Concentration fluctuation function

FIGURE 7.10 KBIs for the mixture model X = 4/5 as a function of the weak-water mole fraction. Full line is for G w. dotted for G , and dashed for G ( W stands for SPC/E water and w for weak water). Dots are for data obtained from integrating of the RDFs and the lines from the KB theory. The inset shows the concentration fluctuation function D x) for a simple model (dotted) and correct version (full line). 

According to Eq. 2.71, the concentration fluctuation function, R q), can be linearized by plotting R(q)lq versus q. The linearity provides evidence of the SD mechanism independently of the scale of the phase separation. 

It follows from the last two equations that away from the spinodal, Rk.so is small and increases as the configurative point approaches the spinodal. Lisnyanski (1961, 1966), Lisnyajiski and Vuks (1962, 1964, 1969), Vuks (1977) have proposed to characterize the level of concentration fluctuations by a concentration fluctuation function / ... 

A model must be introduced to simulate fast chemical reactions, for example, flamelet, or turbulent mixer model (TMM), presumed mapping. Rodney Eox describes many proposed models in his book [23]. Many of these use a probability density function to describe the concentration variations. One model that gives reasonably good results for a wide range of non-premixed reactions is the TMM model by Baldyga and Bourne [24]. In this model, the variance of the concentration fluctuations is separated into three scales corresponding to large, intermediate, and small turbulent eddies. 

In this approach to quantitatively analyzing the distribution of carrier concentrations, it was noted that the spatial length scale of dopant concentration fluctuations was an area for future exploration [207]. It is certainly clear that if each crystallite possessed a different dopant and thus carrier concentration, then this approach would be valid to the extent that the Knight shift followed an nj3 functional dependence as expected for a parabolic band. 

HANNA, S.R. The exponential probability density function and concentration fluctuations in smoke plumes, Boundary Layer Met., 29, 361-375, 1984. 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

Fig. 6.25 Relaxation rates as function of Q in semidilute PDMS/toluene solutions. A collective concentration fluctuation seen in normal contrast, C single chain motion as seen in zero average contrast. B Zimm regime of local chain relaxations, equal in both contrasts. (Reprinted with permission from [325]. Copyright 1991 EDP Sciences)...

When crosslinks are introduced to these polymer solutions, the concentration fluctuations are perturbed due to the presence of crosslinks. The exact solution for the scattering function from gels has not been found yet because of the... 

As long as the concentration of the small molecule is low (<5%), the scattered intensity due to concentration fluctuations will be negligible relative to the density or anisotropy fluctuations. In polystyrene, the HV spectrum will not have any contribution due to concentration fluctuations, but in principle there could be a contribution due to the diluent anisotropy. The average relaxation time will be determined by the longest time processes and thus should reflect only the polymer fluctuations. The data were collected near the end of the thermal polymerization of styrene. Average relaxation times were determined as a function of elapsed time during the final stages of the reaction... 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

The variability of environmental data must also be regarded as being dependent on space and/or time. As an example, the temporal variability is demonstrated for the occurrence of volatile chlorinated hydrocarbons in river water (Fig. 1 -5). The very different pattern for the time functions of the selected volatile chlorinated hydrocarbons at two sampling locations 40 km apart shows that the concentration fluctuations are quite random. 

Figure 4.13 Concentration fluctuations and the spatial autocorrelation function.

An alternative explanation of the observed turbidity in PS/DOP solutions has recently been suggested simultaneously by Helfand and Fredrickson [92] and Onuki [93] and argues that the application of flow actually induces enhanced concentration fluctuations, as derived in section 7.1.7. This approach leads to an explicit prediction of the structure factor, once the constitutive equation for the liquid is selected. Complex, butterfly-shaped scattering patterns are predicted, with the wings of the butterfly oriented parallel to the principal strain axes in the flow. Since the structure factor is the Fourier transform of the autocorrelation function of concentration fluctuations, this suggests that the fluctuations grow along directions perpendicular to these axes. 

Figure 11 Measured exchange rate R (experimental points) as a function of temperature for the solvation layer contaning N 5 molecules that contribute to the dephasing of the probing molecule CH3I the data are deduced from the mixture with molar fraction x = 0.515 using the Knapp-Fischer model. The solid line is estimated for jump diffusion from available viscosity data. Rapid concentration fluctuations are found leading to dephasing in the intermediate regime.

Figure 10 Width (FWHM) of the isotropic Raman line of the sym-methyl stretch in CH3I as a function of concentration in CDCI3 ( ). Voight fits give Lorentzian (O) and Gaussian (A) contributions to the line shape. The Lorentzian component is consistent with a concentration independent fast-modulation process. The Gaussian component suggests an additional contribution from slow concentration fluctuations. (From Ref. 4.)...

---