Fluctuations: critical density

Fluctuations in density and composition produce opalescence, a recognized feature of the critical region. 

As can be seen from Fig. 6, liquid-liquid demixing clearly precedes crystallization in case Cl. Moreover, crystallization in this case occurs at a higher temperature than in cases C2 and C3. Apparently, the crystallization takes place in the dense disordered phase (which has a higher melting temperature than the more dilute solution Fig. 5). In case C2, the crystallization temperature is close to the expected critical point of liquid-liquid demixing, but higher than in case C3. This suggests that even pre-critical density fluctuations enhance the rate of crystal nucleation. 

Critical Behavior of Gels. In 1977, the critical phenomena were discovered in the light scattered from an acrylamide gel in water [18]. As the temperature was lowered, both the scattered intensity and the fluctuation time of the scattered light increased and appeared to diverge at —17 °C. The phenomenon was explained as the critical density fluctuations of polymer networks although the polymers were crosslinked [19, 20]. 

In spite of the constant density of the gel, the friction of the poly(N-isopropylacrylamide) gel reversibly decreases by three orders of magnitude and appears to diminish as the gel approaches a certain temperature. This phenomenon should be universal and may be observed in any gel under optimal experimental conditions of the solvent composition and the temperature because the unique parameter describing the friction is the correlation length which tends to diverge in the vicinity of the volume phase transition point of gels. The exponent v for the correlation length obtained from the frictional experiment is far from the theoretical value. It will, therefore, be important to study a poly(N-isopropylacrylamide) gel prepared at the critical isochore where the frictional property of gel may be governed by the critical density fluctuations of the gel. 

The use of photon correlation spectroscopy to study the dynamics of concentration fluctuations in polymer solutions and gels is now well established. In bulk polymers near the glass transition there will be slowly relaxing fluctuations in density and optical anisotropy which can also be studied by this technique. In this article we review the development of the field of photon correlation spectroscopy from bulk polymers. The theory of dynamic light scattering from pure liquids is presented and applied to polymers. The important experimented considerations involved in the collection and analysis of this type of data are discussed. Most of the article focuses on the dynamics of fluctuations near the glass transition in polymers. All the published work in this area is reviewed and the results are critically discussed. The current state of the field is summarized and many suggestions for further work are presented. 

It should be emphasized that the comparatively large change obtained in more recent work is mainly caused by the application of finite-size scaling. Under these circumstances, one certainly needs to reconsider how far the results of analytical theories, which are basically mean-field theories, should be compared with data that encompass long-range fluctuations. For the van der Waals fluid the mean-field and Ising critical temperatures differ markedly [249]. In fact, an overestimate of Tc is expected for theories that neglect nonclassical critical fluctuations. Because of the asymmetry of the coexistence curve this overestimate may be correlated with a substantial underestimate of the critical density. 

Second, predictions of p are substantially improved when account is made for ion pairs. The increase of the critical density is easily understood A certain free-ion density is needed for driving criticality. If pairs are formed, this free-ion density can only be achieved at a higher overall ion density. Nevertheless, all theories yield too low values if assessed by the more recent MC data. As mentioned, one reason for low critical densities may result from comparison with MC data that encompass long-range fluctuations. It will, however, be shown in the subsequent section that all available analytical theories seem to overestimate the degree of dissociation. Such an overestimate almost invariably leads to an underestimate of the critical density. 

CO -benzene, and CO -n-decane. The critical densities and the corresponding compositions are plotted in Figure 1. The three hydrocarbons in order of higher to lower solubility in C0 were heptane, benzene, and decane. The measured binary diffusion coefficients or the decay rates of the order-parameter fluctuations at various temperatures and pressures are listed in Tables I, II, and III for CO -heptane, CO -benzene, and CO -decane systems respectively. In Figure 2, the critical lines of the three binary systems in the dilute hydrocarbon range are shown in the pressure-temperature space. dP/dT along the critical lines of CO.-heptane and CO -benzene systems are similar and lower than dP/dT along the critical line of CO -decane system, which indicates that C02 and decane form more asymmetric mixtures relative to CO with heptane or benzene. 

Figure 2 shows the transient absorption spectrum of PB in CI%H at 5.7 MPa. The pattern of the transient sp>ectrum is almost the same as those in methanol. The hole broadening occurs mostly within 0.8 ps, and the bleach is recovered with two different time constants in a similar manner as in the liquid solvents, although the recovery after 2 ps is much slower in the fluid near the critical density (about 40 ps). It is also to be noted that the bleach signal after 2 ps is narrow banded in comparison with the equilibrium absorption. This can be interpreted by the overlapping of the excited state absorption, and/or, the small inhomogeneity remained after 2 ps due to the long time density fluctuation. 

Topics not covered in this bibliography here are, for example, the hydrodynamic theory fluctuations of density, anisotropy, concentration fluid shear waves and critical phenomena. 

In this paper, some recent experimental results regarding the density fluctuations in pure SCF are used to show that the local density enhancement in dilute SCR mixtures is mainly due to the near critical fluctuations in the solvent and an explanation is suggested for the negative partial molar volnme of the solute. This conclusion was also strengthened by a discussion, presented in the following section, based on the Kirkwood—Buff (KB) theory of solution. First, the problem will be examined in the framework of the Kirkwood—Buff theory of solution. Second, nsing experimental results about the near critical fluctuations in pure SCF, it will be shown that the density enhancement in dilnte SCR mixtures is mainly caused by the near critical density fluctuations in pure SCF. 

Tsunekawa S, Ito S, More T, Ishikawa K, Li Z-Q, Kawazoe Y (2000a) Critical size and anomalous lattice expansion in nanocrystalline BaTiOs particles. Phys Rev B 62 3065-3070 Tsunekawa S, Ishikawa K, Li Z-Q, Kawazoe Y, Kasuya A (2000b) Origin of anomalous lattice expansion in oxide nanoparticles. Phys Rev Letters 85 3440-3443 Turkovic A, Ivanda M, Popovic S, Tonejc A, Gotic M, Dubcek P, Music S (1997) Comparative Raman, XRD, HREM and SAXS studies of grain sizes in nanophase Ti02. J Molec Struct 410/411 271-273 ten Wolde PR, Frenkel D (1997) Enhancement of protein crystal nucleation by critical density fluctuations. Science 277 1975-1978... 

The Kerr constant around cf> v0.1 becomes very large for ATB microemulsions and the decay curve becomes exponential.The measured decay time t is in fairly good agreement with the lifetime tg of a critical density fluctuation of size . is the correlation length of the density fluctuations previously measured in ATB microemulsion by light scattering and 3... 

Fig. 10. Scaling of the critical density with chain length. The filled triangles show our results for the bond-fluctuation model, the circles are results reported by Wilding et al. [26], the diamonds show our results for the cubic lattice model, the open triangles are results reported by Frauenkron and Grassberger [38], and the squares are results reported by Pana-giotopolous and Wong [25]. The curves are fits to our simulation results (using the functional form 0c( ) = (hi -I- The uncertainty in the critical density is comparable to the size...

Figure 9. Heat capacities near their critical temperatures for systems with different numbers of particles N. Once again the points come from canonical averaging using Eq. (2.13), the line segments join values obtained by finding numerically the second temperature derivative of the free energy. The data exhibit the expected peak near the critical density, corresponding to the enhanced fluctuations in that region, and they show the expected strong size dependence.

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