Scaling theory density fluctuations

With the intensive development of ultrafast spectroscopic methods, reaction dynamics can be investigated at the subpicosecond time scale. Femtosecond spectroscopy of liquids and solutions allows the study of sol-vent-cage effects on elementary charge-transfer processes. Recent work on ultrafast electron-transfer channels in aqueous ionic solutions is presented (electron-atom or electron-ion radical pairs, early geminate recombination, and concerted electron-proton transfer) and discussed in the framework of quantum theories on nonequilibrium electronic states. These advances permit us to understand how the statistical density fluctuations of a molecular solvent can assist or impede elementary electron-transfer processes in liquids and solutions. 

The scaled particle theory of fluids developed by Reiss, Lebowitz, Helfand and Frisch > " need concern itself [in the case of hard spheres by virtue of Eq. (26)] only with calculating g a). To accomplish this we focus our attention on a spherical cavity of radius at least r centered about a fixed point in the fluid. A cavity is defined as a region of space devoid of molecular (hard sphere) centers (see Fig. 8). Such a cavity can be formed spontaneously in our fluid as a result of a local density fluctuation. 

In the framework of the scaling theory, the corona of a spherical micelle can be envisioned [53-56] as an array of concentric spherical shells of closely packed blobs. The blob size, (r) = r/grows as a function of the radial distance r from the center of the core. Each blob comprises a segment of the chain within the local correlation length of the monomer density fluctuations [57], and corresponds to a contribution to the free energy of steric repulsion between the coronal chains. After calculating the total number of blobs in the micellar corona, one finds fhe free energy (per coronal chain) as ... 

Modem scaling theory is also a powerful theoretical tool (applicable to liquid crystals, magnets, etc.) that has been well established for several decades and has proven to be particularly useful for multiphase microemulsion systems (46). Scaling theory relies on the hypothesis that diverse physical systems exhibit large compositional and density fluctuations and essentially behave the same near their critical points. Hence, the only factors that determine their critical properties are the dimensionality of the space and dimensionality of the order parameter. For example, the shape of the critical scaling theory. The temperature dependence is given by... 

The above scaling laws are valid in semidilute solutions where density fluctuations are dominant. What is the crossover behavior between the semidilute solutions and the dense solutions where the Flory-Huggins theory is adequate In addition, a knowledge of the various numerical prefactors is lacking in the scaling results. This problem has yet to be solved convincingly but we offer some background here. 

When an atomic system is cooled below its glass temperature, it vitrifies, that is, it forms an amorphous solid [1]. Upon decreasing the temperature, the viscosity of the fluid increases dramatically, as well as the time scale for structural relaxation, until the solid forms concomitantly, the diffusion coefficient vanishes. This process is observed in atomic or molecular systems and is widely used in material processing. Several theories have been developed to rationalize this behavior, in particular, the mode coupling theory (MCT) that describes the fluid-to-glass transition kinetically, as the arrest of the local dynamics of particles. This becomes manifest in (metastable) nondecaying amplitudes in the correlation functions of density fluctuations, which are due to a feedback mechanism that has been called cage effect [2],... 

Ya.B. s theory explains the appearance of the largest-scale inhomogeneities from initially small fluctuations in the original velocity and density field. 

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. 

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