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# System prepended metadata

title: Phase-field model

---

# Phase-field model
When building a phase-field model, we introduce a scalar phase parameter $\xi$ as a variable of the free energy of the system. The free energy of this system can be described in the form of Ginzburg–Landau theory:
\begin{equation}
    U=\int_{\Omega} \left[ \frac{\kappa_0}{2} |\nabla \xi|^2 + W(\xi) + f^{elec}_v\right]dV,
\end{equation}
where $\kappa_0$ is the lithium surface diffusion, $W(\xi)=\omega_0\xi^2(1-\xi)^2$, and $f^{elec}_v$ represents the change of the energy density subjected to the electrochemical reactions. The equation of f is
\begin{equation}
    f^{elec}_v = \Delta F \cdot h(\xi),
\end{equation}
where $h=\xi^3(10-15\xi+6\xi^2)$ and is an artificially assumed interpolation function, and $\Delta F$ represents the driving force of the system and is related to the current density $i$ in this system.
According to the principle of least action, the system evolves towards the direction of minimum free energy. Hence, the time-evolution of the phase-field model can be obtained from the minimum free energy by using variational principle:
\begin{equation}
    \frac{\delta U}{\delta \xi}=\frac{\partial U}{\partial \xi} - \nabla \cdot \frac{\partial U}{\partial (\nabla \xi)}=0.
\end{equation}
Because the phase parameter $\xi$ is a non-conserved field, its evolution equation is obtained by Allen-Cahn equation:
\begin{equation}
\begin{aligned}
    \frac{\partial \xi}{\partial t} = -M_{\eta}\frac{\partial U}{\partial \xi}
    &= -M_{\xi}\frac{\partial}{\partial \xi} \left[ \int_{\Omega} \left[ \frac{\kappa_0}{2} |\nabla \xi|^2 + W(\xi) + f^{elec}_v\right]dV \right]\\
    &= M_{\xi}\left( \kappa_0 \nabla^2 \xi - W'(\xi) - h(\xi)'\cdot \Delta F \right) \label{con:AC}
\end{aligned}    
\end{equation}
When the linear qpproximation is taken around i=0, f=~m, ththe above equation will be transform into 
By taking the linear approximation around $i=0$, which means $-\Delta F\approx M_{\eta}i$, eq.\ref{con:AC} can therefore also be represented by
\begin{equation}
\begin{aligned}
    \frac{\partial \xi}{\partial t} &= M_{\sigma}\left[ \kappa_0 \nabla^2 \xi - W'(\xi)\right] + M_{\eta}\left[ h'(\xi)i \right]
\end{aligned}    
\end{equation}

# Butler-Volmer equation
Butler-Volmer equation is introduced to describe the relationship between current density $i$ and over-potential $\eta$ at the anode/electrolyte interface, and is represented as
\begin{equation}
    i=i_0 \left[ \frac{c^+}{c_0}e^{\alpha\eta/\Phi_T}-e^{-(1-\alpha)\eta/\Phi_T}\right]
\end{equation}
where $i_0$ is the exchange current density, $\alpha$ is the charge-transfer coefficient and $\Phi_T$ is the thermal voltage. It is defined as $\Phi_T=RT/F$, where $R$ is the gas constant, $T$ is the absolute temperatureand $F$ is the Faraday constant. $c^+$ and $c_0$ represent the lithium ionic concentration at the anode/electrolyte interface and in the bulk. Note that the equation is composed by current density toward the cathode and that toward the anode. And the former subtracts the latter.