Triangle: ground-state degeneracyTheoretical determination of ground statesThe presence of the short-range L-J interaction results in two isotopic unit-cell dimer structures with exactly the same potential energy, splitting from the original one for the dense packing scheme of the hard regular triangle, as illustrated in Fig. 2a. Either type of the isotopes can be spatially duplicated to form the crystalline morphology in Fig. 2b, in which each unit-cell is composed of two neighboring monomers colored differently according to their body-orientations, as shown in the upper part of Fig. 2c. The spatial arrangement of unit-cells is shown in the lower part of Fig. 2c, which is a rhombic crystal phase29,32 (\({\alpha }_{1}\ne {\alpha }_{2}\), \({\alpha }_{2}\) is smaller than \(\pi /2\) but larger than \(\pi /3\)) different from the crystalline structure of the hard triangular system25 due to the existence of the L-J balls.Fig. 2: Ground-state degeneracy of the triangular ball-stick system.a Two isotopic structures of the unit-cell, splitting from the original space-filling scheme for the hard triangle. b Crystalline morphology. c Spatial arrangement of unit-cells. One of \({\alpha }_{1}\) and \({\alpha }_{2}\), which are the angles formed by two unit-cell primitive vectors, is larger than \(\pi /2\), corresponding to a rhombic crystal structure. Each point represents a unit-cell composed of two monomers with opposite body-orientations, as shown at the top of this picture. d Three stable local structures composed of the two isotopes with the local clusters in the three configurations all composed of six atoms, which are circled by red, purple, and green frames, respectively. e Three typical small hand-made structures constructed based on the three local structures in d. The top one is a part of crystal, noted as ‘crystal’, the middle one is a mixture of the first two local configurations in d, noted as ‘perfect-mixing’, and the bottom one is formed by mixing all the three local configurations in d, noted as ‘tower-like’. The local clusters are framed with the same color as in d. f Potential energies (\({E}_{{{{\rm{p}}}}}\)) of different structures under various cutoff radii (\({r}_{{{{\rm{c}}}}}\)) with the solid lines representing \({E}_{{{{\rm{p}}}},{{{\rm{lowest}}}}}\) and the dashed lines representing \({E}_{{{{\rm{p}}}},{{{\rm{avg}}}}}\). Physical quantities are expressed in L-J units.These two isotopes can also mix with each other arbitrarily, forming an amorphous state without any geometrical conflicts. In order to have a comprehensive understanding on all the global structures, it is necessary to enumerate the possible local configurations at first. As shown in Fig. 2d, the stacking of the two types of isotopes can form three non-equivalent local clusters. The exhaustive geometrical enumeration (details in SI) suggests that the first two local configurations in Fig. 2d can fill in the whole space by their own to construct a crystalline structure or by mixing with the other one to form an amorphous state, while the third one cannot fill in the whole space by itself and must mix with the first two to avoid the creation of geometrical defects.If we count in only the L-J interaction from nearest-neighboring atoms, all the global structures without geometrical conflicts, including both crystalline and amorphous ones, have exactly the same potential energy when the system is so large that the energy cost on the boundary can be ignored, attributed to the fact that the atom clusters in all three of the above stable local structures, which are framed respectively by red, purple, and green in Fig. 2d, have 9 atom pairs. Therefore, a huge number of possible global structures with identical potential energy are all the ground states at \(T=0\), i.e., the triangular system has a ground-state degeneracy, which is very similar to the ground-state degeneracy in the spin-ice system42, where the oxygen atoms adopt a fixed lattice arrangement while hydrogen atoms can be either ordered or disordered without altering the potential energy.When the interaction exceeds nearest neighbors, the potential energy is difficult to be analytically calculated, so we instead calculate it numerically. We first construct three typical structures with 30 monomers, as shown in Fig. 2e, where the top one composed of solely the first local structure in Fig. 2d, the middle one containing both the first and the second structures, and the bottom one formed by mixing all the three local structures together. For comparison, we calculate two quantities related to the potential energy \({E}_{{{{\rm{p}}}}}\) for the three structures in Fig. 2e: (1) \({E}_{{{{\rm{p}}}},{{{\rm{avg}}}}}\), the potential energy averaged over nine clusters, as circled in each structure; (2) \({E}_{{{{\rm{p}}}},{{{\rm{lowest}}}}}\), the lowest potential energy of the clusters. The first one quantifies the global property while the second one provides the information on the local stability. By combining these two quantities, we can have a good estimation on the ground state(s). These two quantities are calculated with the cutoff radius (\({r}_{{{{\rm{c}}}}}\)) ranging from \(3\sigma\) which is approximately the size of the atom cluster, to \(7\sigma\) at which the L-J interaction has a very small value about \({10}^{-5}\), where \(\sigma\) is the van der Waals (vdW) diameter, and the results are shown in Fig. 2f. These three structures have nearly the same potential energy, as the difference of the potential energy varies with \({r}_{{{{\rm{c}}}}}\) but remains always on the order of \({10}^{-4}\). Therefore, we affirm that the ground state in the triangular system is degenerate, with two crystalline morphologies purely formed by one of the isotopes and a huge amount of amorphous configurations formed by mixing different local structures in Fig. 2d. At any finite temperatures, the vast amount of possible ways to mix the isotopes results in a very large configurational entropy, which allows the amorphous state to have a lower free energy than the crystal state.Confirmation of the ground-state degeneracy by simulationTo verify the above structural origin of the crystalline instability, we performed a replica exchange MD (REMD) simulation on 625 ball-stick triangles with 26 temperature replicas ranging from 2.55 to 6.1. An energy minimization on the final structure of the replica at the lowest simulated temperature has been performed after the REMD simulation, which is theoretically the most stable configuration at low temperatures43. All the three local structures in Fig. 2d appear in the optimized structure, whose representative snapshot is shown in Fig. 3a. Moreover, benefiting from the fact that the bond length (1.5\(\sigma\)) is just a little larger than the equilibrium distance of \(\root 6\of{2}\sigma\), at finite temperatures, there exist some ‘meta-space-filling’ structures (colored in blue), where at least two atoms in the same cluster are connected by bond directly, leading to small geometrical defects and a little higher potential energy than the space-filling ones at zero temperature. As a result, they should vanish at \(T=0\) but could appear at a slightly higher temperature for they further enlarge the configurational entropy.Fig. 3: Simulation evidence for the ground-state degeneracy.a A small part of the representative snapshot for the most stable structure determined by the replica exchange molecular dynamics simulation. The green and red frames mark different local configurations in Fig. 2d, and the blue frames illustrate the existence of meta-space-filling structures at finite temperatures. Red and green have the same meaning as in Fig. 2d, while we do not distinguish red and purple here. b Amorphization process characterized by the decay of bond-orientational order parameter \(|{\varPsi }_{4}|\) at three temperatures below the melting point. The inset is a small part of the snapshot at T = 2.86 after amorphization. c Caloric curve (potential energy \({E}_{{{{\rm{p}}}}}\) versus temperature T) and density curve (density \(\rho\) versus T), both with the discontinuous changes located at T = 2.88‒2.89. The error-bars represent standard deviations, resulting from the thermal fluctuations in one simulation trajectory. d Time evolution of the unit-cell order parameter d in the simulation starting from a locally disordered crystalline structure at T = 2.83, 2.86, and 2.88. Each curve plotted in b and d is based on the data from a single simulation. Due to the randomness of the dynamical process, the starting point and the rate of amorphization may take different values in different trajectories. Physical quantities are expressed in L-J units.As the amorphous state is more stable than crystal at finite temperatures, we expect that, driven by entropy, the crystal state will spontaneously destabilize into the amorphous state. To check this, MD simulations starting from crystalline morphology are performed at pressure \(P=0\) and various temperatures. As quantified by the bond-orientational order parameter \({\varPsi }_{4}\) (see Methods for the definition) plotted in Fig. 3b, which initially keeps a high value (larger than 0.8) corresponding to the rhombic crystal state29,32 and then decays after certain simulation steps until a very low value due to its amorphous nature, the amorphization takes place at \(T\ge 2.85\), lower than the melting point \(T=2.88 \sim 2.89\) manifested by the sharp changes in the caloric and density curves plotted in Fig. 3c. The non-equilibrium amorphous state whose potential energy and density are close to those of crystal is featured by the mixing of two isotopes, as shown in the inset of Fig. 3b, in agreement with our theoretical prediction described above. At lower temperatures, the fact that amorphization does not occur within the finite simulation time should be attributed to the existence of relatively high energy barrier(s) between the crystalline and amorphous states. To verify this, we manually introduce a very small disordered region (about 30 monomers) into the crystalline configuration (see Methods for technical details) and run NPT simulations at \(T=2.83\), 2.86, and 2.88, respectively. The evolution of the two monomers in each unit-cell is traced by the unit-cell order parameter defined as \(d(t)=\frac{1}{n} < {\sum}_{{{{\rm{unit}}}}-{{{\rm{cell}}}}}\varTheta ({l}_{{{{\rm{c}}}}}-l(t)) > \), where the number of unit-cells n equals to half of the number of monomers, l(t) is the distance between the center-of-masses (COMs) of the two monomers originally in the same unit-cell at time t, \({l}_{{\mathrm{c}}}\) =5.77 is the cut-off value of l, and \(\varTheta\) is the Heaviside step function. This order parameter quantifies the fraction of remaining unit-cells at time t, taking a value close to 1 in crystal and 0 in liquid. The time evolution of d in \(6.4\times {10}^{7}\) steps after a relaxation of \(4\times {10}^{6}\) steps is shown in Fig. 3d, demonstrating that, once the energy barrier is overcome, the system will rearrange into the amorphous state continuously. We expect that d would finally drop to approximately zero after sufficiently long time, as a result of reorganization of the whole system. Although regular MD simulations at much lower temperatures would suffer from very slow dynamics, we have managed to verify that the amorphous state is still more stable than the crystalline state even at a temperature as low as T = 1.5 by running an additional REMD simulation focusing on the low-temperature region (see Supplementary Fig. S4 in SI). Moreover, we never observe an inverse pathway from the amorphous state back to the crystal, as appeared in the simulation for the hard-triangle system25, in agreement with the fact that the amorphous state is more stable than the crystal benefiting from the large configurational entropy.Glassy nature of the spin-ice-like amorphous stateDetailed analysis on dynamics provides a more comprehensive understanding on this amorphous state. As shown in Fig. 4a, in the amorphous state at \(T=2.83\), 2.86, and 2.88, the calculated mean-squared displacements (MSDs) \({\varDelta }^{2}\) of monomers grow in a strongly frustrated way, clearly distinguished from the one for crystal or liquid shown in Supplementary Fig. S5. We further plot the non-Gaussian parameter for MSD in Fig. 4b, defined as \(\alpha=\frac{3 < {(r(t)-r(0))}^{4} > }{5 < {(r(t)-r(0))}^{2}{ > }^{2}}-1\), which is expected to be 0 for the Brownian motion and larger than 0 when some monomers move faster than others. The calculated \(\alpha\) for this amorphous state is higher than in the crystal or liquid state, indicating a strong dynamical heterogeneity and the fact that the amorphous state is a glassy state from the dynamical perspective44.Fig. 4: Physical nature of the non-equilibrium amorphous state and the phase transition.a Mean-squared displacements (MSDs) (Δ2) in the glassy state exhibiting frustrated diffusive behavior. b Non-Gaussian parameters for MSD \(\alpha\) vs. temperature T. The two values at T = 2.83 correspond to the crystalline state (smaller one) and the glassy state (larger one). c Schematic illustration of the spin-ice-like state defined on a topologically triangular lattice with the ‘4-in-2-out’ ice-rule. The left panel shows the six neighboring atom-clusters (circled in blue) of the central cluster (circled in red) while the right one illustrates that the six atoms in the cluster exhibit the ‘4-in-2-out’ feature along the direction indicated by the orange arrows. d Bond-orientational order parameter \({\varPsi }_{6,{\mathrm{atom}}}\) vs. T, whose discontinuous change suggests a first-order nature of the melting of the spin-ice-like state. The inset is a small portion of the snapshot exhibiting the local packing scheme, where the purple lines and circles highlight the six neighboring atoms of the central atom shadowed in purple arranged in a nearly dense-packing way. The error-bars in this figure represent standard deviations, resulting from the thermal fluctuations in one simulation trajectory. Physical quantities are expressed in L-J units.Overall, considering both the structural and dynamical characteristics, the 2D ball-stick triangular system is in a spin-ice-like glassy state at any finite temperatures below melting point45, as a result of the strong confliction between interaction and shape. As shown in Fig. 4c, each stable local atom cluster has six neighboring clusters and contains six atoms with four close to each other while the other two away from them, so our system shares some common features with a spin-ice system defined on a triangular lattice with the ‘4-in-2-out’ ice rule46. However, the ball-stick triangular system is more complicated than the spin-ice system due to the following two features: (1) Not only the pair interaction from nearest neighbors is considered; (2) The triangular lattice is defined topologically without well-defined lattice structure (no on-site oxygen atoms), so the spatial structure is totally disordered.The detailed examination on Fig. 3a, b suggests that no matter the isotopes mix or not, each atom has six atomic nearest neighbors arranging in a nearly dense-packing way. This local structure is shown clearer in the inset of Fig. 4d, where the atom shadowed in purple has six neighbors circled by purple frames. Inspired by the spin-glass system47,48 whose magnetization order parameter characterizes the order-disorder transition while the Edwards-Anderson order parameter measures the glass-liquid transition, here it is also possible to find an order parameter other than \({\varPsi }_{4}\) to distinguish the glassy state from the liquid one. We therefore employ the bond-orientational order parameter \({\varPsi }_{6,{{{\rm{atom}}}}}\) associated with all atoms indistinguishably of the presence of bonds instead of molecular COMs. The discontinuous drop of \({\varPsi }_{6,{{{\rm{atom}}}}}\) at T = 2.88~2.89 in Fig. 4d demonstrates that the spin-ice-like state melts into liquid via a first-order phase transition, in agreement with the caloric and density curves in Fig. 3c.Furthermore, the MD simulation at P = 10 (details in SI) manifests that the crystalline morphology rearranges into a spin-ice-like state at finite temperatures and the melting transition from spin-ice-like to liquid is still a first-order phase transition, indicating that the revealed phase behavior of the 2D ball-stick triangular system is robust with respect to different thermal conditions.Square: distorted square latticeBased on the result from REMD simulation and the consideration on symmetry, the initial state was established as a square lattice shown in Fig. 5a, where the body-orientation of each monomer is not perpendicular to the primary vector of lattice to ensure a higher packing fraction. MD simulations at various temperatures were performed at P = 0. The apparent discontinuous changes in the caloric curve, density curve, and \({\varPsi }_{4}-T\) plotted in Fig. 5b manifest a first-order phase transition from solid to liquid with the melting point located between 2.92 and 2.93.Fig. 5: Simulation results of the ball-stick square system.a Initial crystalline morphology. The two orange arrows are the primitive lattice vectors. b Caloric curve (potential energy \({E}_{{{{\rm{p}}}}}\) versus temperature T) and density curve (density \(\rho\) versus T). The inset is the bond-orientational order parameter \({\varPsi }_{4}\) at different temperatures. All the discontinuous changes of these three curves locate at T = 2.92~2.93. The error-bars represent standard deviations, resulting from the thermal fluctuations in one simulation trajectory. c The spatial arrangement of center-of-masses (the black dots) at T = 2.92. The polygon corresponding to each point is decided by the Delauney tessellation considering the periodic boundary condition and is colored according to its number of edges. d Bond-orientational correlation function (correlation \({{{{\rm{C}}}}}_{\psi }\) as a function of distance r) at the melting point for N = 2496 monomers and N = 22,500 monomers, indicating no decay compared to the brown line corresponding to an algebraical decaying behavior. e Translational correlation functions (correlation \({{{{\rm{C}}}}}_{g}\) as a function of r) at melting points for N = 22,500 monomers with different block sizes n (number of monomers in each block), decaying faster than algebraically (the black line) locally but slower on the long range. The distance is rescaled by the characteristic length of the coarse-graining a. Physical quantities are expressed in L-J units.The visual examination of Fig. 5c tells us that, in the solid state, although each monomer has four neighbors aligned nearly orthogonally, also reflected by the diffraction pattern shown in Supplementary Fig. S7a, it apparently distorts from the normal square lattice. This is evidenced by the fact that the Voronoi cells are close to squares but mostly have 6 edges. Moreover, the concentration of topological defects, defined as the fraction of Voronoi cells with more or less than 6 edges, is as high as about 0.42 (data in Supplementary Fig. S8b), significantly larger than the typical value of crystal (about 0.05) and even hexatic (about 0.123)25,49,50. The quantitative characterization of this state exhibits two seemingly contradict facts: (1) The long-range bond-orientational correlation can be retained until melting, indicated by the correlation function in Fig. 5d, as well as the heatmap in Supplementary Fig. S7c colored according to the argument of \({\psi }_{4,i}\), where the whole picture holds almost the same color, in agreement with its very narrow distribution; (2) The translational correlation function exhibits a decaying behavior slightly faster than algebraical, as shown in Supplementary Fig. S8e.To clarify this paradox, we perform simulations on a larger system with N = 22,500 monomers, whose results are consistent with the above (details in SI). With the larger simulation scale, we identify that, the translational correlation decays relatively fast locally but exhibits a typical algebraical decaying behavior in the long range, as illustrated by the red line in Fig. 5e, indicating that this solid state is still a crystal. Combined with the above features, we regard it as a distorted square lattice. The translational symmetry is further investigated by a ‘renormalization’ process, in which we divide the monomers into blocks, with n (=1, 4, 9, 25, 100, 225) monomers in each block, and then calculate the translational correlation of the block COMs. After rescaling the distance by the characteristic length of the block, which is just the distance of the first peak in the correlation function, different curves collapse together, as shown in Fig. 5e, exhibiting the so-called self-similarity, which guarantees the existence of the translational symmetry. Furthermore, considering the locally disordered nature, this self-similarity strongly implies that it is a critical point for the ordered and disordered solid states51,52, neither totally disordered as the triangular system nor perfectly ordered as a standard crystalline structure for pentagon, hexagon, or octagon. Since this holds at different temperatures (e.g., also T = 2.8 in Supplementary Fig. S8f), the critical point is defined in the shape space in terms of a critical edge number, rather than in the thermal-parameter space.When cooling down the distorted square lattice, the Voronoi cells have less dispersive areas but surprisingly higher concentrations of topological defects (Supplementary Fig. S8b, c), even for the case as low as T = 0.01. This supports the mechanism that the local distortions are originated from the intrinsic properties of the system, in terms of interaction and shape rather than thermal fluctuations, emphasizing its critical nature in the shape space.We also perform simulations at P = 5 and P = 10. Under different pressures, the square system still follows a solid-liquid transition without the appearance of the tetratic phase and the solid structures also share similar features (see SI).Pentagon, hexagon, and octagon: entropy-dominant versus enthalpy-dominantPhase diagramsFor these three polygons, the ground-state crystalline structures: striped phase for pentagon, triangular lattice for both hexagon and octagon, can be stabilized until their melting temperatures, attributed to the even weaker confliction between interaction and shape. The snapshots of these crystalline configurations are shown in Fig. 6a, b, and g, respectively. Below we describe the melting scenarios of each polygon under various pressures, and the readers are referred to the Methods section and SI for more details on the calculation procedure and results.Fig. 6: Crystalline morphologies of pentagon, hexagon, and octagon systems.a Striped phase of the pentagon system, in which monomers have two body-orientations comprising alternating lines. b Rotator crystal phase of the pentagon system, in which monomers have random body-orientations. c Body-orientation distributions for the two crystal phases of the pentagon system, where \(\theta\) represents the orientation of monomer with respect to the x-axis of the simulation box, ranging from 0 to \(2\pi /n\) for n-gons, and \(P(\theta )\) is the corresponding probability density. d Triangular lattice crystal phase of the hexagon system. e Rotator crystal phase of the hexagon system, in which monomers have two preferred body-orientations. f Body-orientation distributions for the two crystal phases of the hexagon system. g Triangular lattice crystal phase of the octagon system. h Rotator crystal phase of the octagon system. i Body-orientation distribution for the two crystal phases of the octagon system, where the y axes are scaled differently to exhibit the trimodal distribution in the rotator crystal phase.Our MD simulations indicate that the striped phase is the most stable phase at low temperatures for the ball-stick pentagon system, which is also theoretically the closest packing mode for hard pentagon53. With increasing temperature, the striped phase first experiences a first-order solid-solid phase transition into the rotator crystal phase and then melts into the liquid phase via another first-order phase transition, the same as the melting scenario of hard pentagon28. In the striped phase, monomers are aligned with two body-orientations to form alternative lines, as shown in Fig. 6a, rendered as two peaks in the body-orientation distribution (the blue line in Fig. 6c). In the rotator crystal phase shown in Fig. 6b, monomers have a totally random body-orientation, demonstrated by the flat distribution of body-orientation (the orange line in Fig. 6c), indicating that this rotator crystal phase is a continuous one54. The phase behavior of the pentagon system is summarized in the phase diagram in Fig. 7a, in which the striped phase, rotator crystal phase, and liquid phase appear in sequence as temperature increases at any pressures.Fig. 7: Pressure (P)-Temperature (T) phase diagrams of different polygons.a Pentagon. b Hexagon. c Octagon. The striped phase (S) and the triangular lattice crystal phase (TX) are colored in orange, the rotator crystal phase (RX) is colored in light blue, the hexatic phase (H) is colored in black (very narrow for hexagon and octagon), and the fluid phase (F) (liquid or gas) is colored in pink. Physical quantities are expressed in L-J units.The hexagon system exhibits a rich phase behavior with increasing pressure. At P = 0, the crystalline structure sublimates without the appearance of the liquid state, evidenced by a nearly-zero potential energy and an RDF curve with only one peak shown in Supplementary Fig. S11 in SI. At P = 0.5 and 1.0, the crystalline structure melts into the liquid state via a first-order phase transition. At P = 1.5, 2.5, and 5, a rotator crystal phase (Fig. 6e) appears between the crystal state and the liquid state, and both of the two phase transitions are discontinuous. The monomer COMs in the rotator crystal phase form a triangular lattice, but monomers have two body-orientations identified by the two broad peaks in Fig. 6f, so this rotator crystal phase is regarded as a discontinuous one54. When the pressure increases to P = 7.5, the hexatic phase appears between the rotator crystal phase and the liquid phase. In this case, both transitions from the triangular lattice crystal to the rotator crystal and from the hexatic phase to the liquid phase are discontinuous, while the one from the rotator crystal phase to the hexatic phase is continuous, basically following the hard-disk-like behavior21. Even at P = 10, the melting scenario still follows a hard-disk-like behavior, instead of the KTHNY theory followed by the hard hexagon or the L-J beads hexagon25,40, attributed to the appearance of the rotator crystal phase. The phase diagram is shown in Fig. 7b, in which the rotator crystal phase only appears at relatively high pressures and the hexatic phase emerges at even higher pressures.The octagon system stabilizes at the triangular lattice up to a finite temperature when it transforms into a rotator crystal phase discontinuously. The rotator crystal of octagon has roughly three preferred body-orientations of monomers, as shown in Fig. 6g, h, similar to the case for hard octagon54. However, the body-orientation distribution is very rough and the ranges between peaks and valleys are much smaller than that of the discrete rotator crystal phase for hexagon. The melting scenario follows a regular solid-liquid one at relatively low pressures (P = 0–5). The hexatic phase appears at higher pressures of P = 5–8 and leads to a hard-disk-like behavior, qualitatively the same as the L-J disk40. The phase diagram of the octagon system is drawn in Fig. 7c, showing that there are always two solid states, and the hexatic region is very narrow at high pressures and disappears at low pressures.Phase stabilitiesTo gain a deeper understanding on the mechanism of phase stabilities, we plot the melting points of different crystalline structures under different pressures in Fig. 8. The melting point increases with pressure and is higher for polygons with more edges at relatively high pressures, which is regarded as the entropy-dominant regime. Meanwhile, the melting-temperature curves for hexagon and octagon have a crossover point located between P = 1 and P = 1.5. At relatively low pressures (\(P\le 1\)), hexagon stays in the triangular-lattice crystal phase before melting into liquid, and has a higher melting point than octagon. By contrast, at relatively high pressures (\(P\ge 1.5\)), hexagon is in the rotator crystal phase before melting into liquid, and has a melting point lower than octagon. Therefore, the crossover of the melting point locates in the pressure interval where the rotator crystal phase for hexagon appears. Detailed examination on the local structure of triangular-lattice crystal morphology suggests that, as shown in Fig. 6d and much clearer in Fig. 9d, each atom has two neighboring atoms belonging to two different molecules, and these three atoms form a regular triangular local structure, which is the most stable structure of an L-J cluster composed of three atoms. In other words, the crystal phase for hexagon has the intermolecular interaction perfectly matches the molecular shape, leading to an ultra-stability at low pressures when the phase stability is dominated by enthalpy.Fig. 8: Melting point (\({T}_{{{{\rm{m}}}}}\)) vs. pressure (P).Lines in different colors represent different polygons, and different symbols represent different solid states: square for close packing (CP) and triangle for rotator crystal (RX). Physical quantities are expressed in L-J units.Fig. 9: Schematic illustration of the confliction between shape and interaction.a Triangle, in which two monomers colored differently belong to the same unit-cell with different orientations. b Square. c Pentagon. d Hexagon. For each polygon, there is a non-zero torsional angle between the connecting line of neighboring center-of-masses determined by shape only (i.e., dense-packing pattern, represented by the dark purple line) and the one in the ground state of ball-stick polygon (represented by the orange line).The above phenomenon helps us to deepen our understanding on the competition between the entropy and enthalpy. At relatively low pressures, the interaction plays a leading role in phase stability, i.e., whether the local structure is interaction-favored determines the phase stability, which is regarded as ‘enthalpy-dominant’. Conversely, at relatively high pressures, the molecular shape of a polygon determines its phase stability, corresponding to an ‘entropy-dominant’ mechanism. The striped phase for pentagon and the triangular lattice crystal for octagon do not show the same ultra-stability as hexagon because the interaction and shape there do not perfectly match each other.
