Inverse-square behavior

The FvdM as well as the BMVW model neglects thermal fluctuation effects both are T = 0 K theories. Pokrovsky and Talapov (PT) have studied the C-SI transition including thermal effects. They found that, for T 0 K the domain walls can meander and collide, giving rise to an entropy-mediated repulsive force of the form F where I is the distance between nearest neighbor walls. Because of this inverse square behavior, the inverse wall separation, i.e. the misfit m, in the weakly incommensurate phase should follow a power law of the form... 

The asymptotic behavior of the coefficients / and mn is determined by the singularities nearest to the real axis in the complex 0 plane. These singularities are square root and inverse square root branch points at i0 = In A. From this it follows that... 

The main problem that the early workers identified with the DLVO theory was that it could not explain how the (i-value varied with the electrolyte concentration c. The observed behavior was that d was inversely proportional to the square root of c, whereas the DLVO theory predicted a much more rapid variation. The origin of this fault is that DLVO theory explains the stable interplate separation in the gel phase as arising from a balance between an electrostatic repulsion, which does depend on c as the inverse square root, and a van der Waals attraction that is more or less independent of the salt concentration. I started studying the gels by neutron diffraction and extended the earlier results to a much wider range of salt concentrations. There was no way the DLVO theory could be made to fit the data, even when the van der Waals force was introduced with an adjustable parameter. 

To illustrate the effect of an irreversible follow-up chemical reaction on the tip feedback response, Figure 4 shows calculated chronoamperometric behavior for log d/a values of 1.0 and 0.4. To emphasize the short-time behavior, the data are presented as normalized current versus the inverse square root of normalized time. At the shortest times (largest r 1/2), the /T response is as for the simple electron transfer case discussed above, showing no dependence on K or dla. On this time scale, the current is governed by... 

The relevance to the present work is that the scale of contacts in the mica platelets studied here is in the order of 1 to 10 pm. This is the intermediate zone referred to by Johnson in which the frictional stress depends upon the inverse square root of the grain size in the manner of a Mode II crack. It has been previously shown that estimations of yield stress for a range of platelet sizes based on an inverse square root law yielded predictions of macroscopic behavior that agreed well with experimentally observed behavior and that the macroscopic yield stress was determined to be in the order of 1 GPa. A yield stress in the order of 1 GPa gives a value of So in the order of 500 MPa, precisely within the intermediate zone described above. The nature of the observed Mode II behaviour on the microstructural scale is thus explained. 

Fig. 6 Inverse susceptibility S"i(0) and inverse square of the correlation length versus inverse temperature for the critical LCST high molar volume blend dPS/PVME. The solid lines represent a fit of the corresponding crossover functions. The mean field approximation is visible in the insets, the Ising behavior by the dashed lines...

Newton s law of attraction states that the force of interaction of particles is inversely proportional to the square of the distance between them. However, in a general case of arbitrary bodies the behavior of the force as a function of a distance can be completely different. 

R. S. Berry. Known as the Berry pseudorotation, the mechanism involves the trigonal bipyramid (D)fl) passing through a square based pyramid (C4v) as shown in Figure 14.8. This behavior is somewhat similar to the inversion of the ammonia molecule (C3 ) as it passes through a planar (D)fl) structure. 

However, a series of such experiments showed that ku is not constant but is inversely proportional to the square root of the total organolithium concentration. This behavior can be contrasted to that of isoprene adding to styryllithium and styrene adding to isoprenyllithium in cyclohexane 263) where the apparent rate constants were constant. 

The critical heat release rate following the Frank-Kamenetskii theory (see Section 13.4), which describes the passive behavior of the reactor without fluid circulation, when heat transfer occurs by thermal conduction only. The critical heat release rate is the highest power that does not lead to a thermal explosion and varies with the inverse of the squared radius ... 

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