Length scales in polymer solutions

The primary length scale in polymer solutions, even at elevated polymer concentrations, is apparently the size of the complete chain. In some sense, this residt should have been expected. The beads on a polymer chain may move in all sorts of different ways, but they are obliged to stay connected to each other, guaranteeing that there is a significant length scale corresponding to the volume of space (partially) occupied by the beads of a polymer chain. 

The scaling concept was applied for the analysis of chain conformations and static properties of the semidilute polymer solutions. The unique characteristic length scale in dilute solution imposes a unique characteristic concentration of the solution, which coincides with the intramolecular concentration c in an isolated coil. All the properties of the semidilute solution can be derived from those of the dilute solution by scaling procedure with the aid of proper crossover functions of a single dimensionless variable c/c. These crossover functions are universal, that is, independent of any details of chemical stmcture of the chains, and exhibit power-law asymptotic behavior at c/c 1. 

The essential requisite of the scaling approach is an assumption that both below and above the overlap threshold, a polymer solution is characterized by a single characteristic length scale in dilute solution, this is the size of the polymer coil,... 

Distinct butterfly patterns have also been observed when inhomogeneous gels are deformed by uniaxial deformation. The length scales in such systems, however, are usually much smaller that those encountered for polymer solutions subject to flow, and small-angle... 

The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

We now consider the large -limit, which is the regime where along the binodals exceeds unity, so we have semi-dilute polymer solutions. The characteristic length scale in semi-dilute polymer solutions is which scales as (4.15), (j)p. For colloid-polymer mixtures this expression can be rewritten in... 

Polyelectrolyte chains in a poor solvent for the polymer backbone adopt neddace-like conformations. There are three different length scales in the necklace globule the length of the string the bead size Db, and the thermal blob size fx determining the length scale of density fluctuations inside beads. Thus, all three different length scales will determine the properties of semidilute polyelectrolyte solutions in a poor solvent for the polymer backbone. 

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