Modeling Solid Electrolytes and Solid-State Batteries

Solid-State Batteries

Solid-state batteries are composed of a solid anode, a solid cathode, and a solid electrolyte. The electrolyte also acts as a separator as shown in Figure 1.

Solid-state lithium ion batteries are manufactured using thin film methods, with the thicknesses of different regions in the order of a few micrometers. Since the electrodes and electrolyte are solid, all the electrochemical reactions happen at the anode-electrolyte and cathode-electrolyte interfaces. In this section, for consistency, the electrolyte region can be interchangeably referred to as the separator. The anode is typically made of metallic lithium, while the cathode is based on Lithium Cobalt Oxide ( $L i C o O_{2}$ ). The electrolyte is solid and might consist of an amorphous Lithium Phosphate ( $L i_{3} P O_{4}$ ), which can be N-doped and commonly referred to as LiPON. Current collectors might be placed on the outer sides of each electrode.

During a discharge process, a reversible electrochemical reaction at the anode-electrolyte interface converts Li to $L i^{+}$ :

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

Here,

Li - Θ

represents lithium in the solid electrode, which through the deintercalation reaction, transforms into a 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.In the solid electrolyte, only a fraction of

L i^{+}

ions are mobile and denoted by a fraction

δ

. It is assumed that a portion of the mobile lithium ions would kinetically bond with the solid electrolyte and become immobile, which can be expressed as:

L i^{+} + e^{-} ⇌_{k_{d}}^{k_{r}} L i_{0},

where

L i_{0}

refers to the immobile Li bonded with anions in the solid electrolyte and

k_{r}, k_{d}

are the reverse reaction rate and dissociation rate, respectively. The net production rate of

{Li}^{+}

ions that can pass through the solid electrolyte is obtained as:

r = k_{r} [(C_{0} δ)^{2} - C_{L i^{+}}^{2}],

where

C_{0}

is the total amount of lithium. The transport of lithium ion in the solid electrolyte is driven by both diffusion and migration.

At the solid electrolyte-cathode interface, the charge transfer reaction can be expressed as:

L i C o O_{2} ⇌ x L i^{+} + x e^{-} + L i_{(1 - x)} C o O_{2}

where

x

is the stoichiometric coefficient. After the charge transfer reaction at the interface, the lithium intercalates within the cathode through a diffusion process.

Terminology

In the discussions that follow, "species" refers to the chemical species that participates in the primary electrochemical reactions in the battery. It is assumed in this section that the solid electrolyte battery is lithium-based. Therefore, "species" refers to lithium; but, in principle, it could be another chemical species.

An important physical mechanism that is active at the cathode of a solid electrolyte battery is diffusion of the chemical species. The discussion and user interface for the section on cathode behavior refer to "species." However, "species" can be interchanged with lithium for the particular kind of solid electrolyte battery discussed here.

Governing Equations

The following wide range of physical phenomena determine the operation of a solid state battery:

Conduction of electrons in the solid electrodes.
Butler-Volmer charge transfer reactions at the interface of the anode-solid electrolyte.
Diffusion and migration of ions in the solid electrolyte.
Butler-Volmer charge transfer reactions at the interface of the solid electrolyte and the cathode.
Diffusion of species (lithium) in the solid cathode.
Heat generation due to current flow.

Conduction of Electrons in the Solid Electrodes

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

\nabla \cdot I_{s} = 0,

where

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

Here,

K_{s}^{eff}

is the effective electrical conductivity of the solid phase, and

Φ_{s}

is the electric potential in the solid phase.

Interaction at the Anode-Solid Electrolyte Interface

At the interface between the separator and the solid lithium anode, a charge-transfer reaction occurs following a Butler-Volmer kinetics that is expressed using the following equations:

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

Here,

Li - Θ

represents lithium in the solid electrode that, through the deintercalation reaction, transforms into a lithium ion in the electrolyte,

{Li}^{+}

, plus an electron,

e^{-}

, and leaves behind a vacancy (or intercalation site) in the solid,

Θ

.

The general form of the Butler-Volmer current expression at the interface of a solid anode and solid electrolyte is:

I_{B V}^{a - s} = I_{0}^{a - s} (\exp [\frac{α_{a} (z F η)}{R (θ - θ^{z})}] - \exp [\frac{{- α}_{c} (z F η)}{R (θ - θ^{z})}])

and

I_{0}^{a - s} = F k_{a} {(\frac{C_{e}}{C_{e}^{r e f}})}^{α_{a}},

where the overpotential,

η

, is defined as:

η = Φ_{s} - Φ_{e} - E_{ref} (θ) .

In the above equations,

$F$: is Faraday's constant;
$R$: is the universal gas constant;
$z$: is the charge number of the lithium ion battery;
$θ$: is temperature;
$θ^{z}$: is the absolute zero temperature;
$α_{c}, α_{a}$: are the cathodic and anodic transfer coefficients, respectively;
$C_{e}^{r e f}$: is the reference value of lithium ion concentration in the separator;
$C_{e}$: is the concentration of mobile lithium ion;
$E_{ref}$: is the open circuit potential (OCP) as a function of $θ$ (The typical value in a solid anode is zero.);
$ϕ_{s}$: is the electric potential from the anode side; and
$ϕ_{e}$: is the electric potential from the separator side.

The electrochemical interaction at the solid electrode-separator interface is represented on the solid electrode side as a surface current density of magnitude ${- S}_{f a c} I_{B V}^{a - s}$ . The dimensionless “surface factor,” $S_{f a c}$ , captures the effects of surface irregularities that can result in the availability of a larger surface area over which the electrochemical interaction happens. On the separator side of the interface, the electrochemical interaction results in two separate surface fluxes, both applied on the solid electrolyte phase: an electrolyte current density of magnitude $S_{f a c} I_{B V}^{a - s},$ and an ion flux of magnitude $\frac{S_{f a c} I_{B V}^{a - s}}{F}$ . For more details on the electrochemical governing equation, see Conduction of Electrons in the Solid Electrodes and Diffusion and Migration in the Solid Electrolyte. The current surface interaction load is treated as surface flux at the interface of the anode and solid electrolyte.

Diffusion and Migration in the Solid Electrolyte

The transport of lithium ions in the solid electrolyte is by a combination of diffusion and migration, which is modeled using the Nernst-Planck equation and Fick’s second law:

N_{e} = - D_{e} . \nabla C_{e} + (\frac{F}{R (θ - θ^{z})}) D_{e} C_{e} . \nabla Φ_{e},

\frac{d C e}{d t} = - \nabla \cdot N_{e},

where

$N_{e}$: is the flux of lithium ions;
$D_{e}$: is the diffusion coefficient; and
$C_{e}$: is the concentration of lithium ion.

The equilibrium equation for the current density in the electrolyte, $I_{e}$ , is given by:

\nabla \cdot I_{e} = 0,

where

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

Here,

K_{e}^{eff}

is the effective electrical conductivity of the electrolyte, and

Φ_{e}

is the electric potential in the electrolyte phase.

Interaction at the Solid Electrolyte - Cathode Interface

At the solid electrolyte-cathode interface, the charge transfer reaction can be expressed as:

L i C o O_{2} ⇌ x L i^{+} + x e^{-} + L i_{(1 - x)} C o O_{2}

This charge-transfer reaction occurs following a Butler-Volmer kinetics that is expressed using the following equations:

I_{B V}^{s - c} = I_{0}^{s - c} {\exp [\frac{α_{c} (z F η)}{R (θ - θ^{z})}] - \exp [\frac{{- (1 - α}_{c}) (z F η)}{R (θ - θ^{z})}]},

where the overpotential,

η

, is defined as:

η = Φ_{s} - Φ_{e} - E_{ref} (θ),

and $I_{0}^{s - c}$ is

I_{0}^{s - c} = F k_{c} {(\frac{(C_{s, \max} - C_{s}) C_{e}}{(C_{s, \max} - C_{s, \min}) C_{e}^{r e f}})}^{α_{c}} {(\frac{C_{s} - C_{s, \min}}{C_{s, \max} - C_{s, \min}})}^{(1 - α_{c})}

In the above equations,

$F$: is Faraday's constant;
$R$: is the universal gas constant;
$z$: is the charge number of the battery;
$θ$: is temperature;
$θ^{z}$: is the absolute zero temperature;
$α_{c}$: is the cathodic transfer coefficients, respectively;
$C_{e}^{r e f}$: is the reference value of lithium ion concentration in the separator;
$C_{e}$: is the concentration of mobile lithium ion;
$E_{ref}$: is the open circuit potential (OCP) as a function of $θ$ ;
$ϕ_{s}$: is the electric potential from the cathode side;
$ϕ_{e}$: is the electric potential from the separator side;
$C_{s, \max}, C_{s, \min}$: are the maximum and minimum possible concentration of lithium in the cathode; and
$C_{s}$: is the concentration of lithium in the cathode.

The electrochemical interaction at the separator-cathode interface is represented on the cathode side as two separate surface fluxes: a surface current density of magnitude ${- S}_{f a c} I_{B V}^{s - c}$ and a species flux of magnitude $\frac{S_{f a c} I_{B V}^{s - c}}{F}$ . The dimensionless “surface factor,” $S_{f a c}$ , captures the effects of surface irregularities that can result in the availability of a larger surface area over which the electrochemical interaction happens. On the separator side of the interface, the electrochemical interaction results in two separate surface fluxes, both applied on the solid electrolyte phase: an electrolyte current density of magnitude $S_{f a c} I_{B V}^{s - c},$ and an ion flux of magnitude $\frac{S_{f a c} I_{B V}^{s - c}}{F}$ . For more details on the electrochemical governing equations, see Conduction of Electrons in the Solid Electrodes, Diffusion of Lithium within the Cathode, and Diffusion and Migration in the Solid Electrolyte. The current surface interaction load is treated as surface flux at the interface of the cathode and solid electrolyte.

Diffusion of Lithium within the Cathode

The diffusion of lithium within the cathode is solved using a diffusion equation, which is expressed as:

\frac{\partial C_{s}}{\partial t} = \nabla . (D_{s} . (\nabla C_{s})),

where

$D_{s}$: is the diffusion coefficient in the cathode for the species (lithium); and
$C_{s}$: is the species (lithium) concentration in the cathode.

Joule Heating at the Electrode–Solid Electrolyte Interface

The total heat generation at the interface of the solid electrode and the solid electrolyte is attributed to two sources: flow of current at the electrode-separator interface and entropy generation. The amount of energy converted into heat from each of the source terms can be scaled using the conversion factors, $β_{1}, β_{2}$ :

Q_{tot} = β_{1} Q_{1} + β_{2} Q_{2},

where

$Q_{1} = S_{f a c} I_{BV} η$: is the ohmic loss at the electrode-separator interface; and
$Q_{2} = S_{f a c} I_{BV} (θ - θ^{z}) \frac{∂U}{\partial θ}$: 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.

Joule Heating within the Electrodes and Separator

The total heat generation within the electrode and separator is attributed to the contribution of two different sources: flow of current in the solid phase and flow of current in the electrolyte phase. The amount of energy converted into heat from each of the source terms can be scaled using the conversion factors, $β_{3}, β_{4}$ :

Q_{tot} = β_{3} Q_{3} + β_{4} Q_{4},

where

$Q_{3} = K_{s}^{eff} \nabla Φ_{s} . \nabla Φ_{s}$: is the ohmic loss in the electrode; and
$Q_{4} = K_{e}^{eff} \nabla Φ_{e} . \nabla Φ_{e}$: is the ohmic loss in the separator or electrolyte.

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

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

[\begin{matrix} K_{u u} & K_{u φ_{s}} & K_{u θ} & K_{u φ_{e}} & K_{u C_{e}} & K_{u C_{s}} \\ K_{φ_{s} u} & K_{φ_{s} φ_{s}} & K_{φ_{s} θ} & K_{φ_{s} φ_{e}} & K_{φ_{s} C_{e}} & K_{φ_{s} C_{s}} \\ K_{θ u} & K_{θ φ_{s}} & K_{θ θ} & K_{θ φ_{e}} & K_{θ C_{e}} & K_{θ C_{s}} \\ K_{φ_{e} u} & K_{φ_{e} φ_{s}} & K_{φ_{e} θ} & K_{φ_{e} φ_{e}} & K_{φ_{e} C_{e}} & K_{φ_{e} C_{s}} \\ K_{C_{e} u} & K_{C_{e} φ_{s}} & K_{C_{e} θ} & K_{C_{e} φ_{e}} & K_{C_{e} C_{e}} & K_{C_{e} C_{s}} \\ K_{C_{s} u} & K_{C_{s} φ_{s}} & K_{C_{s} θ} & K_{C_{s} φ_{e}} & K_{C_{s} C_{e}} & K_{C_{s} C_{s}} \end{matrix}] {\begin{matrix} Δ u \\ Δ φ_{s} \\ Δ θ \\ Δ φ_{e} \\ Δ C_{e} \\ Δ C_{s} \end{matrix}} = {\begin{matrix} R_{u} \\ R_{φ_{s}} \\ R_{θ} \\ R_{φ_{e}} \\ R_{C_{e}} \\ R_{C_{s}} \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 solid electrolyte and lithium in the cathode. Electrical transient effects are not included in the equations because they are very rapid compared to the characteristic times of thermal and diffusion effects.

Input File Usage

COUPLED THERMAL-ELECTROCHEMICAL, STEADY STATE

COUPLED TEMPERATURE-DISPLACEMENT, 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

COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL

Spurious Oscillations due to Small Time Increments

By default, Abaqus/Standard uses nodal integration of the heat capacity term for first-order elements in a transient analysis, resulting in a lumped treatment for this term. This treatment eliminates nonphysical oscillations; however, it can lead to locally inaccurate solutions, especially in terms of the heat flux for small time increments. If smaller time increments are required, you should use a finer mesh in regions where the temperature changes occur. For more information about time integration involving the temperature degree of freedom, see Spurious Oscillations due to Small Time Increments in Uncoupled Heat Transfer Analysis.

Initial Conditions

By default, the initial values of electric potential in the solid, temperature, electric potential in the electrolyte, ion concentration, and species 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 the initial lithium concentration in the cathode, initial ion concentration in the solid electrolyte, and solid and electrolyte electric potentials in the electrodes. The ion concentration in the electrolyte is typically assumed to be uniform in the cell.

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.
Species concentration in the electrode, $C_{s} = C_{s} (x, t)$ (degree of freedom 34) 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 only of grounding (setting to zero) the solid electric potential at the anode. Thermal boundary conditions vary.

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 ion or species concentrations and on the 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 solid electrolyte framework requires that the material definition contain the complete specification of properties required for the solid electrolyte theory, as described below and in the sections that follow. In addition, the material name must begin with "ABQ_EChemSET_" to enable the fully coupled solution between the various fields at the different electrodes. Special-purpose parameter and property tables of type names starting with “ABQ_EChemSET_” are required in these material definitions (see Parameter Table Type Reference and Property Table Type Reference). In a fully coupled thermal-electrochemical-structural analysis, the material definition must include thermal, electrical, electrochemical, and mechanical properties.

Input File Usage

MATERIAL, NAME=ABQ_EChemSET_name
SOLID ELECTROLYTE THEORY

Defining Properties to Model Solid-State Batteries

You must specify the constants required to define the Butler-Volmer kinetics using different table collections for the anode-separator interface and the cathode-separator interface.These table collections can be used in the surface interaction behavior or in a surface-based load. For the cathode and separator, you must define material properties as defined below.

Interaction at the Anode-Solid Electrolyte Interface

You must define the parameters for the Butler-Volmer kinetics and the open circuit potential for the anode in tabular form as a function of temperature. Optionally, you can specify an Arrhenius form of temperature dependency for $I_{0}^{a - s}$ (see Arrhenius Temperature Dependency.) If Joule heating effects are important, you must also specify the Joule heating losses and the entropy table that is used to define the derivative of the open circuit potential with respect to temperature.

Input File Usage

Use a table collection definition to include all properties (listed below) required to define the Butler-Volmer kinetics:

TABLE COLLECTION

Use the following options to define the coefficients of the Butler-Volmer kinetics:

PARAMETER TABLE, TYPE="ABQ_EChem_LMB_ButlerVolmerInterf"
 $k_{c}$ ,  ,  $α_{c}$ ,  $α_{a}$ ,  $C_{e}^{ref}$ ,  ,  ,  $S_{f a c}$ ,  $z$

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

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

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

PROPERTY TABLE, TYPE="ABQ_EChem_LMB_OCPTabular"
 $E_{ref}$ ,  $θ$

Use the following options to define the Joule heat fraction at the interface:

PARAMETER TABLE, TYPE="ABQ_EChem_LMB_Interface_PowerLoss"
 $β_{1}$ ,  $β_{2}$

PROPERTY TABLE, TYPE="ABQ_EChem_LMB_dUdTEntropy_Tabular"
 $\frac{∂U}{\partial θ}$ ,  $θ$

Separator Material Definition

You must define the solid electrolyte properties for the separator region using the tables specified below.

Input File Usage

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

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemSET_Electrode_Definition", 
LABEL="ABQ_EChemSET_Electrode_Definition"
"Electrode name", "Region Type"

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

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

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

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemSET_Electrolyte_ElecCond_Tabular", 
LABEL="ABQ_EChemSET_<Electrolyte name>_ElecCond_Tabular"
 $K_{e}$ ,  $C_{e}$ , $θ$ ,  $FV$ 
PARAMETER TABLE, TYPE="ABQ_EChemSET_Arrhenius",
LABEL="ABQ_EChemSET_<Electrolyte name>_CondArrhenius"
 $E_{a}$ ,  $θ_{ref}$

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

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemSET_Electrolyte_DiffTabular", 
LABEL="ABQ_EChemSET_<Electrolyte name>_DiffTabular"
 $D_{e}$ ,  $C_{e}$ ,  $θ$ ,  $FV$ 
PARAMETER TABLE, TYPE="ABQ_EChemSET_Arrhenius",
LABEL="ABQ_EChemSET_<Electrolyte name>_DiffArrhenius"
 $E_{a}$ ,  $θ_{ref}$

Use the following option to define the solid transfer reaction:

MATERIAL
*PARAMETER TABLE, TYPE="ABQ_EChemSET_Electrolyte_Solid_TransferReaction", 
LABEL="AABQ_EChemSET_<Electrolyte name>_Solid_TransferReaction"
 $δ$ ,  $K_{r}$ , $C_{0}$

Use the following option to define the Joule heat fraction within the separator region:

PARAMETER TABLE, TYPE="ABQ_EChemSET_Electrode_PowerLoss"
 $β_{3}$ 
,  $β_{4}$

Interaction at the Solid Electrolyte-Cathode Interface

You must define $I_{0}^{s - c}$ either in terms of parameters or in in tabular form as a function of $C_{e}$ and $C_{s}$ . In addition, you must define the open circuit potential for the cathode in tabular form as a function of $C_{s}$ . Optionally, you can specify an Arrhenius form of the temperature dependency for $I_{0}^{s - c}$ (see Arrhenius Temperature Dependency.) If Joule heating effects are important, you must also specify the Joule heating losses and the entropy table that is used to define the derivative of the open circuit potential with respect to temperature.

Input File Usage

Use a table collection definition to include all properties (listed below) required to define the Butler-Volmer kinetics:

TABLE COLLECTION

Use the following options to define the coefficients of the Butler-Volmer kinetics:

PARAMETER TABLE, TYPE="ABQ_EChemSET_ButlerVolmerInterf"
 $k_{c}$ , ,  $α_{c}$ ,  $α_{a}$ ,  $C_{s}^{\min}$ ,  $C_{s}^{\max}$ ,  $C_{e}^{ref}$ ,  $S_{f a c}$ ,  $z$

Use the following options to specify a tabular form of $I_{0}^{s - c}$ as a function of $C_{e}$ and $C_{s}$ :

PROPERTY TABLE, TYPE="ABQ_EChemSET_CurrXchgDens_Tabular"
 $I_{0}^{s - c}$ ,  $C_{e}$ ,  $C_{s}$

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

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

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

PROPERTY TABLE, TYPE="ABQ_EChemSET_OCPTabular"
 $E_{ref}$ ,  $C_{s}$

Use the following options to define the Joule heat fraction at the interface:

PARAMETER TABLE, TYPE="ABQ_EChemSET_Interface_PowerLoss"
 $β_{1}$ ,  $β_{2}$

PROPERTY TABLE, TYPE="ABQ_EChemSET_dUdTEntropy_Tabular"
 $\frac{∂U}{\partial θ}$ ,  $θ$

Cathode Region Material Definition

You must define the properties for the cathode region using the tables specified below.

Input File Usage

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

MATERIAL
PARAMETER TABLE, TYPE="ABQ_EChemSET_Electrode_Definition", 
LABEL="ABQ_EChemSET_Electrode_Definition"
"Electrode name", "Region Type"

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

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemSET_Electrode_ElecCond_Tabular", 
LABEL="ABQ_EChemSET_<Electrode name>_ElecCond_Tabular"
 $K_{s}$ ,  $C_{s}$ ,  $θ$ ,  $FV$

Use the following options to define the species diffusivity of the solid phase:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_EChemSET_Electrode_DiffTabular", 
LABEL="ABQ_EChemSET_<Electrode name>_DiffTabular"
 $D_{s}$ ,  $C_{s}$ ,  $θ$ ,  $FV$

Use the following option to define the Joule heat fraction within the cathode region:

PARAMETER TABLE, TYPE="ABQ_EChemSET_Electrode_PowerLoss"
 $β_{3}$ 
,  $β_{4}$

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

MATERIAL
CONDUCTIVITY
DENSITY
SPECIFIC HEAT

Defining a Butler-Volmer Surface Interaction

Abaqus/Standard offers two options to define the Butler-Volmer surface interaction.

When the different regions (that is, the solid lithium anode, solid electrolyte, and solid cathode) are modeled using matching meshes with shared nodes, you can define the Butler-Volmer interaction as a surface-based load on the facets of the anode and cathode elements, respectively.

If the meshes on the two sides of the interface between the solid anode, separator, and solid cathode are dissimilar and do not share nodes, you can use a surface interaction property to define an interface reaction behavior.

For both definitions you must specify the name of the table collection that includes the corresponding surface interaction properties, as described in the previous section.

Input File Usage

Use the following option to define a surface-based Butler-Volmer load:

STEP,
MULTIPHYSICS LOAD, DEFINITION=BUTLER-VOLMER

Use the following options to define an interface reaction behavior:

SURFACE INTERACTION
INTERFACE REACTION, TYPE=BUTLER-VOLMER
CONTACT
CONTACT PROPERTY ASSIGNMENT

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), ion concentration in the electrolyte (degree of freedom 33), and species concentration in the solid cathode (degree of freedom 34) as nodal variables. Coupled thermal-electrochemical elements are available in Abaqus/Standard only in three dimensions (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 solid electrolyte.
NNCE: Ion concentration in the solid electrolyte.
RECURE: Reaction current in the solid electrolyte.
RFLCE: Reaction ion concentration in the solid electrolyte.
NNCS: Species concentration.
RFLCS: Reaction species concentration.

Element Integration point output variables:

CONCS: Species concentration
MFLS: Species concentration flux.

In addition, in a fully coupled thermal-electrochemical-structural analysis you can request the Abaqus/Standard output variables for mechanical degrees of freedom.

References

Danilov, D., A. H. Niessen , and P. H. L. Notten, “Modeling All-Solid-State Li-Ion Batteries,” Journal of the Electrochemical Society, vol. 158, no. 3, 2011.
Tian, H-K., , and Y. Qi, “Simulation of the Effect of Contact Area Loss in All-Solid-State Li-Ion Batteries,” Journal of the Electrochemical Society, vol. 164, no. 11, 2017.
Raijmakers, L. H. J., D. L. Danilov, R-A. Eichel , and P. H. L. Notten, “An Advanced All-Solid-State Li-Ion Battery Model,” Electrochimica Acta, vol. 330, no. 3, 2020.