Liquid part of the phase diagram and percolation line for two-dimensional Mercedes-Benz water

Monte Carlo simulations and Wertheim's thermodynamic perturbation theory (TPT) are used to predict the phase diagram and percolation curve for the simple two-dimensional Mercedes-Benz (MB) model of water. The MB model of water is quite popular for explaining water properties, but the phase diagram has not been reported till now. In the MB model, water molecules are modeled as two-dimensional Lennard-Jones disks, with three orientation-dependent hydrogen-bonding arms, arranged as in the MB logo. The liquid part of the phase space is explored using grand canonical Monte Carlo simulations and two versions of Wertheim's TPT for associative fluids, which have been used before to predict the properties of the simple MB model. We find that the theory reproduces well the physical properties of hot water but is less successful at capturing the more structured hydrogen bonding that occurs in cold water. In addition to reporting the phase diagram and percolation curve of the model, it is shown that the improved TPT predicts the phase diagram rather well, while the standard one predicts a phase transition at lower temperatures. For the percolation line, both versions have problems predicting the correct position of the line at high temperatures.

Figure 1

MB molecules. Particles form the strongest hydrogen bond when the arms are collinear and the distance between two particles is equal to $r_{\mathrm{HB}}$.

Figure 2

Definition of the excluded volume in the modified version of the TPT.

Figure 3

(a) Density as a function of chemical potential for different temperatures. The jump indicates a liquid-gas phase transition. (b) Pressure as a function of density for different temperatures. Below the critical point, we can see nonmonotonic behavior of the pressure, and with Maxwell construction, the coexistence line can be calculated. (c) Internal energy as a function of density for different temperatures. In (a)–(c) results from computer simulations are plotted with symbols; results from the improved TPT, with lines. (d) An alternative to Maxwell construction is to plot the chemical potential as a function of the pressure in the region of phase transition. Results were calculated by the improved TPT method for different temperatures. The intersection of the curve is where there is coexistence of phases. Green symbols, high temperature, $T^{*}=0.3$; blue symbols, $T^{*}=0.2$; and pink symbols, low temperature, $T^{*}=0.15$.

Figure 4

Vapor-liquid equilibria from MC and TPT. Results from computer simulations are plotted as symbols; results from the TPT, as a solid line; and results from the improved TPT, as a dotted line. (a) $T^{*}-{\rho }^{*}$ liquid-gas coexistence; (b) $T^{*}-p^{*}$ coexistence; and (c) $T^{*}-{\mu }^{*}$ coexistence.

Figure 5

Density dependence of ratios of molecules with zero, one, two, and three hydrogen bonds. Symbols are the results from Monte Carlo computer simulations; lines, from the standard TPT. Results are plotted for different temperatures: (a) $T^{*}=0.18$, (b) $T^{*}=0.22$, and (c) $T^{*}=0.3$.

Figure 6

Fraction of molecules belonging to the biggest cluster as a function of the density for different temperatures. Symbol are the results from Monte Carlo computer simulations. Red symbols, high temperature, $T^{*}=0.4$; green symbols, $T^{*}=0.3$; blue symbols, $T^{*}=0.2$; light-blue symbols, low temperature, $T^{*}=0.18$; and pink symbols, low temperature, $T^{*}=0.15$.

Figure 7

Percolation line from MC and TPT. Results from computer simulations are plotted with symbols; results from the TPT, with the solid line; and results from the improved TPT, with the dashed line.