Ising universality class

Fig. 10.8. The ordering operator distribution for the three-dimensional Ising universality class (continuous line - data are courtesy of N.B. Wilding). Points are for a homopolymer of chain length r = 200 on a 50 x 50 x 50 simple cubic lattice of coordination number z = 26 [48], The nonuniversal constant A and the critical value of the ordering operator Mc were chosen so that the data have zero mean and unit variance. Reprinted by permission from [6], 2000 IOP Publishing Ltd...

Figure 10. The distribution of the density of an LJ fluid at its critical point showing the collapse (given a suitable choice of scale) onto a form characteristic of the Ising universality class. (Taken from Fig. 3a of Ref. 44.)...

On the basis of this brief summary of RPM criticality, one might be tempted to conclude that the problem has been solved all finite-size scaling analysis point towards the Ising universality class. There is, however, one critical phenomenon which does not seem to have been demonstrated unambiguously in the RPM. This is the critical divergence of the constant-volume heat capacity, Cy. Recall that on the critical isochore and close to the critical temperature where the parameter t = (T — Tc)/Tc is small,... 

Up to about 35 K, and the highest possible compression is about 1.4 mono-layers. The orientational disordering transition of the pin wheel phase to the incommensurate orientationally disordered solid (PW -> ID) at about 30-35 K belongs to the two-dimensional Ising universality class [113]. 

Figure 59. Fourth-order cumulants(3.22)forthe head-tail order parameter (5.8) obtained from Monte Carlo simulations of complete monolayer ( /3 x -J3)R30° CO on graphite as a function of temperature (5 = 0.13 A). The dashed line marks the trivial value in the ordered phase, the dotted line marks the universal value U for the two-dimensional Ising universality class [47, 53], and = 11.9 K is determined from the intersection point. Symbols for the linear dimension of the L x L system L = 18 (asterisks), 24 (diamonds), 30 (triangles), 42 (squares), and 60 (circles). (Adapted from Fig. 2 of Ref. 215.)...

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 respectively) which is defined for 4) = 4>Acru from a linear extrapolation of the inverse scattering intensity S iiUq = 0) with temperature to the point where S n(q = 0) = 0 must be offset from the actual critical temperature Te (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] orv 0.63 [71], and also the exponent describing the decay of correlations at T has been estimated [215], t] 0.047 0.004. These numbers are in fair agreement with the best numerical values proposed for the Ising universality class [274], y = 1.24, V = 0.63 and q = 0.039, respectively. 

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. 

The critical behavior of density fluctuations in microemulsions with a droplet structure can be treated analogously to simple fluids, because the radius is virtually constant throughout the phase separation and the droplet density may be regarded as an order parameter. Because of the nature of the droplet systems, its critical behavior is expected to belong to the 3D-Ising universality class. However, the observed critical exponents do not always coincide with the exact values of the 3D-Ising model. In particular, the well-known ternary system (WDA), consisting of an oil-rich mixture of water, n-decane, and AOT (dioctyl sulfosucdnate sodium salt) has been the subject of... 

The new SmA2-SmA(j phase boundary terminates at a critical point C where the jump in wavevector goes to zero. Although the fluctuating parameter is a scalar (layer thickness) it will be shown in this chapter, Sec. 6.1.7.4 that the new critical point C is not expected to belong to the Ising universality class [85]. 

From a statistical mechanics viewpoint, molecular systems share several properties with Ising spin systems if density fluctuations are substituted with spin flips. Polymer melts and blends without long-range interactions typically belong to the 3d-Ising universality class [92, 93], and critical points can be determined with techniques that were originally derived for spin systems [94]. To this extent we can calculate second- or fourth-order cumulants [94] ... 

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