Concentration fluctuations structure factors

For a binary blend, the partial structure factors are related to the concentration fluctuation structure factor measured in small-angle neutron scattering experiments via... 

In fact this basic idea allows us to intuitively connect the structure factor S q = 0) and the change in osmotic pressure. If we take q = 0 in Equation (5.28) we can see that it contains terms that are fluctuations in number concentration ... 

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. 

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. 

The simulation result for the time evolution of structure factors as a function of the scattering vector q for an A/B 75/25 (v/v) binary blend is shown in Fig. 9 where time elapses in order of Fig. 9c to 9a. The structure factor S(q,t) develops a peak shortly after the onset of phase separation, and thereafter the intensity of the peak Smax increases with time while the peak position qmax shifts toward smaller values with the phase-separation time. This behavior suggests that the phase separation proceeds with evolution of periodic concentration fluctuation due to the spinodal decomposition and its coarsening processes occurring in the later stage of phase separation. These results, consistent with those observed in real polymer mixtures, indicate that the simulation model can reasonably describe the phase separation process of real systems. 

There is a host of other intriguing phenomena associated with the structure and dynamics of stars, which we only list here. The inhomogeneous monomer density distribution in Fig. 2 is responsible for temperature and/or solvency variation in analogy to polymer brushes attached on a flat solid surface [198]. In fact, multiarm star solutions display a reversible thermoresponsive vitrification (see also Sect. 5) which, in contrast to polymer solutions, occurs upon heating rather than on cooling [199]. Another effect is the organization of multiarm stars in filaments induced by weak laser light due to action of electrostrictive forces [200]. This effect was recently attributed [201] to local concentration fluctuations which provide localized-intensity dependent refractive index variations. Hence, the structure factor speciflc to the particular material plays a crucial role in the pattern formation. 

The experimental intensities, integrated over the quasi-elastic domain, are shown in Fig. 8a, at different CF4 concentrations. A maximiun for the scattered intensities is observed at the intermediate loading of seven molecules per U.C. After normalization with respect to the number of scattering molecules, one obtains a quantity which is related to the structure factor. The values for S(Q) are reported in Fig. 8b they show a continuous decreasing trend for increasing loadings. The extrapolation of S(Q) at zero Q value is known to be a measure of the fluctuations of the number of particles contained in a given volume [35]... 

Dynamic light scattering (DLS) is an effective technique to measure the collective diffusion coefficients by measuring the time correlation of the concentration fluctuations. These experiments allow us to can determine the collective modes of the system that couple to concentration fluctuations. In binary systems this connection is quite useful since there is only one independent concentration variable. Then one can obtain the collective diffusion coefficient theoretically from the dynamic structure factor in the long wavelength limit. 

In the polymer problem, the validity of this equation is not obvious since the des ption of the polymeric system by c disregards the chain connectivity and, therefore, neglects the entanglement effect. However, as far as the dynamics in the short time-scale is concerned, this will not be a serious problemt since, as we shall discuss later, the topological constraints are not important in the short time-scale dynamics. Indeed it will be shown that the initial slope in the dynamical structure factor is correctly described in this approach. In the long time-scale, on the other hand, the validity of eqn (5.88) is not clear, and it may well be that the theory has to be modified in future. Fortunately, many experiments related to concentration fluctuations are concerned with the short time-scale motion, so that it is worthwhile to pursue the idea in detail. 

Shimada, T. Doi, M. Okano, K. Concentration fluctuation of stiff polymers. I. Static structure factor. J. Chem. Phys. 1988, 88, 2815-2821. 

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