The stability of a crystal with diamond structure for patchy particles with tetrahedral symmetry Eva G. Noya, Carlos Vega, Jonathan P. K. Doye and Ard A. Louis Supplementary material TABLE I: Helmholtz free energies (Asol ) of the solid phases for the 12-6 model, as obtained by the Einstein molecule method. The coupling parameters in the Einstein molecule were chosen as 2 ) = λ /(k T )=30000. The free energy A λt /(kB T /σLJ o B sol and the average potential energy energy (U ) are given in units of N kB T . The uncertainty in Asol and U is about 0.02N kT . (TI) refers to independent simulations run as consistency checks. It can be seen that the free energy obtained by thermodynamic integration along an isotherm coincides with the value calculated from the Einstein molecule method within statistical uncertainty. Structure T∗ p∗ ρ∗ U Asol diamond 0.10 0.3 0.466 -16.30 -5.09 diamond 0.10 1.2 0.513 -14.95 -3.69 diamond (TI) 0.10 1.2 0.513 -3.71 bcc 0.10 3.00 1.035 -15.28 -3.34 bcc 0.10 0.65 0.932 -16.56 -5.11 bcc (TI) 0.10 3.00 1.035 bcc 0.15 0.5 0.900 -9.70 -0.03 fcc-o 0.10 6.08 1.234 -8.31 3.17 fcc-d 0.50 10.0 1.208 5.98 1 -3.35 0.12 TABLE II: Coexistence points for the 12-6 model obtained using the Gibbs-Duhem method. Uncertainties in the densities ρ∗ are of the order of 0.001 and in the potential energy per particle u∗ are smaller than 0.01. Phase 1 Phase 2 p∗ T∗ ρ∗1 u∗1 ρ∗2 u∗2 fluid bcc 2.01 0.20 0.959 -0.30 0.999 -0.99 fluid bcc 1.00 0.18 0.811 -0.32 0.922 -1.28 fluid bcc 0.40 0.16 0.652 -0.34 0.833 -0.92 fluid bcc 0.05 0.13 0.333 -0.35 0.868 -1.51 fluid bcc 0.001 0.10 0.010 -0.05 0.881 -1.63 fluid fcc-d 4.60 0.40 0.972 -0.07 1.072 -0.13 fluid fcc-d 3.30 0.30 0.959 -0.16 1.055 -0.22 fluid fcc-d 1.99 0.20 0.936 -0.31 1.034 -0.38 bcc fcc-o 3.80 0.13 1.061 -1.34 1.164 -0.92 bcc fcc-d 3.60 0.15 1.054 -1.28 1.153 -0.85 bcc fcc-d 2.80 0.18 1.022 -1.18 1.118 -0.42 bcc fcc-d 2.00 0.20 0.981 -1.08 1.030 -0.38 2 TABLE III: Helmholtz free energies (Asol ) of the solid phases, as obtained by the Einstein molecule method, for the 20-10 model. The coupling parameters in the Einstein molecule were chosen as 2 ) = λ /(k T )=70000. The free energy A λt /(kB T /σLJ o B sol and the average potential energy (U ) are given in units of N kB T . (TI) refers to independent simulations run as consistency checks. Uncertainties in the free energies and potential energies are of the order of 0.02 N kT . Structure T∗ p∗ ρ∗ U Asol diamond 0.10 0.473 0.530 -16.57 -4.35 diamond 0.10 0.01 0.515 -16.47 -4.47 diamond (TI) 0.10 0.01 0.515 -4.47 diamond 0.13 0.172 0.521 -11.65 -0.76 bcc 0.10 0.466 1.040 -16.86 -4.26 bcc 0.10 1.300 1.065 -16.93 -4.10 bcc (TI) 0.10 1.300 1.065 bcc 0.13 0.284 1.021 -12.02 -0.58 fcc-o 0.10 5.234 1.234 -10.83 1.78 fcc-o 0.10 8.000 1.272 -9.61 3.35 fcc-o (TI) 0.10 8.000 1.272 3.35 fcc-o 0.13 5.675 1.234 -7.59 4.18 fcc-d 0.50 5.423 1.050 -0.14 4.46 fcc-d 0.50 8.500 1.163 -0.09 5.70 fcc-d (TI) 0.50 8.500 1.163 5.71 3 -4.10 TABLE IV: Coexistence points for the 20-10 model obtained using the Gibbs-Duhem method. Uncertainties in the densities ρ∗ are of the order of 0.001 and in the potential energy per particle u∗ are smaller than 0.01. Phase 1 Phase 2 p∗ T∗ fluid bcc 1.0 0.184 0.806 -0.258 1.019 -1.334 fluid bcc 1.4 0.197 0.837 -0.246 1.023 -1.274 fluid bcc 1.6 0.202 0.886 -0.240 1.032 -1.248 diamond bcc 0.023 0.110 0.514 -1.604 1.021 -1.633 diamond bcc 0.010 0.080 0.519 -1.730 1.033 -1.746 diamond bcc 0.001 0.040 0.524 -1.874 1.046 -1.877 bcc fcc-o 6.45 0.010 1.148 -1.762 1.269 -1.219 bcc fcc-d 5.88 0.060 1.139 -1.658 1.252 -1.146 bcc fcc-d 5.20 0.130 1.129 -1.457 1.227 -1.004 bcc fcc-d 4.00 0.190 1.100 -1.281 1.189 -0.410 bcc fcc-d 2.50 0.212 1.057 -1.187 1.088 -0.325 fluid fcc-d 2.59 0.250 0.937 -0.043 1.051 -0.095 fluid fcc-d 3.84 0.350 0.952 -0.104 1.056 -0.155 fluid fcc-d 5.08 0.450 0.955 -0.182 1.060 -0.247 4 ρ∗1 u∗1 ρ∗2 u∗2 (a) (b) FIG. 1: Analysis of the bcc-diamond interface for the 12-6 model for a simulation box containing 2048 molecules. Initially the simulation box contained 1024 molecules in a bcc solid (i.e., 8×8× 8 unit cells) and 1024 molecules of fluid. As before, (a) and (b) show two different repesentations of the final configuration of the simulation, in which all the fluid has crystallized. (a) The molecules that were in the bcc solid structure in the initial configuration are coloured in blue, whereas those that were fluid molecules in the starting configuration are coloured in red. It can be seen that almost all the fluid has solidified into a diamond crystal. Only one or two incomplete bcc layers form at the two bcc-fluid interfaces. (b) The two sublattices are highlighted by colouring the particles belonging to each sublattice in a different colour, red for one sublattice and blue for the other. In contrast to the example discussed in the manuscript, a different sublattice has grown from each of the bcc-fluid interfaces. Some defects appear when the two lattices meet. It can be observed that one of the sublattices grew much more than the other. 5 FIG. 2: Final configuration for the bcc-diamond interface for the 12-6 model and a simulation box containing 4000 particles. Initially the simulation box contained 2000 molecules in the bcc solid (i.e., 10×10×10 unit cells) and 2000 molecules in the fluid phase. Molecules that were in the bcc solid in the initial configuration are coloured blue and those that were in the fluid phase in the initial configuration are coloured in red. As before, the fluid crystallizes in a diamond crystal with some defects. FIG. 3: Final configuration for the bcc-diamond interface for a simulation box containing 864 molecules for the shorted ranged LJ 20-10 model. Initially the simulation box contained 432 molecules in the bcc solid (i.e., 6×6× 6 unit cells) and other 432 molecules in the fluid phase. Molecules that were in the bcc solid in the initial configuration are coloured blue and those that were in the fluid phase in the initial configuration are coloured in red. 6
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