Multi-Species Multi-Reaction (MSMR) Model

Theoretical foundations and application of the MSMR model for thermodynamically consistent lithium insertion electrode characterization.

The Multi-Species Multi-Reaction (MSMR) model provides a thermodynamically consistent framework for describing lithium insertion electrodes. This guide covers the theoretical foundations and practical applications.

Electrode Model Equations

Thermodynamics

The MSMR model assumes that all electrochemical reactions at the electrode/electrolyte interface in a lithium insertion cell can be expressed as:

$$\text{Li}^+ + \text{e}^- + \text{H}_j \rightleftharpoons (\text{Li--H})_j$$

For each species $j$, a vacant host site $\text{H}_j$ can accommodate one lithium, leading to a filled host site $(\text{Li--H})_j$.

Open-Circuit Voltage

The OCV for each reaction is written as:

$$U_j = U_j^0 + \frac{\omega_j}{f} \log\left(\frac{X_j - x_j}{x_j}\right)$$

where:

• $f = F/(RT)$ with $R$, $T$, and $F$ being the universal gas constant, temperature, and Faraday’s constant
• $X_j$ is the total fraction of available host sites for species $j$
• $x_j$ is the fraction of filled sites occupied by species $j$
• $U_j^0$ is the concentration-independent standard electrode potential
• $\omega_j$ is a dimensionless parameter describing the disorder level

Inverse Form

The equation can be inverted to give:

$$x_j = \frac{X_j}{1 + \exp[f(U - U_j^0)/\omega_j]}$$

The overall electrode stoichiometry is:

$$x = \sum_j x_j = \sum_j \frac{X_j}{1 + \exp[f(U - U_j^0)/\omega_j]}$$

This provides an explicit closed-form expression for the inverse of the OCV, which is the opposite of many battery models that give OCV as an explicit function of stoichiometry.

Kinetics

The kinetics of the insertion reaction are given by:

$$i_j = i_{0,j}[e^{(1-\alpha_j)f\eta} - e^{-\alpha_j f\eta}], \quad i = \sum_j i_j$$

where $\eta = \phi_s - \phi_e - U(x)$ is the overpotential, and the exchange current density is:

$$i_{0,j} = i_{0,j}^{ref} (x_j)^{\omega_j \alpha_j} (X_j - x_j)^{\omega_j(1-\alpha_j)} (c_e/c_e^{ref})^{1-\alpha_j}$$

Solid Phase Transport

Within the MSMR framework, the flux within particles is expressed in terms of the chemical potential gradient:

$$N = -c_T x \frac{D}{RT} \nabla\mu + x(N + N_H)$$

Ignoring volumetric expansion ($N + N_H = 0$), this simplifies to:

$$N = c_T f D x(1-x) \frac{dU}{dx} \nabla x$$

The mass balance becomes:

$$\frac{\partial x}{\partial t} = -\nabla \cdot \left( x(1-x) f D \frac{dU}{dx} \nabla x \right)$$

Boundary Conditions

For a radially symmetric spherical particle:

$$N\big|_{r=0} = 0, \quad N\big|_{r=R} = \frac{i}{F}$$

where $R$ is the particle radius.

To avoid evaluating $U(x)$ and $dU/dx$ explicitly, we transform to use $U$ as the dependent variable, yielding:

$$\frac{dU}{dx} \frac{\partial U}{\partial t} = -\nabla \cdot \left( x(1-x) f D \nabla x \right)$$

Fitting OCP Functions to Data

Fitting OCP models to data presents three key challenges.

The Missing-Data Problem

At any point in time, the measured half-cell terminal voltage can be modeled as:

$$V(t) = U_{\text{ocp}}(x(t)) + \text{hysteresis} - \text{impedance} \times \text{current}$$

As a result, the half cell never lithiates all the way to $U_{\text{ocp}} = V_{\text{min}}$ even though the measured terminal voltage reaches $V_{\text{min}}$. The horizontal gaps between discharge and charge curves at the voltage limits represent the “missing-data problem.”

The Inaccessible-Lithium Problem

Laboratory tests only cycle between $V_{\text{min}}$ and $V_{\text{max}}$, meaning:

• Absolute stoichiometry $\theta_s \neq 1$ at $V_{\text{min}}$
• Absolute stoichiometry $\theta_s \neq 0$ at $V_{\text{max}}$

There are portions of active materials never accessed by OCP tests since we cannot achieve high-enough or low-enough voltages in practical laboratory tests.

MSMR as a Solution

By regressing independent parameter sets for discharge and charge:

$$\{U_j^0, X_j, \omega_j, \theta_{\text{min}}, \theta_{\text{max}}\}_{\text{dis}}$$ $$\{U_j^0, X_j, \omega_j, \theta_{\text{min}}, \theta_{\text{max}}\}_{\text{chg}}$$

We can estimate the underlying OCP by averaging the two parameter sets, which are now expressed in terms of absolute degree of lithiation.

Generating Standard U(x) Curves

Unlike typical OCP models, MSMR gives stoichiometry as a function of voltage. To generate lookup tables:

1. Evaluate the MSMR model over a given voltage range
2. Specify the number of evaluation points
3. Use the resulting arrays to construct an interpolant for other models

Full-Cell Balance

Using half-cell parameters, we perform electrode balance to get the full-cell OCP:

$$U_{\text{cell}}(z) = U_p(\theta_p) - U_n(\theta_n)$$

where $z$ is full-cell SOC and $\theta_n$, $\theta_p$ are electrode stoichiometries.

The conversion between stoichiometry and SOC is:

$$z = \frac{\theta_n - \theta_n^0}{\theta_n^{100} - \theta_n^0} = \frac{\theta_p - \theta_p^0}{\theta_p^{100} - \theta_p^0}$$

Capacity-Based Formulation

In practice, full-cell OCV data is given in terms of capacity. The reformulation:

$$U_{\text{cell}}(q) = U_p(q_p) - U_n(q_n)$$

where $q = Qz$, $q_n = \theta_n Q_n$, $q_p = \theta_p Q_p$.

We fit the “lower excess capacity” and “upper excess capacity” instead of stoichiometries at 0% and 100% SOC, as the data typically does not reach the true endpoints.

Blended MSMR / Experimental OCP

The OCPDataInterpolantMSMRExtrapolation calculation creates a blended OCP interpolant that combines:

• Experimental data (in the measured range)
• MSMR model extrapolation (outside the measured range)

Blending Function

$$V(x) = w(x) \cdot V_{\text{ocp}}(x) + (1 - w(x)) \cdot V_{\text{msmr-corrected}}(x)$$

where $w(x)$ is a $C^\infty$ bump function that transitions from $\approx 1$ inside the data range to $\approx 0$ outside it.

The transition_fraction parameter (typically 0.05-0.15) controls the transition region width, balancing data fidelity and smoothness.

Diffusivity in MSMR

The solid-phase transport equation within MSMR includes a thermodynamic factor:

$$N = c_T f D x(1-x) \frac{dU}{dx} \nabla x$$

The term $\frac{dU}{dx}$ can be computed directly from the MSMR parameters, making MSMR particularly useful for modeling concentration-dependent diffusivity.

SOC-Dependent Diffusivity

Diffusivity often varies significantly with state of charge—by 2-3 orders of magnitude in graphite near phase transitions. For empirical SOC-dependent diffusivity without MSMR, use piecewise interpolation.

Characteristic Diffusion Time

The diffusion time scale determines rate capability:

$$\tau_D = \frac{R^2}{D}$$

where $R$ is particle radius. At 1C, $\tau_D$ should be roughly 1 hour for the diffusion process to keep up with the electrochemistry.

References

1. Verbrugge, Mark, et al. “Thermodynamic model for substitutional materials: application to lithiated graphite, spinel manganese oxide, iron phosphate, and layered nickel-manganese-cobalt oxide.” Journal of The Electrochemical Society 164.11 (2017): E3243.

2. Lu, Dongliang, et al. “Implementation of a physics-based model for half-cell open-circuit potential and full-cell open-circuit voltage estimates: part I. Processing half-cell data.” Journal of The Electrochemical Society 168.7 (2021): 070532.