Fully Coupled Thermal-Electrochemical-Structural Analysis

The primary example of a battery electrochemistry application is the charging and discharging of lithium-ion battery cells. During the charging cycle, the lithium ions are extracted (deintercalated) from the active particles of the positive electrode (cathode). This process results in a reduction of the volume of the active particles. The ions move through the electrolyte by migration and diffusion from the positive electrode to the negative electrode (anode). At the anode, the ions intercalate into the active particles. This process results in an increase of the volume of the active particles and induces significant variation in tortuosities on both electrodes, thus strongly influencing the overall electrochemical behavior. Heat is generated during the flow of current in the solid and liquid phases, flow of current in the solid-liquid interface, and flow of ions in the electrolyte. During discharging, the cycle is reversed.

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 theory is based on a concurrent solution of a highly coupled multiphysics-multiscale formulation. For further details on the thermal-electrochemical analysis, see Coupled Thermal-Electrochemical Analysis.

In some applications, a detailed understanding of the effects of the thermal-electrochemical fields on the mechanical state (deformations) of the lithium-ion cell is important and can have a strong influence on the overall performance of the cell. Battery performance characteristics (such as energy storage capacity and discharge voltages) can degrade with deformations caused by particle swelling and thermal strains. Large deformations can cause failure of the separator, resulting in thermal runaways in batteries. In such applications, the coupling between temperature and displacement fields may be due to temperature-dependent material properties, internal heat generation, or thermal expansion. The volume changes in the active electrode particles resulting from the intercalation/deintercalation of lithium ions are modeled as particle concentration–dependent eigenstrains at the macroscale level. These volume changes affect the displacement field in the cell and porosity and tortuosity evolutions. In addition, they generate a convective transport term in the electrolyte that influences the movement of lithium ions through the electrolyte.

The governing equations for the thermal-electrochemical process are based on the porous electrode theory and are described in detail in Governing Equations. The governing equations for particle swelling are also described in Particle Swelling. The volumetric strain that is computed based on particle swelling affects the macroscale quantities, such as the solid volume fraction, ${\epsilon }_s$ ; porosities; tortuosities; and eigenstrain.

Eigenstrain (also referred to as inherent strain, assumed strain, or "stress-free" strain) is an engineering concept used to account for all sources of inelastic deformation that lead to residual stresses and distortions in manufactured components. For example, thermal strains are an example of eigenstrains. Abaqus solves for mechanical equilibrium of all internal and external forces in the system.

In linear elastic deformation, the stress induced by an eigenstrain can be represented as

where

$\sigma$

is the Cauchy stress;

$D^{el}$

is the elasticity matrix;

$\epsilon$

is the total strain;

${\epsilon }^{*}$

is the eigenstrain; and

${\epsilon }^{el}$

is the elastic strain.

Using constitutive equations (such as that shown above), eigenstrains can be used to compute the stresses coming from mechanical, thermal, and microstructural sources.

It is possible to include the effects of particle swelling in both the coupled thermal-electrochemical and the coupled thermal-electrochemical-structural analyses. In a purely coupled thermal-electrochemical analysis, particle swelling results in convection of the electrolyte, which leads to changes in the porosity and the tortuosity of the electrodes. In a coupled thermal-electrochemical-structural analysis, particle swelling also results in mechanical deformations that are modeled as eigenstrains and results in stresses in the medium. Such mechanical deformations can also have an impact on the performance of a battery cell.

A fully coupled solution scheme is needed when the stress analysis is dependent on the other fields involved in an electrochemical analysis, such as temperature, electric potentials in the solid and electrolyte, and ion concentration. In Abaqus/Standard, the temperature is integrated in time using a backward-difference scheme. The nonlinear coupled system is solved using Newton's method. The coupled thermal-electrochemical-structural analysis in Abaqus uses an exact implementation of Newton’s method, leading to an unsymmetric Jacobian matrix in the form:

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.

*COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL, STEADY STATE

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.

*COUPLED TEMPERATURE-DISPLACEMENT, ELECTROCHEMICAL

The integration procedure Abaqus/Standard uses for the transient heat transfer equation introduces a relationship between the minimum usable time increment and the element size. Time increments below this minimum can lead to spurious oscillations in the temperature field. By default, Abaqus/Standard uses nodal integration for 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 additional details for time integration involving temperature degree of freedom, see Spurious Oscillations due to Small Time Increments in Uncoupled Heat Transfer Analysis.