A statistical mechanical theory for a two-dimensional model of water

Abstract

We develop a statistical mechanical model for the thermal and volumetric properties of waterlike fluids. Each water molecule is a two-dimensional disk with three hydrogen-bonding arms. Each water interacts with 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 the Truskett and Dill ͑TD͒ treatment of the "Mercedes-Benz" ͑MB͒ model. The present model gives better predictions than TD for hydrogen-bond populations in liquid water by distinguishing strong cooperative hydrogen bonds from weaker ones. We explore properties versus temperature T and pressure p. We find that the volumetric and thermal properties follow the same trends with T as real water and are in good general agreement with Monte Carlo simulations of MB water, including the density anomaly, the minimum in the isothermal compressibility, and the decreased number of hydrogen bonds for increasing temperature. The model reproduces that pressure squeezes out water's heat capacity and leads to a negative thermal expansion coefficient at low temperatures. In terms of water structuring, the variance in hydrogen-bonding angles increases with both T and p, while the variance in water density increases with T but decreases with p. Hydrogen bonding is an energy storage mechanism that leads to water's large heat capacity ͑for its size͒ and to the fragility in its cagelike structures, which are easily melted by temperature and pressure to a more van der Waals-like liquid state.

Simple Model of Hydrophobic Hydration

“…If neither the orientation nor the relative position of the two water molecules is such as to form H-bonds or van der Waals contact, a test water molecule makes no interactions with its neighbor and the state is referred as open (panel c ). 54 …”

“…We obtain the partition function Δ j for each of the four states by integrating over the appropriate positions and orientations, $${\mathrm{\Delta }}_j=cfalse(Tfalse)true\int true\int \mathrm{normald}x\mathrm{normald}y{\int }_{-\pi /3}^{\pi /3}\text{exp }true(-\frac{u_j+2/3·p{\upsilon }_j}{kT}true)\mathrm{normald}\theta .$$ Performing this integration results in the individual partition functions for the four states described above (HB, LJ, O, and S) (for more details of this model of pure water, see Ref. 54): $${\mathrm{\Delta }}_{\mathrm{normalH}\mathrm{normalB}}=cfalse(Tfalse){\upsilon }_{\text{eff}}^{\mathrm{normalH}\mathrm{normalB}}\text{ exp }true(\frac{{\epsilon }_{\mathrm{normalH}\mathrm{normalB}}+{\epsilon }_{\mathrm{normalL}\mathrm{normalJ}}-2/3·p{\upsilon }_{\mathrm{normalH}\mathrm{normalB}}}{kT}true)\sqrt{\frac{kT\pi }{k_{\mathrm{s}}}}\text{erf }true(\sqrt{\frac{k_{\mathrm{normals}}{\pi }^2}{9kT}}true),$$ $${\mathrm{\Delta }}_{\mathrm{normalL}\mathrm{normalJ}}=cfalse(Tfalse){\upsilon }_{\text{eff}}^{\mathrm{normalL}\mathrm{normalJ}}\frac{2\pi }{3}\text{exp }true(\frac{{\epsilon }_{\mathrm{normalL}\mathrm{normalJ}}-2/3·p{\upsilon }_{\mathrm{normalL}\mathrm{normalJ}}}{kT}true),$$ $${\mathrm{\Delta }}_{\mathrm{O}}=cfalse(Tfalse)\frac{2\pi }{3}\frac{kT}{p}\text{exp }true(-\frac{2/3·p{\upsilon }_{\mathrm{O}}}{kT}true),$$ $${\mathrm{\Delta }}_{\mathrm{S}}={\mathrm{\Delta }}_{\mathrm{normalH}\mathrm{normalB}}\frac{\text{exp }false(-2/3·p{\upsilon }_{\mathrm{S}}/kTfalse)}{\text{exp }false(-2/3·p}$$…”

“…54 The ensemble average energy, 〈 u j 〉, for each of the four types of water molecule structure, is first needed and can be calculated as $$true\langle u_{j}true\rangle =\frac{{\int }_{-\pi /3}^{\pi /3}{u}_j\text{ exp }true(-\frac{u_j+2/3·p{\upsilon }_j}{kT}true)\mathrm{normald}\theta }{{\int }_{-\pi /3}^{\pi /3}\text{exp }true(-\frac{u_j+2/3·p{\upsilon }_j}{kT}true)\mathrm{normald}\theta }.$$ Inserting equations (1) into equation (14) gives $$true\langle u_{\mathrm{normalH}\mathrm{normalB}}true\rangle =-{\epsilon }_{\mathrm{normalH}\mathrm{normalB}}-{\epsilon }_{\mathrm{normalL}\mathrm{normalJ}}+\frac{kT}{2}-\frac{\sqrt{k_{\mathrm{normals}}\pi kT}\text{ exp }true(-\frac{k_{\mathrm{s}}{\pi }^2}{9kT}true)}{3\text{ erf }true(\sqrt{\frac{k_{\mathrm{normals}}{\pi }^2}{9kT}}true)},$$ $$true\langle u_{\mathrm{S}}true\rangle =true\langle u_{\mathrm{normalH}\mathrm{normalB}}true\rangle +\frac{{\epsilon }_{\mathrm{normalc}}}{6},$$ $$true\langle u_{\mathrm{normalL}\mathrm{normalJ}}true\rangle =-{\epsilon }_{\mathrm{normalL}\mathrm{normalJ}},$$ $$true\langle u_{\mathrm{O}}true\rangle =0.$$ The average energy of a water molecule, summed over the four different water states, can be expressed as $${false\langle \epsilon false\rangle }_{\mathrm{b}}=\frac{3}{2}false\{true\langle u_{\mathrm{normalH}\mathrm{normalB}}true\rangle f_{\mathrm{normalH}\mathrm{normalB}}+true\langle u_{\mathrm{S}}true\rangle f_{\mathrm{S}}+$$…”

“…The theory is capable of reproducing all the relevant thermodynamic and volumetric properties of 2D model of water. 54 …”

“…We start from an analytical UD theory of water. 54 A partition function for a water molecule in the bulk and in the first hydration shell of a hydrophobic solute is then built using the expressions for average energies of different states (hydrogen-bonded, van der Waals and open) that the water molecule can be classified into and upon considering the geometric restrictions through which a solute dictates the formation or breakage of the hydrogen bonds between water molecules in the first solvation shell. Finally, from statistical mechanical and thermodynamical relations we calculate the Δ G , Δ H , T Δ S , and Δ C p .…”