Hierarchy of anomalies in the two-dimensional Mercedes-Benz model of water

We investigate by Monte Carlo simulations density, diffusion, and structural anomalies of the simple two-dimensional Mercedes-Benz (MB) model of water, which is a very simple toy model for explaining the origin of water properties. MB water molecules are modeled as two-dimensional Lennard-Jones disks, with three orientation-dependent hydrogen-bonding arms, arranged as in the MB logo. The model is in a way also a variance of silica-like models. Beside the known thermodynamic anomaly for the model we also found diffusion and structural anomalies and map out the cascade of density, structural, pair entropy, and diffusivity anomalies for MB model. The orientational order parameters with three and six-fold symmetry were determined and maximum for each one observed. The anomalies occur in hierarchy order, which is a slight variation of the hierarchy order in real water. The diffusion anomaly region is the innermost in the hierarchy while for water it is the density anomaly region.

Figure 1

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

Figure 2

(a) The unit cell of the low density ice. (b) The unit cell of the high density ice.

Figure 3

$T^{*}-p^{*}$ phase diagram from MC simulations. Red curve are results for gas-liquid coexistence by Urbic [45] in comparison to the prediction of the phase diagram by Silverstain et al. [29], which is plotted by the black line.

Figure 4

(a) The pressure as a function of temperature for different densities. Red lines correspond to isochores from bottom up for densities 0.8, 0.825, 0.85, 0.875, 0.9, 0.925, 0.95, 0.975, 1.0, 1.025, 1.05, 1.075, 1.1. Black line connects points with maximum density for different temperatures. (b) Pressure as a function of density for different temperatures.

Figure 5

(a) The pseudodiffusion coefficient as a function of the temperature for several isochores and (b) the density for several isotherms. Black lines connect points with maximum and minimum of the pseudodiffusion coefficient.

Figure 6

Ratio of differently bonded water molecules for different isochores for (a) at temperature $T^{*}=0.16$, (b) at $T^{*}=0.20$.

Figure 7

(a) The translational order parameter as a function of the temperature for several isochores and (b) the density for several isotherms. Black lines connect points with maximum and minimum of the translational order parameter.

Figure 8

The distribution of the different order parameters for different isochores for (a) three-fold parameter at temperature $T^{*}=0.15$, (b) at $T^{*}=0.20$, and for (c) six-fold parameter at temperature $T^{*}=0.15$, (d) at $T^{*}=0.20$.

Figure 9

(a) The orientational order parameter with three fold symmetry and (b) the orientational order parameter with six-fold symmetry against density for several isotherms.

Figure 10

(a) The $t-q_3$ order map and (b) the $t-q_6$ plane.

Figure 11

(a) The pair entropy as a function of the temperature for several isochores and (b) the density for several isotherms. Black lines connects points with maximum and minimum of the pair entropy.

Figure 12

The pair correlation functions for different isochores for (a) at temperature $T^{*}=0.15$, (b) at $T^{*}=0.20$.

Figure 13

(a) The temperature-density plane containing all the anomalies found for the 2D MB model of water. Colors of lines are explained by legend. (b) Same as in (a) but for pressure temperature plane.