To make the existing CHInterface and ACInterface kernels flexible enough to handle $$$\kappa$$$ as functions of phase field variables we need to rederive the residual equations.

The phase field equations are

$$$$\begin{eqnarray} \frac{\partial c_i}{\partial t} &= \nabla \cdot M_i \nabla \frac{\delta F}{\delta c_i}\\ \frac{\partial \eta_j}{\partial t} &= - L_j \frac{\delta F}{\delta \eta_j}. \end{eqnarray}$$$$

The free energy $$$F$$$ is defined as

$$$$F = \int_V \big[ f_{loc}(c_1, \ldots,c_N, \eta_1, \ldots, \eta_M) + f_{gr}(c_1, \ldots,c_N, \eta_1, \ldots, \eta_M) + E_{d} \big] \, dV,$$$$

where $$$f_{gr}$$$ is the gradient contribution in question, with

$$$$f_{gr} = \sum_i^N \frac{\kappa_i}{2} |\nabla c_i|^2 + \sum^M_j \frac{\kappa_j}{2} |\nabla \eta_j|^2 = \sum^{N+M}_i \frac{\kappa_i}{2} |\nabla \nu_i|^2,$$$$

where we let $$$\vec \nu = (\vec c, \vec\eta)$$$ be an arbitrary order parameter.

We apply the fundamental lemma of calculus of variations

$$$$\frac{\delta F}{\delta c} = \frac{\partial f}{\partial c} - \nabla \cdot \frac{\partial f}{\partial\nabla c},$$$$

to compute the variational derivatives of the gradient contribution with respect to an arbitrary order parameter $$$\nu_j$$$ and obtain

$$$$\frac{\delta f_{gr}}{\delta \nu_j} = \sum^{N_M}_i \left[ \frac12 \frac{\partial\kappa_i}{\partial\nu_j}|\nabla \nu_i|^2 - \frac12\nabla\left(\frac{\partial\kappa_i}{\partial\nabla\nu_j}|\nabla\nu_i|^2\right) - \delta_{ij}\underbrace{\nabla(\kappa_i\nabla\nu_i)}_{=\kappa_i\nabla^2\nu_i+(\nabla\kappa_i)(\nabla\nu_i)}\right]$$$$