Scaling laws of polymer solutions

In Section 1.6, we referred to the scaling laws for the conformation of a single chain in dilute solutions. Starting with them, we develop in this section scaling laws for the structural and thermodynamic properties of polymer solutions in concentrations that range from dilute to concentrated, and also cover a wide temperature range [24], 

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Rg. 2.20 Scaling laws of polymer solutions shown on the temperature-concentration phase plane. 

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. 

Using this approach SANS has been used to measure the dimension of the Gaussian coil structure of a single chain in melts, solution and blends, provided an affirmation of the screened excluded volume model, and a verification of scaling laws in polymer solutions, determined the structure of diblock copolymer aggregates, and established the relationship between the micro and macroscopic deformation in rubber elasticity. 

Scaling laws of bulk solutions of polymers are closely associated with a... 

When physicists tried to apply the scaling law concept and the principles of renormalization theory to the study of polymer solutions, they encountered serious technical problems. However, previous investigations of Flory and especially of Edwards,37 had already shown that the overlap of polymers in... 

In the case of polymer solutions, the solute molecules are flexible and may have a very large size in comparison to the solvent molecules. In such conditions, can we directly apply the scaling laws whose validity has been established for simple, liquid mixtures in the vicinity of the critical demixtion point Anyway, this is what has been done until now. 

This phenomenon is explained in detail in Section 6.3. Many properties of polymer solutions in the semidilute regime are observed to be described by power-law relationships, and the scaling symbol, reflects this mathematical fact. 

Kosmas and Freed have extended and generalized the SCF theory to include the calculation of the moments of the end-to-end vector distributions The asymptotic long chain limits conform to the fifth power law. In another paper they explain and apply a scaling theory for polymer solutions which does not use the RG, and which eliminates the -extrapolation in the latter. Although the technique appears powerful, some of the results seem difficult to apply in practice. 

What then is the relative importance in the real world of stretched-exponential or power-law concentration and molecular weight dependences For polymer solutions, the overwhelming majority of measurements of each transport coefficient follow stretched exponentials in c and M. Scaling behavior is found only as a rare exception. Theoretical models that lead to exponential behavior are therefore desired. Theoretical models that predict scaling behavior at some crude level of approximation appear to be less than useful. Theoretical models of polymer solutions that simply assume scaling as the normal observable behavior over extended ranges of c or M are not consistent with experiment. 

Baeurle, Sa, Nogovitsin, Ea Challenging scaling laws of flexible polyelectrolyte solutions with effective renormalization concepts. Polymer (Guildf) 48, 4883 899 (2007). doi 10. 1016/j.polymer.2007.05.080... 

The scaling law approach, and renormalization group calculations, have been extensively developed for the static properties of polymer solutions. Many exponents and universal ratios are known very accurately, but the determination of full scaling functions is much harder and one often needs some reexponentiations (non-universal) to get results which are comparable with experiment. 

We shall use the dynamical scaling theory to describe the hydrodynamic properties of polymer solutions, focusing mainly on the expected universal behavior. We use a Flory approximation for the power law behavior, which turns out to be a much easier approach and allows a simple understanding of the important physical features often masked by a heavier formalism. For comparison with experiment we shall sometimes quote more detailed results obtained by renormalization group calculations. We will also discuss briefly the deviations from universal behavior related to crossover effects. 

In Section 7.2 we describe briefly the static scaling laws for polymers both in good and 0 solvents. Section 7.3 is devoted to the discussion of the hydrodynamic properties of dilute solutions, which are often used to characterize polymers. The hydrodynamic properties of semi-dilute solutions are divided into two groups collective properties and single chain properties, which are described in Sections 7.4 and 7.5 respectively. 

Chain statistics and scaling concepts are described in detail in Volume 2, Chapter 1. We shall summarize here the main results which are useful in understanding the hydrodynamics of polymer solutions. All these results are expressed in terms of scaling laws. 

The power of the polymer volume fraction V2 in the above expression has also been verified experimentally [4] with some indication that it holds well into the concentrated regime. The transition from dilute to semi-dilute behavior occurs when the concentration of polymers is sueh that the coils begin to overlap. This concentration has been denoted in the literature as the overlap concentration, c or Vj. For high-molecular-weight polymers, this eoncentration occurs at fairly low values, V2 of the order of 0.01. Solutions are generally considered concentrated when V2 > 0.1. For good solvents, c scales with M [3]. Other scaling laws for polymer systans can be found in Reference 3. 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

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