> ## Documentation Index
> Fetch the complete documentation index at: https://docs.ionworks.com/llms.txt
> Use this file to discover all available pages before exploring further.
# 多種・多反応 (MSMR) モデル
> 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](/ja/guide/calculations/piecewise).
### 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.