Cross-over regimes

The present review mainly, but not exclusively, deals with linear flexible polyions and focuses on single chain properties such as the "electrostatic persistence length of intermolecular interactions of a mainly electrostatic nature, on static and dynamic properties at the dilute-semidilute cross-over regime and, only briefly, on the occurrence of unidentified polyelectrolyte structures, sometimes referred to as extraordinary phase , cluster , association or, according to its most striking phenomenology, slow mode . 

Early MC-simulations studied successfully the cross-over regime from dilute to semi-dilute solutions of charged rigid rod-like polyions [80, 81]. In highly diluted solution (Cp < c ) the rod-like particles resemble pointlike objects and the solution properties are similar to those of charged spheres. With increasing concentration the distribution function g(r, u, U2) increasin y depends on the... 

Of special interest is the case of parabolic barrier (1.5) for which the cross-over between the classical and quantum regimes can be studied in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the parabolic barrier transparency [Landau and Lifshitz 1981],... 

Equation (2.41) describes either damped oscillations (at tls < 2do) or exponential relaxation OItls > 2do). Since tls grows with increasing temperature, there may be a cross-over between these two regimes at such that 2h QoJ Ao)coth P hAo) = 2Aq. If the friction coefficient... 

Fig. 10 Charge transport is observed in a variety of nucleic acid assemblies over a wide distance regime (3.4-200 A). Shown are examples of nucleic acid structures through which charge transport has been examined a B-form DNA b DNA-RNA hybrids c cross-over junctions and d nucleosome core particles. In all assemblies, the charge transport chemistry is extremely sensitive to the structure of the -stacked nucleic acid bases...

The dynamical history of stress-relaxation in a star-linear blend begins life in just the same way as a star-star blend,because when t r gp the linear chain relaxation is dominated by pathlength fluctuation and behaves as a two-arm star with M =Mii /2. So very early Rouse fluctuation (Eq. 25) crosses over to activated fluctuation in self-consistent potentials. These are calculated via the coordinate transformation used in the star-star case above. For example, the effective potential for the star component in this regime is... 

In the dilute region w < 1 the autocorrelation function directly crosses over from Debye-type behavior on scales q2Ji2 < 1 to the large momentum behavior as discussed above. For semidilute systems w intermediate regime, governed by the asymptotics of the Debye function,... 

There are two basic approaches to the theory of atomic helium, depending on whether the nuclear charge Z is small or large. For low-Z atoms and ions, the principal challenge is the accurate calculation of nonrelativistic electron correlation effects. Relativistic corrections can then be included by perturbation theory. For high-if ions, relativistic effects become of dominant importance and must be taken into account to all orders via the one-electron Dirac equation. Corrections due to the electron-electron interaction can then be included by perturbation theory. The cross-over point between the two regimes is approximately Z = 27... 

At low relative pressures p/p0 or thin adsorbate films, adsorption is expected to be dominated by the van der Waals attraction of the adsorbed molecules by the solid that falls off with the third power of the distance to the surface (FHH-regime, Eq. 3a). At higher relative pressures p/p0 or thick adsorbate films, the adsorbed amount N is expected to be determined by the surface tension y of the adsorbate vapor interface (CC-regime, Eq. 3b), because the corresponding surface potential falls off less rapidly with the first power of the distance to the surface, only. The cross-over length zcrit. between both regimes depends on the number density np of probe molecules in the liquid, the surface tension y, the van der Waals interaction parameter a as well as on the surface fractal dimension ds [100, 101] ... 

The reduced value of the scaling exponent, observed in Fig. 29 and Fig. 30a for filler concentrations above the percolation threshold, can be related to anomalous diffusion of charge carriers on fractal carbon black clusters. It appears above a characteristic frequency (O when the charge carriers move on parts of the fractal clusters during one period of time. Accordingly, the characteristic frequency for the cross-over of the conductivity from the plateau to the power law regime scales with the correlation length E, of the filler network. 

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