Abstract
The melting transition of two-dimensional (2D) systems is a fundamental problem in condensed matter and statistical physics that has advanced significantly through the application of computational resources and algorithms. 2D systems present the opportunity for novel phases and phase transition scenarios not observed in 3D systems, but these phases depend sensitively on the system and thus predicting how any given 2D system will behave remains a challenge. Here we report a comprehensive simulation study of the phase behavior near the melting transition of all hard regular polygons with $3\leq n\leq 14$ vertices using massively parallel Monte Carlo simulations of up to one million particles. By investigating this family of shapes, we show that the melting transition depends upon both particle shape and symmetry considerations, which together can predict which of three different melting scenarios will occur for a given $n$. We show that systems of polygons with as few as seven edges behave like hard disks; they melt continuously from a solid to a hexatic fluid and then undergo a first-order transition from the hexatic phase to the fluid phase. We show that this behavior, which holds for all $7\leq n\leq 14$, arises from weak entropic forces among the particles. Strong directional entropic forces align polygons with fewer than seven edges and impose local order in the fluid. These forces can enhance or suppress the discontinuous character of the transition depending on whether the local order in the fluid is compatible with the local order in the solid. As a result, systems of triangles, squares, and hexagons exhibit a KTHNY-type continuous transition between fluid and hexatic, tetratic, and hexatic phases, respectively, and a continuous transition from the appropriate "x"-atic to the solid. [abstract truncated due to arxiv length limitations].
Highlights
The phase behavior of two-dimensional solids is a fundamental, long-standing problem in statistical mechanics
We report a comprehensive simulation study of the phase behavior near the melting transition of all hard regular polygons with 3 ≤ n ≤ 14 vertices using massively parallel Monte Carlo simulations of up to 1 × 106 particles. By investigating this family of shapes, we show that the melting transition depends upon both particle shape and symmetry considerations, which together can predict which of three different melting scenarios will occur for a given n
We show that systems of polygons with as few as seven edges behave like hard disks; they melt continuously from a solid to a hexatic fluid and undergo a first-order transition from the hexatic phase to the isotropic fluid phase
Summary
The phase behavior of two-dimensional solids is a fundamental, long-standing problem in statistical mechanics. The system can exhibit a continuous fluidto-x-atic-to-solid transition. The existence of continuous isotropic fluid-to-solid transitions was predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory [21,22,23] and has been confirmed in experiments with electrons [1] and spherical colloids [2,4,8,9]. The KTHNY theory of two-step melting is based on the behavior of topological defects in the form of dislocations and disclinations. The theory envisions that pairs of dislocations unbind to drive a continuous transition from solid to hexatic.
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