Concentration fluctuation model

Kamath, S. K., R. H. Golby, S. K. Kumar, K. Karatasos, G. Floudas, G. Fytas, and J. E. L. Roovers. 1999. Segmental d5mamics of miscible polymer blends Comparison of the predictions of a concentration fluctuation model to experiment./. Chem. Phys. 111 6121-6128. 

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. 

As seen above, the mean chemical source term is intimately related to the PDF of the concentration fluctuations. In non-premixed flows, the rate of decay of the concentration fluctuations is controlled by the scalar dissipation rate. Thus, a critical part of any model for chemical reacting flows is a description of how molecular diffusion works to damp out... 

In the remainder of this section it is desired to obtain the relative, constant-pressure heat capacity of the liquid at x=j and the concentration fluctuation factor for all compositions. Since the latter equation is complicated, it is not written out in full here. This has been done in Eqs. (37)-(45) of the paper by Liao et al. (1982) for the special case that 14 = 34 = 0 and / l3 is the only nonzero cubic interaction term, i.e., the version of the model applied here to the Ga-Sb and In-Sb binaries. Bhatia and Hargrove (1974) have given equations for the composition fluctuation factor at zero wave number for the special cases of complete association or dissociation and only quadratic interaction coefficients. 

This paper deals with one of the mean-field methods of modeling the connectivity build-up that can be applied to polymerization processes. As in the other mean-field methods of modeling, certain physical features such as concentration fluctuations or fluctuation coupled diffusion control of reaction steps, etc., are neglected. 

This simple model predicts that the structure factor will develop a butterfly pattern and grow along an axis that is at 45° with respect to the flow direction, which is parallel to the principal axis of strain in this flow. Since the structure factor is the Fourier transform of the pattern of concentration fluctuations causing the scattering, the model predicts an enhancement of fluctuations perpendicular to the principal axis of strain. 

Here, CA and CB (upper case) denote the mean molar concentrations of reactants A and B while CA and CB (lower case) denote the local concentration fluctuations that result from turbulence. When the species are perfectly mixed, the second term on the right side containing the correlation of the concentration fluctuations, will approach zero. Otherwise, if the species are not perfectly mixed, this term will be negative and will reduce the reaction rate. Estimating this correlation term is not straightforward and numerous models are available. An excellent discussion on this subject was given by Hannon [1],... 

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 12 Raman-echo data from the sym-methyl stretch in 50 50 CH3LCDCI3 (points). Three models for the dephasing are shown solid, fast modulation only dashed, fast and static modulation dotted, fast and intermediate (5 ps) modulation. The best fit (dotted) is consistent with interaction of the vibration with local-concentration fluctuations. The fits are constrained to be consistent with FID and line shape data. (From Ref. 6.)...

The third model assumes that the concentration fluctuations are long lived but not static. The dotted line in Fig. 12 shows good agreement with the data assuming a lifetime of 5 ps for the concentration fluctuations. A range of lifetimes from 4 to 7 ps is compatible with the data. This model not only agrees with the Raman echo data, it is also matches the FID and Raman line shape and peak position data as well. The lifetime found in the Raman echo implies that the Gaussian component of the line shape (4.25 cm-1) is actually motionally narrowed from the full distribution of frequencies (5.15 cm-1). 

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