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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 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 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 U N ( N) e dr1 ...drN P (r1 ,..., rN ) ZN The probability that 1 is at dr1 … and n is at drn irrespective of the configuration of the other particles is P(n) (r1 ,...,rn ) e U N drn1 ...drN ZN The probability that any particle is at dr1 … and n is at drn irrespective of the configuration of the other particles is N! (n) (r1,...,rn ) (n) P (r1 ,...,rn ) (N n)! 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 N! (n) n (n) P (r ,...,rn ) P (r1 )g (r1 ,...,rn ) 1 (N n)! 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) 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 4pr2g(r)dr is the number of particles between r and r+dr around the central particle g(r) e w(r) 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 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. h(r ) c(r12) c(r )h(r23)dr3 12 13 Closure Equations c(r) htotal (r ) hindire ct (r) gtotal (r) 1 gindire ct (r) 1 g(r) gindire ct (r) w(r ) w(r )u(r) e e e w(r ) 1 e u(r) g(r) 1 e u(r) u(r12 ) u(r13 ) g(r )e 12 1 g(r )[1 e 13 ][g(r23 ) 1]dr3 Percus Yev ick (PY) Equation u(r12 ) g(r )e 12 [g(r13 ) 1 ln g(r ) u(r13 )][g(r23 ) 1]dr3 13 Hypernetted Chain (HNC) Equation 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), UN (r1 ,..., rN ) j u(rij ) , then you can calculate i pressure, chemical potential, energy, etc. of the system. For an isotropic fluid 3 E NkT 2p g(r )u(r)r 2 dr 2 0 2 2p du(r ) P kT 3V r3 0 dr g(r )dr 1 kT ln 4p r2 u(r)g(r; )drd 3 0 0 1 h 2 2 w here, ; is a coupling parameter w hich varies betw een 0 and 1. 2pmkT (Taking a partic le in, = 1, and out, = 0, of the s ystem). 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 i (i , i , i ) . For a linear and non-linear molecule the OZ equation becomes the following, respectively h(r ) c(r12 ) 12 4p c(r13)h(r23)dr3 h(r ) c(r12 ) 2 c(r )h(r23 )dr3 12 8p 13 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 g(rsw sw ) expu(rsw sw ) b(rswsw ) 2 c(rww ' ww' )h(rsw' sw' )drw' d w' 8p This is divided into a dependent and independent part g(rsw sw ) 8p 2 P( sw ;rsw )g 0 rsw g 0 (rsw ) k(rsw )exp u 0 (rsw ) b 0 rsw c 0 (rww ' )h0 (rsw' )drw' e sw sw w(r ) P(sw ;rsw ) 8p k(rsw ) 2 1 e w(r sw sw ) 8p 2 k(rsw ) d More Approximations c 0 (rww ' ) is obtained via using a radial distribution function obtained from MC simulation which uses a spherically- averaged potential. c 0 (rww ' ) 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, E(rsw sw ) w(rsw sw ) ' (rsw ) The Water Models BBL Water: – Water is a hard sphere, with a point dipole = 1.85 D. hs uij uij uSP uij ij potential energy of hard-sphere potential sticky potential used to mimic two dipoles for a hydrogen-bond interactions. given orientation Attractive square-well potential, dependant upon orientation 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. 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-. 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 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? 0 – how to calculate c (rww ww ) and h(rsw sw ) ? ' ' ' ' Thanks: – Dr. Paul