Ising criticality

It should be emphasized that the comparatively large change obtained in more recent work is mainly caused by the application of finite-size scaling. Under these circumstances, one certainly needs to reconsider how far the results of analytical theories, which are basically mean-field theories, should be compared with data that encompass long-range fluctuations. For the van der Waals fluid the mean-field and Ising critical temperatures differ markedly [249]. In fact, an overestimate of Tc is expected for theories that neglect nonclassical critical fluctuations. Because of the asymmetry of the coexistence curve this overestimate may be correlated with a substantial underestimate of the critical density. 

Small exponents. Evidence for Ising criticality can be provided by some properties showing weak divergences, which are absent in the mean-field case. One such case is the specific heat, which diverges with the exponent a. Kaatze and coworkers [112] have indeed shown the presence of such an a anomaly in EtNH3N03 + //-octanol, but as already mentioned, this system shows an anomalous location of the critical point, indicating that non-Coulomb interactions play a considerable role in driving the phase separation. 

Figure 4. Phase diagrams for the AHS fluid with a = 6 (top), a = 4 (center), and a = 3.1 (bottom) coexistence densities from GEMC simulations (open circles) critical parameters from MFFSS analyses assuming Ising criticality (filled circles) fits of the form ft] p A(T — T ) B T — T ) eff (solid fines). For a = 6 the effective order parameter exponent is /left = 0.36, for a = 4 / eff = 0.46, and for a = 3.1 /Jeff = 0.46.

The results of MFFSS analysis and the measurements of Cy in NVT MC simulations are consistent with Ising criticality [37, 38] measurements of Cy in pVT MC simulations are in qualitative agreement, inasmuch as a near-critical peak has been observed [34], but the finite-size scaling has not yet been examined. 

Mixed-field finite-size scaling studies of the CHD fluid - in which cation-anion pairs are fused together - are consistent with Ising criticality. When calculated in NVT MC simulations, Cy along the critical isochore shows no sign of a near-critical peak. 

Mixed-field finite-size scaling studies of the AHS fluid with a = 6 are consistent with Ising criticality. Finite-size scaling analyses of the peak position and peak height in Cy obtained from NVT MC simulations are also consistent with Ising criticality. 

A. Coniglio and W. Klein (1980) Clusters and Ising critical droplets a renormalisation group approach. J. Phys. A 13, pp. 2775-2780... 

Sariban et al. [101, 107, 276, 277] were the first to emphasize that the Ising critical behavior can be seen in polymer mixtures for not too long chains and verified it by their simulations. A consequence of Ising behavior that is easily verified by experiment is that the spinodal temperature (or mean field critical temperature T F, respectively) which is defined for 4>A = < >Acrit from a linear extrapolation of the inverse scattering intensity S fq = 0) with temperature to the point where S, n(q = 0) = 0 must be offset from the actual critical temperature Tc (Fig. 32). This phenomenon has been seen in simulations [92,101,107] as well as in various experiments [69, 71, 215, 216, 278], A detailed analysis of the non-mean field critical behavior has allowed the estimation of critical exponents y = 1.26 0.01 [215-217,69], v = 0.59 + 0.01 [215] or v as 0.63 [71], and also the exponent describing the decay of correlations at Tc has been estimated [215], 0.047 0.004. These numbers are in fair agreement with... 

Schwahn, D., Mortensen, K., and Madeira, H. Y. (1987) Mean-field Uid Ising criticed behavior of a polymer blend, Phys. Rev. Lett. 58, 1544-1546. 

A field-theory based on this simple expansion will already yield three-dimensional Ising critical behavior. Note that the non-trivial critical behavior is related to fluctuations vaW or a- b- Fluctuations in the incompressibility field U or the total density a + are not important. 

Both the statics and the collective dynamics of composition fluctuations can be described by these methods, and one can expect these schemes to capture the essential features of fluctuation effects of the field theoretical model for dense polymer blends. The pronounced effects of composition fluctuations have been illustrated by studying the formation of a microemulsion [80]. Other situations where composition fluctuations are very important and where we expect that these methods can make straightforward contributions to our understanding are, e.g., critical points of the demixing in a polymer blend, where one observes a crossover from mean field to Ising critical behavior [51,52], or random copolymers, where a fluctuation-induced microemulsion is observed [65] instead of macrophase separation which is predicted by mean-field theory [64]. 

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. 

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