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Research Papers

The Influence of Dimensionality on the Rate of Diffusive Escape From an Energy Well

[+] Author and Article Information
L. B. Freund

Department of Materials Science and Engineering,  University of Illinois at Urbana-Champaign, 1304 West Green Street, Urbana, IL 61801

J. Appl. Mech 79(3), 031012 (Apr 05, 2012) (5 pages) doi:10.1115/1.4005895 History: Received July 11, 2011; Revised October 25, 2011; Posted February 13, 2012; Published April 04, 2012; Online April 05, 2012

A commonly used idealization when describing separation of a chemical bond between molecules is that of an energy well which prescribes the dependence of energy of interaction between the molecules in terms of a reaction coordinate. The energy difference between the peak to be overcome and the root of the well is the so-called activation energy, and the overall shape of the well dictates the kinetics of separation through a constitutive assumption concerning transport. An assumption tacit in this description is that the state of the bond evolves with only a single degree of freedom—the reaction coordinate—as the system explores its energy environment under random thermal excitation. In this discussion we will consider several bonds described by one and the same energy profile. The cases differ in that the energy profile varies along a line extending from the root of the well in the first case, along any radial line in a plane extending from the root of the well in a second case, and along any radial line in space extending from the root of the well in a third case. To focus the discussion we determine the statistical rate of escape of states from the well in each case, requiring that the profile of the well is the same in all three cases. It is found that the rates of escape each depend exponentially on the depth of the well but that the coefficients of the exponential vary with depth of the well differently in the three cases considered.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

The profile of energy versus distance which provides the common basis for identification of energy bond wells on one, two, and three dimensions

Grahic Jump Location
Figure 2

Plot of the dependence of the natural log of the nondimensional off-rate versus well depth cb for the two-dimensional well. The curves show the behavior based on asymptotic (solid) and numerical (dashed) evaluation of the integrals defining the parameters A0 and A1 . The open circles show the results of solution of the partial differential equations by means of the numerical finite element method, while the open triangles show the results of discrete random walk analysis. Note that ηa2 /D = η¯(2) = A0 /A1 .

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