Polymer solutions, scaling theory

L. Schafer, T. A. Witten. Renormalization field theory of polymer solutions. I. Scaling laws. J Chem Phys 66 2121-2130, 1977 A. Knoll, L. Schafer, T. A. Witten. The thermodynamic scaling function of polymer solution. J Physique 42 161-m, 1981. 

Computer simulation and analytical methods have both been used, based on diffusion equation, partition function and scaling theory approaches. There are a number of parameters which are common to most of these theories some of these are also relevant to theories of polymer solutions, i.e. 

Non-dilute solutions also allow for theoretical descriptions based on scaling theory [16, 21]. When the number of polymer chains in the solution is high enough, the different chains overlap. At the overlapping concentration c , the long-scale density of polymer beads becomes uniform over the solution. Consequently c can be evaluated as... 

Now we compare the isotropic-liquid crystal phase boundary concentrations for various polymer solution systems with the scaled particle theory for the wormlike spherocylinder. If the equilibrium orientational distribution function f(a) in the coexisting liquid crystal phase is approximated by the Onsager trial... 

Fig. 7. Comparison of experimental phase boundary concentrations between the isotropic and biphasic regions for various liquid-crystalline polymer solutions with the scaled particle theory for wormlike hard spherocylinders. ( ) schizophyllan water [65] (A) poly y-benzyl L-glutamate) (PBLG)-dimethylformamide (DMF) [66-69] (A) PBLG-m-cresoI [70] ( ) PBLG-dioxane [71] (O) PBLG-methylene chloride [71] (o) po y(n-hexyl isocyanate) (PHICH°Iuene at 10,25,30,40 °C [64] (O) PHIC-dichloromethane (DCM) at 20 °C [64] (5) a po y(yne)-platinum polymer (PYPt)-tuchIoroethane (TCE) [33] ( ) (hydroxypropyl)-cellulose (HPC)-water [34] ( ) HPC-dimethylacetamide (DMAc) [34] (N) (acetoxypropyl) cellulose (APC)-dibutylphthalate (DBP) [35] ( ) cellulose triacetate (CTA)-trifluoroacetic acid [72]...

Fig. 8. Comparison of experimental phase boundary concentrations between the biphasic and liquid crystal regions for various liquid crystalline polymer solutions with the scaled particle theory for hard wormlike spherocylinders. The symbols are the same as those in Fig. 7...

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

We do not intend to give an overview over all results of scaling theory here. Rather we concentrate on topics relevant for the bulk behavior of normal polymer solutions. We discuss in particular the concentration dependence, introducing the blob -picture (Sect. 9.1). Temperature dependence is discussed in Sect. 9.2. The results are summarized in the Daoud-Jannink diagram [DJ76] which separates parameter space into several regions, where different characteristic behavior is expected. 

Scaling theory also derives such results in another, more intuitive way, based on some heuristic picture of the internal structure of the polymer solution. Consider some piece of length nB within a chain of length n. It is natural to assume that this piece forms a subcoil, a blob1, of typical extension R, which scales like the coil radius for a polymer molecule of segments Rb nB- Thus the local density of segments due to the blob is estimated as... 

Experimentally the overall size of the polymer chain can be studied by light scattering and neutron scattering. A great deal of theoretical work is present in the literature which tries to predict the properties of mixtures in terms of their components. The analytical model by Rouse-Zimm [85,86] is one of the earliest works to derive fundamental properties of polymer solutions. Advances were made subsequently in dilute and concentrated solutions using perturbation theory [87], self-consistent field theory [88], and scaling theory [89],... 

This review is intended to complement those of Cohen Stuart et al. (1986) and de Gennes (1987). The former details experimental techniques available for probing polymer-particle interactions and the lattice, i.e., mean field, theories that predict, via numerical solutions, segment-density profiles and interaction potentials. The latter constructs a simple and elegant picture of the same phenomena through scaling theories developed for semidilute solutions. 

Universal models of polymer solutions attempt to describe a variety of large-scale properties with a minimum number of phenomenological parameters. Some theories (Flory, 1969) predict these parameters through microscopic models of bond geometry and interactions, and difficult but... 

In practice, the Flory-Huggins theory fails to predict many features of polymers solutions, either qualitatively or quantitatively, but remains widely used because of its simplicity. The Flory parameter x, assumed to be constant, often increases with interaction-energy scaled on kT, often exhibits a more complicated temperature dependence than 1/T (Flory, 1970). Such behavior stems from energetic effects, such as directional polar... 

The basic ideas of the scaling theory for homogeneous polymer solutions have also been used to set up a framework for a theoiy describing the adsorption of polymers from good solvents. The aim Is to derive power laws for q> z), valid in certain regimes in most cases numerical coefficients are ignored. So far, the model has only been formulated for weak adsorption, l.e., small x - We shall treat some more details of this model in the following subsection (5.4c). 

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