Simple lattice-gas model for water

Transcription

1 Simple lattice-gas model for water A. Ciach, W. T. Góźdź and A. Perera Motivation H-bonds and packing of molecules Construction of the lattice model Results in mean-field approximation

2 Water molecule and H-bonds

3 Water anomalies - facts: Phase diagram for p<2000 bar, dp/dt<0 at the solid-liquid coexistence

4 Increasing T) for p<2000 bar, standard behavior for p>2000bar. For p=1 bar maximum of density at T=4C Density of ice is larger than the density of liquid water Density along the phase coexistence Insert molar volume vs. temperature

5 Isothermal compressibility Specific heat Thermal expansivity

6 Motivation: Water is the main component of living matter; Water molecule is very simple, but liquid water has very peculiar properties. The role of the H-bond network in self-assembly of ionic, polar and amphiphilic solutes is not yet explained. Nature of the hydrophobic and hydrophilic interactions still open question. In order to model collective phenomena on the nano- or micrometer length scale we need a mesoscopic model for water, simple enough but capturing the essential physics. There are no simple mesocopic models for water. Advantage of simple models: If some degrees of freedom are neglected, and the predictions of the model are correct, then the origin of the observed phenomena is explained.

7 Structure of ice Crystal I h Schematic representation of the tetrahedral structure in ice

8 Structure of liquid water Cartoon showing clusters of H-bonded molecules, and close-packed regions. Confirmed neither by experiments nor by simulations Lifetime of H-bonds in liquid water ~ 0.1 ps.

9 Statistical-mechanical modeling. I. Lattice gas model for simple fluids Typical configuration for T>T c Phase diagram Empty cell Occupied cell Nearest-neighbor interaction p= e E N Probability of a configuration Grand potential = ktln

10 Low density Empty cell High density II. Lattice gas model for water Typical configurations liquid gas Interaction energies a van der Waals h - H-bond relative density difference volume of the lattice cell v = volume per molecule in ice Probability of a configuration p= e E N L N H 1 2 Density in a configuration = L N L 1 2 N H V

11 Hamiltonian Blume-Emery-Griffith model for a binary mixture H [{ s}]= 1 2 nn [ J l l s x s x ' 4J gl s2 x s 2 x ' 2Q s x s 2 x ' ] x [ s x 1 s 2 x ] s= 1,0,1 s s 2 - concentration - cell occupancy = s 1 s 2 - density { s } Probability of the configuration : J ll =a 2 h 4 p[{ s}]= e H [{ s}] grand potential H [{ s}] = { s} e = pvv = kt ln J gl =a 1 2 h 4 Q=a 1 h 4

12 Ground state (T=0K) dependence on the model parameters Simple fluid Water-like By assuming the coexistence between the high-density and the low-density phases at p=2000 bar we obtain a relation between the model parameters a,h,v,

13 Limiting cases High density Low density s=1 s= 1 s=1 s= 1 H l l [{ s }]= J l l 2 nn s x s x ' 6Q x s x H lg [{ s }]= J lg 2 s x s x' [6J nn gl 1 2 ] x s x Each limiting case is isomorphic with the Ising (lattice gas) model. In each case a critical point is expected. By equating the critical-point temperatures to 180K and 650K we obtain two relations between the model parameters.

14 The model parameters: v=35 A 3 a=3.6 kj/mol h=1.1 kj/mol =0.12 In water: v~33 A 3 a~5.5 kj/mol h= E H= kj /mol=1.2kj/mol 0.1 Mean-field (MF) approximation: each molecule is in the external field resulting from interactions with the remaining molecules in their equilibrium positions. Average values are approximated by the most probable values. In MF the critical point temperature T c is overestimated. Exact result for T c in the Ising model is T c ~4.5/6T MF MF c =0.75 T c

15 Functional of two fields, MF [s x, x ]=H [s x, x ] TS[s x, x ] On the lattice the entropy S has the ideal-mixing entropy form The fields s= s MF, = s 2 MF satisfy the minimum condition MF s =0= MF Spinodal surface: det [ 2 MF ]=0 The density: =[ 1 s] ice

16 Spinodal (dashed) and binodal lines in MF

17 Gas-liquid coexistence in experiment and in the model Anomalous density increase

18 EOS isobars P=100 bar p=1000bar p=2500bar p=10bar Critical pressure in the model is p=653 bar

19 Isothermal compressibility 0.1 kbar 1 kbar We observe minimum of the compressibility, but only for high pressures

20 Constant-pressure specific heat 0.1 kbar 1 kbar

21 Thermal expansivity 0.1 kbar 1 kbar

22 Correlation length at the critical density of the mestastable liquid-liquid critical point

23 For p<2000 bar open structure at low T. T increases mixing of the two forms of water density increases. Further increase of T mixing of the occupied and empty cells -density decreases. For p>2000bar compact structure at low T. T increases - mixing of the two forms of water density decreases.

24 Summary Simple model for water is developed. Water is treated as a special mixture of low and high density components with chemical potentials and. The model predicts correctly the special features of water. It may play a similar role as the lattice gas model of simple fluids. Entropy associated with mixing of closely-packed and open structures is the key reason of water anomalies Quantitative improvement can be obtained in off-lattice models, with a more realistic form of the entropy term.