Crossover temperature model

Figure A2.5.27. The effective coexistence curve exponent P jj = d In v/d In i for a simple mixture N= 1) as a fimction of the temperature parameter i = t / (1 - t) calculated from crossover theory and compared with the corresponding curve from mean-field theory (i.e. from figure A2.5.15). Reproduced from [30], Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 358, figure 3, by pennission of Elsevier Science.

Another class of thermodynamic barrier theories focuses on the large increases in the elastic constants that accompany glass formation. (These theoretical approaches seem especially appropriate to polymer fluids below the crossover temperature Fj.) In particular, the barrier height governing particle displacement in the shoving model [57] is taken to be on the order of the elastic energy GqoVo required to displace a particle on a scale comparable to the interparticle distance,... 

Third, in real ionic solutions, solvophobic and Coulombic interactions may define different length scales. This case is, of course, not covered by the RPM and similar continuum models. Anisimov et al. [322] have argued that such a mechanism may be responsible for the observed shift of the crossover temperature closer to Tc found in solutions of a picrate in a homologous series of alcohols. 

The two-mode model has two characteristic crossover temperatures that correspond to the freezing of each vibration. At temperatures above Tc0 = h(o0/2kB, the rate constant k(T) exhibits its ordinary Arrhenius form, in which the activation energy is determined by the effective barrier height... 

Concerning the slow dynamics below the crossover temperature Tc, the predictive power of the theory seems to be rather limited. In particular, the emergence of intrinsic slow secondary processes, which seems to be associated with the dynamic crossover in the experimental spectra, is not contained even in the extended versions of the theory consequently, the slow dynamics spectrum is not reproduced correctly. In this respect, the extended theory introducing the hopping mechanism for describing the susceptibility minimum below Tc is misleading. On the other hand, the most prominent prediction of MCT below Tc is the anomaly of the nonergodicity parameter, which, as discussed, is found by different model-independent approaches. However, within the framework of MCT, this anomaly is closely connected with the appearance of a so-called knee feature in the spectral shape of the fast dynamics spectrum below Tc. This feature, however, has not been identified experimentally in molecular liquids, and only indications for its existence are observed in colloidal systems [19]. In molecular systems, merely a more or less smooth crossover to a white noise spectrum has been reported in some cases [183,231,401]. Thus, it may be possible that the knee phenomenon is also smeared out. 

Below the crossover temperature, the square root term becomes dominant and the relaxation of the magnetization is no longer exponential. In contrast with the magnetic susceptibility, the mono-disperse and poly-disperse models predict qualitatively different dynamic behaviors. However, the dominant characteristic time r is still given by Eqs. 37 or 38. 

FIGURE 1 Modeled crossover temperatures of the quantum yield for CO2 uptake for Ci and C4 plants as a function of atmospheric CO2 concentrations. The boundary conditions shown are for NADP-me C4 plants (upper boundary) and NAD-me C4 plants (lower boundary). The crossover temperature is defined as the temperature (for a particufar atmospheric CO2 concentration) at which the quantum yiefds for CO2 uptake are equivalent for both the C, and the C4 pfant. Figure is modified from Ehleringer et al. (1997). 

Fig. 1 Thickness (d) dependence of Tg of atactic PS films obtained during the heating process (open circ/es correspond to Mw = 2.8 x 10, and open triang/es to= 1.8 x 10 ). The values of Tg are determined as the crossover temperature between the straight lines characterizing the real part of electric capacitance C T) at 10 kHz below and above Tg. The filled circles (A/ = 1.61 X 10 ) and filled triangles M = 2.2 x 10 ) represent the data obtained for imcapped supported films of ataetie PS by Forrest et al. using ellipsometry for atactic PS [37]. The solid curve was obtained from the layer model [27]...

The crossover Landau model uses a rescaled temperature distance to the critical point... 

---