Modeling Aging in Batteries

Aging Mechanisms

Rechargeable lithium-ion batteries are widely used in a variety of applications, including portable electronic devices and electric vehicles. The performance of a battery is affected strongly by 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 a commonly accepted framework for modeling the charge-discharge behavior of lithium-ion cells. The method is based upon a homogenized Newman-type approach that does not consider the details of the pore geometry. The porous electrode theory is based upon a concurrent solution of a highly coupled multiphysics-multiscale formulation. For more information, see Coupled Thermal-Electrochemical Analysis.

Abaqus/Standard can model the following primary battery aging mechanisms:

Formation and growth of the Solid Electrolyte Interface (SEI) due to electrolyte decomposition and deposition on the electrode surface.
SEI growth in cracks at the particle surface.
Decrease in porosity and active surface area due to SEI growth.
Metallic lithium plating on the particle surface.
Dissolution of the cathode.

The first four of the above mechanisms happen at the anode, while the last degradation mechanism happens at the cathode, as shown in Figure 1

Governing Equations

In addition to intercalation, the chemical reactions corresponding to SEI formation, lithium plating, and SEI growth in cracks contribute to the total current, $I_{T o t}$ , at the particle surface:

$I_{T o t} = I_{B V} + I_{S E I} + I_{L P L} + I_{C r a c k} .$

In the above equation, the expression for the Butler-Volmer current, $I_{B V}$ , is the same as described in Coupled Thermal-Electrochemical Analysis. The quantities $I_{S E I}$ , $I_{L P L}$ , and $I_{C r a c k}$ represent the currents associated with different aging mechanisms; that is, the SEI growth at the particle-electrolyte interfaces, lithium plating, and the SEI growth in the particle surface cracks, respectively. Therefore, aging affects the governing equations of the Neumann framework through a modification of the Butler-Volmer intercalation current at the interface between the active particles and the electrolyte. The total current contributes as source terms for the equations that govern the macroscale solid electric potential, fluid-electric potential, and ion concentration fields (see Coupled Thermal-Electrochemical Analysis) and, therefore, affects the overall performance of the battery over time.

In the absence of any aging effects, $I_{T o t} = I_{B V}$ , and the other currents are zero. However, with aging, the combined effects of the other currents result in a progressive reduction in the magnitude of $I_{T o t}$ and an associated loss in capacity.

The combination of SEI formation and lithium plating also results in the growth of a surface film on the solid particles of the anode, which reduces the effective surface area between the particles and the electrolyte. The discussions that follow outline the constitutive equations for the currents $I_{S E I}$ , $I_{L P L}$ , and $I_{C r a c k}$ and describe the evolution of the surface film at the surface of the active electrode particles.

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 $I_{S E I}$ is as follows:

$I_{S E I} = I_{0, S E I} {\exp [\frac{α_{a, S E I} z F η_{S E I}}{R (θ - θ^{z})}] - \exp [\frac{{- α}_{c, S E I} z F η_{S E I}}{R (θ - θ^{z})}]},$

$I_{0, S E I} = F k_{0, S E I} C_{E C}^{s},$

$η_{S E I} = Φ_{s} - Φ_{e} - E_{S E I} - R_{E, S E I} I_{T o t},$

$R_{E, S E I} = \frac{(δ_{f i lm} + δ_{f i lm}^{0})}{σ_{S E I}},$

where

$I_{0, S E I}$ and

$η_{S E I}$ are the exchange current density and equilibrium potential of SEI formation, respectively, and

$R_{E, S E I}$ is the electrical resistance at the solid electrolye interface.

In the above equations,

$F$: is Faraday's constant;
$R$: is the gas constant;
$z$: is the charge number of the lithium ion battery;
$θ$: is temperature;
$θ^{z}$: is the absolute zero temperature;
$K_{0, S E I}$: is the kinetic rate constant of SEI;
$α_{c, S E I}, α_{a, S E I}$: are the cathodic and anodic transfer coefficients, respectively;
$E_{S E I}$: is the open circuit potential (OCP) as a function of $θ$ ;
$δ_{f i lm}^{0}, δ_{f i lm}$: are the initial thickness and evolving thickness of the SEI film layer;
$σ_{S E I}$: is the electrical conductivity of the SEI film layer;
$ϕ_{s}, ϕ_{e}$: are the electric potentials in the solid and electrolyte phases, respectively; and
$C_{E C}^{s}$: is the concentration of Ethylene Carbonate (EC) on the anode particle surface.

The surface concentration, $C_{E C}^{s}$ , evolves during the analysis based on the bulk concentration of ethylene carbonate in the electrolyte, $C_{E C}^{b u l k}$ , as:

$- D_{E C} \frac{C_{E C}^{s} - C_{E C}^{b u l k}}{δ_{f i lm}} = \frac{- I_{S E I}}{F},$

where $D_{E C}$ is the diffusivity of EC and $δ_{f i lm}$ is the thickness of the SEI film layer on the particle. The computation of $δ_{f i lm}$ 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 $I_{C r a c k}$ defined as:

$I_{S E I_r e f o r m} = {- I}_{0, S E I_r e f o r m} f_{c r a c k} {\exp [\frac{- z F η_{C r a c k}}{R (θ - θ^{z})}]},$

$η_{C r a c k} = Φ_{s} - Φ_{e} - E_{S E I},$

where

$I_{0, S E I_r e f o r m}$ and

$η_{C r a c k}$ are the exchange current density of the SEI reformation and the equilibrium potential of the SEI reformation, respectively. Typically,

$I_{0, S E I_r e f o r m}$ =

$I_{0, S E I}$ .

In the above equation, $f_{C r a c k}$ 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, $ϕ_{s} - ϕ_{e} < 0$ . 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 $I_{L P L}$ and is given as:

$I_{L P L} = {- I}_{0, L P L} {\exp [\frac{{- α}_{c, L P L} z F η_{L P L}}{R (θ - θ^{z})}]},$

$η_{L P L} = Φ_{s} - Φ_{e} - R_{E, S E I} I_{T o t},$

where

$I_{0, L P L}$ and

$η_{L P L}$ are the exchange current density and the equilibrium potential of lithium plating, respectively, and

$R_{E, S E I}$ is the electrical resistance at the solid electrolye interface.

In the above equations, $α_{c, L P L}$ 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:

$\frac{\partial C_{S E I}}{\partial t} = - \frac{a_{s} (I_{S E I} + I_{C r a c k})}{z F} - \frac{a_{s} I_{L P L}}{z F} β,$

$\frac{\partial C_{L P L}}{\partial t} = - \frac{a_{s} I_{L P L}}{F} (1 - β),$

where

$C_{S E I}, C_{L P L}$ are the molar concentrations of the SEI and plated lithium, respectively, per unit volume of the electrode;

$a_{s}$ is the wetted particle surface area per unit volume; and

$β$ is the fraction of plated lithium that can convert to the SEI.

The surface film is characterized by an equivalent thickness, $δ_{f i lm}$ , 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:

$δ_{f i lm} = \frac{1}{a_{s}} (\frac{C_{S E I} M_{S E I}}{ρ_{S E I}} + \frac{C_{L P L} M_{L P L}}{ρ_{L P L}}),$

where

$M_{S E I}, M_{L P L}$ are the molar weights of the SEI and plated lithium, and

$ρ_{S E I}, ρ_{L P L}$ are the molar densities of the SEI and plated lithium.

Using the computed film thickness, one can compute the ionic resistance, $R_{S E I}$ , and electrical resistance, $R_{E, S E I}$ , 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:

$R_{S E I} = ω_{S E I} \frac{δ_{f i lm}}{κ_{S E I}},$

where

$ω_{S E I}$ is the volume fraction of the SEI in the film, and

$κ_{S E I}$ is the ionic conductivity of the SEI layer. The computed

$R_{S E I}$ is used in the Butler-Volmer kinetics as described in Coupled Thermal-Electrochemical Analysis.

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:

$I_{D i s s} = I_{0, D i s s} {\exp [\frac{F η_{D i s s}}{R (θ - θ^{z})}]},$

$η_{D i s s} = Φ_{s} - Φ_{e} - E_{D i s s},$

where

$I_{0, D i s s}$ and

$η_{D i s s}$ are the exchange current density and the equilibrium potential of the cathode dissolution, respectively.

$E_{D i s s}$ is the potential above which the dissolution of active material happens. The cathode dissolution effect is captured as a change in solid volume fraction,

$ϵ_{s}$ , that affects the wetted surface area,

$a_{s}$ , and, therefore, further reduces the capacity of the battery.

$ϵ_{c a t h o d e} = ϵ_{c a t h o d e}^{0} - \frac{I_{D i s s} Δ t}{C_{s, P}^{\max} l_{c a t h o d e} F},$

where

$l_{c a t h o d e}$ is the length of the cathode.

Defining the Properties for the Aging Mechanisms

You must specify the constants required to define each of the aging mechanisms.

Solid Electrolyte Interface Properties

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

Input File Usage

Use the following options to define the properties to model aging due to the solid electrolyte interface growth:

PARAMETER TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI
 $α_{a, S E I}$ , $α_{c, S E I}$ , $K_{0, S E I}$ , $C_{0, S E I}$ , $C_{E C, 0}^{s}$ , $C_{E C}^{b u l k}$ , $β$ , $ρ_{S E I}$ , $M_{S E I}$ , $z$

Use the following options to specify a tabular form of $I_{0, S E I}$ :

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_SEI_Tabular
 $I_{0, S E I}$ ,  $C_{e c}^{S}$

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

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI_OCPTabular
 $E_{S E I}$ ,  $C_{e c}^{S}$

Use the following options to specify a tabular form of $D_{E C}$ :

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_SEI_Tabular
 $D_{E C}$ ,  $C_{e c}^{S}$

Use the following options to specify a tabular form of ionic conductivity, $κ_{S E I}$ :

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI_IonCondTabular
 $κ_{S E I}$ ,  $C_{e c}^{S}$

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 .

Input File Usage

Use the following options to define the properties for reformation of the SEI:

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_CrackSEI_Tabular
 $I_{0, S E I_r e f o r m}$ ,  $C_{E C}^{S}$

Use the following options to define the cracking function in tabular format:

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_SEI_CrackFunction_Tabular
 $f_{c r a c k}$ ,  ${\hat{C}}_{s, P}$

Lithium Plating Properties

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

Input File Usage

Use the following options to define the properties for Butler-Volmer kinetics:

PARAMETER TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_LPL
 $α_{c, L P L}$ , $C_{L P L}^{0}$ , $ρ_{L P L}$ , $M_{L P L}$

Use the following options to specify a tabular form of $I_{0, L P L}$ :

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

Cathode Dissolution Properties

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

Input File Usage

Use the following options to define the properties for Butler-Volmer kinetics:

PARAMETER TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_ElectrodeDissolution
 $l_{c a t h o d e}$

Use the following options to specify a tabular form of $I_{0, D i s s}$ as a function of temperature:

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_CurrXchgDens_ElecDiss_Tabular
 $I_{0, D i s s}$ ,  $θ$

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

PROPERTY TABLE, TYPE=ABQ_EChemPET_Electrode_Particle_ElecDiss_OCPTabular
 $E_{d i s s}$ ,  $θ$

Output

In addition to the output variables available for the coupled thermal-electric procedures, the coupled thermal-electrochemical procedures, the coupled thermal-electrochemical-structural procedures, and the thermal-electrochemical-structural-pore pressure procedures, you can request the following element integration point output variables related to aging in Abaqus/Standard:

CONCSEI_: SEI concentration in the film for particle i.
CONCLPL_: Plated lithium concentration in the film for particle i.
CONCSOL_: Concentration of solute on the surface of particle i.
ECDSEI_: SEI Butler-Volmer current for particle i.
ECDLPL_: Butler-Volmer current for plated lithium in particle i.
ECDCRACKSEI_: Butler-Volmer current for particle i due to reformation of the SEI.
ECDELECDISS_: Butler-Volmer current for particle i due to dissolution.
ECDTOTAL: Total current due to the different aging mechanisms.

References

Yang, X-G., Y. Leng, G. Zhang, S. Ge , and C-Y. Wang, “Modeling of Lithium Plating Induced Aging of Lithium-Ion Batteries: Transition from Linear to Nonlinear Aging,” Journal of Power Sources, vol. 360, no. 28-40, 2017.
Kindermann, F., K. Jonas, A. Frank , and A. Jossen, “A SEI Modeling Approach Distinguishing between Capacity and Power Fade,” Journal of the Electrochemical Society, vol. 164, no. 12, 2017.
Newman, J., , and K. E. Thomas-Alyea, Electrochemical Systems, Wiley-Interscience , Third Edition , 2004.