Dynamics of concentration fluctuations

The use of photon correlation spectroscopy to study the dynamics of concentration fluctuations in polymer solutions and gels is now well established. In bulk polymers near the glass transition there will be slowly relaxing fluctuations in density and optical anisotropy which can also be studied by this technique. In this article we review the development of the field of photon correlation spectroscopy from bulk polymers. The theory of dynamic light scattering from pure liquids is presented and applied to polymers. The important experimented considerations involved in the collection and analysis of this type of data are discussed. Most of the article focuses on the dynamics of fluctuations near the glass transition in polymers. All the published work in this area is reviewed and the results are critically discussed. The current state of the field is summarized and many suggestions for further work are presented. 

Several attempts have been made to explain theoretically the effects of flow on the phase behavior of polymer solutions [112,115-118,123,124]. This has been done by modification of the mean-field free energy. The key point is to include properly the elastic energy of deformation produced by flow. A more rigorous approach originates from Helfand et al. [125, 126] and Onuki [127, 128] who proposed hydrodynamic theories for the dynamics of concentration fluctuations in the presence of flow coupled with a linear stability analysis. 

We conclude this section by drawing attention to various theories considering the dynamics of block copolymer melts rheology of these systems has been considered [340-342], single chain dynamics and selfdiffusion [343, 344], nu-cleation of the ordered phase [61], ordering kinetics [345,346], and dynamics of concentration fluctuations [347]. These topics are not under consideration here, just as other extensions of the theory random copolymer melts [348, 349], multiblock copolymer melts [350] etc. 

Coalescence occurs in shear as well as in quiescent systems. In the latter case the effect can be caused by molecular diffusion to regions of lower free energy, by Brownian motion, dynamics of concentration fluctuation, etc. Diffusion is the mechanism responsible for coalescence known as Ostwald ripening". The process involves diffusion from smaller drops (high interfacial... 

It has been shown both theoretically and experimentally that the kinetics of volume change and the dynamics of concentration fluctuations of a gel are controlled by a diffusive process in which many chains of the network move cooperatively (9-i3). The first part of this chapter describes how the... 

Equation (5.105) is the basic equation for the dynamics of concentration fluctuation. We now study the consequence of this equation. 

Coalescence occurs in quiescent as well as in sheared systems. In the former it starts by molecular diffusion from smaller to larger diameter drops, caused by the difference in surface energy. This mechanism, known since 1896 as Ostwald ripening, was originally proposed for rain drop formation. Small drops may also coalesce by Brownian motion, dynamics of concentration fluctuation, and so on. Shearing enhances the coalescence [291] ... 

The relation between the scattering of r2idiation and diffusion manifests itself directly in the dynamics of concentration fluctuations as well as a result, the diffusion coefficient. 

Light scattering. Dynamics of concentration fluctuations in the critical region. Critical indices... 

The application of the dynamic SCF theory [97] or EPD [29,31,109] to the collective dynamics of concentration fluctuations and the relation between the dynamics of collective concentration fluctuations and the single chain dynamics is an additional, practically important aspect. We have merely illustrated the simplest possible case—the early stages of spontaneous phase separation within purely diffusive dynamics. In applications the hydrodynamic effects [110,111], shear and viscoelasticity [112] might become important. Even deceptively simple situations—like nucleation phenomena in binary polymer blends—still pose challenging questions [113]. Also the assumption of local equilibrium for the chain conformations, which allows us to use the SCE free energy functional, has to be questioned critically. Methods have been devised to incorporate some of these complications [76,96,99, 111, 112] but the development in this area is still in its early stages. 

Hamano, K., Kawazura, T, Koyama, T. Kuwahara, N. (1985). Dynamics of concentration fluctuations for butylcellosolve in water. J. Chem. Phys., 82,2718-2722. 

It is essential to first briefly review the model for the dynamics of concentration fluctuations in the absence of shear, first derived by Cahn and Hilliard [39] for binary alloys, later modified by Cook [40] to include thermal noise, and more recently adapted by deGennes [41] and Pincus [42] for describing phase separation in polymer blends. 

Doi and Onuki [50] (DO), extended the models to polymer blends in which both components are entangled. The key aspect to address is how to incorporate stress into the equation of motion for concentration fluctuations. Effectively, by determining the conditions for force balance, it was shown that the stress enters the equation of motion at the same level as the chemical potential. Such an approach enabled the development of a framework that coupled the dynamics of concentration fluctuations to the flow fields and stress gradients however, only the simplest form of constitutive relation for the stress was treated. In entangled polymer solutions, the tube model predicts that the relaxation of an imposed stress is well described by a single exponential decay, with the characteristic time-scale being that required for... 

On the theoretical side, there has been an effort to understand the dynamics of concentration fluctuations in polymer blends, mostly by de Gennes, Pincus, and Binder. ... 

In the preceding section, we have examined the frictional properties of semi-dilute solutions and gels and the analogy between the two systems. In this section, our interest is focused on the dynamics of concentration fluctuations of the polymer. Let us consider first the case of a gel. The polymer network fluctuating around its equilibrium position is subjected to two driving forces The osmotic force tends to equalize the concentration and the elastic force tends to keep the network in its position. The fluctuations are damped by the frictional force between the polymer network and the solvent. 

The dynamics of concentration fluctuations expressed by eqn [161] (or eqn [34]) of the Omstdn-Zemike type exhibits the diflrrsion coeflrdent D =feT/(6nj 0), eqn [72], which can also be derived from fluctuations of gel-like networks with screening of hydrodynamic interactions. Thus, it is often rderred to as the gd mode. A decrease in mesh size increases the rdaxation rate of the gel mode. Experimentally, DLS for semidilute solutions rrsually exhibits another mode of motions with a very slow decay rate. This slow mode may be assodated with some inhomogeneity depending on solvent quality and sample preparation, or may be rdated to the translational... 

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