First-order and continuous melting transitions in two-dimensional Lennard-Jones systems and repulsive disks

Amir Hajibabaei and Kwang S. Kim

Phys. Rev. E 99, 022145 – Published 28 February 2019

Abstract

In two-dimensional Lennard-Jones (LJ) systems, a small interval of melting-mode switching occurs below which the melting occurs by first-order phase transitions in lieu of the melting scenario proposed by Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY). The extrapolated upper bound for phase coexistence is at density $ρ \sim 0.893$ and temperature $T \sim 1.1$ , both in reduced LJ units. The two-stage KTHNY scenario is restored at higher temperatures, and the isothermal melting scenario is universal. The solid-hexatic and hexatic-liquid transitions in KTHNY theory, even so continuous, are distinct from typical continuous phase transitions in that instead of scale-free fluctuations, they are characterized by unbinding of topological defects, resulting in a special form of divergence of the correlation length: $ξ \approx exp (b | T - T_{c} |^{- ν})$ . Here such a divergence is firmly established for a two-dimensional melting phenomenon, providing a conclusive proof of the KTHNY melting. We explicitly confirm that this high-temperature melting behavior of the LJ system is consistent with the melting behavior of the $r^{- 12}$ potential and that melting of the $r^{- n}$ potential is KTHNY-like for $n \leq 12$ but melting of the $r^{- 64}$ potential is first order; similar to hard disks. Therefore we suggest that the melting scenario of these repulsive potentials becomes hard-disk-like for an exponent in the range $12 < n < 64$ .

Received 23 October 2017
Revised 11 December 2018

DOI:https://doi.org/10.1103/PhysRevE.99.022145

Physics Subject Headings (PhySH)

Continuous phase transition Liquid-solid phase transition Phase diagrams Phase separation Phase transitions

Thin films

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Amir Hajibabaei and Kwang S. Kim^*

Center for Superfunctional Materials, Department of Chemistry and Department of Physics, Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919, Korea

^*kimks@unist.ac.kr

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Vol. 99, Iss. 2 — February 2019

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Images

Figure 1
(a) Phase diagram of the LJ system for $512^{2}$ particles (lines are interpolations). Continuation of the hexatic phase with similar density width is verified up to $T = 10$ . The extrapolated upper bound for phase coexistence is at $ρ \sim 0.893$ and $T \sim 1.1$ . The green+black (light+dark gray) solid curve is $ρ_{0} (T)$ given by Eq. (2). Above the red+black (light+dark gray) dashed curve orientational correlations are long ranged. Above the dotted curve, positional correlation functions are perfectly power law $[g (x, 0) \propto x^{- η}]$ with an exponent $η \leq 1 / 3$ . Between the dashed and dotted curves $g (x, 0)$ decays as a power law up to moderate distances (with $η$ slightly higher than $1 / 3$ ) but slightly faster than power law at the largest distances due to the finite-size effects. The uncertainty in density is 0.005 for the dotted line and 0.002 for all other lines and symbols. (b) The shifted pressure-density equations of states at temperatures, from bottom to top, are 0.5, 0.7, 1, 1.5, 2, 3, 5, 7, and 10 for $512^{2}$ particles. Boundaries of phase coexistence are obtained by Maxwell constructions and marked by $\times$ signs on isotherms [and by blue triangles in (a)]. (c) Scaling of the free energy barrier per particle with the system size at $T = 0.5$ . The solid line is proportional to $1 / \sqrt{N}$ . (d) The volume ( $v = 1 / ρ$ ) interval of phase coexistence disappears at $T \sim 1.1$ .
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Figure 2
Temperature dependence of correlation-derived quantities for $512^{2}$ particles. Plots of raw correlation functions and size dependence analysis are provided in the SM [25]. The color codes for temperatures are given at the right side. The scatter plots verify that, for instance, $ξ_{6} (Δ ρ, T) \to ξ_{6} (Δ ρ)$ at $T > 1.1$ . The correlations at $T = 1$ differ only slightly from that of $T > 1.1$ . (a) Orientational correlation length. (b) Exponent of the algebraic fits to the orientational correlations. (c) Relative error of exponential and power-law fits to the RDF. Error is estimated from variance of the difference between the fit and the raw correlation function at $r > 10$ . (d) Correlation length of the RDF.
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Figure 3
Solid-liquid phase coexistence vs hexatic phase. Both systems are at relative density $Δ ρ = 0.002$ with $512^{2}$ particles but at different temperatures: 0.5 (left) and 2 (right). This snapshot is reached after $3 \times 10^{8}$ MD steps. The systems are divided into $16 \times 16$ subsystems to analyze the fluctuations. The color-bar indicates $ρ (r) - \bar{ρ}$ , and the histograms show the probability distribution function (PDF) of density in subsystems obtained from ensemble averaging. The dashed lines coinciding with the two maxima of PDF, on the left histogram, are the lower and upper densities for solid-liquid coexistence obtained from ECMC simulations. The vectors on top of the color maps show the local orientational order: $(Re (Ψ), Im (Ψ))$ . In coexistence the orientation is uniform in the solid portion of system, while in hexatic phase there are mobile topological defects in the orientational field (see movies in SM [25]).
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Figure 4
Orientational order in hexatic phase. (a, b) Dynamics of orientational order parameter $Ψ = | Ψ | e^{i arg (Ψ)}$ with time step 0.001, at relative density $Δ ρ = 0.002$ , at $T = 2$ , and for $512^{2}$ particles. $t$ is number of MD steps. (c) Finite-size scaling of spontaneous orientational order. $L = 128, 256, 512$ is the linear size of system.
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Figure 5
Pressure-density equations of state for (a) $r^{- 12}$ and (c) $r^{- 64}$ potentials. (b) Power-law finite-size scaling of the bond orientational order at $ρ = 1.002$ for $r^{- 12}$ potential. Notice the logarithmic scale of the horizontal axis in (b) and that $x = ln \sqrt{N}$ in the legend.
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Figure 6
Fluctuations of local density when the global density is fixed at (a) $\bar{ρ} = 1.002$ for the $r^{- 12}$ and (b) $\bar{ρ} = 0.894$ for the $r^{- 64}$ potentials.
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