Critical phenomena point

Molecular systems are challenging from the critical phenomenon point of view. In this section we review the finite-size scaling calculations to obtain critical parameters for simple molecular systems. As an example, we give detailed calculations for the critical parameters for Hj-like molecules without making use of the Born-Oppenheimer approximation. The system exhibits a critical point and dissociates through a first-order phase transition [11],... 

Point c is a critical point known as the upper critical end point (UCEP).y The temperature, Tc, where this occurs is known as the upper critical solution temperature (UCST) and the composition as the critical solution mole fraction, JC2,C- The phenomenon that occurs at the UCEP is in many ways similar to that which happens at the (liquid + vapor) critical point of a pure substance. For example, at a temperature just above Tc. critical opalescence occurs, and at point c, the coefficient of expansion, compressibility, and heat capacity become infinite. 

The relation with the microscopic structure is pointed up. An analysis of the sol-gel transition in terms of a critical phenomenon is proposed. 

There are some similarities between third-phase formation in liquid/liquid extraction and the critical phenomenon of cloud points in aqueous solutions of nonionic polyethoxylated surfactants (12, 91). When a nonionic micellar solution is heated to a certain temperature, it becomes turbid, and by further increasing the temperature,... 

In 1958, Pitzer (141), in a remarkable contribution that appears to have been the first theoretical consideration of this phenomenon, likened the liquid-liquid phase separation in metal-ammonia solutions to the vapor-liquid condensation that accompanies the cooling of a nonideal alkali metal vapor in the gas phase. Thus, in sodium-ammonia solutions below 231 K we would have a phase separation into an insulating vapor (corresponding to matrix-bound, localized excess electrons) and a metallic (matrix-bound) liquid metal. This suggestion of a "matrix-bound analog of the critical liquid-vapor separation in pure metals preceeded almost all of the experimental investigations (41, 77, 91,92) into dense, metallic vapors formed by an expansion of the metallic liquid up to supercritical conditions. It was also in advance of the possible fundamental connection between this type of critical phenomenon and the NM-M transition, as pointed out by Mott (125) and Krumhansl (112) in the early 1960s. 

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,... 

The same data that were plotted in the heat flux/quality form can also be plotted in terms of xcrit against boiling length (Ls)cnl, where boiling length is the length between the point where x = 0 and the point where a critical phenomenon occurs (usually at the end of the channel for uniform heat flux). 

The intensity of light scattered from a fluid system increases enormously, and the fluid takes on a cloudy or opalescent appearance as the gas-liquid critical point is approached. In binary solutions the same phenomenon is observed as the critical consolute point is approached. This phenomenon is called critical opalescence.31 It is due to the long-range spatial correlations that exist between molecules in the vicinity of critical points. In this section we explore the underlying physical mechanism for this phenomenon in one-component fluids. The extension to binary or ternary solutions is not presented but some references are given. 

Family [9] considered the conformations of statistical branched fractals (which simulate branched polymers) formed in equilibrium processes in terms of the Flory theory. Using this approach, he found only three different states of statistical fractals, which were called uncoiled, compensated, and collapsed states. In particular, it was found that in thermally induced phase transitions, clusters occur in the compensated state and have nearly equal fractal dimensions ( 2.5). Recall that the value df = 2.5 in polymers corresponds to the gelation point this allows gelation to be classified as a critical phenomenon. 

In order to clarify the relation between the phase behavior, interactions between droplets, and the Ginzburg number, we have undertaken further SANS studies of critical phenomenon in a different three-component microemulsion system called WBB, consisting of water, benzene, and BHDC (benzyldimethyl-n-hexadecyl ammonium chloride). This system also has a water-in-oil-type droplet structure at room temperature and decomposes with decreasing temperature. Above the (UCST) phase separation point, critical phenomena have been investigated by Beysens and coworkers [9,10], who obtained the critical indexes, 7 = 1.18 and v = 0.60, and concluded that their data could be interpreted within the 3D-Ising universality. However, Fisher s renormalized critical exponents were not obtained. 

The now popular concept of percolation has proved quite successful in many fields in which a macroscopic property depends on the existence of a connected path within a two-phase discrete medium, most often a regular 2D or 3D arrangement of sites (site lattice percolation). Typical features related to percolation are the existence of a critical phenomenon, for instance, a threshold concentration of conducting sites when conduction is considered, and of a power law dependence with respect to the critical quantity in the close vicinity of the critical point. Both the site percolation thresholds and the power exponents have well-established theoretical values for any given lattice geometry and coimection rules [193],... 

Although it was not mentioned in this section, the gel point is regarded as an application example of a critical phenomenon or an example of a phase transition [29-31], In these studies, it was expected that a universal exponential law exists between the viscous and elastic properties of gels and the equivalent quantity that describes the deviation from the gel point. However, this exponential law is not well understood from the molecular structural point of view. The gel point may appear to be an especially simple property among various properties of gels. However, there are many unanswered problems and future progress in this area is desired. 

The difference between the FS model and percolation model is in the critical phenomenon. As summarized in Table 1, if the statistical values are normalized by the equivalent distance e(= 1 — a/a ) from the gel point (the critical point), there is a significant difference in critical index for flie FS model and percolation model. This difference reflects the difference in size distribution (see Fig. 1 [6]). The difference of the structure in flie model is reflected on the fractal dimension D of the fraction that has a certain degree of polymerization x. If the radius of a sphere that corresponds to the volume of the branched polymer fiaction with the degree of polymerization x is R, the relationship between x and R is fimm the fiactal dimension D... 

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