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

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

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

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  1. On independence of the solvation of interaction sites of a water molecule On independence of the solvation of interaction sites of a water molecule M. Předota1, A. Ben-Naim2, I. Nezbeda1,31Institute 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: ivonez@icpf.cas.cz M. Předota1, A. Ben-Naim2, I. Nezbeda1,31Institute 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: ivonez@icpf.cas.cz

  2. On independence of the solvation of interaction sites of a water molecule On independence of the solvation of interaction sites of a water molecule M. Předota1, A. Ben-Naim2, I. Nezbeda1,31Institute 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: ivonez@icpf.cas.cz M. Předota1, A. Ben-Naim2, I. Nezbeda1,31Institute 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: ivonez@icpf.cas.cz

  3. 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, 165 02 Prague, Czech Republic 2Department of Physical Chemistry, Hebrew University, Jerusalem, Israel 3Department of Physics, J. E. Purkyně University, 400 96 Ústí n.Lab., Czech Republic E-mail: ivonez@icpf.cas.cz

  4. Aim • Support 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 , where • AW solvation free energy of a water molecule • Acoresolvation free energy of the core (typically LJ sphere) •  Asitesolvation free energy of an interaction site •  Asite=-logexp[-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

  5. Extended primitive models of water • Short-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 models • EPM4 = 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)

  6. EPM4 and EPM5 primitive models of water M O H H EPM4 EPM5 • Number of sites • Core + 3 off-center sites Core + 4 off-center sites • Parent model • TIP4P ST2 • Geometry • Planar, tetrahedral angle HOH, Off-center sites arranged • M site on bisector tetrahedrally on a sphere • Role of H and M sites • Single M site plays the role of doubly Full symmetry of H and M sites • degenerated bonding site Directionality of hydrogen bonds • Combination of M site attraction and H site dictated by the arrangement of • repulsion essential for hydrogen bonding sites • Sites can form multiple bonds Maximum 1 bond per site O M H H M

  7. Solute molecules descending from EPM5 water molecule • 6 different molecules obtained by removal (turning off) of some of the interaction sites of EMP5 water molecule • Other combinations symmetrical by exchanging all H and M sites • Labeled by active sites HH hard sphere H HHMM=EPM5 HM HHM

  8. Solute molecules descending from EPM4 water molecule • 6 different molecules obtained by removal (turning off) of some of the interaction sites of EMP4 water molecule • No symmetry of H and M sites • Labeled by active sites hard sphere H HH HM HHM=EPM4 M

  9. Bonding • Both models prevent double bonding between two water molecules • Each 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 bonds • Since M site plays the role of a degenerated (geometrically collapsed) double site, it ordinarily forms 2 bonds • Molecule 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 nwould be binomial • where Nsites is the number of sites of the solute and p is half of solute’s average energy

  10. Definitions • 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 • Energies • Total 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

  11. Simulation method • Monte 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/kT • EPM4=6 , EPM5=5 • 5  105 equilibration cycles, 18  106 productive cycles • Preferential sampling • f(rSW )=(1+D)/(rSW2+D); f(L/2)=0.1 • Properties observed • Average energy of water molecule (solvent) • Average energy of solute molecule • Average 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

  12. Average energy and number of bonds of EPM5 • Probability 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

  13. n H M HH HM HHM n 0 0.25 H M — HH 0.24 HM 0.22 0.22 HHM 1 0 0.25 0.75 — — 0.24 0.76 0.22 0.78 0.77 0.22 1 2 0.75 0.01 — — 0.01 0.76 0.01 0.78 0.01 0.77 Average 2 0.76 0.01 — — 0.01 0.76 0.01 0.79 0.01 0.79 0 Average — 0.76 0.06 — — 0.76 0.79 0.05 0.04 0.79 0 1 — — 0.38 0.06 — — 0.36 0.05 0.34 0.04 1 2 — — 0.55 0.38 — — 0.57 0.36 0.34 0.60 2 3 — — 0.02 0.55 — — 0.57 0.02 0.01 0.60 3 Average — — 0.02 1.53 — — 0.02 1.56 1.59 0.01 Average 0 — 0.23 0.05 1.53 — 0.05 0.01 1.56 1.59 0.003 0 1 0.78 0.23 0.35 0.05 0.35 0.05 0.12 0.01 0.003 0.04 1 2 0.78 — 0.35 0.60 0.35 0.60 0.41 0.12 0.18 0.04 3 2 — — — 0.60 — 0.60 0.47 0.41 0.42 0.18 4 3 — — — — — — 0.47 — 0.36 0.42 0 4 0.25 — — 0.06 — 0.05 0.01 — 0.36 0.003 0 1 0.25 0.75 0.38 0.06 0.36 0.05 0.01 0.12 0.04 0.003 1 2 0.75 0.01 0.55 0.38 0.36 0.57 0.40 0.12 0.04 0.17 2 3 0.01 — 0.55 0.02 0.57 0.01 0.45 0.40 0.40 0.17 3 4 — — — 0.02 0.01 0 0.02 0.45 0.37 0.40 4 5 — — — — 0 — 0.02 0 0.02 0.37    =-2ES 5 0.760.02 — — 1.530.03 — 1.540.04 2.340.03 0 0.02 3.150.04 -4ES /Nsites    =-2ES 0.760.02 1.520.03 1.530.03 1.530.03 1.540.04 1.540.04 1.560.02 2.340.03 3.150.04 1.570.02 -EW /NW -4ES /Nsites 1.540.01 1.520.03 1.530.03 1.540.01 1.540.04 1.550.01 1.560.02 1.560.01 1.570.02 1.550.01 -EW /NW 1.540.01 1.540.01 1.550.01 1.560.01 1.550.01 Average energy and number of bonds of EPM4 • The probabilities of creation of nbonds from simulation are given separately for the individual sites, PHsim (n) and PMsim (n),and for the entire solute,Psim (n). Thetheoretical prediction Pth(n) is given by thebinomial distribution with p=0.775

  14. Probabilities of a creation of n bonds for the H site and M site in different solutes descending from the EPM4 water molecule • The probabilities follow binomial distribution with p=0.775 • Proved that M site acts as degenerated double site

  15. Angular distribution of bonded molecules around different solutes EPM4 • Little correlations of the peaks • EPM5 • The peaks are independent of the presence of other sites

  16. 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

  17. Solute-solventpair correlation function EPM4 • Turning on sites forces molecules to approach each other closer • Behavior originates from the position of M site closer to the central of molecules • EPM5 • Turning on sites changes the PCF from hard-sphere like to water-like • Molecules cannot approach close each other because of site-site repulsions

  18. Conclusions • Independence of bonding of individual sites of water molecule proved for both EPM4 and EPM5 models • For 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 molecule • Support 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 sites • Additional information on the behaviour of water • M. Předota, A. Ben-Naim, I. Nezbeda, J. Chem. Phys.118, 6446-6454 (2003) Reference:

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