Metastic potential

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

This is Kramers escape problem. Since no analytic solution is known for any metastable potential of the shape in fig. 40 the quest is for suitable approximation methods. This problem has received an extraordinary amount of attention from physicists, chemists and mathematicians.5 0 We describe the main features - all present already in the seminal paper by Kramers. 

In his seminal work [109], Kramers considered the noise-induced flux from a single metastable potential well i.e. he considered a Brownian particle... 

Fig. 4.7. (a) Rectangular potential model (b) metastable potential model. 

In particular, the analytical solution of the Smoluchowski equation for a metastable potential well depicted in Fig. 4.7b was found taking as initial condition a uniform distribution in the potential... 

This expression is nothing but the Bohr-Sommerfeld quantization rule (see, e.g., Landau and Lifshitz [1981]). In the metastable potential of Figure 3.3 there are also imaginary-time periodic orbits satisfying (3.41) that develop between the turning points inside the classically forbidden region. It is these trajectories that are responsible for tunneling [Levit et... 

The term including zero-point energy (o0l2. In the classically accessible region near the minima x0 one may use the harmonic oscillator approximation. The factor N is determined in a manner similar to the route described previously for a metastable potential. The tunneling splitting calculated from (A.23) is... 

Consider a metastable potential well like that in Figure 3.3. In the vicinity of the parabolic well (r->0) at the instanton solution (3.55)... 

Any general increase in supersaturation, brought about by increasing concentration or decreasing temperature, raises the local metastable potential and smoothes away the potential barrier to spontaneous nucleation, shown... 

Figure A2.5.8. Constant temperaPire isothenns of reduced chemical potential p. versus reduced density p. for a van der Waals fluid. Full curves (including the horizontal two-phase tie-lines) represent stable siPiations. The dashed parts of the smooth curve are metastable extensions, while the dotted curves are unstable regions.

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

In the potential range catliodic to one frequently observes so-called metastable pitting. A number of pit growtli events are initiated, but tire pits immediately repassivate (an oxide film is fonned in tire pit) because the conditions witliin tire pit are such that no stable pit growtli can be maintained. This results in a polarization curve witli strong current oscillations iU [Pg.2728]

Fig 5 3 Changes in the potential energy of a static mechanical system tell us whether it is in a stable, unstable or metastable state. 

Here the nucleation barrier AO is the excess thermodynamic potential needed to form the critical embryo within the uniform metastable state, while the prefactor Jq is determined by the kinetic characteristics for the embryo diffusion in the space of its size a. Expressions for both AO and Jo given by Zeldovich include a number of phenomenological parameters. 

Using the constructed potentials the y-surface for the (111) plane was calculated. (For more details see Girshick and Vitek 1995). T e lowest energy minimum on this surface corresponds to the ideal Llo structure. However, there are three different metastable stacking fault type defects on (111) the antiphase boundary (APB), the complex stacking fault (CSF) and the superlattice intrinsic stacking fault (SISF). The displacements... 

Fig. 1.18 Modified potential-pH diagram for the Ni-H20 system the curves showing the stability of the nickel oxides have been extrapolated into the acid region to indicate the formation of metastable oxides (after De Gromoboy and Shreir")...

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