A variational approach to surface evolution

A partial differential equation governing the evolution of the surface of a strained elastic solid due to surface diffusion appears in (9.9). Given the instantaneous elastic state of the deformed solid and the instantaneous shape of the evolving surface, this partial differential equation specifies the rate 

If the evolving shape itself must be determined over the course of time as part of the solution, the underlying boundary value problem is no longer linear. Analysis of surface evolution in such situations must rely on approximate methods, in general. The two most common approaches are (i) to determine approximate solutions to the governing partial differential equations by numerical methods and (ii) to express the evolving shape in terms of a small number of modes, each with its own time-dependent amplitude. In the former approach, the solution proceeds incrementally. An elasticity 

For either numerical solution of the field equations by means of the finite element method or determination of a system of ordinary differential equations for modal amplitudes, the existence of a variational statement or weak form of the field equations is essential. For the complementary aspect of the problem concerned with the elastic field for a fixed boundary configuration, the powerful minimum potential energy theorem is available (Fung 1965). The purpose here is to introduce a variational principle as a basis for describing the rate of shape evolution for a fixed shape and a fixed elastic field. 

Suppose that the mass conservation equation is used to replace in (9.47) by its equivalent form in terms of j. Then, after application of a vector identity in the integrand, 

This result implies that any choice of surface mass flux j on S derived from a chemical potential results in a reduction in free energy, provided only that ms 0. This as an essential feature of spontaneous surface evolution, and the result also provides an important ingredient for a variational statement. 

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Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

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