The pair potential used in this study is an isotropic step potential with hard-core diameter σ. is the soft-core distance, and is the attractive distance limit. r is the distance between two particles and ε is the bond energy.
Illustration of the phases modelled. (a) The liquid (L), here shown as a small portion of a simulation in which distinct local packing environments are visible. (b) The square crystal (S). (c) The low-density triangular crystal (LDT). (d) The high-density triangular crystal (HDT). (e) The A crystal. (f) The Z crystal. Line segments for the crystal phases indicate a bond with energy −ε and a dashed line segment one with energy −ε/2.
Equations of state of the liquid (circles) and S crystal (diamonds) at k B T/ε = 0.55. The curves show fits according to Eqs. (3) and (4) , with a l = −0.457818, b l = −9.18372, and c l = 33.4007 for the liquid (inset) and a s = 479.035, b s = −686.583, and c s = 246.067 for the crystal.
Determination of a coexistence P between the L and S phases at k B T/ε = 0.55. Panel (a) shows the chemical potential isotherms for the liquid (solid curve) and the square crystal (dashed curve). Inset shows a closeup of the crossing. In panel (b) we show the difference in chemical potential Δμ between the two phases over the entire range of P for which the equations of state overlap, with dashed lines indicating upper and lower uncertainty estimates.
Snapshot configuration obtained from an NVT simulation for 10 000 particles at k B T/ε = 0.55 and ρ = 0.786567. Black symbols represent particles belonging to the S phase, while grey symbols represent the L phase.
(a) Enthalpy per particle for the liquid (circles) and LDT crystal (diamonds) along Pσ2/ε = 0.05. Here we have subtracted the ideal gas contribution to the energy. Inset shows data for a larger system of approximately 4000 particles (filled symbols). (b) The corresponding chemical potential difference between the L and LDT phases for the entire range in T of metastability. (c) The chemical potential difference between the L and S phases at Pσ2/ε = 0.15.
Sample time series of the number density near the L-HDT coexistence curve, with N = 986, k B T/ε = 5.0, and P 0 = 50.0ε/σ2. The system samples both the (lower density) liquid and the HDT crystal.
Conditional Gibbs free energy as a function of ρ. At k B T/ε = 5.0 and a pressure P 0 = 50.0ε/σ2 slightly above coexistence (solid curve, circles), the high-density basin (HDT crystal) has a lower free energy than the low-density (liquid) basin. Through Eq. (16) , an appropriate shift in the pressure locates the coexistence pressure, i.e., transforms the P 0 curve so that the liquid and HDT minima are at the same level to within precision of the data (dashed line, squares).
Phase diagram of the 2D model in the P-T plane, showing the liquid (L), gas (G), and crystal phases HDT, S, LDT, A, and Z (see Fig. 2 ). The panels show portions of the phase diagram at (a) high, (b) medium, and (c) low P. The liquid-gas coexistence line terminates at a critical point at k B T c /ε = 0.533 and P C σ2/ε = 0.0185 (filled circle). Dashed lines in (b) are metastable coexistence lines assuming the absence of the gas phase. Initial coexistence points, i.e., starting points for Gibbs-Duhem integration, are indicated by circles, while ×'s show repeated coexistence calculations done as checks on the Gibbs-Duhem integration.
Phase diagram in the ρ-T plane. (a) The points along the G-L coexistence lines indicate results from Gibbs Ensemble simulations and the large filled orange circle shows our estimate of the G-L critical point based on an extrapolation described in the text. Panel (b) shows the phase diagram in the absence of the gas phase. The filled black circle shows the location of the obscured L-L critical point discussed in Ref. 39 .
P-T phase diagram obtained at high pressure, near the L-S-HDT triple point. The grid of points is obtained from three sets of simulations. Each of L, S, and HDT is used to initialize a simulation set with N = 986, 992, and 986, respectively. The final phase adopted from each set at each state point is indicated by a symbol: S, square; HDT, triangle; L, ×. For example, at low P and high T, both L and HDT transform to S, while near the triple point, each phase retains metastability.
Comparison of our phase diagram with previously reported system properties. Red curves are taken from Ref. 39 and represent crystallization lines (+), locus of temperatures of maximum density along isobars (TMD, circles), pressures of maximum diffusivity along isotherms (D max, diamonds), maxima of isothermal compressibility (K Tmax, squares), and G-L coexistence. Also shown are the G-L critical point (filled red circle) and the obscured L-L critical point (large hashed circle). Dotted lines show pressure along isochores. All other symbols as in Fig. 10 . We note that we determine the location of the G-L critical point from an extrapolation of data above k B T/ε = 0.50, while the one reported in Ref. 39 is based on inflection points of pressure isotherms. The previously reported crystallization lines fall within the presently calculated crystal stability fields.
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