Statistical Models of Solvation

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 rdr around
  the central particle

6
Correlation Functions

• Pair Correlation Function, h(r12), is a measure
  of the total influence 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
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