Ising criticality fluid models

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

According to RG theory [11, 19, 20], universality rests on the spatial dimensionality D of the systems, the dimensionality n of the order parameter (here n = 1), and the short-range nature of the interaction potential 0(r). In D = 3, short-range means that 0(r) decays as r p with p>D + 2 — tj = 4.97 [21], where rj = 0.033 is the exponent of the correlation function g(r) of the critical fluctuations [22] (cf. Table I). Then, the critical exponents map onto those of the Ising spin-1/2 model, which are known from RG calculations [23], series expansions [11, 12, 24] and simulations [25, 26]. For insulating fluids with a leading term of liquid metals [27-29] the experimental verification of Ising-like criticality is unquestionable. 

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

In view of these observations, one would like to establish the Ising-like nature of the critical point by an RG treatment. Unfortunately, lattice models, as successfully applied to describe the criticality of nonionic fluids, may be of little help in this regard, because predictions for the Coulomb gas have proved to be surprisingly different from those for the continuum RPM. Discretization effects—and, more generally, the relevance of the results of lattice models with respect to the fluid—still need to be explored in detail. On the other hand, an RG treatment of the RPM or UPM is still lacking and, as Fisher [278] notes, the way ahead remains misty. 

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. 

Symmetry.—The Ising model, like the phenomenon of ferromagnetism which it was designed to simulate, has as an essential feature an exact symmetry. All the thermodynamic functions are symmetric or antisymmetric with respect to an axis of zero magnetic field. Transcription of this model to that of a one-component fluid leads to the highly artificial lattice-gas in which the critical density is exactly half the density po of the close-packed lattice. 

In Table 2, typical values of the correlation length, and of the ratio of the compressibility to that of an ideal gas at the same density, are shown for carbon dioxide and water. Parameters and asymptotic power laws are those from Ref. [ 10. The correlation length is to be compared with a molecular dimension, which is 0.15 nm for carbon dioxide, and 0.13 nm for water, of the order of the average molecular radius. The second and third columns show how large the ratio becomes in fluids and the Ising model as the critical point is approached. In two dimensions, fluctuations are even more pronounced. 

Therefore, only two amplitudes are independent. It has been established theoretically [1, 5] and verified experimentally [6, 7] that all fluids and fluid mixtures, regardless of variety and complexity in their microscopic structure, belong to the same universality class, i.e. they have the same universal values of the critical exponents (Table 2) and of the critical-amplitude ratios (Table 1) as those of the 3-dimensional Ising model. The physical reason of the critical-point universality originates from the divergence of the order-parameter fluctuations near the critical point. 

The two-term crossover Landau model has been successfully applied to the description of the near-critical thermodynamic properties of various systems, that are physically very different the 3-dimensional lattice gas (Ising model) [25], one-component fluids near the vapor-liquid critical point [3, 20], binary liquid mixtures near the consolute point [20, 26], aqueous and nonaqueous ionic solutions [20, 27, 28], and polymer solutions [24]. 

The lattice gas (Ising model), the simplest model that describes condensation of fluids, has played an important role in the theory of critical phenomena [1] providing crucial tests for most basic theoretical concepts. Recently, accurate numerical results for the crossover from asymptotic (Ising-like) critical behavior to classical (mean-field) behavior have been reported both for two-dimensional [29, 30] and three-dimensional [31] Ising lattices in zero field with a variety of interaction ranges. The Ginzburg number, as defined by Eq. (36), depends on the normalized interaction range R = as... 

Povodyreveta/. (1997) have developedasix-term Landau expansion crossover scaling model to describe the thermodynamic properties of near-critical binary mixtures, based on the same model for pure fluids and the isomorphism principle of the critical phenomena. The model describes densities and concentrations at vapor-liquid equflibrium and isochoric heat capacities in the one-phase region. The description shows crossover from asymptotic Ising-hke critical behavior to classical (mean-field) behavior. This model was applied to aqueous solutions of sodium chloride. 

Fisher and Wortis have shown that Tohnan s length is zero for symmetric fluid coexistence and non-zero for asymmetric fluid coexistence. " Symmetric fluids are represented by the lattice-gas (Ising) model in which the shape of the coexistence curve is perfectly symmetric with respect to the critical isochore. Real fluids always possess some degree of asymmetryAsymmetry in the vapour-liquid coexistence in helium, especially in He, is very small, but not zero. In the mean-field approximation, the asymmetry in the vapour-liquid coexistence is represented by the rectilinear diameter ... 

The simplest theoretical prototype of the critical vapour-liquid transition is the lattice gas which is a reformulation of the 3D Ising model in terms of fluid variables. For the lattice gas the scaling fields are ... 

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