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On independence of the solvation of interaction sites of a water
molecule M. Pedota1, 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. Purkyn University, 400 96 st n. Lab., Czech
RepublicE-mail: [email protected]
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AimSupport simplifying assumptions used in analytic theories of
aqueous systemsJustify previously used speculative approximations
for the calculation of the solvation Helmholtz free energy of a
water moleculeLend support to the first order thermodynamic
perturbation theory of WertheimExamine 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 moleculeAcore solvation free energy of the core
(typically LJ sphere) Asitesolvation free energy of an interaction
site Asite=-logexp[-Bsite]W+coreCalculation 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
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Extended primitive models of waterShort-range model of water,
interactions on the simplest levelHard core and like site
repulsions as hard sphere repulsionHydrogen bonding resulting from
unlike site attraction as square-well attractionGeometry 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)
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EPM4 and EPM5 primitive models of waterNumber of sitesCore + 3
off-center sitesCore + 4 off-center sitesParent modelTIP4PST2
GeometryPlanar, tetrahedral angle HOH,Off-center sites arrangedM
site on bisectortetrahedrally on a sphereRole of H and M
sitesSingle M site plays the role of doublyFull symmetry of H and M
sitesdegenerated bonding siteDirectionality of hydrogen
bondsCombination of M site attraction and H sitedictated by the
arrangement ofrepulsion essential for hydrogen bondingsitesSites
can form multiple bonds Maximum 1 bond per site
EPM5EPM4MHHOMMHOH
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Solute molecules descending from EPM5 water molecule6 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 sitesLabeled by active
siteshard sphereHHHHHMHHMM =EPM5HM
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Solute molecules descending from EPM4 water molecule6 different
molecules obtained by removal (turning off) of some of the
interaction sites of EMP4 water moleculeNo symmetry of H and M
sitesLabeled by active sites
hard sphereHHHHHM =EPM4MHM
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BondingBoth models prevent double bonding between two water
moleculesEach site of EPM5 can form no more than one bondMolecule
can create maximum of 4 bondsH 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 maximumIf
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 solutes average energy
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DefinitionsDefinition 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
onto the reference plane of the soluteIn-plane moleculesLying close
to the HOH plane of the soluteBonded mostly to H sites of the
soluteEnergiesTotal 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 bondsE = EWW +
ESW = - NWW - NSWSplitting the total energy E into the energy of
water molecules, EW, and the energy of the solute, ESE = EW + ESEW
= EWW + 1/2ESW ; ES = 1/2ESW
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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+ 1EPM4=0.35 , EPM5=0.3
Temperature =1/kTEPM4=6 , EPM5=55 105 equilibration cycles, 18 106
productive cyclesPreferential samplingf(rSW )=(1+D)/(rSW2+D);
f(L/2)=0.1Properties observedAverage energy of water molecule
(solvent)Average energy of solute moleculeAverage number of bonds
of each site of the soluteProbability distribution of the solute to
form n bondsAngular distribution of water molecules around the
soluteAll (i.e. both bonded and nonbonded) and only bonded to the
solute studied separatelySolute-solvent pair correlation
function
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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
5(10-4
4(10-5
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
4
0.72
0
0.08
0.008
0.004
0.001
3(10-5
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
4
0.74
(n
=-2ES
0.92((0.01
1.83(0.02
1.83(0.02
2.76(0.02
3.70(0.02
-4ES /Nsites
1.83(0.02
1.83(0.02
1.83(0.02
1.84(0.02
1.85(0.01
-EW /NW
1.842(0.002
1.843(0.002
1.845(0.002
1.843(0.002
1.845(0.002
_1110636137.unknown
_1110636186.unknown
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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 PMsim (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
HH
HM
HHM
0
0.25
0.24
0.22
0.22
1
0.75
0.76
0.78
0.77
2
0.01
0.01
0.01
0.01
Average
0.76
0.76
0.79
0.79
0
0.06
0.05
0.04
1
0.38
0.36
0.34
2
0.55
0.57
0.60
3
0.02
0.02
0.01
Average
1.53
1.56
1.59
0
0.23
0.05
0.05
0.01
0.003
1
0.78
0.35
0.35
0.12
0.04
2
0.60
0.60
0.41
0.18
3
0.47
0.42
4
0.36
0
0.25
0.06
0.05
0.01
0.003
1
0.75
0.38
0.36
0.12
0.04
2
0.01
0.55
0.57
0.40
0.17
3
0.02
0.01
0.45
0.40
4
0
0.02
0.37
5
0
0.02
(nPsim(n)=-2ES
0.76((0.02
1.53(0.03
1.54(0.04
2.34(0.03
3.15(0.04
-4ES /Nsites
1.52(0.03
1.53(0.03
1.54(0.04
1.56(0.02
1.57(0.02
-EW /NW
1.54(0.01
1.54(0.01
1.55(0.01
1.56(0.01
1.55(0.01
_1110636019.unknown
_1110636073.unknown
_1110684524.unknown
_1110968138.unknown
_1110636039.unknown
_1110635977.unknown
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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=0.775Proved that
M site acts as degenerated double site
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Angular distribution of bonded molecules around different
solutesEPM5The peaks are independent of the presence of other
sites
EPM4Little correlations of the peaks
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Angular distribution of EPM5 water molecules (bonded and
nonbonded) around different solutesComplex behavior resulting from
the combination of additive distribution of bonded molecules and
nonadditive distribution of nonbonded moleculesCombination of
water-like and hard-sphere-like structure
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Solute-solvent pair correlation functionEPM5Turning on sites
changes the PCF from hard-sphere like to water-likeMolecules cannot
approach close each other because of site-site
repulsionsEPM4Turning on sites forces molecules to approach each
other closerBehavior originates from the position of M site closer
to the central of molecules
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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 smallFully 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 WertheimAssumption of independence of
bonding justified for practical applicationsReduction of the
calculation of average quantities over up to quadruplet
distribution function to calculations of averages over pair
distributions onlyDrastic simplification which we hope will render
the development of an analytical theory of water (and aqueous
systems in general) feasibleStudied 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. Pedota, A. Ben-Naim, I. Nezbeda, J. Chem. Phys. 118,
6446-6454 (2003)Reference:
