[Show abstract][Hide abstract] ABSTRACT: An integral equation [Rasaiah and Zhu, J. Chem. Phys. 98, 1213 (1993)] for the survival probabilities of electron transfer (ET) between thermally equilibrated reactants in solution is extended to include quantum effects on the ligand vibration and ET from a nonequilibrium initial state. We derive the kernel of the integral equation using a Green’s function technique and demonstrate that it is determined by the solvent dynamics, the relative contributions of ligand and solvent reorganization energies, and the barrier heights for electron transfer. The extension of the theory to ET from a nonequilibrium initial state modifies the integral equation to provide the survival probabilities for the reactants that are not necessarily kinetically of first order, but can be directly compared with experiment. The long time rate, however, shows a simple exponential time dependence that is analyzed in terms of a rate constant with a diffusive solvent controlled component and a remainder. The effect of solvent dynamics on the diffusive part is governed by the same factors that determine the kernel. We find that the fast diffusive mode (small relaxation time) affects the rate of ET reactions with high barriers, while the slow diffusive part (large relaxation times) influences the rate when the barriers are low. Quantum corrections to these effects are calculated using the semiclassical approximation. The theory is used to analyze the ET kinetics of betaine-30 in glycerol triacetate (GTA) over a 100° temperature range and the influence of the details of solvent dynamics on the rates of electron transfer is elucidated. An appendix discusses improved saddle point approximations for the rates of electron transfer reactions calculated using the golden rule.
The Journal of Chemical Physics 01/1994; 101:9966-9981. · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The solution to an integral equation [J. Zhu and J. C. Rasaiah, J. Chem. Phys. 96, 1435 (1992)] for the survival probabilities in the Sumi-Marcus model of reversible electron transfer (ET) reactions, in which ligand vibrations and fluctuations in the solvent polarization play important roles, is obtained numerically using a simple computer program suitable for use on a PC. The solutions depend on the time correlation function A(t)
of the reacting intermediates along the reaction coordinate which is shown to be equal to the time correlation function of the Born free energy of solvation of these intermediates even in discrete molecular solvents provided its response is linear. This enables A(t) to be determined accurately from time-delayed fluorescence Stokes shift experiments or from dynamical theories of ion solvation; it is usually an exponential (Debye solvent) function of time or a sum of such exponentials (non-Debye solvent). The solutions to the integral
equation, which can be obtained numerically for any given A(t), are found to predict the electron-transfer dynamics successfully over a wide range of model parameters. They can also be approximated by single or multiexponential interpolation formulas in which the thermally equilibrated rate constants are modified by a factor which reflects the relative importance of ligand (or inner-sphere solvent) vibration and outer-sphere solvation dynamics. The use of an effective longitudinal relaxation time in calculations of ET rates in solution is shown to be a poor assumption in some solvents. The theory is compared with an experiment in the inversion region, and its extension to include high-frequency vibrational modes that lead to an increased ET rate in other experiments is discussed.
The Journal of Chemical Physics 06/1993; 98:1213-1227. · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The general solutions obtained earlier [J. Chem. Phys. 95, 3325 (1991)] for the coupled diffusion‐reaction equations describing reversible electron transfer reactions in Debye solvents, governed by Sumi–Marcus free energy surfaces, are extended to non‐Debye solvents. These solutions, which depend on the time correlation function of the reaction coordinate Δ(t), are exact in the narrow and wide window limits for Debye and non‐Debye solvents and also in the slow reaction and non‐diffusion limits for Debye solvents. The general solution also predicts the behavior between these limits and can be obtained as the solution to an integral equation. An iterative method of solving this equation using an effective relaxation time is discussed. The relationship between Δ(t) and the time correlation function S(t) of Born solvation energy of the reacting intermediates is elucidated.
The Journal of Chemical Physics 01/1992; 96(2). · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The dynamics of reversible electron transfer reactions in Debye solvents are studied by employing two coupled diffusion–reaction equations with the rate constants depending on the reaction coordinate. The equations are solved analytically in four limiting cases: fast and slow reactions as well as wide and narrow reaction windows. A general solution for the survival probabilities is obtained by employing a decoupling approximation similar to the one used by Sumi and Marcus [J. Chem. Phys. 84, 4896 (1986)] for nonreversible reactions; our solution verifies the existence of four limiting cases and also predicts the behavior between these limits. Interpolation between long and short time approximations to the general solution, leads to survival probabilities with a single exponential time dependence and rate constants ki satisfying the relation k1/k2=exp(−βΔG0), where ΔG0 is the standard free energy change for the reaction. Multiexponential behavior of the survival probabilities is exhibited when higher order terms are included in the evaluation of the general solution, but this deteriorates to a single exponential, governed by a first order rate constant, at long times. In the narrow reaction window limit the multiexponential solution is exact when both the forward and reverse reactions are barrierless, and the behavior at long times is determined by a rate constant k=0.83 τ−1L where τL is the longitudinal relaxation time. Similar behavior is found when the forward reaction alone is barrierless and the barrier for the reverse reaction is large (βΔG@B|1=0, βΔG∗2≫1), except that the forward rate constant k1≊τ−1L [0.6+(π/βΔG@B|2)1/2]−1 depends on the barrier height for the reverse reaction which has a small rate constant. Our solutions reduce to those of Sumi and Marcus when the reverse reaction is ignored. They are also compared with numerical solutions to the diffusion reaction equations. The extension to non‐Debye solvents is briefly discussed.
The Journal of Chemical Physics 09/1991; 95(5). · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 1. INTRODUCTlON The sticky electrolyte model (SEM) has been studied by in a series of papers. In this model for weak electrolytes, ion association is introduced in the Hamiltonian through a delta function interaction between oppositely charged ions at a L distance which is less than the sum of the radii of the ions. Our studies so far are confined to symmetrical electrolytes in which the ions have the same charge magnitudes and their sizes are the same. The Omstein-Zernike equations were solved analytically in the mean spherical approximation (MSA) and numer-ically in the hypemetted chain (HNC) approximation for different values of L. The solvent effect in this model has also been investigated, and it was found that a hard sphere solvent has a strong packing effect on association while a dipolar solvent has 'both a packing effect due to the hard cores and a screening effect attributed to the dipoles. When L < o/2, where o is the hard core diameter, the hard core repulsion between ions of the same sign ensures that polymerization is sterically inhibited so that the only associated species present are expected to be dimers. By adjusting the coefficient of ,the delta function interaction it is possible to ensure that all. of the ions are paired> then the theory already developed for weak electrolytes can be applied to these dimers, which are extended dipoles. In particular the analytic solutions for the energy of these dipolar fluids in the mean spherical approximation have obtained for L = o/n with n = 2,3,4 and 5. In this paper we begin the study of sticky electrolytes in which the sizes of the associating ions may be different and the magnitudes of the charges on them are not neces- sarily the same. This is a more realistic model for weak electrolytes but the mathematical development is more complicated than it is for symmetrical sticky electrolytes. We begin our discussion in general terms with the bonding distance L < Ri + Ri, where Ri and Ri are ionic radii, but our detailed analysis is confined only to adhesion between oppositely charged ions. This is similar to the model first introduced by Baxter7(') and studied by Barboy and Tenne7(b) for a mixture of adhesive hard spheres of un- equal size; the difference lies in the presence of charges on the spheres and the allowance for adhesion only between unlike ions. The special case of adhesion between oppo- sitely charged ions of the same size has already been stud- ied by us4?' in the MSA and the results for the more general case presented here reduce to those found earlier in the limit of equal ion sizes. The extension of our studies to mixtures of charged ions, aside from its immediate rele- vance to the aggregation of charged particles and colloids, also provides the means to investigate the properties of the double layer at charged surfaces when preferential adsorp- tion or adhesion of one or more ions plays an important role.' This may be realized by taking the "wall limit" of our model in which the density of one species (the aspiring electrode or charged surface) is allowed to tend to zero while its radius becomes infinitely large. Our system consists of at least two kinds of ions of opposite charge; ion i has density pi, diameter oi and charge z,e, where z is the valence and e is the magnitude of the electronic charge. Throughout this paper, we also use subscripts 1 or 2 to denote the two species of a single electrolyte. Electroneutrality implies that
The Journal of Chemical Physics 01/1991; 94:3141-3149. · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The Ornstein–Zernike (OZ) equations are solved for the sticky electrolyte model (SEM) with a hard sphere solvent using the hypernetted chain (HNC) approximation for the stickiness and the mean spherical (MS) approximation for the electrical interactions. Relations among the coefficients of Baxter’s q functions and the equation for the excess internal energy are given in the MS approximation for L≤σ, where σ is the molecular diameter, and L is the distance at which the oppositely charged ions can stick. The analytical results for L=σ in the HNC/MS and PY/MS approximations are presented in detail. When the charges are switched off, the results automatically lead to those of the sticky hard sphere system; when the stickiness is turned off and the discrete solvent is changed to a continuum, the results lead correctly to those of the restricted primitive model (RPM). The thermodynamic properties of the SEM in a hard sphere solvent for L=σ are calculated and compared with the properties in a continuum solvent; special attention is paid to the derivation of the osmotic coefficient in the McMillan–Mayer system for the SEM and for the corresponding uncharged system. By switching off the charge and the stickiness, the osmotic coefficient of an isotopic solute–solvent system is also obtained. The numerical results show that the hard sphere solvent has a strong packing effect on the structural and thermodynamic properties of the electrolyte and the association of the oppositely charged ions is greatly enhanced by the hard sphere solvent. The influence of a discrete solvent on the osmotic coefficient is quite subtle: for the charged system, the solvent tends to raise the osmotic coefficient; for the sticky hard sphere system, the solvent has just the opposite effect.
The Journal of Chemical Physics 01/1990; 92. · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Weak electrolytes and other association reactions are modeled as sticky spheres. An analysis of the density expansion, including the bridge diagrams, of the cavity functions yAB(L) for sticky hard spheres (charged or uncharged spheres binding at a distance L) leads to an approximation which provides the degree of association α as the solution to a simple quadratic equation determined by the association constant K0 and the cavity function y0AB(L) for the reference system in which the chemical bonding between the reacting species has been turned off. Similar relations are assumed to hold when the bonding is directional and specific enough to lead only to the formation of dimers. Applications to the determination of the reference cavity functions for acetic acid and monochloro acetic acid from experimental data of the degree of association are discussed. In a discrete solvent, the approximation scheme for α remains the same, except that the reference cavity function is scaled differently. Solvent medium effects on the association constant are shown to be related to the cavity function of the undissociated dimer in a pure solvent. An exponential approximation for the reference cavity function y0AB(L) is derived when the associating species are of the same size and the bonding is spherically symmetric. Expressions for the changes in the thermodynamic functions due to association are obtained analytically in terms of the degree of association and the reference cavity functions. The magnitude of the degree of association, calculated from the exponential approximation for y0AB(L), and its effect on the thermodynamic properties are different from what was previously observed using the hypernetted chain (HNC) approximation. The thermodynamics of weak 1–1 electrolytes are discussed using the new method and a comparison is made between the new and old methods for 2–2 electrolytes.
The Journal of Chemical Physics 01/1990; 92:7554-7564. · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Theoretical and computer simulation studies of annihilation reactions with traps on two and in three dimensional lattice systems are reported for the following reactions: (1) Bimolecular trapping/annihilation: A+A↠*; A+T↠AT; A+AT↠T; (2) unimolecular trapping/annihilation: A+A↠*; A↠AT; A+AT↠*. The mean field analysis and combinatorial calculations of the rate constants given previously for a square lattice are generalized to lattices in two and three dimensions. It is found that the kinetics of trapped A’s can be described by mean field theory for bimolecular but not for unimolecular trapping reactions. The kinetics of free A’s obeys mean field theory at short times, but at longer times and at low trap densities the free A population decays as a stretched exponential at when large density fluctuations dominate the reaction. This stretched exponential behavior of the Donsker–Varadhan from A(t)∼exp(−td/(d+2)), where d is the dimensionality, already found for the reactant decay in A–A annihilation reactions with traps on a square lattice [Rasaiah etal., J. Phys. Chem. 94, 652 (1990)] was tested for universality by studying triangular and hexagonal lattices in two dimensions (2D) and a cubic lattice in three dimensions (3D). The same behavior is also observed when the free particle annihilation is turned off. The effect of a finite staying probability ps on the kinetics of these reactions are also investigated.
The Journal of Chemical Physics 01/1990; 93. · 3.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The sticky electrolyte model in a dipolar solvent is studied in this paper. A detailed separation of the Ornstein–Zernike (OZ) equations and their solution in the mean spherical (MS) approximation for binding (or stickiness) at L=σ are given. The results derived earlier by Adelman and Deutch, Blum et al. and by Høye et al. in this approximation are reproduced when the stickiness is switched off. Also when the density of the solvent goes to zero, the results reduce to those of the sticky electrolyte model (SEM) in a continuum solvent. It is found that the PY/MS approximation gives negative solutions for the association parameter λ, while the HNC/MS approximation works in a narrow interval of the sticky potential well depth ϵ2 between the positive and negative ions. As expected, the ion association increases when sticky potential well becomes deeper, but the dipole moment of the solvent is found to have a strong screening effect on this association. The study of the radial distribution functions of this system shows that the probability of a free ion appearing near a counter ion is greatly decreased when binding occurs between the oppositely charged ions at the contact; the opposite happens for ions of the same sign. The absolute value of the ion solvation energy becomes smaller as the electrolyte concentration increases and when stickiness between oppositely charged ions is introduced.
The Journal of Chemical Physics 07/1989; 91(1). · 3.12 Impact Factor