Density-functional theory for fluid-solid and solid-solid phase transitions

We develop a theory to describe solid-solid phase transitions. The density functional formalism of classical statistical mechanics is used to find an exact expression for the difference in the grand thermodynamic potentials of the two coexisting phases. The expression involves both the symmetry conserving and the symmetry broken parts of the direct pair correlation function. The theory is used to calculate phase diagram of systems of soft spheres interacting via inverse power potentials $u\left(r\right)=\epsilon {\left(\sigma /r\right)}^n$, where parameter $n$ measures softness of the potential. We find that for $1/n<0.154$ systems freeze into the face centered cubic (fcc) structure while for $1/n\ge 0.154$ the body-centred-cubic (bcc) structure is preferred. The bcc structure transforms into the fcc structure upon increasing the density. The calculated phase diagram is in good agreement with the one found from molecular simulations.

Figure 1

Variation of $\frac{\mathrm{\Delta }{W}_{fc}}{N_c}$ as a function of ${\alpha }_c$ and $\mathrm{\Delta }{\rho }^{*}$ for $1/n=0.20$ is shown. The minimum of $\frac{\mathrm{\Delta }{W}_{fc}}{N_c}$ and the value of ${\alpha }_c$ at which this minimum occurs depend on $\mathrm{\Delta }{\rho }^{*}$. In (a) the variation is shown for ${\gamma }_f=3.296$ for three values of $\mathrm{\Delta }{\rho }^{*}$. in all cases the minimum is greater than zero; the lowest minimum is for $\mathrm{\Delta }{\rho }^{*}=0.0039$. In (b) we show the plot for ${\gamma }_f=3.297$ where the condition $\frac{\mathrm{\Delta }{W}_{fc}}{N_c}=0$ is satisfied for $\mathrm{\Delta }{\rho }^{*}=0.0039$ and ${\alpha }_c=16.3$. In (c) the variation of minimum of $\frac{\mathrm{\Delta }{W}_{fc}}{N_c}$ for ${\gamma }_f=3.297$ as a function $\mathrm{\Delta }{\rho }^{*}$ is shown.

Figure 2

Phase diagram of systems interacting via IPPs. The dashed line represents the theoretical values for the fluid-solid phase boundary and the full line represents those for the bcc-fcc phase boundary. Open squares and full circles represent, respectively, simulation values [7] for the fluid-solid (fcc or bcc) and the bcc-fcc transition points.

Figure 3

Variation of $\mathrm{\Delta }f$ as a function of ${\alpha }_l$ for different ${\gamma }_l$. The minimum found at a value of ${\alpha }_l$ for each ${\gamma }_l$ (shown in figure by full circle) is $\mathrm{\Delta }f({\gamma }_l,{\alpha }_l)$ [see Eq. (2.19)].

Figure 4

Variation of $\frac{\mathrm{\Delta }{W}_{l-h}}{N_h}$ as a function of ${\alpha }_h$ and $\mathrm{\Delta }{\rho }^{*}$ for $1/n=0.20$ is shown. In (a) the variation is shown for ${\gamma }_{bcc}=3.459$ for three values of $\mathrm{\Delta }{\rho }^{*}$. The lowest minimum occurs at $\mathrm{\Delta }{\rho }^{*}=0.0026$. In (b) the variation is for ${\gamma }_{bcc}=3.46$, where the condition $\frac{\mathrm{\Delta }{W}_{l-h}}{N_h}=0$ is satisfied for $\mathrm{\Delta }{\rho }^{*}=0.0026$ and ${\alpha }_h=32.0$. In (c) the variation of $\frac{\mathrm{\Delta }{W}_{l-h}}{N_h}$ for ${\gamma }_{bcc}=3.46$ as a function of $\mathrm{\Delta }{\rho }^{*}$ is shown.

Figure 5

The density-temperature phase diagram of the IPP for $n=5$ (or $1/n=0.20$). The dashed lines represent simulation values [7] and the full lines theoretical values.