examples.elphf¶

The following examples exhibit various parts of a model to study electrochemical interfaces. In a pair of papers, Guyer, Boettinger, Warren and McFadden [22] [23] have shown that an electrochemical interface can be modeled by an equation for the phase field \(\xi\)

(1)¶\[ \underbrace{ \frac{1}{M_\xi}\frac{\partial \xi}{\partial t} \vphantom{\sum_{j=1}^{n} C_j} }_{\text{transient}} = \underbrace{ \kappa_{\xi}\nabla^2 \xi \vphantom{\sum_{j=1}^{n} C_j} }_{\text{diffusion}} - \underbrace{ \overbrace{ \sum_{j=1}^{n} C_j \left[ p'(\xi) \Delta\mu_j^\circ + g'(\xi) W_j \right] }^{\text{phase transformation}} + \overbrace{ \frac{\epsilon'(\xi)}{2}\left(\nabla\phi\right)^2 \vphantom{\sum_{j=1}^{n} C_j} }^{\text{dielectric}} }_{\text{source}}\]

a set of diffusion equations for the concentrations \(C_j\), for the substitutional elements \(j = 2,\ldots, n-1\)

(2)¶\[\begin{split}\underbrace{ \frac{\partial C_j}{\partial t} }_{\text{transient}} &= \underbrace{ D_{j}\nabla^2 C_j \vphantom{\frac{\partial C_j}{\partial t}} }_{\text{diffusion}} \\ & \qquad + \underbrace{ D_{j}\nabla\cdot \frac{C_j}{1 - \sum_{\substack{k=2\\ k \neq j}}^{n-1} C_k} \left\{ \overbrace{ \sum_{\substack{i=2\\ i \neq j}}^{n-1} \nabla C_i }^{\text{counter diffusion}} + \overbrace{ C_n \left[ p'(\xi) \Delta\mu_{jn}^{\circ} + g'(\xi) W_{jn} \right] \nabla\xi \vphantom{\sum_{\substack{i=2\\ i \neq j}}^{n-1} \nabla C_i} }^{\text{phase transformation}} + \overbrace{ C_n z_{jn} \nabla \phi \vphantom{\sum_{\substack{i=2\\ i \neq j}}^{n-1} \nabla C_i} }^{\text{electromigration}} \right\} }_{\text{convection}}\end{split}\]

a diffusion equation for the concentration \(C_{\text{e}^{-}}\) of electrons

(3)¶\[\begin{split}\underbrace{ \frac{\partial C_{\text{e}^{-}}}{\partial t} \vphantom{\left\{ \overbrace{ \left[p'(\xi)\right] }^{\text{phase transformation}} \right\}} }_{\text{transient}} = \underbrace{ D_{\text{e}^{-}}\nabla^2 C_{\text{e}^{-}} \vphantom{\left\{ \overbrace{ \left[p'(\xi)\right] }^{\text{phase transformation}} \right\}} }_{\text{diffusion}} \\ + \underbrace{ D_{\text{e}^{-}}\nabla\cdot C_{\text{e}^{-}} \left\{ \overbrace{ \left[ p'(\xi) \Delta\mu_{\text{e}^{-}}^{\circ} + g'(\xi) W_{\text{e}^{-}} \right] \nabla\xi }^{\text{phase transformation}} + \overbrace{ z_{\text{e}^{-}} \nabla \phi \vphantom{\left[p'(\xi)\right]} }^{\text{electromigration}} \right\} }_{\text{convection}}\end{split}\]

and Poisson’s equation for the electrostatic potential \(\phi\)

(4)¶\[\underbrace{ \nabla\cdot\left(\epsilon\nabla\phi\right) \vphantom{\sum_{j=1}^n z_j C_j} }_{\text{diffusion}} + \underbrace{ \sum_{j=1}^n z_j C_j }_{\text{source}} = 0\]

\(M_\xi\) is the phase field mobility, \(\kappa_\xi\) is the phase field gradient energy coefficient, \(p'(\xi) = 30\xi^2\left(1-\xi\right)^2\), and \(g'(\xi) = 2\xi\left(1-\xi\right)\left(1-2\xi\right)\). For a given species \(j\), \(\Delta\mu_j^{\circ}\) is the standard chemical potential difference between the electrode and electrolyte for a pure material, \(W_j\) is the magnitude of the energy barrier in the double-well free energy function, \(z_j\) is the valence, and \(D_{j}\) is the self diffusivity. \(\Delta\mu_{jn}^{\circ}\), \(W_{jn}\), and \(z_{jn}\) are the differences of the respective quantities \(\Delta\mu_{j}^{\circ}\), \(W_{j}\), and \(z_{j}\) between substitutional species \(j\) and the solvent species \(n\). The total charge is denoted by \(\sum_{j=1}^n z_j C_j\).

Although unresolved stiffnesses make the full solution of this coupled set of equations intractable in FiPy, the following examples demonstrate the setup and solution of various parts.

Modules

`examples.elphf.diffusion`
`examples.elphf.input`	This example adds two more components to `examples/elphf/input1DphaseBinary.py` one of which is another substitutional species and the other represents electrons and diffuses interstitially.
`examples.elphf.phase`	A simple 1D example to test the setup of the phase field equation.
`examples.elphf.phaseDiffusion`	This example combines a phase field problem, as given in `examples.elphf.phase`, with a binary diffusion problem, such as described in the ternary example `examples.elphf.diffusion.mesh1D`, on a 1D mesh
`examples.elphf.poisson`	A simple 1D example to test the setup of the Poisson equation.
`examples.elphf.test`

Last updated on Jun 26, 2024. Created using Sphinx 7.1.2.