Mean-field regime

The problem of the dynamic crossover from the asymptotic critical to the mean field regime has been treated by Jacob et al. [94] and by Kostko et al. [95], Kostko et al. derived a decomposition of D = Db + AD into a background contribution Db and an enhancement AD of the form... 

Experiments have shown that, at least for PDMS/PEMS blends of equal weight fraction, K(T) indeed depends only weakly on temperature and is independent of the molar mass of the constituents [97], Consequently, the different values of the Soret coefficient in the classical mean field regime are almost exclusively caused by the variation of the static structure factor. 

The validity of the linear theory observed for the early stage of spinodal decomposition is chiefly related to the large size of the chain molecules. As shown above, characteristic quantities as the time t or the wavelength Am(0) of the fastest growing fluctuation are proportional to Ro and Rg, respectively. Furthermore, the Landau-Ginzburg criterion (cf. condition 2)) ensures that the mean-field regime is sufficiently extended. 

The amplitude A is the same below and above the gel point in the mean-field regime e s> g, while the values below and above gel point are quite different in the critical gelation regime. Propose a way of matching these differences at the Ginzburg point eq. 

Consider vulcanization of a melt of A-mers. At some extent of reaction — ip Pc)/Pc above the gel point and beyond the Ginzburg point (in the mean-field regime) the network strands become entangled with each other. 

In this section, we summarize our results and discuss their physical implications on the reaction kinetics in the bicontinuous phases. The RG analysis confirmed our mean-field arguments and thereby predicts that the dynamic fluctuations ofthe microemulsion will be irrelevant in impacting the hydrodynamic behavior (long length and time scale) ofthe system. Thus, the kinetics ofthe reaction A B Q is expected to follow the anomalous mean-field regime (AMF), one wherein concentrations decay as ca,cb (nQyi (Dt) l at long times. This constitutes the central result of this part of the article. 

Fig. 31. Log-log plot of mN,/2 at T = Tc vs L/N, for the bond fluctuation model at <)>, = 0.5. Different symbols show various chain lengths N, as indicated. The dashed line illustrates the power law expected in the mean-field regime of finite size scaling M >ocL-3, for L/N 1) and the dash-dotted line in the non-mean-field regime K M >ocL-p/v, with p/v 0.515 for Ising exponents [274]. From Binder and Deutsch [275].

Phase transitions can be first-order or second-order (or continuous), and critical energy fluctuations quite often have a significant impact on thermal parameters. First-order transitions are characterized by discontinuous jumps in the first derivatives of the free energy, resulting in finite density p and enthalpy H differences between two distinct coexisting phases at the transition temperature Tp.. For a second-order transition there are no discontinuities in the density or the enthalpy but the specific heat capacity Cp will exhibit either a discontinuous-jump (for mean-field regime) or a critical anomaly... 

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