Geometrical defects in two-dimensional melting of many-particle Yukawa systems

We present a theoretical polygon construction analysis of two-dimensional melting and freezing transitions in many-particle Yukawa systems. Two-dimensional melting transitions can be characterized as proliferation of geometrical defects—nontriangular polygons, obtained by removing unusually long bonds in the triangulation of particle positions. A liquid state is characterized by the temperature-independent number of quadrilaterals and linearly increasing number of pentagons. We analyze specific types of vertices, classified by the type and distribution of polygons surrounding them, and determine temperature dependencies of their concentrations. Solid-liquid phase transitions are followed by the peaks in the abundances of certain types of vertices.

Figure 1

Triangulation map of a polycrystalline Yukawa solid, with topological defects marked as squares and triangles (a). By removing the bonds facing unusually large angles (marked by bold lines), the polygon construction (b) is obtained. In general there is no one-to-one correspondence between topological and geometrical defects.

Figure 2

Twelve types of vertices, frequently observed in the polygon construction of two-dimensional Yukawa systems. Vertices are classified by the number and relative distribution of polygons around a vertex.

Figure 3

Averaged orientational order parameters ${\psi }_6$ and ${\psi }_{\left|6\right|}$ of a 2D Yukawa system during the heating and cooling simulation cycle.

Figure 4

Topological $\mathrm{DF}$ as a function of the kinetic temperature $kT$.

Figure 5

Typical arrangements of topological defects during the melting transition of a 2D Yukawa system. Triangles here represent particles with five closest neighbors, while squares mark vertices with the coordination number equal to seven. The corresponding temperatures are $kT=0.00245$ (a), $0.0026$ (b), $0.0027$ (c), and $0.0045$ (d).

Figure 6

Time series for the topological $\mathrm{DF}$ at the prescribed temperature of $k{T}_{\mathrm{ref}}=0.00243$.

Figure 7

Typical arrangements of geometrical defects during the gradual heating stage in a 2D Yukawa system: solid phase at $t=45002$ (a), configurations during (b) and right after (c), the solid-liquid transition and high-temperature liquid phase at $t=120000$ (d). The corresponding temperatures are $kT=0.0018$ (a), $0.0026$ (b), $0.0027$ (c), and $0.0045$ (d).

Figure 8

Time series for order parameters $P_3$ (a), $P_4$ (b), $P_5$ (c), and $P_6$ (d), characterizing the abundances of triangles, quadrilaterals, pentagons, and hexagons in the polygon construction of the 2D Yukawa system. Time series of the kinetic temperature $kT$ is presented in panel (e). Vertical lines mark the beginning and ending points of the melting transition as well as corresponding points during the crystallization.

Figure 9

Time series for vertex fractions and the temperature $kT$. The time dependence of a vertex fraction $X_{\mathrm{F}}$ is virtually identical to the one for vertex E and therefore is not shown here. Vertical lines mark the beginning and ending points of the rapid order-disorder transition and similarly peak positions during the crystallization.