Pseudo-2D Battery Modeling Equations

Since R2026a

The pseudo-2D (P2D) model is a common approach for simulating lithium-ion battery behavior with high fidelity. The model captures the essential electrochemical processes occurring within the battery, including diffusion in the solid and electrolyte phases, charge transport, and electrochemical reactions at the active material particle surfaces. The model is called pseudo-2D because it combines a 1D macroscopic dimension (across the cell sandwich from the anode to the cathode) with a 1D microscopic dimension (radial direction within the active material particles). This approach provides a balance between computational efficiency and physical accuracy, making it suitable for battery design, optimization, and control applications.

Partial Differential Equation Toolbox™ solves the P2D battery modeling equations by using the finite element method (FEM).

Symbols and Notations Used in P2D Model Equations

The following tables contain all the variables and parameters used in the P2D modeling equations. The variables and parameters are specified in SI units of measurements.

Dependent Variables

Dependent variables are the quantities that the model solves for.

Symbol	Description	Units
c_e	Electrolyte concentration	mol·m⁻³
c_s	Solid-phase lithium concentration	mol·m⁻³
Φ_e	Electrolyte potential	V
Φ_s	Solid-phase potential	V
j^d	Reaction ionic flux in domain d	mol·m⁻²·s⁻¹

Independent Variables

Independent variables are the coordinates used for solving the model.

Symbol	Description	Units
t	Time	s
x	Spatial coordinate across the cell sandwich	m
r	Radial coordinate within particles	m

Parameters

Parameters are the physical properties and constants that define the battery system.

Symbol	Description	Units
ϵ^d	Porosity of domain d	Not applicable
D^d_eff	Effective diffusion coefficient in the electrolyte	m²/s
D^d_bulk	Bulk diffusion coefficient in electrolyte	m²/s
D^d_s	Solid-phase diffusion coefficient	m²/s
σ^d_eff	Effective electronic conductivity	S·m⁻¹
κ^d_eff	Effective ionic conductivity	S·m⁻¹
κ^d_bulk	Bulk ionic conductivity of electrolyte	S·m⁻¹
brugg^d	Bruggman’s coefficient of domain d	Not applicable
a^d	Specific interfacial area	m^-1
R^d_p	Particle radius in domain d	m
υ^d_f	Active material volume fraction in domain d	Not applicable
L^d	Thickness of domain d	m
t₊	Transference number	Not applicable
F	Faraday constant	C·mol⁻¹
R	Universal gas constant	J·mol⁻¹·K⁻¹
T	Temperature	K
k	Reaction rate constant	m^2.5·mol^−0.5·s⁻¹
c_s^max,d	Maximum solid-phase concentration	mol·m⁻³
c_s^surf,d	Surface concentration in solid particles	mol·m⁻³
η^d	Overpotential	V
α^d	Charge transfer coefficient of domain d	Not applicable
U^d_ocp	Open-circuit potential function	V
I_app	Applied current density	A·m⁻²
V_app	Applied voltage	V

Subscripts and Superscripts

These notations show the domain or phase to which a variable or parameter applies.

Notation	Description
d (superscript)	Domain: a = anode, sep = separator, c = cathode
eff (subscript)	Effective property (adjusted for porosity using Bruggeman approximation)
max (superscript)	Maximum value
surf (superscript)	Particle surface concentration
s (subscript)	Solid phase
e (subscript)	Electrolyte phase
p (subscript)	Particle
bulk (subscript)	Intrinsic bulk property (before porosity adjustment)
ocp (subscript)	Open-circuit potential
app (subscript)	Applied boundary condition

General P2D Equations

The P2D model consists of a coupled system of partial differential equations that describe the conservation of species, charge, and electrochemical kinetics.

Conservation of Lithium in Electrolyte

This equation describes how lithium ions move through the electrolyte. The left side accounts for accumulation and diffusion, and the right side represents the source or sink term due to electrochemical reactions. Because electrochemical reactions occur in the electrodes, the equation for the anode and cathode domains is as follows:

$^{ϵ^{d} \frac{\partial c_{e}}{\partial t} - \frac{\partial}{\partial x} [D_{e f f}^{d} \frac{\partial c_{e}}{\partial x}] = a^{d} (1 - t_{+}) j^{d}, d \in {a, c}}$

Because there is no active material in the separator, and therefore, no electrochemical reactions occur, the right side of the equation is zero.

$^{ϵ^{s e p} \frac{\partial c_{e}}{\partial t} - \frac{\partial}{\partial x} [D_{e f f}^{s e p} \frac{\partial c_{e}}{\partial x}] = 0}$

The specific interfacial area a^d is a derived quantity computed from the particle radius and volume fraction:

$a^{d} = \frac{3 \cdot υ_{f}^{d}}{R_{p}^{d}} d \in {a, c}$

Here, υ^d_f is the active material volume fraction, and R^d_p is the particle radius in domain d.

The effective diffusion coefficient D^d_eff is computed from the bulk diffusion coefficient using Bruggman’s relation:

$D_{e f f}^{d} = D_{b u l k}^{d} \cdot {(ϵ^{d})}^{b r u g g^{d}} d \in {a, s e p, c}$

Here D^d_bulk is the bulk diffusion coefficient in the electrolyte and brugg is Bruggman’s coefficient that accounts for the tortuosity of the porous medium.

You specify the bulk diffusion coefficient, porosity, and Bruggman’s coefficient as properties of the electrolyte and domains. The effective diffusion coefficient is computed internally by the model.

Conservation of Charge in Electrolyte

This equation governs the electric potential in the electrolyte phase. The first term represents Ohm’s law, the second term accounts for concentration-dependent conductivity effects, and the right side represents the current generation or consumption due to electrochemical reactions. Because electrochemical reactions occur in the electrodes, the equation for the anode and cathode domains is as follows:

$- \frac{\partial}{\partial x} [k_{e f f}^{d} \frac{\partial Φ_{e}}{\partial x}] + \frac{\partial}{\partial x} [k_{e f f}^{d} \frac{2 (1 - t_{+}) R T}{F} \frac{\partial \ln c_{e}}{\partial x}] = a^{d} F j^{d}, d \in {a, s e p, c}$

Because there is no active material in the separator, and therefore, no electrochemical reactions occur, the right side of the equation is zero.

$- \frac{\partial}{\partial x} [k_{e f f}^{s e p} \frac{\partial Φ_{e}}{\partial x}] + \frac{\partial}{\partial x} [k_{e f f}^{s e p} \frac{2 (1 - t_{+}) R T}{F} \frac{\partial \ln c_{e}}{\partial x}] = 0$

Similar to the effective diffusion coefficient, the effective ionic conductivity κ^d_eff is computed using Bruggman’s relation:

$κ_{e f f}^{d} = κ_{b u l k}^{d} \cdot {(ϵ^{d})}^{b r u g g^{d}} d \in {a, s e p, c}$

Here κ^d_bulk is the bulk ionic conductivity of the electrolyte.

You specify the bulk ionic conductivity in electrolyte. You specify porosity and Bruggeman's coefficient in anode and cathode domains. The effective ionic conductivity is computed internally by the model.

Conservation of Charge in Solid Phase

This equation describes how electric charge moves through the solid electrode materials. This equation applies only to the anode and cathode domains, because the separator does not conduct electrons.

$\frac{\partial}{\partial x} [σ_{e f f}^{d} \frac{\partial Φ_{s}}{\partial x}] = a^{d} F j^{d}, d \in {a, c}$

Unlike the effective diffusion coefficient and ionic conductivity, the effective electrical conductivity σ^d_eff cannot be computed using bulk properties, porosity, and Bruggman’s coefficient. The electrical conductivity of the electrode is a complex function of the composite structure consisting of active material, conductive additives, and binders. Therefore, you specify σ^d_eff explicitly, based on experimental measurements or literature values for the specific electrode composition.

This equation is not defined in the separator region because the separator is an electronic insulator and does not contain any electronically conductive material. The specific interfacial area a^d and reaction current density j^d are only defined in the anode and cathode regions where active material is present.

Lithium Diffusion in Active Material Particles

This equation models how lithium diffuses within the spherical active material particles. The equation uses spherical coordinates, where r is the radial coordinate within the particle. This microscopic diffusion process occurs in both the anode and cathode particles.

$\frac{\partial c_{s}}{\partial t} = \frac{1}{^{r^{2}}} \frac{\partial}{\partial r} [r^{2} D_{s}^{d} \frac{\partial c_{s}}{\partial r}], d \in {a, c}$

The diffusion coefficient D^d_s is assumed to be the bulk diffusion coefficient of the active material itself, as each particle is modeled as being composed entirely of the active material. Unlike the electrolyte phase properties, no correction for porosity or tortuosity is required here since you are modeling diffusion within a solid particle rather than through a porous medium. Specify this value explicitly as a property of the active material.

Electrochemical Reaction Kinetics

This is the general form of the Butler-Volmer equation, where α^a and α^c are the anodic and cathodic charge transfer coefficients, respectively.

$j^{d} = k \sqrt{c_{e} (c_{s}^{\max, d} - c_{s}^{s u r f, d}) c_{s}^{s u r f, d}} [\exp (\frac{α^{a} F η^{d}}{R T}) - \exp (\frac{α^{c} F η^{d}}{R T})], d \in {a, c}$

For a symmetric reaction on the anode and cathode, the Butler-Volmer equation is of the form:

$j^{d} = 2 k \sqrt{c_{e} (c_{s}^{\max, d} - c_{s}^{s u r f, d}) c_{s}^{s u r f, d}} \sinh (\frac{0.5 F η}{R T}), d \in {a, c}$

The overpotential, which drives the electrochemical reaction, is defined as:

$η^{d} = Φ_{s} - Φ_{e} - U_{o c p}^{d} (c_{s}^{s u r f, d}) d \in {a, c}$

Here, U^d_ocp is the open-circuit potential, which is a function of the surface concentration of lithium in the active material particles.

Boundary Conditions for P2D Model

The P2D model requires appropriate boundary conditions to fully define the problem. These boundary conditions reflect the physical constraints at the interfaces between different domains and at the current collectors.

Solid Phase Potential Boundary Conditions

The solid potential at the anode current collector interface is set to 0 as a reference point: Φ_s = 0 at x = 0.

Current-Controlled Simulation

For current-based simulations, a Neumann boundary condition is applied at the cathode current collector:

$- σ_{e f f}^{c} \frac{\partial Φ_{s}}{\partial x} = I_{a p p}$

Here, I_app is the applied current density (A·m⁻²), which is computed as a product of the normalized current specified in the cycling step and the areal capacity: I_app = NormalizedCurrent × arealCapacity.

The areal capacity (Ah·m⁻²) is computed as:

arealCapacity = υ_f × c^max,a_s × (stoUpper – stoLower) × F × L^a/3600

The division by 3600 converts from seconds to hours for C-rate normalization. Here:

υ_f is the active material volume fraction in the anode.
c^max,a_s is the maximum solid concentration in the anode active material.
stoUpper and stoLower are the upper and lower stoichiometric limits of the anode active material.
F is the Faraday constant.
L^a is the anode thickness.

Voltage-Controlled Simulation

For voltage-based simulations, a Dirichlet boundary condition is applied at the cathode current collector: Φ_s = V_app. Here, V_app is the applied voltage specified in the cycling step. This boundary condition directly sets the potential difference across the battery terminals. The resulting current distribution is then calculated as part of the solution.

Particle Boundary Conditions

At the center of each spherical particle (r = 0), symmetry requires:

$\frac{\partial c_{s}}{\partial r} = 0$

At the surface of each particle (r = R_p), the flux of lithium is determined by the electrochemical reaction rate:

$- D_{s}^{d} \frac{\partial c_{s}}{\partial r} = j^{d} d \in {a, c}$

This condition balances the diffusive flux of lithium within the solid particle with the ionic flux due to the electrochemical reaction at the particle surface.

Default Boundary Conditions

For all other variables in the model, zero Neumann boundary conditions (zero flux) are applied by default at both current collector interfaces. These conditions reflect that lithium ions and ionic current cannot penetrate the current collectors.

$\frac{\partial c_{e}}{\partial x} = 0, \frac{\partial Φ_{e}}{\partial x} = 0$

At the interfaces between different domains (anode-separator and separator-cathode), continuity conditions are automatically imposed by the finite element method:

$\begin{array}{l} c_{e}^{-} = c_{e}^{+} \\ Φ_{e}^{-} = Φ_{e}^{+} \\ D_{e f f}^{-} \frac{\partial c_{e}^{-}}{\partial x} = D_{e f f}^{+} \frac{\partial c_{e}^{+}}{\partial x} \\ κ_{e f f}^{-} \frac{\partial Φ_{e}^{-}}{\partial x} = κ_{e f f}^{+} \frac{\partial Φ_{e}^{+}}{\partial x} \end{array}$

These conditions ensure that both the variables and their fluxes are continuous across domain interfaces, reflecting the physical reality that there are no discontinuities in concentration or potential within the electrolyte.

References

[1] Torchio, Marcello, et al. “LIONSIMBA: A Matlab Framework Based on a Finite Volume Model Suitable for Li-Ion Battery Design, Simulation, and Control.” Journal of The Electrochemical Society, vol. 163, no. 7, 2016, pp. A1192–205. DOI.org (Crossref), https://doi.org/10.1149/2.0291607jes.