Statistical Models of Solvation Eva Zurek Chemistry Final Presentation

Methods n Continuum models: macroscopic treatment of the solvent; inability to describe local solute-solvent interaction; ambiguity in definition of the cavity n Monte Carlo (MC) or Molecular Dynamics (MD) Methods: computationally expensive n Statistical Mechanical Integral Equation Theories: give results comparable to MD or MC simulations; computational speedup on the order of 10 2

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

Radial Distribution Function n If the distances between n particles increase the correlation between the particles decreases. n In the limit of |r i -r j |  the n-particle probability density can be factorized into the product of single-particle probability densities. n If this is not the case then n In particular g (2) (r 1,r 2 ) is important since it can be measured via neutron or X-ray diffraction n g (2) (r 1,r 2 ) = g(r 12 ) = g(r)

Radial Distribution Function n g(r 12 ) = g(r) is known as the radial distribution function n it is the factor which multiplies the bulk density to give the local density around a particle If the medium is isotropic then 4  r 2  g(r)dr is the number of particles between r and r+dr around the central particle

Correlation Functions n Pair Correlation Function, h(r 12 ), is a measure of the total influence particle 1 has on particle 2 h(r 12 ) = g(r 12 ) - 1 n Direct Correlation Function, c(r 12 ), arises from the direct interactions between particle 1 and particle 2

Ornstein-Zernike (OZ) Equation n In 1914 Ornstein and Zernike proposed a division of h(r 12 ) into a direct and indirect part. n The former is c(r 12 ), direct two-body interactions. n 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.

Closure Equations

Thermodynamic Functions from g(r) n 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. n For an isotropic fluid

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

Integral Equation Theory for Macromolecules n 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  dependent and independent part

More Approximations n is obtained via using a radial distribution function obtained from MC simulation which uses a spherically- averaged potential. n is used to calculate b 0 (r sw ) for SSD water. n For BBL water b 0 (r sw ) = 0, giving the HNC-OZ. n 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,

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

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

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

Conclusions n 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. n Problems: –which solvent model? –which closure? –how to calculate and ? n Thanks: –Dr. Paul