Modeling Aging in Batteries

Solid Electrolyte Interface

The basic form of the Butler-Volmer kinetics is used to characterize the current at the interface of the particle and the electrolyte, where the SEI layer forms and grows progressively. The expression for $I_{SEI}$ is as follows:

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;

$K_{0,SEI}$

is the kinetic rate constant of SEI;

${\alpha }_{c,SEI},{\alpha }_{a,SEI}$

are the cathodic and anodic transfer coefficients, respectively;

$E_{SEI}$

is the open circuit potential (OCP) as a function of $\theta$ ;

${\delta }_{fi\mathrm{lm}}^0,{\delta }_{fi\mathrm{lm}}$

are the initial thickness and evolving thickness of the SEI film layer;

${\sigma }_{SEI}$

is the electrical conductivity of the SEI film layer;

${\varphi }_s,{\varphi }_e$

are the electric potentials in the solid and electrolyte phases, respectively; and

$C_{EC}^s$

is the concentration of Ethylene Carbonate (EC) on the anode particle surface.

The surface concentration, $C_{EC}^s$ , evolves during the analysis based on the bulk concentration of ethylene carbonate in the electrolyte, $$\begin{gathered} C_{EC}^{bulk \\ } \end{gathered}$$ , as:

where $D_{EC}$ is the diffusivity of EC and ${\delta }_{fi\mathrm{lm}}$ is the thickness of the SEI film layer on the particle. The computation of ${\delta }_{fi\mathrm{lm}}$ is explained in Surface Film Growth.

SEI Reformation

During the regular charging-discharging of a lithium battery, the graphite particle at the anode can expand or contract based on litheation or delitheation of ions. This change in volume can cause cracking to occur on the particle and SEI layer. The cracked surfaces are then exposed to the electrolyte, which leads to the formation of a new SEI layer. The effects of this additional SEI layer enter the formulation through the current $I_{Crack}$ defined as:

In the above equation, $f_{Crack}$ is a cracking function dependent on the intercalation in the particle.

Lithium Plating

Lithium plating occurs only when the electric potential at the anode is negative; that is, ${\varphi }_s-{\varphi }_e<0$ . This process is assumed to be irreversible, which means that stripping of lithium in the subsequent discharge is neglected. The lithium plating contribution to the total current is $I_{LPL}$ and is given as:

In the above equations, ${\alpha }_{c,LPL}$ is the cathodic transfer coefficient.

Lithium plating typically dominates a narrow portion of the anode near the separator after a certain number of cycles when the potential is negative. The lithium plating is associated with a drop of anode porosity associated with SEI film growth, which can increase the local electrolyte potential gradient in the anode. After hundreds of cycles of cyclic charging, there can be an exponential increase of lithium plating, as well as local pore clogging near the anode/separator interface.

Surface Film Growth

SEI and lithium metal together form the surface film covering the active particles at the anode. Within the film, the time evolution of the concentrations of SEI and plated lithium are governed by the following equations:

The surface film is characterized by an equivalent thickness, ${\delta }_{fi\mathrm{lm}}$ , which is defined as the ratio of the total volume of the SEI and lithium metal to the specific surface area per unit volume of electrode, $a_s$ . The equivalent thickness of the surface film is determined by the concentrations of the SEI and plated lithium as:

Using the computed film thickness, one can compute the ionic resistance, $R_{SEI}$ , and electrical resistance, $R_{E,SEI}$ , of the SEI layer. Since this model neglects the stripping of lithium metal, the plated lithium is considered isolated from the main electron-conduction matrix. Therefore, the ionic resistance is determined only by SEI as:

Cathode Dissolution

The dissolution of cathode happens above a certain prescribed voltage where, in the case of lithium ion batteries, the hydrofluoric acid starts to dissolve the active cathode material, resulting in a reduction of active surface area and solid volume fraction of the cathode. The dissolution mechanism is governed by the following equations:

Solid Electrolyte Interface Properties

You must define tables of properties required to activate the computation of the SEI current.

PARAMETER TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI
${\alpha }_{a,SEI}$,${\alpha }_{c,SEI}$,$K_{0,SEI}$,$$\begin{gathered} C_{0,SEI \\ } \end{gathered}$$,$C_{EC,0}^s$,$$\begin{gathered} C_{EC}^{bulk \\ } \end{gathered}$$,$\beta$,${\rho }_{SEI}$,$M_{SEI}$,$z$

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_SEI_Tabular
$I_{0,SEI}$, $C_{ec}^S$

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI_OCPTabular
$E_{SEI}$, $C_{ec}^S$

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_SEI_Tabular
$D_{EC}$, $C_{ec}^S$

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI_IonCondTabular
${\kappa }_{SEI}$, $C_{ec}^S$

SEI Reformation Properties

You must define tables of properties required to activate the SEI and to compute the SEI reformation after a crack on the particle .

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_CrackSEI_Tabular
$$\begin{gathered} I_{0,SEI\_reform \\ } \end{gathered}$$, $C_{EC}^S$

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI_CrackFunction_Tabular
$f_{crack}$, ${\hat{C}}_{s,P}$

Lithium Plating Properties

You must define tables of properties required to activate the computation of the lithium plating current.

PARAMETER TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_LPL
${\alpha }_{c,LPL}$,$C_{LPL}^0$,${\rho }_{LPL}$,$M_{LPL}$

TABLE COLLECTION
PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_LPL_Tabular
$I_{0,LPL}$, $C_{LPL}$

Cathode Dissolution Properties

You must define tables of properties required to activate the computation of the cathode dissolution.

PARAMETER TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_ElectrodeDissolution
$l_{cathode}$

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_ElecDiss_Tabular
$I_{0,Diss}$, $\theta$

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_ElecDiss_OCPTabular
$E_{diss}$, $\theta$