Entropy scaling close to criticality: From simple to metallic systems

Entropy has recently drawn considerable interest both as a marker to detect the onset of phase transitions and as a reaction coordinate, or collective variable, to span phase transition pathways. We focus here on the behavior of entropy along the vapor-liquid phase coexistence and identify how the difference in entropy between the two coexisting phases vary in ideal and metallic systems along the coexistence curve. Using flat-histogram simulations, we determine the thermodynamic conditions of coexistence, critical parameters, including the critical entropy, and entropies along the binodal. We then apply our analysis to a series of systems that increasingly depart from ideality and adopt a metal-like character, through the gradual onset of the Friedel oscillation in an effective pair potential, and for a series of transition metals modeled with a many-body embedded-atoms force field. Projections of the phase boundary on the entropy-pressure and entropy-temperature planes exhibit two qualitatively different behaviors. While all systems modeled with an effective pair potential lead to an ideal-like behavior, the onset of many-body effects results in a departure from ideality and a markedly greater exponent for the variation of the entropy of vaporization with temperature away from the critical temperature.

Figure 1

Argon: Behavior for the entropy density $\mathcal{S}$ (top left), for the entropy density of vaporization $\mathrm{\Delta }\mathcal{S}$ (top right), for the entropy $S$, and for the entropy of vaporization $\mathrm{\Delta }S$. Results obtained for the entropy density are shown in black, while results for the entropy are shown in blue. Experimental data are shown as squares [65], while dashed lines show fits to the data using a scaling law with the 3D-Ising exponent $\beta =0.326$ and the solid lines are obtained with an exponent of 0.47.

Figure 2

EWL results for $X=0.9$ and $T=1$. The grand-canonical partition function $\mathrm{\Theta }(\mu ,V,T)$ is shown in (a), with $Q(N,V,T)$ displayed as the inset. The number distribution $p\left(N\right)$ is plotted at coexistence in the top panel of (b), with the potential energy of the system shown as a function of $N$ in the bottom panel of (b).

Figure 3

(a) Dependence of the difference between the coexisting densities upon temperature for $X=0.9$ and $T=1$. EWL simulation results are shown as open squares and the estimate for the critical point is plotted as a filled square. The dashed line is a power-law fit to the simulation results using an Ising-type exponent of 0.326. (b) Variation of the pressure at coexistence with temperature. Same legend as in (a), with the dashed line corresponding to an exponential fit to the EWL simulation results.

Figure 4

Entropy for the two phases at coexistence for $X$ ranging from 0.7 to 1 (LJ system), $X=0.9$. In (a), the entropy scaled with respect to the critical entropy $S_c$ is plotted against the scaled pressure (with respect to $P_c$), while (b) shows a scaled entropy-temperature plot.

Figure 5

Entropy for the two phases at coexistence for a series of metals modeled with the qSC-EAM many-body potential (Ag: black; Ir: red; Ni: green; and Pd: blue). (a) Scaled entropy-pressure plot and (b) scaled entropy-temperature plot.

Figure 6

Scaled entropy difference–pressure plot. The graph shows results for both the effective pair potential as squares ($LJ$: violet, $X=0.9$: red, $X=0.8$: green and $X=0.7$: blue) and for the many-body potential as circles (Ag: black, Ir: red, Ni: green and Pd: blue). Results for Cu from prior simulation work [63] using a many-body potential are shown as cyan circles.

Figure 7

Left: Entropy density difference–scaled temperature plot (lines are power-law fits with an exponent of 0.326). Right: Entropy difference–scaled temperature plot (lines are power-law fits with an exponent of 0.47 for $X=1$ and 0.67 for Cu). The graph shows results for the effective pair potential ($X=1$) in purple and for the many-body potential (Cu) in cyan.

Figure 8

Scaled entropy difference–temperature plot. The graph shows results for the effective pair potential as squares and for the many-body potential as circles. Same legend as in Fig. 6.