Stochastic Theory of Desorption Reactions (Brownian Motion)
Lightfoot, Edwin James
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https://hdl.handle.net/2142/69762
Description
Title
Stochastic Theory of Desorption Reactions (Brownian Motion)
Author(s)
Lightfoot, Edwin James
Issue Date
1985
Department of Study
Chemical Engineering
Discipline
Chemical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Chemical
Abstract
A Brownian motion model is used to study the escape of a molecule from a physisorbed state into a dense fluid.
In the limit of a high dynamical friction coefficient, the Smoluchowski diffusion equation, with an annihilation boundary condition, is used to predict the concentration profile, the rate constant for desorption, and the mean time to desorption. A general asymptotic solution to the Smoluchowski equation is obtained which demonstrates that within the restrictions of the annihilation assumption the concentration profile is given by the Boltzmann distribution multiplied by a depletion factor determined exclusively by the asymptotic form of the attraction to the surface. Two methods of determining the rate constant by direct integration are given--one a variational approach and the other a pseudo-steady-state approach introduced by Kramers. Both methods are used to estimate the rate constant for the escape from a piecewise parabolic potential and from a potential of the Lennard-Jones type (r('-2n) - r('-n)). For both potentials the two methods give essentially identical results. Over a broad range of conditions (including a dynamical friction coefficient which rises near the surface), the mean time to escape is found to approach asymptotically a linear function of the distance between the surface and the annihilation point. Asymptotic estimates are made of both the slope and intercept of this linear function; numerical integration confirms the accuracy of these estimates.
For moderate values of the dynamical friction, the Fokker-Planck phase space diffusion equation, popularized by Kramers and Chandrasekhar, is used to estimate the rate of equilibration in a Lennard-Jones type double well (in the limit of high friction this is equivalent to the annihilation formalism used with the Smoluchowski equation). Again, the mean time to escape from a Lennard-Jones type potential well is found to approach asymptotically a linear function of the distance from the surface to the center of the double well. The slope of this linear relationship is found to reduce to the same slope found for the Smoluchowski equation as either the friction constant or the width of the double well becomes large. The predicted dependence of the rate constant on the dynamical friction is confirmed by numerical integration of the Fokker-Planck equation.
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