Mean field-Ising crossover

Critical phenomena and the Ising—mean field crossover... 

Martensitic traasfonnation Master ec[uations Mean field crossover to Ising Mechanical properties Metallic alloys Metallic glasses Metastable alloys Microhardness test Microscopic theory of nucleation... 

From a global assessment of these results, it seems inescapable to conclude that mean-field behavior does not remain valid asymptotically close to the critical point. Rather, ionic systems seem to show Ising-to-mean-field crossover. Such a crossover has been a recurring result observed near liquid-liquid consolute points in Coulombic electrolyte solutions, in ternary aqueous electrolyte solutions containing an organic cosolvent, and in binary aqueous solutions of NaCl near the liquid-vapor critical line. 

Kleemeier, M., Wiegand, S., Derr, T., Weiss, V., Schroer, W., and Weingartner, H. Critical viscosity and Ising-to-mean-field crossover near the upper consolute point of an ionic solution. Ber. Bunsenges. Phys. Chem., 1996, 100, p. 27-32. 

Monte Carlo simulations very early demonstrated the effect of thermal composition fluctuations in low molecular blends. Studies by Sariban et al. [16] exclusively found Ising critical behavior in blends of molar volume up to about 16000 cm /mol and no indications of a crossover to mean field behavior. Such a mean field crossover was later detected by Deutsch et aL [ 17] in blends with an order of magnitude larger chains. These results and the techniques of Monte Carlo simiflations have been extensively reviewed by Binder in [4]. 

Jacob J, Kumar A, Anisimov M A, Povodyrev A A. and Sengers J V 1998 Crossover from Ising to mean-field critical behavior in an aqueous electrolyte solution Phys. Rev. E 58 2188... 

There are other scenarios for an apparent mean-field criticality [15, 17]. The most likely one is crossover from asymptotic Ising behavior to mean-field behavior far from the critical point, where the critical fluctuations must vanish. For the vicinity of the critical point, Wegner [43] worked out an expansion for nonasymptotic corrections to scaling of the general form... 

Clearly, the simultaneous presence of crossover from Ising to mean-field criticality, a transition from two-component behavior of solutions to one-component behavior, and the possible presence of Fisher renormalization renders any analysis difficult. 

We recall that comparatively sharp and even nonmonotonous crossover from Ising to mean-field behavior has been deduced from experiments for a diversity of ionic systems. We note that this unusually sharp crossover is a striking feature of some other complex systems as well we quote, for example, solutions of polymers in low-molecular-weight solvents [307], polymer blends [308-311], and microemulsion systems [312], Apart from the fact that application of the Ginzburg criterion to ionic fluids yields no particularly... 

Crossover. Generally, crossover from an Ising-like asymptotic behavior to mean-field behavior further away from the critical point [86, 87] may be expected. Such a behavior is also expected for nonionic fluids, but occurs so far away, that conditions close to mean-field behavior are never reached. Reports about crossover [88] and the finding of mean field criticality [14—16] suggest that in ionic systems the temperature distance of the crossover regime from the... 

Critical Behavior and the Crossover from Ising to Mean-Field... 

Note that all these formulas also contain the result for the limiting case of short chains dynamics described by the Rouse model [139,140] if we formally put Ne N in these equations. As will be discussed later (Sect. 2.5), there occurs a crossover in the static critical behavior from mean-field-like behavior where ocR e-1/2 with e= 1 — x/X rit> Scon(0)ccN e to the nonclassical critical behavior with Ising model [73, 74] critical exponents cce-v, S, ii(0) oceT, vw0.63, 1.24. This crossover occurs, as predicted by the Ginzburg... 

A different way of testing the crossover from Ising to mean field behavior is based on the use of Eq. (123), see Fig. 30. For small N distinct deviations from... 

The best evidence for the crossover from Ising to mean field theory comes from an analysis of crossover phenomena in the context of finite size scaling [92, 275]. For a detailed analysis of this problem we refer to the literature, but rather mention here only that the critical value of the order parameter in a finite box at Tc is predicted to vary as [275]... 

There are still two caveats that must be mentioned, however all measurements refer to an analysis of the collective scattering function ScoM(q) in the one phase region of the blends, thus the precise value of Tc has to be treated as a fitting parameter and does not result from an independent measurement. Secondly, there are problems with understanding the temperature where the crossover from mean field behavior to Ising-like critical behavior occurs, as already discussed in the last section. 

Some compounds exhibit a spin crossover of the form that the fraction xHs of molecules in the HS state increases with temperature in two steps a plateau of a few K exists between these steps. This behaviour can be explained by considering two sublattices (A and B) containing the same number of molecules [26], The Ising-like Hamiltonians corresponding to the respective lattices, in the mean field approach, are defined as follows... 

These data are replotted in a different form in Figure 12, on the assumption that the order parameter (the coexistence density gap) for the LJ system should behave in an Ising-like manner. This is reflected in the nearly straight-line behavior of much of the data very close to the critical points the data deviate from linearity, becoming mean-field-like because of the limitation on fluctuations in a finite system. The precision of the results puts us in position to study this finite-size crossover and also other nonuniversal properties of the critical behavior of fluid phase transitions. 

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