Crossover behavior, phase transitions

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

The results substantiate earlier observations for the liquid-liquid phase transition of Na + NH3. This system shows a transition to metallic states in concentrated solutions but in dilute solutions and near criticality, ionic states prevail [98], and the gross phase behavior seems to be in accordance with a Coulombic transition [37]. Crossover was found at f — 10-2 [46], and it seems to be much more abrupt than in the picrate systems. However, much depends on the subtle details of the data evaluation. Das and Greer [99] could smoothly represent the data by a Wegner series. 

Phase separations could also be observed for Bu4NPic in alkanediols and glycerol, but the latter solvents possess a high cohesive energy density, and the phase transition seems to be of solvophobic nature [72]. In fact, Narayanan and Pitzer [108-110] showed that addition of 1,4-butanediol to 1-dodecanol shifts the crossover region away from Tc. For pure 1,4-butanediol, plain Ising behavior is observed. 

Figure 21. The characteristic relaxation times for Process B. On the figure are marked the three phase transitions and the critical crossover temperature, T — 354 K, between Ahrrenius and VFT behaviors. The circles are the experimental data and the lines are the fitting functions Ahrrenius, VFT, and Saddle-like [78,179]. (Reproduced with permission from Ref. 257. Copyright 2005, Elsevier Science B.V.)...

Some complexes show a strong interdependence between crystal structure and spin-transition features. In the series of compounds [Fe(Rtz)6](BF4)2 (Rtz = 1-alkyltetrazole) the spin crossover behavior varies with the substituent R and is strongly influenced by cooperative effects. For example, the propyl derivative shows a quantitative spin transition, which is accompanied by a first-order crystallographic phase transition in the methyl and ethyl derivatives the Fe11 complexes occupy two nonequivalent lattice sites, only one of which shows a thermal spin transition.29... 

These data are replotted in a different form in Figure 12, on the assumption that the order parameter (the coexistence density gap) for the LJ system should behave in an Ising-like manner. This is reflected in the nearly straight-line behavior of much of the data very close to the critical points the data deviate from linearity, becoming mean-field-like because of the limitation on fluctuations in a finite system. The precision of the results puts us in position to study this finite-size crossover and also other nonuniversal properties of the critical behavior of fluid phase transitions. 

P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley, Phase behavior of metastable water. Nature, 360 (1992), 324—328 L. Xu, P. Kumar, S. V. Buldyrev, et ah. Relation between the Widom line and the dynamic crossover in systems with a liquid-liquid phase transition. Proc. Natl. Acad. Sci. USA, 102 (2005), 16558-16562. 

The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

The crossover from BCS to BEC behavior has recently attracted a great deal of interest, in particular with respect to the nature of superfluid pairing, transition temperature, and elementary excitations. This type of crossover has been earlier discussed in the literature in the context of superconductivity [34-37] and in relation to superfluidity in two-dimensional films of He [38,39]. The idea of resonant coupling through a Feshbach resonance for achieving a superfluid phase transition in ultracold two-component Fermi gases has been proposed in Refs. [40] and [41], and for the two-dimensional case it has been discussed in Ref. [42]. 

Singh, S. K., Singh, J. K., Kwak, S. K., and Deo, G. 2010b. Chem. Phys. Lett. Phase transition and crossover behavior of coUoidal fluids under confinement. 494 182. 

In water-in-oil droplet microemulsion systems undergoing a phase separation, we observed the crossover behavior of Belyakov and Kiselev and could determine the Ginzburg temperature Tq which is smaller in LCST case than and equal in UCST case to that of gas-liquid-phase transitions of low molecular weight as expected. 

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. 

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