Two-dimensional (2D) systems have attracted significant interests in condensed matter physics due to their novel physical properties and diverse applications, for which comprehensive understandings on phase behaviours of 2D systems are necessary. Having been theoretically studied under the framework of statistical mechanics since 1930s, it is now well known that the continuous symmetry in 2D typically corresponds to a topological phase transition, first proposed by Kosterlitz and Thouless1, who won the Nobel Prize in 2016. Theoretically, the so-called 2D crystals have only quasi-long-range translational orders along with long-range bond-orientational orders at finite temperatures. They melt into the liquid state via one of the following three pathways with possible participation of an intermediate phase called hexatic2: the KTHNY theory, hard-disk-like behaviour, or regular solid-liquid phase transition.Since analytical theories focus on topological defects regardless of microscopic details, “How do 2D molecular systems melt?” still remains elusive. Thanks to the fast development of experimental and simulation techniques, extensive researches have been performed and achieved comprehensive understandings for 2D systems on both the enthalpy-limited (interactive point-like particles) case and entropy-limited (hard polygons) case. However, these studied systems are still too simple to understand real 2D molecular systems containing compatible entropy and enthalpy effects. The ball-stick polygons composed of covalent bonds and interactive balls are better models since on the one hand, they contain more features of real molecules, while on the other hand, their results can be conveniently compared with those for the previously well-studied simpler model systems.
Fig. 1 An illustration of the phase stabilities of the crystalline morphologies with increasing edge number at finite temperatures.
By molecular dynamics simulations, we revealed a critical edge number nc = 4 for the phase stabilities of the crystalline morphologies in the regular ball-stick polygon family (triangle, square, pentagon, hexagon, and octagon)3, as shown in Fig. 1. First, the polygon system with less than 4 edges, i.e., triangle, stabilizes at a spin-ice-like glassy state at any finite temperatures before melting into liquid via a discontinuous phase transition. The structure shown in Fig. 2a is featured by the mixing of the two isotopic unit-cell structures, and can be mapped to a spin-ice system with ‘4-in-2-out’ ice-rules defined on a topological triangular lattice when considering the atomic clusters, as illustrated in Fig. 2b. Second, the square system with exact 4 edges forms a distorted square lattice as shown in Fig. 2c, quantified by a self-similar translational symmetry but very high concentration of topological defects, as evidenced by the Delauney tessellation shown in Fig. 2d, where each Voronoi cell is coloured according to its edge number. At higher temperatures, this distorted square lattice melts into liquid directly without the appearance of tetratic phase, which has been observed in the hard square system. Finally, polygons with 5 or more edges have normal crystalline morphologies at finite temperatures. By comparing the melting points of pentagon, hexagon, and octagon crystals, we found that a polygon with more edges always has a higher melting point at high pressures, but hexagon is ultra-stable at low pressures attributed to the perfect match of the shape and interactions. This indicates two competitive mechanisms for phase stabilities: ‘entropy-dominant’ at high pressures and ‘enthalpy-dominant’ at low pressures.
Fig. 2 Spin-ice-like state formed by ball-stick triangles and distorted square lattice formed by ball-stick squares. The frames with different colours in (a) highlight the two isotopic unit-cell structures. By focusing on the central atomic clusters, the system can be mapped to a spin-ice system defined on a topological triangular lattice since each cluster has six neighbouring clusters, as shown in the left panel in (b), and the ‘4-in-2-out’ ice-rule is illustrated by the right panel in (b). The distorted square lattice in (c) clearly exhibits the 4-fold symmetry with a very high concentration of topological defects, as demonstrated in (d).
Whether there is a hexatic phase is a central focus for the study on 2D melting. A phenomenological rule has been summarized from the studies on hard regular polygons, which suggests that the hexatic phase appears when the symmetries in liquid and solid states are matched or the effect of orientational entropy is relatively weak4. Here we have found that this rule is generally effective for ball-stick polygons with normal crystalline morphologies (pentagon, hexagon, and octagon) but fails for polygons with intrinsically disordered features (triangle and square).
All the results may be understood under the theoretical framework considering the competition between entropy and enthalpy. Since the interactive balls locate on the vertices, the symmetries of shape and interaction spatially conflict with each other. The confliction is stronger for polygons with less edges, which are more difficult to form ordered structure. Entropy and enthalpy also compete with each other thermodynamically, leading to different features of phase stabilities. Although this theoretical framework comes from relatively simple systems studied here, it may be valuable for understanding more complex systems and more general cases of 2D melting scenarios.
Finally, ball-stick polygon systems, though still simpler than real molecular systems, capture two essential factors: shape and interaction. With these two factors, the building blocks on different scales, ranging from atoms to polymers, share qualitatively the same assembled structures, even for some out-of-equilibrium structures5.Therefore, our work is anticipated to facilitate the design of various 2D materials. For instance, the topological spin-ice-like state should have some unexpected features beyond the well-studied spin-ice models, and the distorted lattice may be used for tuning mechanical properties or electronic band structures of 2D materials.
References
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C: Solid State Physics 6, 1181 (1973)
Strandburg, K. J. Two-dimensional melting. Rev. Mod. Phys. 60, 161 (1988)
Zhu, R. & Wang, Y. A critical edge number revealed for phase stabilities of two-dimensional ball-stick polygons. Nat. Commun. 15, 6389 (2024)
Anderson, J. A. et al. Shape and symmetry determine two-dimensional melting transitions of hard regular polygons. Phys. Rev. X 7, 021001 (2017)
Whitelam, S. et al. Common physical framework explains phase behavior and dynamics of atomic, molecular, and polymeric network formers. Phys. Rev. X 4, 011044 (2014)