Metastable minimum

Figure 2. A schematic of the free energy density of an aperiodic lattice as a function of the effective Einstein oscillator force constant a (a is also an inverse square of the locahzation length used as input in the density functional of the liquid). Specifically, the curves shown characterize the system near the dynamical transition at Ta, when a secondary, metastable minimum in F a) begins to appear as the temperature is lowered. Taken from Ref. [47] with permission.

Figure 15. DMBE potential111 for an 0(3P) atom moving around a OH(X 2/7) molecule fixed at the equilibrium geometry, with the center of the bond at the origin. Note the presence (contour K on the right) of the metastable minimum for the C., hydrogen-bonded species OH - O. The contours in this figure (and the following ones till Fig. 19) are equally spaced by 0.01 t, starting at A = — 0.211 Eh (close to the equilibrium Cs geometry of the H02 molecule).

However, there remains the definite possibility of a metastable minimum that corresponds to the LDL state. Such a possibility is supported by at least two... 

Attempts to locate the state of OF2 revealed stability issues that are not found in calculations for the SF2(a Bj) state. A metastable minimum is found at the RCCSD(T)/AVTZ level, but no Bj minimum is obtained at the MRCI-fQ/AVTZ level. At both levels of theory, a weakly bound complex with A" symmetry exists between OF(X n) and F( P). At the RCCSD(T)/AVTZ level, the O-F distances are... 

The first one corresponds to a metastable minimum and the second one to a maximum. These solutions disappear when... 

Depending on the value of q, the A G (r) dependence can be monotonicaUy increasing (large q, nucleation forbidden), or with a metastable minimum (intermediate values of q), or with a stable minimum (small q, nucleation possible) as demonstrated in Figure 4.17. Crossover to possible nucleation (second minimum at zero level) means dAG... 

A similar behaviour is expected when the flow is gradually reduced and the power maintained constant. The flow diversion from one channel into another may lead to local dryout. Therefore, as a first approximation, the metastable minimum and maximum point in the characteristic curve indicate the onset of flow diversion and dryout, respectively. It has indeed been observed that CHF could occur in the negative slope region between the minimum and maximum of the pressure drop vs. flow characteristic curve. 

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... 

As compared to ECC produced under equilibrium conditions, ECC formed af a considerable supercooling are at thermodynamic equilibrium only from the standpoint of thermokinetics60). Indeed, under chosen conditions (fi and crystallization temperatures), these crystals exhibit some equilibrium degree of crystallinity at which a minimum free energy of the system is attained compared to all other possible states. In this sense, the system is in a state of thermodynamic equilibrium and is stable, i.e. it will maintain this state for any period of time after the field is removed. However, with respect to crystals with completely extended chains obtained under equilibrium conditions, this system corresponds only to a relative minimum of free energy, i.e. its state is metastable from the standpoint of equilibrium thermodynamics60,61). 

Plotting U as a function of L (or equivalently, to the end-to-end distance r of the modeled coil) permits us to predict the coil stretching behavior at different values of the parameter et, where t is the relaxation time of the dumbbell (Fig. 10). When et < 0.15, the only minimum in the potential curve is at r = 0 and all the dumbbell configurations are in the coil state. As et increases (to 0.20 in the Fig. 10), a second minimum appears which corresponds to a stretched state. Since the potential barrier (AU) between the two minima can be large compared to kBT, coiled molecules require a very long time, to the order of t exp (AU/kBT), to diffuse by Brownian motion over the barrier to the stretched state at any stage, there will be a distribution of long-lived metastable states with different chain conformations. With further increases in et, the second minimum deepens. The barrier decreases then disappears at et = 0.5. At this critical strain rate denoted by ecs, the transition from the coiled to the stretched state should occur instantaneously. 

Position A represents many cases of practical stability it does not represent the condition of minimum energy and is more accurately referred to as metastable, because it pertains to a condition of comparative rather than absolute stability. 

Verwey and Hamaker (10) have modified the Morse curve to take into account the approach of two charged bodies, as shown in Figure 4. Here, as one moves outward toward increasing distance of separation, an electrostatic repulsion is encountered because the charges are similar. A secondary minimum is then encountered as a result of the concentration of counter ions around each charged particle. The shallowness of the secondary minimum shows that the deflocculated system is metastable. The importance of the Verwey and Hamaker concept lies in its ability to show graphically the correlation between the secondary minimum and the metastable position. A, Figure 1. 

Initially, an overdamped Brownian particle is located in the potential minimum, say somewhere between x and X2- Subjected to noise perturbations, the Brownian particle will, after some time, escape over the potential barrier of the height AT. It is necessary to obtain the mean decay time of metastable state [inverse of the mean decay time (escape time) is called the escape rate]. 

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