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Statistical Models of Solvation

Statistical Models of Solvation. Eva Zurek Chemistry 699.08 Final Presentation. Methods. Continuum models: macroscopic treatment of the solvent; inability to describe local solute-solvent interaction; ambiguity in definition of the cavity

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Statistical Models of Solvation

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  1. Statistical Models of Solvation Eva Zurek Chemistry 699.08 Final Presentation

  2. Methods • Continuum models: macroscopic treatment of the solvent; inability to describe local solute-solvent interaction; ambiguity in definition of the cavity • Monte Carlo (MC) or Molecular Dynamics (MD) Methods: computationally expensive • Statistical Mechanical Integral Equation Theories: give results comparable to MD or MC simulations; computational speedup on the order of 102

  3. Statistical Mechanics of Fluids • A classical, isotropic, one-component, monoatomic fluid. • A closed system, for which N, V and T are constant (the Canonical Ensemble). Each particle i has a potential energy Ui. • The probability of locating particle 1 at dr1, etc. is • The probability that 1 is at dr1 … and n is at drn irrespective of the configuration of the other particles is • The probability that any particle is at dr1 … and n is at drn irrespective of the configuration of the other particles is

  4. Radial Distribution Function • If the distances between n particles increase the correlation between the particles decreases. • In the limit of |ri-rj| the n-particle probability density can be factorized into the product of single-particle probability densities. • If this is not the case then • In particular g(2)(r1,r2) is important since it can be measured via neutron or X-ray diffraction • g(2)(r1,r2) = g(r12) = g(r)

  5. Radial Distribution Function • g(r12) = g(r) is known as the radial distribution function • it is the factor which multiplies the bulk density to give the local density around a particle • If the medium is isotropic then 4pr2rg(r)dr is the number of particles between r and r+dr around the central particle

  6. Correlation Functions • Pair Correlation Function, h(r12), is a measure of the totalinfluence particle 1 has on particle 2 h(r12) = g(r12) - 1 • Direct Correlation Function, c(r12), arises from the direct interactions between particle 1 and particle 2

  7. Ornstein-Zernike (OZ) Equation • In 1914 Ornstein and Zernike proposed a division of h(r12) into a direct and indirect part. • The former is c(r12), direct two-body interactions. • The latter arises from interactions between particle 1 and a third particle which then interacts with particle 2 directly or indirectly via collisions with other particles. Averaged over all the positions of particle 3 and weighted by the density.

  8. Closure Equations

  9. Thermodynamic Functions from g(r) • If you assume that the particles are acting through central pair forces (the total potential energy of the system is pairwise additive), , then you can calculate pressure, chemical potential, energy, etc. of the system. • For an isotropic fluid

  10. Molecular Liquids • Complications due to molecular vibrations ignored. • The position and orientation of a rigid molecule i are defined by six coordinates, the center of mass coordinate ri and the Euler angles • For a linear and non-linear molecule the OZ equation becomes the following, respectively

  11. Integral Equation Theory for Macromolecules • If s denotes solute and w denotes water than the OZ equation can be combined with a closure to give • This is divided into a W dependent and independent part

  12. More Approximations • is obtained via using a radial distribution function obtained from MC simulation which uses a spherically-averaged potential. • is used to calculate b0(rsw) for SSD water. • For BBL water b0(rsw) = 0, giving the HNC-OZ. • The orientation of water around a cation or anion can be described as a dipole in a dielectric continuum with a dielectric constant close to the bulk value. Thus,

  13. potential energy of two dipoles for a given orientation hard-sphere potential sticky potential used to mimic hydrogen-bond interactions. Attractive square-well potential, dependant upon orientation The Water Models • BBL Water: • Water is a hard sphere, with a point dipole m = 1.85 D. • SSD Water: • Water is a Lennard-Jones soft-sphere, with a point dipole m = 2.35 D. Sticky potential is modified to be compatible with soft-sphere.

  14. Results for SSD Water • Position of the first peak, excellent agreement. • Coordination number, excellent agreement except for anions which differ ~13-16% from MC simulation. • Solute-water interaction energy for water differs between ~9-14% and for ions/ion-pairs ~1-24%. Greatest for Cl-.

  15. Results for BBL Water Radial distribution function around five molecule cluster of water from theory (line) and MC simulation (circles) Twenty-five molecule cluster of water

  16. Conclusions • Solvation models based upon the Ornstein-Zernike equation could be used to give results comparable to MC or MD calculations with significant computational speed-up. • Problems: • which solvent model? • which closure? • how to calculate and ? • Thanks: • Dr. Paul

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