Ising-like model

It should also be noted that the LIESST phenomenon has been recently observed on these materials [53-55]. This discovery may lead to a new wave of photomagnetic investigations of these bistable materials in view of potential applications. The shape of the relaxation curves after LIESST could be modelled within the framework of a revised ID Ising like model [56]. 

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. 

Although the asymptotic critical regime with the Ising-like scaling exponents has been neglected in this description, the fit curves in Fig. 8 are a reasonable parameterization for all three coefficients in the one-phase regime. This parameterization then serves as input for the numerical model. A more detailed discussion of the whole procedure can be found in [100],... 

Before presenting the measurements, we summarize a simplified spin Hamiltonian describing the [Mn4]2 dimer [48], Each Mn4 SMM can be modeled as a giant spin of S -9/2 with Ising-like anisotropy [Eq. (1)]. The corresponding Hamiltonian is given by... 

T. Tome and R. Dickman, Ziff-Gulari-Barshad model with CO desorption an Ising-like nonequilibrium critical point, Phys. Rev. E, 47 (1993) 948. 

Zunger A, Wang LG, Hart GLW, Sanati M (2002) Obtaining Ising-like expansions for binary alloys from first-principles. Modell Simul Mater Sci Eng 10 685... 

Kawamoto T, Abe S. Thermal hysteresis loop of the spin-state in nanoparticles of transition metal complexes Monte Carlo simulations on an Ising-like model. Chemical Communications. 2005 No. 31, 3933-3935. DOI 10.1039/b506643c. 

Atitoaie A, Tanasa R, Enachescu C. Size dependent thermal hysteresis in spin crossover nanoparticles reflected within a Monte-Carlo based Ising-like model. Journal of Magnetism and Magnetic Materials. 2012 324 1596-1600. DOI 10.1016/j.jmmm. 2011.12.011. 

A quantitative comparison between the mean field prediction and the Monte Carlo results is presented in Fig. 15. The main panel plots the inverse scattering intensity vs. xN. At small incompatibility, the simulation data are compatible with a linear prediction (cf. (48)). From the slope, it is possible to estimate the relation between the Flory-Huggins parameter, x, and the depth of the square well potential, e, in the simulations of the bond fluctuation model. As one approaches the critical point of the mixture, deviations between the predictions of the mean field theory and the simulations become apparent the theory cannot capture the strong universal (3D Ising-like) composition fluctuations at the critical point [64,79,80] and it underestimates the incompatibility necessary to bring about phase separation. If we fitted the behavior of composition fluctuations at criticality to the mean field prediction, we would obtain a quite different estimate for the Flory-Huggins parameter. 

The label a is a generalized index to indicate the energy of a particular excitation associated with a given configuration. We note that the present discussion emphasizes the local atomic-level relaxations and how they can be handled within the context of the Ising-like model introduced earlier. However, there are additional effects due to long-range strain fields that require further care (they are usually handled in reciprocal space) which are described in Ozoli s et al. (1998). 

Table 3 Parameters of the Ising-like binuclear model, treated in the mean-field approximation.

Ising-like model with molecular vibrations... 

Within the Ising-like model the Hamiltonian of a dinuclear system formed of a symmetric pair of metallic centres is... 

Figure 2. Finite-size effects in the vapor-liquid coexistence curve of -octane (TraPPE model) obtained from simulations in the Gibbs ensemble [89]. Open circles and crosses depict results for simulations with N = 200 and 1600, respectively. The upper sets of points use a mean-field exponent (j3 = 0.5) and the lowers ones use an Ising-like exponent (fi — 0.32). The estimated critical temperatures for the Ising-like exponent are also shown.

In the AOT-water-oil system, when the temperature is increased sufficiently a cloud point (critical point) is reached. At temperatures well below this transition temperature the viscosity data are in agreement with a simple hard-sphere model. Upon approaching the critical point one first notices a relatively moderate increase of viscosity by about a factor of 4 followed by a critical divergence of viscosity very close to the cloud point. The critical divergence of such a system (with decane as oil) at the critical temperature was studied and was shown to scale almost Ising-like according to rj [ Tc - T)/Tc] with a critical exponent of 0.03 [69]. 

Upon increasing distance from the critical point, this crossover model provides a continuous transformation from Ising-like behavior asymptotically close to the critical point to mean-field behavior far away from the critical point. Due to the critical fluctuations, the position of the actual critical temperature is shifted with respect to the mean-field critical temperature Tc. The critical temperature shift Tg = (Tc — Tc)/Tc can be estimated from different properties such as the inverse susceptibility or the order parameter. These different estimates of Tg are all proportional to a unique combination of the crossover parameters (rg uA /ct). The transformation is generally controlled by two physical parameters the coupling constant u and the ratio A/k or, equivalently, by the ratio of the correlation length over the microscopic characteristic length d-... 

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... 

Qualitatively, a sharp crossover to mean-field behavior has been reported earlier for metal-ammonia solutions [50, 51]. Crossover between Ising-like asymptotic behavior and mean-field classical behavior has also been reported for polymer blends [42-46] and for a microemulsion system [52]. Solutions of polymers in low-molecular-weight solvents exhibit sharp nonmonotonic crossover behavior when the correlation length of the critical fluctuations and the polymer molecular size, as specified by the radius of gyration, are of the same order [24]. Hence, a description of this crossover phenomenon requires two independent parameters associated, respectively, with intramolecular and intermolecular correlations [53]. It has been demonstrated that the two-term crossover Landau model is indeed... 

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