3d-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... 

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] ... 

Certainly the most important models for the development of modem scaling theory of critical phenomena have been the discrete Ising model of ferromagnetism and its antipode - the continuum van der Waals model of fluid. The widespread belief is that real fluids and the lattice-gas 3D-model belong to the same universality class but the absence of any particle-hole-type symmetry in fluids requires the revised scaling EOS. The mixed variables were introduced to modify the original Widom EOS and account the possible singularity of the rectilinear diameter. 

Phase transitions of confined fluids were extensively studied by various theoretical approaches and by computer simulations (see Refs. [28, 278] for review). The modification of the fluid phase diagrams in confinement was extensively studied theoretically for two main classes of porous media single pores (stit-Uke and cylindrical) and disordered porous systems. In a slit-like pore, there are true phase transitions that assume coexistence of infinite phases. Accordingly, the liquid-vapor critical point is a true critical point, which belongs to the universality class of 2D Ising model. Asymptotically close to the pore critical point, the coexistence curve in slit pore is characterized by the critical exponent of the order parameter = 0.125. The crossover from 3D critical behavior at low temperature to the 2D critical behavior near the critical point occurs when the 3D correlation length becomes comparable with the pore width i/p. 

Thermal fluctuations can be described within the Gaussian approximation at sufficiently high temperatures above the critical temperature. For these situations, the system fulfills the conditions of mean field approximation [9]. On the other hand, thermal composition fluctuations become strong near the critical temperature, leading to non-Hnear effects which asymptotically close to the critical temperature imply that the system obeys the universality class of 3D-Ising critical behavior. Thermal fluctuations are described by the Ginzburg-Landau Hamiltonian which is written as a fimctional of the spatially varying order parameter

[Pg.21]

Figure 29 Crossover from 3D Ising to isotropic Lifshitz class of critical universality. From Pipich, V. Schwahn, D. Willner, L. Phys. Rev. Lett. 2005, 94,117801 J. Chem. Phys. 2005, 123,124904-124916 ...

---