Semidilute and concentrated solutions

For the purpose of the discussion, polymer solutions in good solvents can be divided into three regions dUute, semidilute, and concentrated (see Fig. 5.1). 

A dilute solution is defined as one of sufficiently low concentration that the polymers are separated from each other each polymer on avmnge occupying a spherical region of radius Rg. (Fig. S.lo) In this solution, the polymer-polymer interaction has only a small effect, and any physical property is expressed as a power series with respect to the polymer concentration p (weight of polymer in unit volume). Consider for example the osmotic pressure II and the viscosity ri. In the dilute limit, these are written as 

The parameters A2 and k are called the second virial coeffident and the Huggins coeffident, respectively. 

As the concentration increases, the polymer coils come closer and start to overlap each other. Since the number of polymers in a unit volume is pNJM the concentration p at which the overlap starts is estimated as 

Note that the concentration p can be quite low. As Rg is proportional to M , p depends on the molecular weight as 

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The plots of log 4 and log against log C for semidilute and concentrated solutions of polystyrene in tri-m-tolyl phosphate, shown in Figure 8.26, give straight lines with a slope of -2, in agreement with the relationships given above. 

It is expected that the same picture that gives a good account of the linear viscoelastic behavior of polymer melts should also hold for semidilute and concentrated solutions. In the case of semidilute solutions some conclusions can be drawn from sealing arguments (19,3, p. 235). In this way, concentration dependence of the maximum relaxation time tmax the zero shear rate viscosity r Q, and the plateau modulus G% can be obtained, where t is the viscosity of the solvent. The relevant parameters needed to obtain Xmax as a function of concentration are b, c, N, kgT, and Dimensional analysis shows that... 

In this chapter, our understanding of ideal chain conformations (Chapter 2), real chain conformations (Chapter 3), and thermodynamics (Chapter 4) will be combined to describe the conformations of polymer solutions at all concentrations and temperatures. In this chapter, the focus is on semidilute and concentrated solutions that span the large range of concentrations between dilute solutions and melts.-------------... 

Semidilute and concentrated solutions are characterized by a correlation length C the scale at which a given chain starts to find out about other chains. This correlation length is... 

Recall from Chapter 5 that the crossover concentration (p Ki jb [Eq. (5.36)] denotes the boundary between semidilute and concentrated solutions. For 0 > 0 chains are nearly ideal in concentrated solutions, whereas for 0 < 0 chains are swollen on intermediate scales. Network modulus and equilibrium swelling depend on the relative value of preparation and fully swollen concentrations (0o and l/Q) with respect to the crossover concentration 0. Since the swollen concentration is always lower than the preparation concentration (l/Q < 0o) there are three... 

Scattering techniques for measuring various static and thermodynamic properties of polymers, such as molar mass, size, conformations, interaction parameters, etc. were described in experimental sections of Chapters 1-5. In addition to static properties, scattering can provide important information about dynamic properties of polymeric systems. This section focuses on dynamic scattering from dilute solutions, but similar methods are used in semidilute and concentrated solutions."... 

The length scales a, and R are plotted as a function of concentration for a "typical good solvent in Fig. 9.7. All three length scales change their con-centration dependences from athermal to ideal at the concentration separating semidilute and concentrated solutions. 

Although not within the scope of this chapter, the theoretical description of simple polymers in dilute, semidilute, and concentrated solutions discussed by Flory, DeGennes, Doi, Edwards, and others is of great value to the synthetic chemist when tailoring more complicated water-soluble polymers. The parameters just discussed as well as associations of a chemical nature must be considered. These chain-chain enthalpic and entropic in-... 

Fluorescence anisotropy studies are popular in biological and biochemical research of lipid membranes [16-18], proteins [19-22], etc. and also in polymer science. They have been performed for monitoring the conformations and flexibility of polymer chains in dilute, semidilute and concentrated solutions [23-27], in polymer melts and blends [28-31], and also for studying polymer self-assembly [32-34]. Nowadays, steady-state and time-resolved fluorescence anisotropy are currently used methods in polymer chemistry. 

Therefore in both the semidilute and concentrated solutions, the contribution from the form birefringence decreases with the concentration. Experimental results are sununarized in refs 62 and 63. 

So far we have been considering polymer melts. It is expected that the same picture will hold in semidilute and concentrated solutions. Though this is generally believed to be the case, a few remarks must be made. 

The properties of crosslinked gels are considered in detail in section 3.5. Here w e note the analogy of the dynamic properties of concentration fluctuations in semidilute and concentrated solutions with the screening length and in polymer networks with the same distance between entanglements. 

Unlike a crosslinked gel, in semidilute and concentrated solutions, entanglements with the distance arise for a certain characteristic period, which were denoted by de Geiincs (1971) as Tr. Then, provided that... 

I he hydrodynamic interactions of segments in a maeromolecule is screened by other macromolecules on increasing the polymer concentration in solution. In semidilute and concentrated solutions of the polymer, there occur fluctuation entanglements with the scale and existence time Tr. With Dq > T the solution behaves as a gel in respect lo its dynamic properties. 

Polymer melts and semidilute and concentrated solutions of polymer are highly viscous. Even at a concentration of 1 wt %, solutions of polymer with a molecular weight greater than several million g/mol can flow only slowly. Their behaviors are even elastic like rubber at accessible time and frequency ranges. These exquisite properties had eluded researchers for decades until the tube model and the reptation theory elegantly solved the mystery. The tube model and the reptation theory were introduced by de Gennes." They were refined and applied to the viscoelasticity of semidilute solutions of polymers and polymer melts in the late 1970s by Doi and Edwards." Until then, there had been no molecular theory to explain these phenomena. We will leam the tube model and the reptation theory here. 

Detailed explanations on theoretical tools are given. Emphasis is on viscoelastic properties. The book contains a few chapters on dilute, semidilute, and concentrated solutions of rodlike molecules. The following is a simphfied version M. Doi, Introduction to Polymer Physics, Oxfor Univ. Press Clarendon, 1996. 

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