Modeling Solid Electrodes in Lithium Metal Batteries

A typical lithium metal battery has an anode that is made of solid lithium while the cathode is a porous electrode. The anode and cathode regions are separated by an electrically insulating porous separator, which allows ions to move across from anode to cathode and vice versa. Figure 1 shows a typical configuration of a solid lithium battery. Lithium metal batteries tend to have a higher capacity and reduced size compared to batteries with all porous electrodes.

Figure 1. Schematic diagram of a solid lithium metal battery.

Since the anode is not porous, the Newman porous electrode theory framework is not required to model the anode. Instead, the behavior of the anode can be characterized using classical thermal-electrical-structural physics. The separator and cathode regions are modeled as porous electrodes as described in Coupled Thermal-Electrochemical Analysis. The Abaqus lithium metal battery models assume that the interface between the separator and the anode can be characterized using a surface-based Butler-Volmer load (see Figure 1) that accounts for fluid electric potential and ion concentration on the separator side and the solid electric potential on the anode side.

The governing equations for the thermal-electrochemical process are based on the porous electrode theory and are described in detail in Governing Equations in Coupled Thermal-Electrochemical Analysis. 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:

The general form of the Butler-Volmer current expression is:

In the above equations,

$F$

is Faraday's constant;

$R$

is the gas constant;

$z$

is the charge number of the lithium ion battery;

$\theta$

is temperature;

${\theta }^z$

is the absolute zero temperature;

${\alpha }_c,{\alpha }_a$

are the cathodic and anodic transfer coefficients, respectively;

$C_e^{\mathrm{ref}}$

is the reference value of lithium ion concentration in the electrolyte;

$C_e$

is the ion concentration in the electrolyte from the separator side;

$E_{\mathrm{ref}}$

is the open circuit potential (OCP) as a function of $\theta$ . The typical value in a solid anode is zero.

${\varphi }_s$

is the electric potential from the anode side; and

${\varphi }_e$

is the fluid 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 ${-S}_{fac}{I}_{BV}$ . $S_{fac}$ represents a dimensionless “surface factor” that 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 electrolyte phase: a fluid current density of magnitude $S_{fac}{I}_{BV}$ and an ion flux of magnitude $\frac{S_{fac}{I}_{BV}}{F}$ , where $F$ is the Faraday constant. For more details on the electrochemical governing equation, see Conduction of Electrons in the Solid Phase and Diffusion and Migration of Lithium Ions in the Liquid Phase. However, an important distinction between the referenced governing equation sections and the current surface interaction load is that the conduction and the diffusion/migration equations in the referenced sections treat the effects of the Butler-Volmer kinetics as a volumetric source. Here, they contribute as surface fluxes for the solid lithium electrode battery.

The total heat generation at the interface of the solid electrode and the fluid electrolyte is attributed to two sources: flow of current at the solid-liquid interface and entropy generation. The amount of energy converted into heat from each of the source terms can be scaled using the conversion factors, ${\beta }_1,{\beta }_2$ :

$Q_1=S_{fac}{I}_{\mathrm{BV}}\eta$

is the ohmic loss at the solid-liquid interface; and

$Q_2=S_{fac}{I}_{\mathrm{BV}}(\theta -{\theta }^z)\frac{\mathrm{\partial U}}{\partial \theta }$

is the entropy generation, where $\frac{\mathrm{\partial U}}{\partial \theta }$ is the derivative of the open circuit potential with respect to temperature that you can specify in tabular form.

You must specify the constants required to define the Butler-Volmer kinetics for a given surface interaction using a table collection. The name of the table collection must be included in the definition of the corresponding Butler-Volmer surface interaction (Defining a Butler-Volmer Surface Interaction) .

You must define $\overline{I_0}$ and the open circuit potential for the anode in tabular form as a function of temperature. Optionally, you can specify the temperature dependency of $\overline{I_0}$ using an Arrhenius definition (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.

TABLE COLLECTION

PARAMETER TABLE, TYPE="ABQ_EChem_LMB_ButlerVolmerInterf"
, , ${\alpha }_c$, ${\alpha }_a$, $C_e^{\mathrm{ref}}$, , , $S_{fac}$, $z$

PROPERTY TABLE, TYPE="ABQ_EChem_LMB_CurrXchgDens_Tabular"
$\overline{I_0}$, $\theta$

PARAMETER TABLE, TYPE="ABQ_EChem_LMB_Arrhenius"
$E_a$, ${\theta }_{\mathrm{ref}}$

PROPERTY TABLE, TYPE="ABQ_EChem_LMB_OCPTabular"
$E_{\mathrm{ref}}$, $\theta$

PARAMETER TABLE, TYPE="ABQ_EChem_LMB_Interface_PowerLoss"
${\beta }_1$, ${\beta }_2$

PROPERTY TABLE, TYPE="ABQ_EChem_LMB_dUdTEntropy_Tabular"
$\frac{\mathrm{\partial U}}{\partial \theta }$, $\theta$

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

When the two regions (namely, the solid lithium anode and the porous separator) 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 elements.

If the meshes on the two sides of the interface between the solid anode and the separator 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 a table collection that includes the corresponding Butler-Volmer kinetics properties for the interface (Defining Properties for Butler-Volmer Kinetics).

STEP,
MULTIPHYSICS LOAD, DEFINITION=BUTLER-VOLMER

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