Cavity stability criteria

Modeling of tensile failure leads to the expression of a stability criterion in terms of the normalized drawdown pressure gradient, gpn, at the cavity wall (24). For cylindrical geometry this is expressed as... 

Figure 2.3 Stability criterion for open optical resonators. The curves plot the ratio of mirror spacing L to radius of curvature R of the two mirrors making up the cavity in this case Rj = In the hatched sectors and inside the curve boundaries the stability criterion is not met (see text) resulting in increased loss from the cavity and reduced Q (Adapted from Yariv and Pozar )...

For concave mirrors this cavity stability condition may be stated in the form the cavity will be stable if either both centres of curvature lie inside the cavity and overlap or both centres of curvature lie outside the cavity. In more general cases it is convenient to recast the stability criterion in the form (Problem 12.6) ... 

Fig.12.6. Stability diagram for optical resonators. Shaded areas are those for which the stability criterion 0 < (1-L/R )(I-L/R2) s 1 is violated. In these regions the cavity has a high loss.

The more incisive calculation of Springett, et al., (1968) allows the trapped electron wave function to penetrate into the liquid a little, which results in a somewhat modified criterion often quoted as 47r/)y/V02< 0.047 for the stability of the trapped electron. It should be noted that this criterion is also approximate. It predicts correctly the stability of quasi-free electrons in LRGs and the stability of trapped electrons in liquid 3He, 4He, H2, and D2, but not so correctly the stability of delocalized electrons in liquid hydrocarbons (Jortner, 1970). The computed cavity radii are 1.7 nm in 4He at 3 K, 1.1 nm in H2 at 19 K, and 0.75 nm in Ne at 25 K (Davis and Brown, 1975). The calculated cavity radius in liquid He agrees well with the experimental value obtained from mobility measurements using the Stokes equation p = eMriRr], with perfect slip condition, where TJ is liquid viscosity (see Jortner, 1970). Stokes equation is based on fluid dynamics. It predicts the constancy of the product Jit rj, which apparently holds for liquid He but is not expected to be true in general. 

When one thinks in terms of the many fused-ring isomers with unsatisfied valences at the edges that would naturally arise from a graphite fragmentation, this result seems impossible there is not much to choose between such isomers in terms of stability. If one tries to shift to a tetrahedral diamond structure, the entire surface of the cluster will be covered with unsatisfied valences. Thus a search was made for some other plausible structure which would satisfy all sp valences. Only a spheroidal structure appears likely to satisfy this criterion, and thus Buckminster Fuller s studies were consulted (see, for example, ref. 7). An unusually beautiful (and probably unique) choice is the truncated icosahedron depicted in Fig. 1. As mentioned above, all valences are satisfied with this structure, and the molecule appears to be aromatic. The structure has the symmetry of the icosahedral group. The inner and outer surfaces are covered with a sea of v electrons. The diameter of this C o molecule is 7 A, providing an inner cavity which appears to be capable of holding a variety of atoms. ... 

The next selection criterion concerns the polarity of the selected cavity. In the most favorable case, it should contain hydrophobic residues to favor the design of lipophilic inhibitors. The addition of hydrophobic substitutions (taking care to ensure their solubility) is an effective way of improving the potency of an inhibitor thanks to the hydrophobic effect. It has been shown that electrostatic interactions are important for the rate of association, but not for the stability of protein complexes [20], Furthermore, electrostatic interactions are weakened by the high dielectric constant of water. It might therefore be more difficult to identify inhibitors that bind tightly to the target cavity when it is essentially polar. 

These criteria describe different states of local excitation and deformation of chain segments. The stress bias criterion [86, 139] refers implicitely to two mechanisms cavitation in a dilatational stress field and stabilization of cavities through a deviatoric stress component. These mechanisms have been more explicitely considered in the mathematical model of cavity expansion in a rigid plastic by Haward et al. [137] and in the molecular models by Argon [152] and Kausch [11]. 

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