Bulk Chemical Potential - \(\mu\)

The bulk chemical potential \(\mu(c)\), is defined as the derivative of the bulk free energy density \(f(c)\), w.r.t the order parameter \(c\). In SpinDec2, the user defines the form of \(f(c)\) by inputting polynomial coefficents for a \(N-1\) degree polynomial, therefore the bulk chemical potential can be readily found from the rules of differntaition:

$$ \mu\left(c_{i,j}\right) = \sum_{m = 0}^{N-1} m\,a_{m} \left(c_{i,j}\right)^{m-1}, $$

where \(a\) are the polynomial coefficents, and \(c_{i,j}\) is the order paramter at grid point \(i,j\).

Total Chemical Potential - \(Q\)

The total chemical potential (which we now denote as \(Q\)), is defined by the Euler-Lagrange equation of the total free energy. To simplfy the finite differnce method, we evaluate \(Q\) before the time evoloution step, such that we are only dealing with second order derivatives in space. he finite difference method will solve this numerically for each node in the 2D grid. The Laplacian of the concentration will be discretised using the conventional central difference for the second derivative, which has second-order accuracy:

$$ \frac{\delta F}{\delta c} = Q \approx \mu\left(c_{i,j}\right) - {\kappa}\left[\frac{c_{i+1, j} - 2 c_{i,j} + c_{i-1, j}}{\left(\Delta x\right)^2} + \frac{c_{i, j+1} - 2 c_{i,j} + c_{i, j-1}}{\left(\Delta y\right)^2}\right] + \mathcal{O}\left(\Delta x^{2}\right) + \mathcal{O}\left(\Delta y^{2}\right), $$

where \(\Delta x\), \(\Delta y\) are the spatial grid spacings in \(x\) and \(y\) axes and the shorthand \( c_{i \pm 1, j \pm 1} = c( x \pm \Delta x, y \pm \Delta y) \). It should be highlighted here that periodic boundary conditions will be followed to get the concentration Laplacians at the boundary. This will mean that out of bound encounters at the boundaries for finite difference like \(c_{N+1,j}\) will be equivalent to \(c_{1, j}\) and so on.