Coupled Thermal-Electrochemical Analysis

Typical Application

The primary example of a battery electrochemistry application is the charging and discharging of lithium-ion battery cells. During a charging cycle, the lithium ions are extracted from the active particles of the positive electrode (cathode). The ions move through the electrolyte by migration and diffusion from the positive electrode to the negative electrode across the separator. At the negative electrode (anode), the ions intercalate into the active particles. Heat is generated during the flow of electrons, ion migration, and the intercalation process. During discharge, the cycle is reversed.

Coupled Thermal-Electrochemical Analysis

Rechargeable lithium-ion batteries are widely used in a variety of applications, including portable electronic devices and electric vehicles. The performance of a battery highly depends on the effects of repeated charging and discharging cycles, which can cause the degradation of the battery capacity over time. The porous electrode theory (Newman et al., 2004) is commonly accepted as the leading method for modeling the charge-discharge behavior of lithium-ion cells. The method is based on a homogenized Newman-type approach that does not consider the details of the pore geometry. The porous electrode is based on the concurrent solution of a highly coupled multiphysics-multiscale formulation:

At macroscale, the porous electrode is modeled as a homogenized medium consisting of the superposition of an active solid electrode particle phase and a liquid electrolyte solution phase, with known volume fractions.
At microscale, a collection of spherical particles is assumed for which a (nonlinear) lithium-ion diffusion model is solved. The solid particles are connected by a conductive binder and together form the solid electrode phase.

You can use the coupled thermal-electrochemical procedure to model other battery electrochemical processes that use the governing equations described in the sections below. Although the discussion in the following sections is written in terms of lithium cells, the theory is general and can be used for other cell types.

A full stack lithium-ion battery cell consists of an anode collector, porous anode, porous separator, porous cathode, and cathode collector. The porous electrode can be described as a material with a solid skeleton and a distribution of interconnected pores that are filled with the electrolyte. The solid skeleton consists of a large collection of active spherical particles that are in contact with the electrolyte. The spherical particles are electrically conductive and are in electrical contact among themselves, but they do not allow for interparticle diffusion. The spherical particles are also chemically active and allow for intercalation and deintercalation of lithium ions.

Figure 1 shows a typical jelly roll configuration of a cylindrical battery. The porous parts of the battery are immersed in an electrolyte bath that facilitates the movement of ions from the cathode to the anode during the charging cycle (Figure 2). The porous electrodes are connected to nonporous collectors that are connected to the external terminal of the battery. The external terminals of the battery are connected to a power source during the charging cycle. The electrons move in the solid phase, while the ions move in the liquid phase.

At the cathode solid-liquid interface, lithium ions can move out of the solid electrode particles and enter the electrolyte through a deintercalation (or extraction) process that can be written as:

Li - Θ \leftrightarrow {Li}^{+} + e^{-} + Θ .

Here, $Li - Θ$ represents lithium in the solid electrode, which through the deintercalation reaction, transforms into lithium ion in the electrolyte, ${Li}^{+}$ , plus an electron, $e^{-}$ , and leaves behind a vacancy (or intercalation site) in the solid, $Θ$ . The reverse reaction corresponds to the intercalation of lithium into the solid electrode.

During the charging cycle, deintercalated lithium ions at the cathode solid-liquid interface move in the electrolyte through the separator and reach the anode. At the anode, the lithium ions are intercalated (inserted) into the solid electrode particles. The separator is porous and allows ions to pass through; however, it is an electrically insulated material. Therefore, the electrons cannot pass through, which avoids the possibility of a short circuit in the cell.

The governing equations used to model lithium-ion cells span both multiple scales and multiple physics and involve solving a fully coupled thermal and electrochemistry problem. The primary solution variables that are solved for at the macroscale (or continuum scale) are the electric potential in the solid phase ( $Φ_{s}$ ), temperature ( $θ$ ), electric potential in the electrolyte ( $Φ_{e}$ ), and the lithium ion concentration ( $C_{e}$ ). At microscale, the model solves for the concentration of lithium ( $C_{s}$ ) in the particle.

Governing Equations

The porous electrode theory consists of five main parts:

Intercalation at the solid-liquid interface.
Conduction of electrons in the solid phase.
Diffusion and migration of lithium ions in the liquid phase.
Diffusion of lithium in the microscale particle.
Coupling of the continuum scale and microscale.

An optional sixth contribution is associated with Joule heating generation.

Intercalation at the Solid-Liquid Interface

The intercalation process at the solid-liquid interface is governed by electrochemical kinetics and is defined by the Butler-Volmer equation. The Butler-Volmer equation describes the relationship between the electric current density through the solid electrode-electrolyte interface, $I_{BV}$ , and the voltage difference between the electrode and electrolyte according to:

I_{BV} = I_{0} {\exp [\frac{α_{a} (z F η - σ_{m} Ω)}{R (θ - θ^{z})}] - \exp [\frac{{- α}_{c} (z F η - σ_{m} Ω)}{R (θ - θ^{z})}]},

where $I_{0}$ and $η$ are the exchange current density and overpotential, respectively; $σ_{m}$ is the mean stress or hydrostatic pressure; and $Ω$ represents the partial molar volume of lithium in the electrode material. The term $σ_{m} Ω$ captures the effects of mechanical stress on the electrical response of the electrode material. The overpotential, $η$ , is defined as:

η = Φ_{s} - Φ_{e} - E_{ref} ({\hat{C}}_{s}^{surf}, θ) - R_{S E I} I_{B V},

where

R_{S E I}

is the solid electrolye interface resistance.

The exchange current density, $I_{0}$ , can be written as:

I_{0} = F k_{c}^{α_{a}} k_{a}^{α_{c}} C_{s}^{\max} {(1 - {\hat{C}}_{s}^{surf})}^{α_{a}} {({\hat{C}}_{s}^{surf})}^{α_{c}} {(\frac{C_{e}}{C_{e}^{ref}})}^{α_{a}} .

In the above equations,

$F$: is Faraday's constant;
$R$: is the gas constant;
$z$: is the charge number of the lithium ion battery;
$θ$: is temperature;
$θ^{z}$: is the absolute zero temperature,
$k_{c}, k_{a}$: are the cathodic and anodic rate constants, respectively;
$α_{c}, α_{a}$: are the cathodic and anodic transfer coefficients, respectively;
$C_{s}^{\max}$: is the theoretical maximum lithium capacity;
${\hat{C}}_{s}^{surf}$: is the normalized surface lithium concentration in the microscale particle;
$C_{e}^{ref}$: is the reference value of lithium ion concentration in the electrolyte; and
$E_{ref}$: is the open circuit potential (OCP) as a function of ${\hat{C}}_{s}^{surf}$ and $θ$ .

The normalized concentration, ${\hat{C}}_{s}$ , is defined as ${\hat{C}}_{s} = \frac{C_{s}}{C_{s}^{\max}}$ , where $C_{s}$ is the lithium concentration in the particle. The normalized surface concentration, ${\hat{C}}_{s}^{surf}$ , is defined as ${\hat{C}}_{s}^{surf} = \frac{C_{s}^{surf}}{C_{s}^{\max}}$ , where $C_{s}^{surf}$ is the lithium concentration on the surface of the particle.

Conduction of Electrons in the Solid Phase

The equilibrium equation for the current density, $I_{s}$ , in the solid phase is given by:

\nabla \cdot I_{s} = - a_{s} I_{BV},

where

I_{s} = - K_{s}^{eff} \cdot \nabla Φ_{s} .

$K_{s}^{eff}$ is the effective electrical conductivity of the solid phase computed using the Bruggeman relationship for tortuosity ( $τ_{s} = ε_{s}^{-α}$ ). Tortuosity represents an intrinsic property of a porous material characterizing the ratio of the actual flow path length to the straight distance between the ends.

For isotropic tortuosity:

K_{s}^{eff} = \frac{ε_{s}}{τ_{s}} K_{s} I = ε_{s}^{(1 + α)} K_{s} I .

For anisotropic tortuosity, the above expression can be written as:

K_{s}^{eff} = [\begin{matrix} ε_{s}^{(1 + α_{x})} & 0 & 0 \\ 0 & ε_{s}^{(1 + α_{y})} & 0 \\ 0 & 0 & ε_{s}^{(1 + α_{z})} \end{matrix}] K_{s},

where

$K_{s}$: is the electric conductivity of the bulk solid material (which can be a function of normalized concentration);
$τ_{s}$: is the tortuosity of the solid porous electrode;
$ε_{s}$: is the volume fraction of the solid phase;
$I$: is the unit tensor;
$α$: is the Bruggeman exponent for isotropic tortuosity;
$α_{x}, α_{y}, α_{z}$: are the Bruggeman exponents for anisotropic tortuosity; and
$a_{s}$: is the wetted particle surface area per unit volume, computed by default as $a_{s} = \frac{3 ε_{s}}{R_{p}}$ , where $R_{p}$ is the outer radius of the spherical particle associated with the microscale model.

Diffusion and Migration of Lithium Ions in the Liquid Phase

The current density in the electrolyte, $I_{e}$ , and the lithium ion concentration, $C_{e}$ , in the liquid phase are solved for using the following governing equations:

\nabla \cdot I_{e} = a_{s} I_{BV},

where

I_{e} = - K_{e}^{eff} \cdot [\nabla Φ_{e} - \frac{2 R θ}{F} (1 + \frac{d ln f_{\pm}}{d ln C_{e}}) (1 - t_{+}) \nabla \ln C_{e}],

\frac{\partial (ε_{e} C_{e})}{∂t} = - \nabla \cdot J_{e} + \frac{a_{s} I_{BV}}{F},

J_{e} = - D_{e}^{eff} \cdot \nabla C_{e} + \frac{t_{+}}{F} I_{e} .

In the above equations,

$K_{e}^{eff}$: is the effective electrical conductivity in the electrolyte phase,
$D_{e}^{eff}$: is the effective ion diffusivity in the electrolyte phase,
$t_{+}$: is the transference number that defines the fraction of the total electrical current that is carried by lithium ions in the electrolyte, and
$(\frac{{d ln f}_{\pm}}{{d ln C}_{e}})$: is the molar activity coefficient that accounts for deviations from the ideal behavior in a mixture of chemical substances.

$K_{e}^{eff}$ and $D_{e}^{eff}$ are computed using the Bruggeman relationship for tortuosity.

For isotropic tortuosity:

K_{e}^{eff} = \frac{ε_{e}}{τ_{e}} K_{e} I = ε_{e}^{(1 + α)} K_{e} I,

D_{e}^{eff} = \frac{ε_{e}}{τ_{e}} D_{e} I = ε_{e}^{(1 + α)} D_{e} I .

For anisotropic tortuosity, the above expression can be written as:

K_{e}^{eff} = [\begin{matrix} ε_{e}^{(1 + α_{x})} & 0 & 0 \\ 0 & ε_{e}^{(1 + α_{y})} & 0 \\ 0 & 0 & ε_{e}^{(1 + α_{z})} \end{matrix}] K_{e},

D_{e}^{eff} = [\begin{matrix} ε_{e}^{(1 + α_{x})} & 0 & 0 \\ 0 & ε_{e}^{(1 + α_{y})} & 0 \\ 0 & 0 & ε_{e}^{(1 + α_{z})} \end{matrix}] D_{e},

where

$K_{e}, D_{e}$: are the electric conductivity and ion diffusivity of the electrolyte, respectively;
$I$: is the unit tensor;
$τ_{e}$: is the tortuosity of the liquid (electrolyte) phase;
$ε_{e}$: is the volume fraction of the liquid phase;
$α$: is the Bruggeman exponent for isotropic tortuosity; and
$α_{x}, α_{y}, α_{z}$: are the Bruggeman exponents for anisotropic tortuosity.

Diffusion of Lithium in the Microscale Particle

The active material at the electrodes is assumed to consist of spherical microparticles. The particles are assumed to have electrical contact between themselves but no interparticle diffusion. It is assumed that the microparticle material is a good electronic conductor (transport number=1) and has spherical symmetry. The conservation of lithium inside the particle is governed by Fick’s second law of diffusion, written in spherical coordinates:

\frac{\partial C_{s}}{\partial t} = - \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} j_{s}),

j_{s} = D_{s} (C_{s}) \frac{\partial C_{s}}{\partial r},

where

$r$: is the radial coordinate of the spherical particle,
$C_{s}$: is the lithium concentration in the particle, and
$D_{s} (C_{s})$: is the concentration-dependent diffusion constant of the solid material.

The time evolution of lithium concentration in the particle is determined by diffusion and the intercalation current density, $I_{BV}$ , from the electrochemical model in the continuum scale. The microscale particle is modeled as a sphere over which a one-dimensional finite element mesh in spherical coordinates is created internally by Abaqus, and the diffusion problem is solved. You can specify the discretization for the internal mesh and the type of meshing over the microscale particle (see Particle Layer Mesh).

Coupling of the Continuum Scale and Microscale

The coupling of the microscale with the continuum scale happens through the Butler-Volmer current density, $I_{BV}$ , from the electrochemical model:

j_{s} = \frac{I_{BV}}{F}, at r = R_{P} .

Heat Transfer and Joule Heating

The total heat generation in a battery is attributed to the contribution of five different sources: flow of current at the solid-liquid interface, flow of current in the solid phase, flow of current in the liquid phase, flow of ions, and entropy generation. The amount of energy converted into heat from each of the source terms can be scaled using the conversion factors, $β_{α}$ ( $α$ =1–5):

Q_{tot} = β_{1} Q_{1} + β_{2} Q_{2} + β_{3} Q_{3} + β_{4} Q_{4} + β_{5} Q_{5},

where

$Q_{1} = a_{s} I_{BV} η$: is the ohmic loss at the solid-liquid interface;
$Q_{2} = a_{s} I_{BV} θ \frac{∂U}{\partial θ}$: is the entropy generation, where $\frac{∂U}{\partial θ}$ is the derivative of the open circuit potential with respect to temperature that you can specify in tabular form;
$Q_{3} = K_{s}^{eff} \nabla Φ_{s} \cdot \nabla Φ_{s}$: is the ohmic loss in the electrode;
$Q_{4} = K_{e}^{eff} \nabla Φ_{e} \cdot \nabla Φ_{e}$: is the ohmic loss in the electrolyte; and
$Q_{5} = \frac{2 R θ}{F} K_{e}^{eff} (1 + \frac{d ln f_{\pm}}{d ln C_{e}}) (1 - t_{+}) \nabla \ln C_{e} ⋅∇ Φ_{e}$: is the ohmic loss due to ion diffusion.

Fully Coupled Solution Scheme

The coupled thermal-electrochemical analysis in Abaqus uses an exact implementation of Newton’s method, leading to an unsymmetric Jacobian matrix in the form:

[\begin{matrix} K_{φ_{s} φ_{s}} & K_{φ_{s} θ} & K_{φ_{s} φ_{e}} & K_{φ_{s} C_{e}} \\ K_{θ φ_{s}} & K_{θ θ} & K_{θ φ_{e}} & K_{θ C_{e}} \\ K_{φ_{e} φ_{s}} & K_{φ_{e} θ} & K_{φ_{e} φ_{e}} & K_{φ_{s} C_{e}} \\ K_{C_{e} φ_{s}} & K_{C_{e} θ} & K_{φ_{e} φ_{s}} & K_{φ_{e} φ_{e}} \end{matrix}] {\begin{matrix} Δ φ_{s} \\ Δ θ \\ Δ φ_{e} \\ Δ C_{e} \end{matrix}} = {\begin{matrix} R_{φ_{s}} \\ R_{θ} \\ R_{φ_{e}} \\ R_{C_{e}} \end{matrix}} .

Steady-State Analysis

Steady-state analysis provides the steady-state solution by neglecting the transient terms in the continuum scale equations. It can be used to achieve a balanced initial state or to assess conditions in the cell after a long storage period.

In the thermal equation, the internal energy term in the governing heat transfer equation is omitted. Similarly, the transient term is omitted in the diffusion equations for the lithium ion concentration in the electrolyte. Electrical transient effects are not included in the equations because they are very rapid compared to the characteristic times of thermal and diffusion effects. A steady-state analysis has no effect on the microscale solution; the transient terms are always considered in the solution of the lithium concentration in the solid particle.

Input File Usage

COUPLED THERMAL-ELECTROCHEMICAL, STEADY STATE

Transient Analysis

In a transient analysis, the transient effects in the heat transfer and diffusion equations are included in the solution. Electrical transient effects are always omitted because they are very rapid compared to the characteristic times of thermal and mass diffusion effects.

Input File Usage

COUPLED THERMAL-ELECTROCHEMICAL

Initial Conditions

By default, the initial values of electric potential in the solid, temperature, electric potential in the electrolyte, and ion concentration of all nodes are set to zero. You can specify nonzero initial values for the primary solution variables (see Initial Conditions).

The typical set of initial conditions includes uniform but different values for the two electrodes. The initial lithium concentration in the particles and solid and fluid electric potentials in the electrodes are chosen such that the overpotential in the electrodes is zero. The ion concentration in the electrolyte is typically assumed to be uniform in the cell.

The initial condition for the microscale spherical particle is:

C_{s} (r) = C_{s}^{initial}, at t = 0 .

You can specify the initial concentration of the spherical particle using the parameter table "ABQ_EChemPET_Electrode_Particles".

Boundary Conditions

You can prescribe the following boundary conditions:

Electric potential in the solid, $φ_{s} = φ_{s} (x, t)$ (degree of freedom 9).
Electric potential in the electrolyte, $φ_{e} = φ_{e} (x, t)$ (degree of freedom 32).
Temperature, $θ = θ (x, t)$ (degree of freedom 11).
Ion concentration in the electrolyte, $C_{e} = C_{e} (x, t)$ (degree of freedom 33) at the nodes.

You can specify boundary conditions as functions of time by referring to amplitude curves.

A boundary without any prescribed boundary conditions corresponds to an insulated (zero flux) surface.

The typical boundary condition consists of only grounding (setting to zero) the solid electric potential at the anode. Thermal boundary conditions vary.

At the microscale, the boundary condition for the spherical particle is:

\frac{\partial C_{s}}{\partial r} = 0, at r = 0 .

Abaqus applies the microscale boundary conditions automatically.

Loads

You can apply thermal, electrical, and electrochemical loads in a coupled thermal-electrochemical analysis.

You can prescribe the following types of thermal loads (as described in Thermal Loads):

Concentrated heat flux.
Body flux and distributed surface flux.
Convective film and radiation conditions.

You can prescribe the following types of electrical loads on the solid (as described in Electromagnetic Loads):

Concentrated current.
Distributed surface current densities and body current densities.

You can prescribe the following types of electrical loads on the electrolyte (as described in Electromagnetic Loads):

Concentrated current.
Distributed surface current densities and body current densities.

You can prescribe the following types of ion concentration loads (as described in Thermal Loads):

Concentrated flux.
Distributed body flux.

The typical loads include specification of a solid electric flux (current) at the cathode. Thermal boundary conditions vary but typically include convective film on the exterior surfaces. Customarily, no loads are applied on the concentrations and electrolyte potential.

Predefined Fields

Predefined temperature fields are not allowed in coupled thermal-electrochemical analyses. You can use boundary conditions to specify temperatures. You can specify other predefined field variables in a coupled thermal-electrochemical analysis. These values affect only field variable–dependent material properties.

Material Options

The thermal and electrical properties for both the solid and the electrolyte are active in a coupled thermal-electrochemical analysis. All mechanical behavior material models (such as elasticity and plasticity) are ignored in a coupled thermal-electrochemical analysis. The electrochemistry framework requires that the material definition contain the complete specification of properties required for the porous electrode theory, as described below and in the sections that follow. In addition, the material name must begin with "ABQ_EChemPET_" to enable the coupled micro-macro solution at the different electrodes. Special-purpose parameter and property tables of type names starting with “ABQ_EChemPET_” are required in these material definitions (see Parameter Table Type Reference and Property Table Type Reference).

Input File Usage

MATERIAL, NAME=ABQ_EChemPET_name
POROUS ELECTRODE THEORY

Thermal Material Properties

You must define thermal conductivity for the heat transfer portion of the analysis. In addition, you must define the specific heat for transient problems. Thermal expansion coefficients are not meaningful in a coupled thermal-electrochemical analysis because the deformation of the structure is not considered. You can specify internal heat generation.

Input File Usage

CONDUCTIVITY
DENSITY
SPECIFIC HEAT

Electrical Properties for the Solid Electrode Material

You must define the electric conductivity for the solid electrode portion of the analysis. The electrical conductivity is defined as a function of the average normalized concentration of the particle, ${\hat{C}}_{s}$ (and, optionally, of temperature and user-defined field variables).

Input File Usage

Use the following options to define the electric conductivity of the solid phase:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrode_ElecCond_Tabular", 
LABEL="ABQ_EChemPET_<Electrode name>_ElecCond_Tabular"
 $K_{s}$ ,  ${\hat{C}}_{s}$ ,  $θ$ ,  $FV$

Electrical and Ion Diffusion Properties for the Electrolyte Material

You define the electrolyte name and the ion charge number (z). The charge number for the lithium ion is 1.0.

A label with the electrolyte name is used to identify all subsequent definitions required to specify the electrical and ion diffusion properties of the electrolyte.

You define the electrical conductivity, diffusion coefficient, molar activity coefficient, and the transference (migration). The electrical conductivity and diffusion coefficients are functions of the lithium ion concentration and the electrolyte porosity in the electrolyte. The molar activity coefficient and the transference are functions only of the lithium ion concentration. In addition, all of the electrolyte properties can depend on temperature and field variables. You can define Arrhenius temperature dependency for the electrical conductivity and diffusivity. When you define temperature dependence using a property table, the Arrhenius definition is ignored. For more details about the Arrhenius dependency, see Arrhenius Temperature Dependency.

Input File Usage

Use the following options to define the electrolyte name and charge number:

MATERIAL
*PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrolyte", 
LABEL="ABQ_EChemPET_Electrolyte"
"Electrolyte name",  $z$

Use the following options to define the electric conductivity of the electrolyte:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrolyte_ElecCond_Tabular", 
LABEL="ABQ_EChemPET_<Electrolyte name>_ElecCond_Tabular"
 $K_{e}$ ,  $C_{e}$ ,  $ε_{e}$ ,  $θ$ ,  $FV$ 
PARAMETER TABLE, TYPE="ABQ_EChemPET_Arrhenius",
LABEL="ABQ_EChemPET_<Electrolyte name>_CondArrhenius"
 $E_{a}$ ,  $θ_{ref}$

Use the following options to define the diffusion coefficient of the electrolyte:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrolyte_DiffTabular", 
LABEL="ABQ_EChemPET_<Electrolyte name>_DiffTabular"
 $D_{e}$ ,  $C_{e}$ ,  $ε_{e}$ ,  $θ$ ,  $FV$ 
PARAMETER TABLE, TYPE="ABQ_EChemPET_Arrhenius",
LABEL="ABQ_EChemPET_<Electrolyte name>_DiffArrhenius"
 $E_{a}$ ,  $θ_{ref}$

Use the following options to define the molar activity coefficient of the electrolyte:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrolyte_MolarActivityCoeff_fPM", 
LABEL="ABQ_EChemPET_<Electrolyte name>_MolarActivityCoeff_fPM"
 $\frac{{d ln f}_{\pm}}{{d ln C}_{e}}$ ,  $C_{e}$ ,  $θ$ ,  $FV$

Use the following options to define the transference for the electrolyte:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrolyte_Transference", 
LABEL="ABQ_EChemPET_<Electrolyte name>_Transference"
 $t_{+}$ ,  $C_{e}$ ,  $θ$ ,  $FV$

Defining Material Properties for the Porous Electrode and Microscale Particle

The following sections describe the material properties used to define the multiscale nature of the electrodes and separator progressing from the macroscale electrode level to the microscale particle level.

Electrode Definition

You must define various properties for the electrode at the macroscale, including a unique region name; a region identifier characterizing the electrode as either an anode, cathode, or separator; the volume fractions of the various phases in the electrode; and the tortuosity factors in the three material directions.

You can specify a utilization fraction to account for inaccessible regions in the electrode (see Ecker, 2015).

When lithium ions intercalate into a particle, the particle can swell resulting in a convection of the electrolyte away from that region. If this effect is significant and of interest to you, you can specify a factor of one for the convection coefficient. By default, the value for the convection coefficient is zero.

When particle swelling effects are included, the binder volume fraction, $ε_{b}$ , and solid volume fraction, $ε_{s}$ , are used to compute and update the electrolyte volume fraction, $ε_{e}$ .

Input File Usage

Use the following options to define the electrode name; region type (anode, cathode, or separator); and other electrode properties:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Definition", 
LABEL="ABQ_EChemPET_Electrode_Definition"
"Electrode name", "Region Type",  $ε_{s}$ ,  $ε_{e}$ ,  $ε_{b}$ ,  $ε_{inactive}$ , "Utilization Fraction", ,  $α_{z}$ ,  $α_{x}$ ,  $α_{y}$ ,  ${Conv}_{coef}$

Particle Definition

For each electrode you must define particle-specific properties such as the particle radius, the initial concentration of the particle, and the contribution factor of the particle when multiple particles are present. You can define multiple particle types within the same electrode.

Input File Usage

Use the following options to define the name and properties of the particle inside an electrode:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Particles", 
LABEL="ABQ_EChemPET_<Electrode name>_Particles"
"Particle name",  $R_{p}$ ,  $ε_{sParticle}$ ,  $C_{s_Init}$ , "Formulation Type"

Repeat the data line as often as necessary to define a list of particles within the same electrode.

Particle Layer Definition

Each particle can have one layer or multiple concentric layers. All layers within a particle must have the same material. For each layer, you must define a name for the layer and a weight percentage per layer.

Input File Usage

Use the following options to define one or more layers:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_Layers", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_Layers"
"Layer name", "Weight Fraction of Layer"

Repeat the data line as often as necessary to define additional layers.

Use the following options to define the open circuit potential for a layer in tabular format:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_Layer_OCPTabular", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_OCPTabular"
 $E_{ref}$ ,  ${\hat{C}}_{s}^{surf}$ ,  $θ$ ,  $FV$

Use the following options to define the open circuit potential for a layer in spline format:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_CubicSplineC2", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_CubicSplineC2"
 $X_{i}$ ,  $X_{i+1}$ ,  $C_{3}$ ,  $C_{2}$ ,  $C_{1}$ ,  $C_{0}$

Repeat the data line as often as necessary to cover the full range of the independent variable, $X$ .

Particle Layer Mesh

The microscale particle is modeled as a sphere over which a one-dimensional finite element mesh in spherical coordinates is created and the diffusion problem is solved. You must specify the discretization ( $N_{elemPerLayer}$ ) to use on each layer of the microscale particle. The total number of elements in a particle can be between 1 and 100. Typically, a mesh of 25 elements on a single particle gives good results. The meshing can be uniform or biased toward the outside of the sphere using a quadratic or equal volume approach.

Input File Usage

Use the following options to define the mesh discretization within a particle layer:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_Layer_Discretization", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_Discretization"
 $N_{elemPerLayer}$ , "Mesh Type"

Particle Layer Diffusion

You must define the theoretical maximum lithium capacity, $C_{s}^{\max}$ , of each layer material in the particle and the diffusion coefficient to use in the solution of the microscale lithium diffusion equations. Several formats are available to define the diffusion coefficient as a function of the normalized concentration, ${\hat{C}}_{s}$ : tabular format, logarithmic tabular format, and spline format.

For a tabular diffusion model, the diffusion coefficient is defined as a piecewise linear function of the normalized particle layer concentration.
For the logarithmic definition of the layer diffusion, you specify the base 10 logarithm of the diffusion coefficient as a tabular function of the normalized particle layer concentration.
The spline input form for the diffusion curve requires the diffusion curve as a function of the normalized concentration to be split into equal intervals along the x-axis. Each interval is then curve fit with a cubic spline equation. The parameter table has entries ordered as the start and end of the interval range and the four polynomial coefficients that correspond to the curve fit within that range.

In addition, the diffusion coefficient can be a function of temperature. For the spline format, only the Arrhenius form of temperature dependency is supported.

Input File Usage

Use the following options to define the theoretical maximum lithium capacity and the diffusion model type for a given particle layer:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_Layer_Diffusion", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_Diffusion"
 $C_{s}^{\max}$ , "Diffusion Data Type", "Chemical Diffusion Type"

Use the following options to define the diffusion coefficient using a tabular format:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_Layer_DsTabular", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_DsTabular"
 $D_{s}$ ,  ${\hat{C}}_{s}$

Use the following options to define the diffusion coefficient using a logarithmic tabular format:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_LogScale_Tabular", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_DsLogTabular"
 ${Log}_{10} D_{s}$ ,  ${\hat{C}}_{s}$

Use the following options to define the diffusion coefficient for a layer in spline format:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_CubicSplineC2", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_CubicSplineC2"
 $X_{i}$ ,  $X_{i+1}$ ,  $C_{3}$ ,  $C_{2}$ ,  $C_{1}$ ,  $C_{0}$

Repeat the data line as often as necessary to cover the full range of the independent variable, $X$ .

Use the following options to define the coefficients of the Arrhenius model:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Arrhenius",  
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_DiffArrhenius"
 $E_{a}$ ,  $θ_{ref}$

Particle Swelling

The intercalation/deintercalation process of lithium atoms can lead to significant changes in the volume of the spherical particles, which increases with the average lithium concentration within the particle. Particle swelling also results in the local displacement of the electrolyte (convection effects) to accommodate the new particle volume. While elastic deformation is not modeled explicitly by the thermal-electrochemical procedure, the effects of particle swelling on the electrochemical process are accounted for by considering its influence on the effective volume fractions of the solid and liquid phases, as well as on the wetted particle surface area per unit volume $(a_{s})$ .

The volume change in the particle due to a change in average concentration is characterized by a swelling coefficient, $Ω$ , as:

\frac{∆ V_{p}}{V_{p}^{0}} = Ω ∆ c_{s}^{avg} .

The solid volume fraction is defined as

ε_{s} = \frac{n_{p} V_{p}^{0}}{V_{macro}},

where $n_{p}$ is the number of particles in the macroscopic volume, and $V_{p}^{0}$ is the particle initial volume.

Assuming that the local particle volume change is accommodated by displacing away the electrolyte, with no overall macroscopic volume change, then

Δ ε_{s} = \frac{n_{p} Δ V_{p}}{V_{macro}} = \frac{n_{p} V_{p}^{0} Ω Δ c_{s}^{avg}}{V_{macro}} = ε_{s} Ω Δ c_{s}^{avg},

and, for the electrolyte

Δ ε_{e} = - Δ ε_{s} = - ε_{s} Ω Δ c_{s}^{avg} = - (1 - ε_{e}) Ω Δ c_{s}^{avg} .

Similarly, the active surface area per volume, $a_{s},$ is also affected by swelling with

Δ a_{s} = \frac{2}{3} a_{s} Ω Δ c_{s}^{avg} .

The electrochemical governing equations are modified accordingly to incorporate the effects of particle swelling on $ε_{s}$ , $ε_{e}$ , and $a_{s}$ .

You can include the effects of particle swelling in the simulation by specifying a swelling coefficient.

Input File Usage

Use the following options to define the swelling coefficient as a function of normalized particle concentration:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_Layer_SwellingTabular", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_<Layer name>_SwellingTabular"
 $Ω$ ,  ${\hat{C}}_{s}$

Butler-Volmer Definition

You must define the constants required to compute the current density using the Butler-Volmer equation. You can choose whether the particle surface area per unit volume, $a_{s}$ , is computed within Abaqus or specified directly as a single value. By default, Abaqus computes the current exchange density, $I_{0}$ . You can specify $\bar{I_{0}}$ in tabular form as a function of the normalized particle concentration and temperature. $\bar{I_{0}}$ is defined as follows for tabular definitions and does not include the electrolyte concentration terms:

\bar{I_{0}} = F k_{c}^{α_{a}} k_{a}^{α_{c}} C_{s}^{\max} {(1 - {\hat{C}}_{s}^{surf})}^{α_{a}} {({\hat{C}}_{s}^{surf})}^{α_{c}} .

Input File Usage

Use the following options to define the parameters used in the Butler-Volmer computation:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_ButlerVolmer", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_ButlerVolmer"
 $k_{c}$ ,  $k_{a}$ ,  $α_{c}$ ,  $α_{a}$ ,  $C_{e}^{ref}$ ,  $R_{SEI}$ ,  ${Stress}_{coef}$ ,  $a_{s}$

Use the following options to specify a tabular form of $\bar{I_{0}}$ as a function of the normalized surface particle concentration and, optionally, temperature and field variables:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_CurrXchgDens_Tabular", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_CurrXchgDens_Tabular"
 $\bar{I_{0}}$ ,  ${\hat{C}}_{s}^{surf}$ ,  $θ$ ,  $FV$

Use the following options to define the coefficients of the Arrhenius model:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Arrhenius",  
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_BVArrhenius"
 $E_{a}$ ,  $θ_{ref}$

Specifying the Joule Heat Fraction in the Different Phases

You specify the contribution of the different phases to the overall joule heat fraction using two parameter tables, one for the macroscale and one for the microscale. The entropy table is used to define the derivative of the open circuit potential with respect to temperature. This term is one of the five terms that contribute to the overall joule heat fraction.

Input File Usage

Use the following options to define the joule heat fraction in the macroscale and microscale:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_PowerLoss", 
LABEL="ABQ_EChemPET_<Electrode name>_PowerLoss"
 $β_{3}$ ,  $β_{4}$ ,  $β_{5}$

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_PowerLoss", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>_PowerLoss"
 $β_{1}$ ,  $β_{2}$

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemPET_Electrode_Particle_dUdTEntropy_Tabular", 
LABEL="ABQ_EChemPET_<Electrode name>_<Particle name>dUdTEntropy_Tabular"
 $\frac{∂U}{\partial θ}$ ,  ${\hat{C}}_{s}$

Arrhenius Temperature Dependency

An Arrhenius form of temperature dependence is also supported for the definition of many of the material properties. The Arrhenius form of temperature dependence provides a scaling of the corresponding material property given as:

X (θ) {= X}_{0} e^{\frac{E_{a}}{R} (\frac{1}{(θ - θ^{z})} - \frac{1}{(θ_{ref} - θ^{z})}),}

where

$E_{a}$: is the activation energy,
$R$: is the universal gas constant,
$θ^{z}$: is the absolute zero temperature,
$θ_{ref}$: is the reference temperature,
$X_{0}$: represents the value of material property at the reference temperature, and
$X$: is the resulting value after accounting for Arrhenius temperature dependency.

The Arrhenius form is ignored when you specify an explicit tabular temperature dependency in the property table definitions for the different properties (using the temperature parameter).

Input File Usage

Use the following options to define the coefficients of the Arrhenius model:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemPET_Arrhenius"
 $E_{a}$ ,  $θ_{ref}$

Universal Constants

You must define universal constants such as the Faraday number, gas constant, Avogadro's number, Boltzmann number, absolute zero temperature, and the elementary charge number.

Input File Usage

PHYSICAL CONSTANTS

Include File for Property and Parameter Table Definitions

The type definitions for the parameter and property tables described in the previous sections are described in Parameter Table Type Reference and Property Table Type Reference and available as an include file (see Including Model or History Data from an External File). You can use the abaqus fetch utility to obtain the file containing the type definitions of the parameter and property tables used by the electrochemistry framework.

abaqus fetch job=ABQ_EchemPET_Types.inp

Solution-Dependent State Variables

The electrochemistry solution framework requires the use of solution-dependent state variables at each integration point for the microscale problem at the anode and cathode. Seventy-five solution-dependent state variables are required to transfer information between the microscale and macroscale solutions. At the microscale level, the number of solution-dependent state variables that are required per particle is 822. Depending on the number of particles, $N_{p}$ , that are defined, the total number of state variables that must be declared is $75 + N_{p} \times 822$ . Only the anode and cathode regions of the battery require the definition of solution-dependent state variables. Because the microscale problem is not solved in the separator region, you are not required to define state variables for the separator region of the battery.

Input File Usage

Use the following options to define the solution-dependent state variables:

MATERIAL, NAME=ABQ_EchemPET_matName
DEPVAR
 $75 + N_{p} \times 822$

Elements

The simultaneous solution in a coupled thermal-electrochemical analysis requires the use of elements that have electric potential in the solid (degree of freedom 9), temperature (degree of freedom 11), electric potential in the electrolyte (degree of freedom 32), and ion concentration in the electrolyte (degree of freedom 33) as nodal variables. The coupled thermal-electrochemical elements are available in Abaqus/Standard in three dimensions only (see Coupled Thermal-Electrochemical Elements).

Output

In addition to the output quantities available for the coupled thermal-electric procedure, you can request the following output variables in a coupled thermal-electrochemical analysis.

Nodal output variables:

EPOT: Electric potential in the solid phase.
EPOTE: Electric potential in the fluid (electrolyte).
NNCE: Ion concentration in the fluid (electrolyte).
RECURE: Reaction current in the fluid (electrolyte).
RFLCE: Reaction ion concentration in the fluid (electrolyte).

Element integration point variables:

ECDE: Magnitude and components of electrical current density in the electrolyte.
MFLE: Ion flux rate in the electrolyte.
CONCE: Ion concentration in the electrolyte.

ELECPOTE: Electric potential in the electrolyte.

ELECPOT: Electric potential in the solid.

CONCGE: Magnitude and components of ion concentration gradient in the electrolyte.

EPGE: Magnitude and components of electrical potential gradient in the electrolyte.

For the microscale, the following subset of solution-dependent variables are most relevant:

SDV1: Volume-averaged concentration on the surface of the microscale based on the number of spherical particles defined in the model.
SDV2: Volume-averaged Butler-Volmer current based on the number of particles defined in the model.
SDV10: Sum of entropy generation and ohmic loss heat terms at the microscale (see Specifying the Joule Heat Fraction in the Different Phases).
SDV16: Volume fraction of the electrolyte (see Particle Swelling).
When convection effects are modeled, the volume fraction of the electrolyte changes at each integration point in the element.

If there is more than one particle, for the first particle:

SDV72: Concentration on the surface of the first particle.
SDV74: Average concentration in the particle.
SDV76: Butler-Volmer current computed for the first particle.

For the $N^{t h}$ particle, with $I = (N - 1) \times 822$ :

SDV(I+72): Concentration on the surface of the particle.
SDV(I+74): Average concentration in the particle.
SDV(I+76): Butler-Volmer current computed for the particle.

Input File Template

HEADING
*INCLUDE, INPUT=ABQ_EchemPET_Types.inp
*MATERIAL, NAME=ABQ_EChemPET_matName
*DEPVAR
*CONDUCTIVITY
*DENSITY
*SPECIFIC HEAT
*POROUS ELECTRODE THEORY
*PARAMETER TABLE
*PROPERTY TABLE
…
*STEP
*COUPLED THERMAL-ELECTROCHEMICAL
*BOUNDARY
*CECURRENT
*FILM
*END STEP

References

Ebner, M., , and V. Wood, “Tool for Tortuosity Estimation in Lithium-Ion Battery Porous Electrodes,” Journal of the Electrochemical Society, vol. 162, no. 2, 2015.
Ecker, M., S. Kabitz, I. Laresgoiti , and D. Sauer, “Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery II. Model Validation,” Journal of the Electrochemical Society, vol. 162, no. 9, 2015.
Ecker, M., T. K. D. Tran, P. Dechent, S. Kabitz, A. Warnecke , and D. Sauer, “Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery I. Determination of Parameters,” Journal of the Electrochemical Society, vol. 162, no. 9, 2015.
Kumaresan, K., G. Sikha , and R E. White, “Thermal Model for a Li-Ion Cell,” Journal of the Electrochemical Society, vol. 155, no. 2, 2008.
Liu, B., X. Wang, H. S. Chen, S. Chen, H. Yang, J. Xu, H. Jiang, B. Liu , and D. N. Fang, “A Simultaneous Multiscale and Multiphysics Model and Numerical Implementation of a Core-Shell Model for Lithium-Ion Full-Cell Batteries,” Journal of Applied Mechanics, vol. 86, 2019.
Newman, J., , and K. E. Thomas-Alyea, “Electrochemical Systems,” Wiley-Interscience, Third Edition, 2004.
Wu, B., “Modeling and Design of Lithium-Ion Batteries: Mechanics and Electrochemistry,” Doctoral Dissertation, University of Michigan, 2019.