Fluctuations of density

Nucleation in a pure liquid. According to the kinetic theory for pure gases and liquids, there are local fluctuations of densities, which are clusters of molecules in a gas and holes (or vapor clusters) in a liquid. Frenkel (1955) established the population distribution of such holes of phase B in a liquid of continuum phase A by Boltzmann s formula,... 

The depolarization of light by dense systems of spherical atoms or molecules has been known as an experimental fact for a long time. It is, however, discordant with Smoluchowski s and Einstein s celebrated theories of light scattering which were formulated in the early years of this century. These theories consider the effects of fluctuation of density and other thermodynamic variables [371, 144]. 

In the amorphous pattern we observe a spatial fluctuation of density as well as different repulsive and attractive forces. Describing the amorphous structure with (n — m)/n = x and using the 6—12-potential of Eq. (5) we obtain an internal shearing stress... 

These regions provoke anomalous fluctuations of density and entropy, from which, at least at a qualitative level, an explanation can be derived for the increase of the isothermal compressibility and specific heat in the metastable region. 

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

The most elegant and clear exposition of the ideas of Ornstein and Zernike is that of Klein and Tisza. The ideas of Ornstein and Zernike will be understood to mean generally the assumption of a linear direct (or short-range) coupling between fluctuations of density in different volume elements, in an expression for the free energy in an inhomogeneous system. 

Porod predicted that, for an ideal lamellar system of two phases (Fig. 19.5) in which neither fluctuations of density within phases nor interfacial thickness of finite wide are present, the intensity of dispersion diminishes proportionally to the reciprocal of the fourth power of q, which is mathematically expressed as... 

The attempts to linearise the dependence (1 - a) on t for system EPS-l/DDM with the use of Equations (10.8) and (10.9) have not resulted in success. It means that the pointed nonhomogeneous reaction proceeds in the reaction medium with large fluctuations of density, but it is not described by the Equations (10.8) and (10.9) for low-molecular substances. Thus, the assumption about the connection of curing reaction type with a level of fluctuation of density of reaction medium is confirmed. 

The neglect of small-scale fluctuations of density except when multipUed by gravity (the so-called Boussinesq approximation). 

Figure 1.15. State diagram of a gas-liquid system the projection of the surface f P,g,T) = 0 on the plane P = 0. Temperature dependences of the correlation length of fluctuations of density T > T ) and of an order parameter analogue Ap = pi — Pg (T < 7 c) are described by power functions...

If we perform a scattering experiment (using X-rays or neutrons) on a dense chain system that is made of identical monomers, we obtain some information on the local fluctuations of density, or concentration. If we look for universal properties, we must probe the fluctuations at wavelengths much larger than Ae monomer size. (More precisely, if q is the scattering wave vector, we want qa < 1.) However, at this scale, the density in a melt is essentially fixed—i.e., there is no scattering intensity. 

At the critical point, the size of fluctuations of density or concentration, diverges and the time of their relaxation becomes infinite. The slow relaxation near the critical point is known as critical slowing down and the theory that describes this phenomenon is known as dynamic scaling .The basic idea of dynamic scaling is simple a fluctuation the size of the correlation length has a lifetime proportional to the volume of the fluctuation ... 

Local fluctuations of density occur at all points even in a macroscopi-cally homogeneous fluid, but are suflidently small in magnitude and in Ae range of their correlations for them properly to be neglected in many calculations in equilibrium statistical mechanics. Fluids near critical points are obvious exceptions and, as was mentioned in 3.5, so is the fluid at the gas-liquid surface. We saw there that the van der Waals theory was justifiable only if certain constraints were imposed on the fluctuations at the interface. We now use results established earlier in this chapter to study these fluctuations in more detail. 

A simple but natural way of discussing fluctuations of density at the surface is to suppose that there is a spectrum of capillary waves imposed on an intrinsic or bate profile (denoted by a subscript b) given by... 

Pure silica was studied by Bacon et al. [22] they gave the density as a function of the temperature calculated from surface tension of molten silica. The results are reported in Table 3.4.4. These results are compared with our calculations using a NPT ensemble (Figure 3.4.3). Up to 2400 K, the calculated density follows the same trend that Bacon et al. [22] obtained. For a temperature higher than 2400 K, some fluctuations of density and differences with experimental data appear [22]. At temperatures higher than 2400 K, the difference becomes significant and can be explained by a deficiency of the BKS potential model [5]. This deficiency has already been shown by Saika-Voivod et al. [25]. However, the observed discrepancies between our calculations and the measurements of Bacon et al. [22] might be also ascribed to ... 

This system has been studied mostly for those cases where the electrons are weakly bound and when the mobility of the excess electron is high. In such cases the mobility can be rationalized by considering that the medium is microscopically inhomogeneous in the manner discussed earlier. In some regions it is localized due to the optimal fluctuations of density or molecular conformations, while in other regions it can not be localized. If those traps are very shallow and the electron density very diffuse one can use the semi-classical percolation pictures. A very naive theory was discussed by Kestner and Jortner (1973) and shows the correct trends versus V. Others such as Holroyd (1987) have analyzed this simple theory in detail and have pointed out its deficiencies. 

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