Semidilute solution length scales

Recently the wall-PRISM theory has been used to investigate the forces between hydrophobic surfaces immersed in polyelectrolyte solutions [98], Polyelectrolyte solutions display strong peaks at low wavevectors in the static structure factor, which is a manifestation of liquid-like order on long lengths-cales. Consequently, the force between surfaces confining polyelectrolyte solutions is an oscillatory function of their separation. The wall-PRISM theory predicts oscillatory forces in salt-free solutions with a period of oscillation that scales with concentration as p 1/3 and p 1/2 in dilute and semidilute solutions, respectively. This behavior is explained in terms of liquid-like ordering in the bulk solution which results in liquid-like layering when the solution is confined between surfaces. In the presence of added salt the theory predicts the possibility of a predominantly attractive force under some conditions. These predictions are in accord with available experiments [99,100]. 

On length scales larger than the correlation length, the excluded volume interactions are screened by the overlapping chains. The semidilute solution on these length scales behaves as a melt of chains made of correlation blobs and the polymer conformation is a random walk of correlation blobs ... 

Chains in semidilute solutions are random walks on their largest length scales and their size is thus proportional to the square root of the number of monomers R Therefore... 

In the final expression, Eq. (5.23) was used for the correlation length The scaling prediction for osmotic pressure is significantly different from the mean-field prediction because the exponents for the concentration dependence differ (2.3 instead of Z). t he scaling prediction is in excellenT agreement with experiments, as demonstrated in Fig. 5.7 (the high concentrations are described by IT/c c ). Equation (5,49) demonstrates that the osmotic pressure provides a direct measure of the correlation length in semidilute solutions. 

In Section 5.3, the static correlation length was defined for semidilute solutions. This correlation length separates single-chain (dilute-like) conformations at shorter length scales (r < 0 from many-chain (melt-like) statistics at longer length scales (for r > 0. The concentration correlation blob of size contains g monomers of a chain, with conformation similar to dilute solutions ... 

In semidilute solutions, both statics and dynamics are similar to dilute solutions on length scales shorter than the screening length. For short distances from a given monomer (r < ). essentially all other monomers... 

In semidilute solutions, hydrodynamic interactions are not screened on scales smaller than the correlation length Each mode involves... 

However, for non-dilute systems, the diffusion coefficient obtained from the low q time dependence of S q, t) may not be the diffusion coefficient of the polymers. For example, in semidilute solutions the dominant decay in S q, t) corresponds to correlations disappearing at the scale of the correlation length. In such cases, the diffusion coefficient is called the cooperative diffusion coefficient. 

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. 

Explain the length scales over which the reptation, Rouse, and Zimm models describe dynamics in semidilute entangled solutions of linear polymers. 

Fig. 9 Schematic representation of (sections of) multi-arm star polymers in semidilute solution in good solvent. The three different length scales, the radius ofthe star R, the coat ri, hard core are indicated. The open circles denote the correlation length blob size)...

In a semidilute solution there are three different, and in principle, independent length scales the mesh size the... 

Renormalization group theory (see, e.g., [35]) lies at the heart of this theory, justifying the use of scaling laws in the asymptotic limit, i.e., for infinitely long polymer chains and for dilute solutions. For semidilute solutions, however, this criterion is not so crucial because the polymer chains are overlapping and many properties, e.g., osmotic pressure, are independent of the chain length. 

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