Ising-like Models with Energy-Volume Coupling

We consider a regular assembly of singly occupied cells with two accessible volumes. Coupled to cell volumes are interaction energies between nearest neighbors that lead to a phase transition with a critical point. We find that these compressible cell models can serve as Ising-like prototypes of the one-component liquid-liquid and isostructural solid-solid phase transitions that originate in the short-range features of the intermolecular potential. The mean-field solutions provide hints concerning the analytical form of the equation of state of liquid water.

Figure 1

Single-cell ($+$) and ($-$) states for two situations, (a) and (b). Pictures are two-dimensional for the sake of simplicity, and the shaded (blue) areas are the free volumes a particle can explore in its cell. The relative proportions of particle sizes, volumes, and free volumes are schematic. In (a), we choose—because we have the freedom to do so—to consider that the free volume of particles in ($+$) cells is restricted so that $\lambda <1$. In (b), two particle sizes are used while $\lambda >1$.

Figure 2

Isotherms in the pressure-volume plane calculated from Eqs. (5) and (6) for the model in Fig. 1 with parameters $c=6$, $\delta \epsilon =1000\mathrm{J}{\mathrm{mol}}^{-1}$, $v_0=2\times {10}^{-5}{\mathrm{m}}^3{\mathrm{mol}}^{-1}$, $\delta v=0.5\times {10}^{-5}{\mathrm{m}}^3{\mathrm{mol}}^{-1}$, and $\lambda =0.2$. According to Eq. (7), these parameters yield $T_c=184.4\mathrm{K}$ and $p_c=1.17\mathrm{k}\mathrm{bar}$. The straight line is the $c\delta ϵ(v-v_0)/\delta v^2$ contribution, while the dashed lines are the $Tg(v)$ contributions for $T=150\mathrm{K}$ (bold dashed line, red) and $T=200\mathrm{K}$ (thin dashed line, blue), with the two remaining solid curves representing, according to Eq. (5), the resulting $p(v)$ values; hence, the bold (red) line corresponds to $T=150\mathrm{K}$ and the thin (blue) line to $T=200\mathrm{K}$. The inset shows that the coexistence curve in the pressure-temperature plane has a negative slope and ends at a critical point.

Figure 3

Temperature dependence of the isobaric heat capacity $C_p$ (in $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$), the isobaric thermal expansivity ${\alpha }_p$ (in ${10}^{-3}{\mathrm{K}}^{-1}$), and the isothermal (${\kappa }_T$) and isentropic (${\kappa }_S$) compressibilities (in ${10}^{-12}{\mathrm{Pa}}^{-1}$) for the waterlike model in Fig. 1, with the model parameters and critical coordinates specified in the caption of Fig. 2, at $-500\text{bar}$ (bold solid line, red), 1 bar (thin solid line), and 500 bar (thin dashed line, blue). Data for compressibilities correspond to 1 bar, with the literature experimental data in the lower-right panel obtained from Refs. [10] and [30].