On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim 2, I. Nezbeda 1,3 1 Institute of Chemical Process Fundamentals,

On independence of the solvation of interaction sites of a water molecule

M. Předota1, A. Ben-Naim2, I. Nezbeda1,3 1Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, Czech Republic;2Department of Physical Chemistry, Hebrew University, Jerusalem, Israel; 3Department of Physics, J. E. Purkyně University, Ústí n. Lab., Czech Republic; E-mail: [email protected]








M. Předota1, A. Ben-Naim2, I. Nezbeda1,3

1Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, 165 02 Prague, Czech Republic2Department of Physical Chemistry, Hebrew University, Jerusalem, Israel3Department of Physics, J. E. Purkyně University, 400 96 Ústí n. Lab., Czech Republic

E-mail: [email protected] Institute of Chemical Process Fundamentals

AimSupport simplifying assumptions used in analytic theories of aqueous systems

Justify previously used speculative approximations for the calculation of the solvation Helmholtz free energy of a water molecule†

Lend support to the first order thermodynamic perturbation theory of Wertheim‡

Examine correlations in the bonding of the individual sites of a water molecule using two qualitatively different extended primitive models

Implication: AW= Acore+Nsites Asite , whereAW solvation free energy of a water molecule

Acore solvation free energy of the core (typically LJ sphere)

Asite solvation free energy of an interaction site Asite=-logexp[-Bsite]W+core

• Calculation of solvation free energy reduces to the calculation of the average energy of the individual interaction sites

† A. Ben-Naim, Solvation thermodynamics (Plenum Press, New York, 1987), A. Ben-Naim, Statistical thermodynamics for chemists and biochemists, (Kluwer-Plenum, New York, 1992)

‡ M. S. Wertheim, J. Stat. Phys. 42, 459 (1986)

Assumption:

Interaction sites of a molecule act independently

rH-M

uH-M

0rH-H rM-M

uH-H

0R

M

M

O

H eeMe

H ee

rM-M

rH-M

hard-sphere repulsion

square-wellattraction

uM-M

O

H H

M

O

H H M

O

H H M

rH-H rH-M

(hydrogen bonding)

M

M

O

H eeMe

H ee

M

M

O

H eeMe

H ee

Extended primitive models of waterShort-range model of water,

interactions on the simplest level†

Hard core and like site repulsions as hard sphere repulsion

Hydrogen bonding resulting from unlike site attraction as square-well attraction

Geometry of modelsEPM4 = 0.7 , EPM4 = 0.8

|OM| = 0.15, OO = 1.0

EPM5 = 0.4 , EPM4 = 0.8

|OM| = |OM| = 0.5 , OO = 1.0

† I. Nezbeda, J. Mol. Liq. 73-74, 317 (1997)

Number of sitesCore + 3 off-center sites Core + 4 off-center sites

Parent modelTIP4P ST2

GeometryPlanar, tetrahedral angle HOH, Off-center sites arrangedM site on bisector tetrahedrally on a sphere

Role of H and M sitesSingle M site plays the role of doubly Full symmetry of H and M sitesdegenerated bonding site Directionality of hydrogen bondsCombination of M site attraction and H site dictated by the arrangement ofrepulsion essential for hydrogen bonding sitesSites can form multiple bonds Maximum 1 bond per site

EPM4 and EPM5 primitive models of water

EPM5EPM4

M

H

H

O M

M

H

OH

6 different molecules obtained by removal (turning off) of some of the interaction sites of EMP5 water moleculeOther combinations symmetrical by exchanging all H and M sites

Labeled by active sites

Solute molecules descending from EPM5 water molecule

hard sphere HHH

HHM

HHMM=EPM5HM

6 different molecules obtained by removal (turning off) of some of the interaction sites of EMP4 water moleculeNo symmetry of H and M sites

Labeled by active sites

Solute molecules descending from EPM4 water molecule

hard sphereH

HH

HHM=EPM4

M HM

BondingBoth models prevent double bonding between two water

moleculesEach site of EPM5 can form no more than one bond

Molecule can create maximum of 4 bonds

H site of EPM4 can form up to 2 bonds and M site up to 3 bondsSince M site plays the role of a degenerated (geometrically

collapsed) double site, it ordinarily forms 2 bondsMolecule can forms up to 5 bonds, 6 bonds maximum

If the sites acted independently, the probability of the number of bonds of the solute to be n would be binomial

where Nsites is the number of sites of the solute and p is half of solute’s average energy

nNn ppn

NnP

sites1sitesth

M

(solute) O

H eeM

H eeM

M

O (solvent)

eeM

Definition of angular distribution of molecules around the solute defined as the angle between the OH vector of the solute molecule and the

projection of the solute-solvent OO vector ontothe reference plane of the solute

In-plane molecules• Lying close to the HOH plane of the solute

• Bonded mostly to H sites of the solute

EnergiesTotal internal energy E is given by the water-water interaction, EWW, and by

the solute water interaction, EWS, which are given directly by the number of corresponding bonds

• E = EWW + ESW = - NWW - NSW

Splitting the total energy E into the energy of water molecules, EW, and the energy of the solute, ES

• E = EW + ES

• EW = EWW + 1/2ESW ; ES = 1/2ESW

Definitions

Simulation methodMonte Carlo simulation of NW=215 water molecules and a single solute – molecule

originating from water molecule when some of its interaction sites are removed (turned off)

Packing fraction =(/6)(N/V)W3 ; N=NW+ 1

EPM4=0.35 , EPM5=0.3

Temperature =1/kTEPM4=6 , EPM5=5

5 105 equilibration cycles, 18 106 productive cyclesPreferential sampling

f(rSW )=(1+D)/(rSW2+D); f(L/2)=0.1

Properties observedAverage energy of water molecule (solvent)Average energy of solute moleculeAverage number of bonds of each site of the solute

• Probability distribution of the solute to form n bonds

Angular distribution of water molecules around the solute• All (i.e. both bonded and nonbonded) and only bonded to the solute studied separately

Solute-solvent pair correlation function

Average energy and number of bonds of EPM5Probability distributions of different solutes to form n bonds with the

solvent molecules, and average energies for the EPM5 solvent. Pth(n) is the the binomial distribution with p=0.9225, and Psim(n) is the simulation result; ES is the average energy of the solute, ES /Nsites is the average energy per site of the solute, and EW is the average energy of solvent per water molecule

n H HH HM HHM HHMM 0 0.08 0.006 0.006 510-4 410-5 1 0.92 0.14 0.14 0.02 0.002 2 — 0.85 0.85 0.20 0.03 3 — — — 0.79 0.24

)(th nP

4 — — — — 0.72 0 0.08 0.008 0.004 0.001 310-5 1 0.92 0.15 0.17 0.02 0.003 2 — 0.84 0.83 0.22 0.03 3 — — — 0.76 0.23

)(sim nP

4 — — — — 0.74 n )(sim nP =-2ES 0.920.01 1.830.02 1.830.02 2.760.02 3.700.02

-4ES /Nsites 1.830.02 1.830.02 1.830.02 1.840.02 1.850.01 -EW /NW 1.8420.002 1.8430.002 1.8450.002 1.8430.002 1.8450.002

Average energy and number of bonds of EPM4The probabilities of creation of n bonds from simulation are given separately for

the individual sites, PHsim (n) and PM

sim (n), and for the entire solute, Psim (n). The theoretical prediction Pth(n) is given by the binomial distribution with p=0.775

n H M H H H M H H M 0 0 . 2 5 — 0 . 2 4 0 . 2 2 0 . 2 2 1 0 . 7 5 — 0 . 7 6 0 . 7 8 0 . 7 7 2 0 . 0 1 — 0 . 0 1 0 . 0 1 0 . 0 1

)(simH nP

A v e r a g e 0 . 7 6 — 0 . 7 6 0 . 7 9 0 . 7 9 0 — 0 . 0 6 — 0 . 0 5 0 . 0 4 1 — 0 . 3 8 — 0 . 3 6 0 . 3 4 2 — 0 . 5 5 — 0 . 5 7 0 . 6 0 3 — 0 . 0 2 — 0 . 0 2 0 . 0 1

)(simM nP

A v e r a g e — 1 . 5 3 — 1 . 5 6 1 . 5 9 0 0 . 2 3 0 . 0 5 0 . 0 5 0 . 0 1 0 . 0 0 3 1 0 . 7 8 0 . 3 5 0 . 3 5 0 . 1 2 0 . 0 4 2 — 0 . 6 0 0 . 6 0 0 . 4 1 0 . 1 8 3 — — — 0 . 4 7 0 . 4 2

)(thM nP

4 — — — — 0 . 3 6 0 0 . 2 5 0 . 0 6 0 . 0 5 0 . 0 1 0 . 0 0 3 1 0 . 7 5 0 . 3 8 0 . 3 6 0 . 1 2 0 . 0 4 2 0 . 0 1 0 . 5 5 0 . 5 7 0 . 4 0 0 . 1 7 3 — 0 . 0 2 0 . 0 1 0 . 4 5 0 . 4 0 4 — — 0 0 . 0 2 0 . 3 7

)(sim nP

5 — — — 0 0 . 0 2

n P s i m ( n ) = - 2 E S 0 . 7 6 0 . 0 2 1 . 5 3 0 . 0 3 1 . 5 4 0 . 0 4 2 . 3 4 0 . 0 3 3 . 1 5 0 . 0 4 - 4 E S / N s i t e s 1 . 5 2 0 . 0 3 1 . 5 3 0 . 0 3 1 . 5 4 0 . 0 4 1 . 5 6 0 . 0 2 1 . 5 7 0 . 0 2

- E W / N W 1 . 5 4 0 . 0 1 1 . 5 4 0 . 0 1 1 . 5 5 0 . 0 1 1 . 5 6 0 . 0 1 1 . 5 5 0 . 0 1

n0 1 2

PHsim(n)

0.0

0.2

0.4

0.6

0.8

HHHHMHHMbinomial

n0 1 2 3

PMsim(n)

0.0

0.2

0.4

0.6

MHMHHMbinomial

Probabilities of a creation of n bonds for the H site and M site in different solutes

descending from the EPM4 water moleculeThe probabilities follow binomial distribution with p=0.775

• Proved that M site acts as degenerated double site

All

In-plane

0 45 90 135 180 225 270 315 360

0

1

2

3

4

5

6

HHHHMHHMHHMM

0 45 90 135 180 225 270 315 360

0

1

2

All

In-plane

0 45 90 135 180 225 270 315 360

0

1

2

3

4

50 45 90 135 180 225 270 315 360

0

1

2

3

4

5

6

7

8

H MHHHMHHM

Angular distribution of bonded molecules around different solutes

EPM5The peaks are independent of

the presence of other sites

EPM4Little correlations of the

peaks

0 45 90 135 180 225 270 315 3603

4

5

6

7

8

HHHHMHHMHHMM

0 45 90 135 180 225 270 315 360

0

1

2

All

In-plane

Angular distribution of EPM5 water molecules (bonded and nonbonded) around different solutes

Complex behavior resulting from the combination of additive distribution of bonded molecules and nonadditive distribution of nonbonded molecules

Combination of water-like and hard-sphere-like structure

r

1.00 1.25 1.50

gSW

1

2

3

hard sphereHHHHMHHMHHMM

Solute-solvent pair correlation function

EPM5Turning on sites changes the PCF

from hard-sphere like to water-like

Molecules cannot approach close each other because of site-site repulsions

r1.00 1.25 1.50

gSW

1

2

3

4

5

6hard sphereHMHH HM HHM

EPM4Turning on sites forces

molecules to approach each other closer

Behavior originates from the position of M site closer to the central of molecules

ConclusionsIndependence of bonding of individual sites of water molecule proved for both

EPM4 and EPM5 modelsFor EPM5 independence exactly, for EPM4 it does not hold exactly but

correlations are very small

Fully justified previously used speculative approximations for the calculation of the solvation Helmholtz free energy of a water moleculeSupport to the first order thermodynamic perturbation theory of Wertheim

Assumption of independence of bonding justified for practical applications• Reduction of the calculation of average quantities over up to quadruplet distribution

function to calculations of averages over pair distributions only

• Drastic simplification which we hope will render the development of an analytical theory of water (and aqueous systems in general) feasible

Studied not only fully interacting water molecules (considered as a solute) but also a series of other solutes made from the water molecule by turning off some of its interaction sitesAdditional information on the behaviour of water

M. Předota, A. Ben-Naim, I. Nezbeda, J. Chem. Phys. 118, 6446-6454 (2003)

Reference:

Documents

On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim 2, I. Nezbeda 1,3 1 Institute of Chemical Process Fundamentals,