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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 ( N) P e U N dr1 ...drN (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 (r1 ,...,rn ) (n) 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) (r1 ,...,rn ) N! (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 (r1 ,...,rn ) P (r1 )g (r1 ,..., rn ) (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(r12 ) c(r12 ) c(r13 )h(r23 )dr3 Closure Equations c(r) htotal(r ) hindirect (r) gtotal (r) 1 gindirect(r) 1 g(r) gindirect(r) e w(r ) w(r )u(r) e e w(r ) 1 eu(r) g(r) 1 e u(r12 ) g(r12 )e u(r) 1 g(r13 )[1 e u(r13 ) ][g(r23 ) 1]dr3 Percus Yevick (PY) Equation u(r12 ) g(r12 )e [g(r13 ) 1 ln g(r13 ) u(r13 )][g(r23 ) 1]dr3 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 u(rij ) , then you can calculate pairwise additive), UN (r1,...,rN ) i j 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 2p 2 3 du(r ) P kT r g(r )dr 3V 0 dr 1 kT ln 4p r2 u(r)g(r; )drd 3 0 0 1 2 h where, ; is a coupling parameter which varies between 0 and 1. 2pmkT 2 (Taking a particle in, = 1, and out, = 0, of the system). 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(r12 ) c(r12 ) c(r13 )h(r23 )dr3 4p h(r12 ) c(r12 ) 2 c(r13 )h(r23 )dr3 8p 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 ) exp u(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' ew(rsw sw ) P(sw ;rsw ) 2 8p k(rsw ) 1 w(rsw sw ) k(rsw ) 2 e d 8p More Approximations c 0 (rww ' ) is obtained via using a radial distribution function obtained from MC simulation which uses a sphericallyaveraged 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, w(rsw sw ) E(rsw sw ) ' (rsw ) The Water Models BBL Water: – Water is a hard sphere, with a point dipole = 1.85 D. uij uijhs uSP u ij ij hard-sphere potential sticky potential used to mimic hydrogen-bond interactions. Attractive square-well potential, dependant upon orientation potential energy of two dipoles for a given 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 ' ' '