Crossover between exponents

Figure 8.3 also shows clearly that caution is needed when using Mark-Houwink equations from the literature that have intermediate exponents in the range 0.5 < a < 0.76. Such intermediate exponents correspond to the crossover between regimes and are only valid for the range of molar masses they were measured in. 

Another possibility is that the crossover between mean field and Ising critical behavior, which is spread out over many decades in 1 — TITc (68,69), also causes the exponents xi, X2, X3 in equation 10 to be effective exponents, which show a significant variation when one studies B N),cQ ), etc. over many decades in N. Usually experiments and simulations have only 1 to 2 decades in N at their disposal, and therefore all conclusions on the validity of equations 10 and 11 are still preliminary. However, experiments do allow a study of enough decades in 1 - T/Tcl to confirm the theoretical expectation that the critical exponents etc. take the values of the Ising universality class. 

Fig. 13. Longest and segmental relaxation time ratio as a function of the molecular weight at 250 K. The vertical axis is scaled by to emphasize the transition from the Rouse to the intermediate regime. The solid line corresponds to the description of the data with a sharp crossover between two power law regimes with different exponents. Reprinted with permsision from Riedel et al, Rheologica Acta 49,507-512 Copyright 2010 Springer...

The study of stacked hexatics has yielded a complete classification of the large variety of thermotropic liquid crystal phases. It has also presented a single experiment which yields a simultaneous measurement of many crossover anisotropy exponents, previously obtained from separate complex experiments. It is also an excellent system to reveal dimensional crossover and the interplay between two- and three-dimensional physics. For very thin layers, the stacked hexatic layers can be used to extract the hexatic stiffness, or Frank constant, and thus test predictions of defect mediated transition theories. 

Below it will be shown that field-cycling NMR relaxometry studies unambiguously reveal a crossover between high-frequency and low-frequency dispersion regimes that can be identified with the high-mode-number and low-mode-number limits of the renormalized Rouse models. Moreover, the variation of the power law exponents closely corresponds to that predicted by the renormalized Rouse models. These dynamic regimes cannot be ex-... 

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

Unfortunately, the slmulational data " are neither conclusive, suggesting scaling exponents between 4 and 6. Obviously, finite-size and finite-time effects are important, giving rise to gradual crossovers (similar to what we observed in the evaporation case ). In addition, details of the dynamics, for instance, fast or slow terrace diffusion (as discussed above), should be viewed carefully. 

The boundaries between the various domains shown in Fig. 4.14 need not be very sharp. They represent regions wherein a crossover occurs in the scaling exponents. The value of C2, for example, might be calculated to lie in the range (3Af/47iiV / j C2 < (Af/W R, ). Here the lower bound assumes that the polymer molecules behave like hard spheres whereas the upper bound assumes that they occupy cubic cells. Actually, there is evidence to suggest that... 

From (5.5.20) and (5.5.21) it is seen that Beet, where = 2vj — v, and D oc fK If V = 2, B should be finite at the transition temperature. However, experimentally, it appears that B at the transition is almost vanishingly small within experimental limits. Few measurements are available on D to draw any definite conclusions. In any case, as pointed out earlier, the exponents are neither universal nor do they agree with the predictions of any of the theoretical models. Vithana et a/. have suggested that the widely differing values of the exponents for the different compounds may be a consequence of the fact that one is measuring effective values associated with crossover effects between the XY class and a tricritical point. A further complication is that the experiments of Evans-Lutterodt et appear to indicate that the occurrence of different... 

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