First-principles order-parameter theory of freezing

A first-principles order-parameter theory of the fluid-solid transition is presented in this paper. The thermodynamic potential $\Omega$ of the system is computed as a function of order parameters ${\lambda }_i(={\lambda }_{{\overrightarrow{\mathrm{k}}}_i})$ proportional to the lattice periodic components of the one-particle density $\rho \left(\overrightarrow{\mathrm{r}}\right)$, ${\overrightarrow{\mathrm{K}}}_i$'s being the reciprocal-lattice vectors (RLV) of the crystal. Computation of $\Omega \left(\right\{{\lambda }_i\left\}\right)$ is shown to require knowing $\Omega$ for a fluid placed in lattice periodic potentials with amplitudes depending on ${\lambda }_i$. Using systematic nonperturbative functional methods for calculating the response of the fluid to such potentials, we find $\Omega \left(\right\{{\lambda }_i\left\}\right)$. The fluid properties (response functions) determining it are the Fourier coefficients $c_i(=c_{{\overrightarrow{\mathrm{K}}}_i})$ and $c_0(=c_{\overrightarrow{\mathrm{q}}=0\mathrm{}})$ of the direct correlation function $c\left(\overrightarrow{\mathrm{r}}\right)$. The system freezes when at constant chemical potential $\mu$ and pressure $P$, locally stable fluid and solid phases [i.e., minima of $\Omega \left(\right\{{\lambda }_i\left\}\right)$ with $\left\{{\lambda }_i\right\}=0\mathrm{}$ and $\left\{{\lambda }_i\right\}\ne 0$, respectively] have the same $\Omega$. The order-parameter mode most effective in reducing $\Omega \left(\right\{{\lambda }_i\left\}\right)$ corresponds to ${\overrightarrow{\mathrm{K}}}_j$ being of the smallest-length RLV set (${\mathrm{c}}_{\overrightarrow{\mathrm{q}}}$ is largest for $\left|\overrightarrow{\mathrm{q}}\right|\simeq \left|{\overrightarrow{\mathrm{K}}}_j\right|$). In some cases one has to consider a second order parameter ${\lambda }_n$ with a RLV ${\overrightarrow{\mathrm{K}}}_n$ lying near the second peak in $c_{\overrightarrow{\mathrm{q}}}$. The effect of further order-parameter modes on $\Omega$ is shown to be small. The theory can be viewed as one of a strongly first-order density-wave phase transition in a dense classical system. The transition is a purely structural one, occurring when the fluid-phase structural correlations (measured by $c_j$, etc.) are strong enough. This fact has been brought out clearly by computer experiments but had not been theoretically understood so far. Calculations are presented for freezing into some simple crystal structures, i.e., fcc, bcc, and two-dimensional hcp. The input information is only the crystal structure and the fluid compressibility (related to $c_0$). We obtain as output the freezing criterion stated as a condition on $c_j$ or as a relation between $c_j$ and $c_n$, the volume change $V$, the entropy change $\Delta s$, and the Debye-Waller factor at freezing for various RLV values. The numbers are all in very good agreement with those available experimentally.