The physics-based battery swelling model is based on the Newman’s P2D model (see Newman’s P2D Model) and includes the swelling effects. The Newman’s P2D model resolves the lithium transport and the lithium insertion/desertion reactions at the scale of the anode-separator-cathode sandwich layer. The swelling effects are modeled by considering the deformation of the electrode active particles during the lithium insertion/desertion reactions and the elastic deformation of the sandwich layer under externally applied pressure. Consequently, during battery charging, the geometric configuration of the P2D model deviates from the initial (or reference) geometric configuration due to deformation as shown schematically in Figure 19.6: Physics-Based Battery Swelling Model: Reference and Deformed Configurations.

Figure 19.6: Physics-Based Battery Swelling Model: Reference and Deformed Configurations

In the swelling battery model, the P2D equations are solved in the reference configuration. For this purpose, the transformation laws based on the continuum mechanics theory are applied to account for the deformation effects. The transformation laws are originally formulated using high-order tensors with the assumptions that the deformation is one-dimensional in the electrode layer and isotropic in the particles. All terms related to deformation degenerate from tensors into scalars. The resulting enhanced P2D model equations are formulated as follows:

where

= scalar reduced from the deformation gradient inverse of the P2D particle domain

= scalar reduced from the deformation gradient inverse-transpose of the P2D particle domain

= determinant of the deformation gradient of the P2D particle domain

= scalar reduced from the deformation gradient inverse of the P2D electrode domain

= scalar reduced from the deformation gradient inverse-transpose of the P2D electrode domain

= determinant of the deformation gradient of the P2D electrode domain

The remaining nomenclature is the same as in Newman’s P2D Model.

A more comprehensive description of the swelling model theory can be found in [730].

With deformation taken into account via the transformation terms, the enhanced P2D model equations are solved in the reference domain (that is, using the original P2D mesh). The solid-state lithium concentration in the above model equations represents the value in the reference configuration. It is transformed to the value of the deformed configuration (that is, its true value) when used to calculate any battery properties, such as the transfer current in the Butler-Volmer equation (Equation 19–18 - Equation 19–20). The transformation of the solid phase lithium concentration from the reference domain into the deformed domain is expressed as:

(19–68)

where

= reference domain value

= deformed domain value

Both the particle lithium insertion and external mechanical load contribute to the domain deformations, while the effects of particle surface film growth are not included. The following transformation terms are used to account for the deformation effects:

(19–69)

(19–70)

(19–71)

(19–72)

(19–73)

Equation 19–69 quantifies the coefficient of particle domain volume change , which varies linearly depending on the state of lithiation (that is, the lithium concentration averaged over the particle domain normalized by the theoretical maximum concentration of lithium ) of the electrode active material. Here, is the swelling coefficient that describes how fast the particles swell during lithium insertion, and is the lithium concentration in the swell-neutral state, which acts as an offset in a linear formulation that marks the swell-neutral state, that is, where there is no change in the particle volume. Both and are user-specified model parameters.

In Equation 19–72, is the active material porosity at the swell-neutral state, is the mechanical pressure exerted normal to the battery electrode sandwich layer, and is Young's modulus. The second and third terms in Equation 19–72 are the electrode deformation induced by the particles swelling and the electrode elastic deformation, respectively. and are user-specified parameters. See Coupling between the Swelling and FSI Structural Models for more details about .

In addition to deformation, the model formulation also includes a change in the electrode porosity. In Equation 19–64-Equation 19–67, with the exception of some reference properties (marked with the superscript ), all the other porosity-dependent properties (for example, the effective conductivity and diffusivity in Equation 19–21 and Equation 19–22) are obtained with the following modified porosities that include the swelling effects:

As in the case of the swell-neutral state lithium concentration in Equation 19–69, the swell-neutral state porosities in Equation 19–74 and Equation 19–75 are also user-specified model parameters.