Double metastability

Fig. 17. Simulation of fronts in the region of double metastability. In (a) the faster front overruns the slower one for slightly different parameters (b) it is decelerated and a bound state (pulse) forms.

This behavior ( double metastability [101-103]) is also obtained in model calculations. As long as the two fronts propagate with significantly different velocities, the faster eventually overruns the slower one (Figure 17a). However, for comparable velocity, a bound state forms (Figure 17b), corresponding to a stable pulse in one, and a circular wave or spiral in two spatial dimensions. The spiral waves behave qualitatively the same as in the excitable region. 

Spiral waves have been studied in most detail on Pt(l 10), both under conditions of excitability and of double metastability [108]. Under identical external conditions the spirals did not exhibit a fixed period and wavelength, rather a continuous distribution of these quantities was observed (Figure 18). It is well known that spirals can be pinned to artificial nonexcitable cores [109, 110]. Such artificial cores can be formed by surface defects consequently the continuous distribution of rotation periods simply reflects the fact that defects of various size exist on the surface. Comparison of the dispersion relation computed from the model [111] with the experimentally observed periods and wave velocities (Figure 19) allows conclusions to be drawn about the size distribution of surface defects by using the Tyson-Keener formula [112], which gives the relationship between core size and period. For the data... 

At higher temperatures the stable form is valentinite, which consists of infinite double chains. The orthorhombic modification is metastable below 570 °C however, it is sufficientiy stable to exist as a mineral. Antimony trioxide melts in the absence of oxygen at 656°C and partially sublimes before reaching the boiling temperature, 1425°C. The vapor at 1500°C consists largely of Sb O molecules, but these dissociate at higher temperatures to form Sb202 molecules. 

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. 

Needless to say, tunneling is one of the most famous quantum mechanical effects. Theory of multidimensional tunneling, however, has not yet been completed. As is well known, in chemical dynamics there are the following three kinds of problems (1) energy splitting due to tunneling in symmetric double-well potential, (2) predissociation of metastable state through... 

He(ls ) S and also from the metastable level Nr (ls2s)-1- He(ls ) in order to take care of the fraction of metastable ion in the incident beam, as well as the double-electron capture process from both ground and metastable ions. 

In order to rationalize the catalyst-dependent selectivity of cyclopropanation reaction with respect to the alkene, the ability of a transition metal for olefin coordination has been considered to be a key factor (see Sect. 2.2.1 and 2.2.2). It was proposed that palladium and certain copper catalysts promote cyclopropanation through intramolecular carbene transfer from a metal carbene to an alkene molecule coordinated to the same metal atom25,64. The preferential cyclopropanation of terminal olefins and the less hindered double bond in dienes spoke in favor of metal-olefin coordination. Furthermore, stable and metastable metal-carbene-olefin complexes are known, some of which undergo intramolecular cyclopropane formation, e.g. 426 - 427 415). 

The arrows show the isotherm evolution for continual addition of dissolved Me. The initial isotherm with the slope of 1 (in the double logaritmic plot) corresponds to a Langmuir isotherm (surface complex formation equilibrium). [Me]S0 = solubility concentration of Me for the stable metal oxide [Me]p = solubility concentration of Me for a metastable precursor (e.g., a hydrated Me oxide phase). 

Lacey, M.J. Macdonald, C.G. Interpreting Metastable Peaks From Double Focusing Mass Spectrometers. Org. Mass Spectrom. 1980,15, 484-485. 

K. Pays, J. Kahn, P. Pouligny, J. Bibette, and F. Leal-Calderon Double Emulsions A Tool for Probing Thin Film Metastability. Phys. Rev. Lett. 87, 178304 (2001). 

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