On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim...
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Transcript of On independence of the solvation of interaction sites of a water molecule M. Předota 1, A. Ben-Naim...
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]
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]
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]
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]
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, 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: