Analytical model for three-dimensional Mercedes-Benz water molecules

Abstract

We developed a statistical model which describes the thermal and volumetric properties of water-like molecules. A molecule is presented as a three-dimensional sphere with four hydrogen-bonding arms. Each water molecule interacts with its neighboring waters through a van der Waals interaction and an orientation-dependent hydrogen-bonding interaction. This model, which is largely analytical, is a variant of a model developed before for a two-dimensional Mercedes-Benz model of water. We explored properties such as molar volume, density, heat capacity, thermal expansion coefficient, and isothermal compressibility as a function of temperature and pressure. We found that the volumetric and thermal properties follow the same trends with temperature as in real water and are in good general agreement with Monte Carlo simulations, including the density anomaly, the minimum in the isothermal compressibility, and the decreased number of hydrogen bonds upon increasing the temperature.

FIG. 1

Lattice of the model showing both the hexagon of the ice-like structure and a pair interaction used for bookkeeping to avoid triple counting. Presented is only one layer.

FIG. 2

Three model states: (1) hydrogen-bonded, (2) van der Waals, and (3) nonbonded.

FIG. 3

Temperature dependence of density at p^* = 0.20: Theory (line) vs Monte Carlo simulation of MB model [46] (symbols). Notice also that density is plotted in the same units as in Dias’s work [46].

FIG. 4

Temperature dependence of molar volume coefficient; legend otherwise as for Fig. 3.

FIG. 5

Temperature dependence of thermal expansion coefficient; legend otherwise as for Fig. 3.

FIG. 6

Temperature dependence of isothermal compressibility; legend otherwise as for Fig. 3.

FIG. 7

Temperature dependence of heat capacity at constant pressure; legend otherwise as for Fig. 3.

FIG. 8

(a) Temperature dependence of populations f_i of the different type of hydrogen bonds, at constant pressure, p^* = 0.19. The population of strong hydrogen bonds (long dashed line), weak hydrogen bonds (solid line), no hydrogen bonds (short dashed line). (b) Experimental populations of OH states in liquid water vs temperature T_r [48] along its saturation curve, from ir spectroscopic data (adapted from Fig. 5 of Luck [57]).

FIG. 9

Temperature dependence of molar volume (a), heat capacity (b), isothermal compressibility (c), and thermal expansion coefficient (d) at p^* = 0.2: theory (solid line) and van der Waals 3D gas (dashed line).

FIG. 10

Temperature dependence of reduced density at different values of pressure, p^* = 0.2 solid line, p^* = 0.16 long dashed line, p^* = 0.12 dashed line, and p^* = 0.08 dash-dotted line.

FIG. 11

Density dependence of excess entropy at different temperatures, T^* = 0.25 solid line, T^* = 0.20 long dashed line, T^* = 0.15 dashed line, and T^* = 0.12 dash-dotted line.