Separating Energy and Entropy Contributions to the Hexatic-Liquid Transitions in Two-Dimensional Repulsive Systems

TL;DR

The paper investigates why two-dimensional hexatic–liquid transitions are highly sensitive to interaction details. It analyzes Hertzian, Gaussian, and inverse-power-law repulsive models in 2D, calculating the isothermal equation of state $P(v)$ and the Helmholtz free energy $F(v)$ via thermodynamic integration, using a Maxwell construction to diagnose curvature through $ riangle F(v)=F(v)-F_L(v)$. By decomposing $F$ into $E$ and $-T S$, with $E=E_{ m IS}+E_{ m vib}$ and $S=S_{ m vib}+S_{ m conf}$, the study reveals a robust separation: $E$ is convex (dominated by $E_{ m IS}$) while entropy is concave, with $ riangle(-T S_{ m vib})$ driving concavity in first-order cases and both $ riangle(-T S_{ m vib})$ and $ riangle(-T S_{ m conf})$ concave in continuous cases. Shannon entropies of defects ($S_6$, $S_{ m def}$, $S_v$, $S_{ m clus}$) scale with configurational entropy and show curvature $ riangle(-T S_{ m Shannon})$ that mirrors $ riangle(-T S_{ m conf})$, linking defect proliferation to configurational disorder. At $T\to 0$, entropic effects vanish and the Mayer–Wood loop disappears, predicting the first-order hexatic–liquid transition vanishes; the results offer a general curvature-based framework for 2D melting and point to future work on attractive interactions and 3D behavior.

Abstract

Over the past decades, research on two-dimensional melting has established that both first-order and continuous hexatic-liquid transitions can occur, influenced by various factors in the potential energy and system details. The fundamental thermodynamic origins of this sensitivity remains elusive. Here, by decomposing the Helmholtz free energy across three representative repulsive systems, we reveal a universal competition between energy and entropy that dictates the melting pathway. The energetic contribution consistently imparts convexity to the free energy, whereas entropy imparts concavity. A first-order transition occurs when concave entropy dominates; otherwise, the transition is continuous. Further decomposition shows that vibrational entropy drives the concave total entropic curvature, while the configurational entropy's curvature switches from convex (first-order) to concave (continuous), mirroring defect proliferation measured by Shannon entropy. The convexity of the energy is dominated by the inherent potential, with minimal vibrational influence. Finally, we predict and verify that the first-order transition becomes continuous at zero temperature, where entropic effects vanish. Our work establishes the curvature of different thermodynamic quantities as a fundamental principle for understanding the nature of two-dimensional melting.

Figures (4)

• Figure 1: Isothermal equation of state and Helmholtz free energy curvature.a, b Equilibrium isothermal equation of state $P(v)$ for the Hertzian system at a low and b high density. Black solid lines are fifth-order polynomial fits. The horizontal dashed line in a indicates the Maxwell construction. c, d Corresponding difference in Helmholtz free energy, $\Delta F = F - F_L$, for c the low-density HertL system and d the high-density HertH system. Here, $F$ is the Helmholtz free energy (see Supplementary Fig. S2), and $F_L$ is a linear reference function connecting the free energy at two boundary states: between $F(v_A)$ and $F(v_B)$ for HertL system, and between $F(v_C)$ and $F(v_D)$ for HertH system. The sign of $\Delta F$ quantifies the curvature of $F(v)$: $\Delta F > 0$ (shaded light blue in c) indicates concave curvature, characteristic of the first-order hexatic–liquid transition; $\Delta F < 0$ (shaded yellow in d) indicates convex curvature, characteristic of the continuous hexatic–liquid transition. Symbols in all panels correspond to different states, as shown in the legend of panel a.
• Figure 2: Decomposed various energy and entropy contributions to curvature.a, b Difference in total energy $\Delta E$ (black squares), inherent energy $\Delta E_{\rm IS}$ (red circles), and vibrational energy $\Delta E_{\rm vib}$ (blue triangles) for a the HertL and b the HertH systems. Analogous to $\Delta F$ in Figs. \ref{['fig:eos']}c and \ref{['fig:eos']}d, the sign of these functions determines the local curvature (concavity or convexity) of the corresponding function. c, d Difference in total entropic $\Delta (-TS)$ (black squares), vibrational entropy $\Delta (-TS_{\rm vib})$ (red circles), and configurational entropy $\Delta (-TS_{\rm conf})$ (blue triangles), for the two systems. The data in all panels are shown only for the transition regions (shaded in Fig. \ref{['fig:eos']}).
• Figure 3: Curvature of various Shannon entropies defined from different defect quantifications. Difference in Shannon entropy $\Delta (-TS_{\rm Shannon})$, for a HertL and b HertH systems. The Shannon entropy is calculated from four different measures: the hexatic order parameter $\Delta (-TS_6)$ (black squares), a binary defect state $\Delta (-TS_{\rm def})$ (red circles), the Voronoi area $\Delta (-TS_v)$ (blue triangles), and the cluster length of defects $\Delta (-TS_{\rm clus})$ (green triangles). Analogous to $\Delta F$ in Figs. \ref{['fig:eos']}c and \ref{['fig:eos']}d, the sign of $\Delta (-TS_{\rm Shannon})$ determines the local curvature (concavity or convexity) of the corresponding Shannon entropic contribution. Data are shown only for the transition regions (shaded in Fig. \ref{['fig:eos']}).
• Figure 4: Melting at zero temperature for the HertL system.a Inherent equation of state for the HertL system. b Probability distribution of the local area per particle for the thermal equilibrium configuration (black open triangles) and the inherent structure (blue solid triangles), at state point ① marked in panel a as well as in Fig. \ref{['fig:eos']}a.