficient: c11 = 0, c12 = 0, c21 = 0, c22 = 0,
  • b.
    Absorption Coefficient: a11 = 0, a12 = 0, a21 = -fk∗charge_parameter, a22 = flampda∗(1∗(rhos>=0.2∗rhomax)+0∗(rhos<0.2∗rhomax))∗(1-charge_parameter).
  • c.
    Source Term: f11 = -2∗J/x, f12 = 0,
  • d.
    Mass Coefficient: e_a11 = 0, e_a12 = 0, e_a21 = 0, e_a22 = 0,
  • e.
    Damping or Mass Coefficient: d_a11 = 1, d_a12 = 1, d_a21 = 0, d_a22 = 1,
  • f.
    Conservative Flux Convection Coefficient: α11 = 0, α12 = 0, α21 = 0, α22 = 0,
  • g.
    Convection Coefficient: β11 = 0, β12 = 0, β21 = 0, β22 = 0.
  • h.
    Conservation Flux Source: γ11 = J, γ12 = 0.
    The general form of the coefficient form of a partial differential equation is

    $$e_a\frac{{\partial }^{2}u}{\partial t^2}+d_a\frac{\partial u}{\partial t}+\nabla ·\left(-c\nabla u-\alpha u+\gamma \right)+\beta ·\nabla u+au=f\text{.}$$ (Equation 6)
    Substituting e_a, d_a, c, α, γ, β, a, f in Equation 6, one obtains the mass transport equation during de-lithiation process by setting charge_parameter equal to 1,

    $$\frac{\partial \text{rhoc}}{\partial t}+\frac{\partial \text{rhos}}{\partial t}+\frac{\partial J}{\partial x}+\frac{2J}{x}=0\text{,}$$ (Equation 7)

    $$\frac{\partial \text{rhos}}{\partial t}=\{\begin{array}{l}flampda·rho\mathrm{s}forrhos>=0.2\ast rhomax\\ 0forrhos<0.2\ast rhomax\end{array}$$ (Equation 8)
• 11.Create a time-dependent study.

Study→Add study→Under the General Studies double click the Time Dependent study→Rename the “Study 2” as “1cycle-delithiation”

• 12.
  Solver parameterization.

  • a.
    1cycle-delithiation→Step 1: Time Dependent→Study Settings→Set the Output times as 0, 0.001∗R0ˆ2/D, 0.005∗R0ˆ2/D, 0.01∗R0ˆ2/D, 0.015∗R0ˆ2/D, 0.025∗R0ˆ2/D.
  • b.
    1cycle-delithiation→Step 1: Time Dependent→Values of Dependent Variables→All the settings of “initial values of variables solved for” are as follows:

    • i.
      Settings as “User controlled”;
    • ii.
      Method as “Solution”;
    • iii.
      Study as “1cycle-lithiation,
    • iv.
      Time Dependent”;
    • v.
      Time as “Automatic” or “Last”.
    • vi.
      Keeping all the settings of “initial values of variables solved for”
    • vii.
      “Store field in output” as default.
  • c.
    1cycle-delithiation→Solver Configurations→Solution 1→Time-Dependent Solver 1→Fully Coupled 1→Method and Termination→set Nonlinear method as “Automatic highly nonlinear (Newton)”→set initial damping factor as 1e-4 and minimum damping factor 1e-8. The final solver configuration can be found in Figure 9.
  • d.
    1cycle-delithiation→press Compute to start analyzing the diffusion induced stress in a spherical nanoparticle during de-lithiation.

Figure 9.

The solver parameterization of de-lithiation

Major step five: Post process for the constant and nonuniform concentration of residual trapped lithium

• 13.
  Post processing for the constant concentration of residual trapped lithium.

  • a.
    Results→Right click Results and create a 1D Plot Group→Right click 1D Plot Group 1 and create Line Graph→Switch to Line Graph Settings panel→Set the Dataset as “1cycle-delithiation/Solution 2”, Solution parameters as “Manual” and Time selection as “From list” →Choose all the times→Choose the 1D symmetrical model in the Selection part→The Expression in the y-Axis Data is set as rhos/rhomax, which indicates the concentration of immobile lithium during the de-lithiation.¹
  • b.
    Results→Right click Export and create Data→Switch to Data Settings panel→Set the Dataset as “1cycle-delithiation/Solution 2”, Solution parameters as “Manual” and Time selection as “From list” →Choose all the times→The Expression is set as rhos/rhomax, which indicates the concentration of immobile lithium during the de-lithiation →Browse the directory of the Filename and create a output file name, and then click Export button to export the data of the concentration of immobile lithium.
• 14.
  Modification of the model into the model with nonuniform concentration of residual trapped lithium.

  • a.
    Change the absorption coefficient as a11 = 0, a12 = 0, a21 = -fk∗charge_parameter, a22 = flampda∗(1∗(rhos>=0.8∗withsol('sol1′,rhos))+0∗(rhos<0.8∗withsol('sol1′,rhos)))∗(1-charge_parameter).
    The general form of the coefficient form of a partial differential equation is.

    $$e_a\frac{{\partial }^{2}u}{\partial t^2}+d_a\frac{\partial u}{\partial t}+\nabla ·\left(-c\nabla u-\alpha u+\gamma \right)+\beta ·\nabla u+au=f\text{.}$$ (Equation 9)
    Substituting e_a, d_a, c, α, γ, β, a, f into Equation 9, one obtains the mass transport equation during lithiation by setting charge_parameter as 1,

    $$\frac{\partial \text{rhoc}}{\partial t}+\frac{\partial \text{rhos}}{\partial t}+\frac{\partial J}{\partial x}+\frac{2J}{x}=0\text{,}$$ (Equation 10)

    $$\frac{\partial \text{rhos}}{\partial t}=\{\begin{array}{l}flampda·rho\mathrm{s}forrhos>=0.8\ast withsol\left(\text{'}sol1\text{'},rhos\right)\\ 0forrhos<0.8\ast withsol\left(\text{'}sol1\text{'},rhos\right)\end{array}$$ (Equation 11)
  • b.
    Results→Right click Results and create a 1D Plot Group→Right click 1D Plot Group 1 and create Line Graph→Switch to Line Graph Settings panel→Set the Dataset as “1cycle-delithiation/Solution 2”, Solution parameters as “Manual” and Time selection as “From list” →Choose all the times→Choose the 1D symmetrical model in the Selection part→The Expression in the y-Axis Data is set as rhos/rhomax, which indicates the concentration of immobile lithium during the de-lithiation.
  • c.
    Results→Right click Export and create Data→Switch to Data Settings panel→Set the Dataset as “1cycle-delithiation/Solution 2”, Solution parameters as “Manual” and Time selection as “From list” →Choose all the times→The Expression is set as rhos/rhomax, which indicates the concentration of immobile lithium during the de-lithiation→Browse the directory of the Filename and create a output file name, and then click Export button to export the data of the concentration of immobile lithium.

Expected outcomes

The preceding section comprehensively outlines the procedures for establishing Partial Differential Equations (PDEs), defining initial/boundary conditions, and specifying mesh parameters for both mechanical equations and diffusion equations. The primary outputs encompass the concentration distribution of mobile lithium and immobile lithium, the spatial distribution of radial and hoop stress, as well as the spatial distribution of strain in the spherical nanoparticle. By following the aforementioned procedure, one can acquire a comprehensive set of numerical results, covering stress, strain, displacement, and the concentration of both mobile and immobile lithium. In this context, we specifically delve into the discussion of the typical results pertaining to the concentration of immobile lithium during the de-lithiation process.

Figure 10 illustrates the settings for the 1D Plot Group (left) and the concentration of immobile lithium during the de-lithiation (right). The blue, green, red, sky blue, pink and yellow lines in Figure 10 represent the spatial distributions of the normalized concentration of immobile lithium with constant concentration of residual trapped lithium at dimensionless time 0.025∗R0ˆ2/D, 0.026∗R0ˆ2/D, 0.03∗R0ˆ2/D, 0.035∗R0ˆ2/D, 0.04∗R0ˆ2/D, and 0.05∗R0ˆ2/D, respectively, which are identical to the numerical results in Figure 3B in our previous study.¹

Figure 10.

Illustrations of the settings for 1D Plot Group (left) and the concentration of immobile lithium during the de-lithiation (right) for the constant concentration of residual trapped lithium

Figure 11 illustrates the settings for 1D Plot Group (left) and the concentration of immobile lithium during the de-lithiation (right). The blue, green, red, sky blue, pink and yellow lines in Figure 11 represent the spatial distributions of the normalized concentration of immobile lithium with nonuniform concentration of residual trapped lithium at dimensionless time 0.025∗R0ˆ2/D, 0.026∗R0ˆ2/D, 0.03∗R0ˆ2/D, 0.035∗R0ˆ2/D, 0.04∗R0ˆ2/D, and 0.05∗R0ˆ2/D, respectively, which are identical to the numerical results in Figure 6A in our previous study.¹ It should be noted that the numerical results for the dimensionless time 0.035∗R0ˆ2/D, 0.04∗R0ˆ2/D, and 0.05∗R0ˆ2/D nearly overlap with each other, resulting in only four visible lines in Figure 11.

Figure 11.

Illustrations of the settings for 1D Plot Group (left) and the concentration of immobile lithium during the de-lithiation (right) for nonuniform concentration of residual trapped lithium

Limitations

This protocol only describes the process of calculating the chemical stress in a largely deformed electrode by considering the effects of lithium trapping. The computational details of a largely deformed electrode² with the effects of defect evolution,³ free volume evolution,⁴ anisotropic deformation⁵ are not included. The computational details of these works are available by contacting the lead contact, Fuqian Yang (fuqian.yang@uky.edu).

Troubleshooting

Most of the problems discussed below are related to numerical calculation.

Problem 1

When performing the numerical calculation, there may be an error information “zero initial conditions and non-integer powers of negatives”.

Potential solution

Add a small initial value in the variables. For example, we have set the Eigen-transformation transformed gradient tensor as (abs(Jc)+0.01)ˆ(1/3) in order to avoid zero initial conditions and non-integer powers of negatives. If the value we added is small enough, it likely has negligible or no effect on the final output. For example, the inelastic volume expansion of a largely deformed electrode (i.e., silicon) is about 3. A small initial value of 0.01 likely has no effect on the final output.

Problem 2

When performing the numerical calculation, there may be an error information “Last time step not Converged”

Potential solution

Change the initial damping factor and minimum damping factor in the solver configuration to 1E-3 and 1D-6 respectively.

Problem 3

The initial condition of the de-lithiation is not the result of the last step of the lithiation.

Potential solution

Check carefully whether the initial values in the time dependent settings are correct. If the “Last” time values of dependent variables do not work, you can choose the exact time of the values of dependent variables (e.g., 15.625 s in this work).

Problem 4

The start time of the de-lithiation process is set as 0, and the numerical results of 0.025∗R0ˆ2/D results would reach the same numerical results.

Potential solution

If we set the initial de-lithiation time as 0, the output time is 0, 0.001∗R0ˆ2/D, 0.005∗R0ˆ2/D, 0.01∗R0ˆ2/D, 0.015∗R0ˆ2/D, 0.025∗R0ˆ2/D; If we set the initial de-lithiation time as 0.025∗R0ˆ2/D, the output time becomes 0.025∗R0ˆ2/D, 0.026∗R0ˆ2/D, 0.03∗R0ˆ2/D, 0.035∗R0ˆ2/D, 0.04∗R0ˆ2/D, 0.05∗R0ˆ2/D.

Problem 5

There are no Solver Configurations in the Study.

Potential solution

Press the “Compute” button to generate the Solver Configuration and simply ignore the pop-up error window.

Problem 6

The final results are evidently incorrect due to the low calculation accuracy. For instance, the lithium concentration at the boundary differs from the prescribed condition for the Dirichlet Boundary Condition.

Potential solution

Set the tolerance in the time-dependent setting to “user-controlled” and use 1e-9 as the relative tolerance.

Resource availability

Lead contact

Further information and requests for resources should be directed to and will be fulfilled by the lead contact, Fuqian Yang (fuqian.yang@uky.edu).

Technical contact

Questions about the technical specifics of performing the protocol should be directed to and will be answered by the technical contact, Yong Li (yongli@sspu.edu.cn).

Materials availability

No new materials were generated in this study.

Data and code availability

No additional data was used. This paper does not report any original code. Any additional information generated from COMSOL PDE for reanalyzing this work is available from the lead contact upon request.