Phase-field model of interfaces in single-component systems derived from classical density functional theory

Phase-field models have been applied in recent years to grain boundaries in single-component systems. The models are based on the minimization of a free energy functional, which is constructed phenomenologically rather than being derived from first principles. In single-component systems, the free energy is a functional of a “phase field,” which is an order parameter often referred to as the crystallinity in the context of grain boundaries, but with no precise definition as to what that term means physically. We present a derivation of the phase-field model by Allen and Cahn from classical density functional theory first for crystal-liquid interfaces and then for grain boundaries. The derivation provides a clear physical interpretation of the phase field, and it sheds light on the parameters and the underlying approximations and limitations of the theory. We suggest how phase-field models may be improved.

Figure 1

Cartoon of a local Fourier transform. The full line shows the actual density profile $\rho \left(\mathbf{r}\right)$ within a system of size $\Omega$ parametrized by an expression such as Eq. (19) which contains a space dependent pseudo-Fourier coefficient. A Fourier transform taken over the small region $V$ marked by the dashed lines around $\stackrel{\mathrm{̃}}{\mathbf{r}}$ produces coefficients that reproduce the density locally (shown as thick line) and can be continued periodically throughout the system (dotted line). The inset shows the two-point direct correlation function $C^{\left(2\right)}\left(\mathbf{r}\right)$ which drops to a negligible value outside the domain $V^{″}$.