Rate constants in spatially inhomogeneous systems

Abstract

We present a theory and accompanying importance sampling method for computing rate constants in spatially inhomogeneous systems. Using the relationship between rate constants and path space partition functions, we illustrate that the relative change in the rate of a rare event through space is isomorphic to the calculation of a free energy difference, albeit in a trajectory ensemble. Like equilibrium free energies, relative rate constants can be estimated by importance sampling. An extension to transition path sampling is proposed that combines biased path ensembles and weighted histogram analysis to accomplish this estimate. We show that rate constants can also be decomposed into different contributions, including relative changes in stability, barrier height and flux. This decomposition provides a means of interpretation and insight into rare processes in complex environments. We verify these ideas with a simple model of diffusion with spatially varying diffusivity and illustrate their utility in a model of ion pair dissociation near an electrochemical interface.Macroscopic rate constants quantify the characteristic timescale for an ensemble of components to transition between two stable states. These states may be the reactants and products of a chemical transformation or the collective reorganizations accompanying a phase transformation. Microscopically, rate constants are related to dynamical fluctuations as codified in timecorrelation functions of populations of the stable states. 1 In the presence of spatial inhomogenieties, such as extended interfaces, dynamical fluctuations need not be the same throughout space, and rate constants may obtain a spatial dependence. 2 With the advent of interfacial sensitive measurements, 3-6 single molecule experiments, 7-9 and precision electrochemical techniques, 10,11 quantifying how reactivity changes spatially at a molecular level is now possible. Theoretical work has trailed behind these advances, with few methods to efficiently study such processes and consequently few guiding principles for understanding how reactivity is altered in inhomogeneous systems. Here, we detail a theory and numerical technique to compute rate constants in the presence of spatial inhomogenieties without assuming that the mechanism of the transition is conserved at different points in space. This theory relies on the relationship between rate constants and trajectory space partition functions, and the method it motivates is general, capable of application to complex environments and processes.Rare dynamical events often take place in environments that are complex and inhomogeneous. Heterogeneous catalysis relies on the increase of the rate of a chemical reaction near an extended fluid-solid interface relative to the rate in the fluid. 12 Moreover, heterogeneities along the interface like defects or grain boundaries can act as active sites for catalysis and have tremendous impact on reactivity. 13 Reactivity can be influenced analogously by extended liquid-vapor interfaces, or by a) Electron...

Ion Dissociation Dynamics in an Aqueous Premelting Layer

“…Analogously, the rate of ion pair dissociation need not be constant and could similarly depend on where in the layer the rare event occurs. 49 Since vertical motion of the ions is so slow, we will assume that dissociation events occur at fixed ionic z coordinates. We have computed transition rates from the CIP to SSP as a continuous function ofz, using the Bennett-Chandler approach.…”

Ion Pair Dissociation Dynamics in an Aqueous Premelting Layer

“…Analogously, the rate of ion pair dissociation need not be constant and could similarly depend on where in the layer the rare event occurs. 46 Since vertical motion of the ions is so slow, we will assume that dissociation events occur at fixed ionic z coordinates. We have computed transition rates from the CIP to SSP as a continuous function of z, using the Bennett-Chandler approach.…”

Structure and Efficiency in Bacterial Photosynthetic Light Harvesting

“…The exact form of the disorder will vary slowly with the motion of the vibration over the lifetime of an exciton. We approximate this time-dependence by generating a new, random Hamiltonian at regular intervals of 0.1 ps, corresponding approximately to a vibrational frequency below which higher harmonic levels become significantly occupied at 300 K. To our knowledge, this is the first time any attempt has been made to include the dynamical effects of slow vibrations in a master equation description of photosynthetic exciton transfer, although a similar technique has been used successfully to describe the photochemistry of small organic molecules [37].…”