Semidilute solution correlation length

As shown in Ref. 48, l is intimately related to the static correlation length In infinitely dilute solutions is proportional to Rg. In semidilute solutions, is proportional to and respectively, in low and high salt limits. In... 

The scattering function g k) is a function of static correlation length as given by Eqs. (225)-(227). For semidilute solutions at high salt concentrations, Dc follows from Eqs. (226) and (282) in the —> 0 limit. 

Fig. 6.24 Comparison of the scattering from a semidilute PDMS solution under normal polymer contrast (I) revealing the correlation length with the single chain scattering (JJ as obtained by a zero average contrast preparation (see text). The line through 4 represents a Debye function with R =7 nm whereas the line through I corresponds to a Lorentzian (Orn-stein-Zernike) with a correlation length f=l nm. (Reprinted with permission from [325]. Copyright 1991 EDP Sciences)...

This small section of a chain is hardly aware that the solution is semidilute--------extends beyond the correlation length. 

The correlation length in a semidilute solution decreases with increasing concentration as a power law ... 

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 ... 

The concentration dependence of polymer size in semidilute solution is determined by substituting the expressions for the concentration dependence of the correlation length [Eq. (5.23)] and the number of monomers in a correlation blob [Eq. (5.24)] into Eq. (5.25) ... 

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. 

The concentration dependence of the correlation length in semidilute 6>-solutions is stronger than that in good solvent ... 

The exponent x is once again determined from the condition that the correlation length in semidilute solution should be independent of chain length A ------------------------—--------------------------------------... 

Only the sections of the chains in this first layer gain energy due to contact with the surface. Sections further away from the surface do not gain attraction energy, but are relaxing the concentration from the high value near the surface to a low value in the bulk of the solution in the most optimal way. The correlation length in a semidilute... 

The concentration in the adsorbed layer decreases away from the surface as (f) (z/b) for exponent 0.588. This power law concentration profile in an adsorbed polymer layer was proposed by de Gennes and is called the de Gennes self-similar carpet. This profile of adsorbed polymer can be described by a set of layers ot correlation blobs with their size of order of their distance to the surface z (see Fig. 5.11). The self-similar concentration profile starts at the adsorption blob ads in the first layer. The self-similar profile ends either at the correlation length of the surrounding solution if it is semidilute or at the chain size Rf bN if the surrounding solution is dilute. 

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... 

This equation holds for theta, good, and athermal solvents. Hence, osmotic pressure or osmotic compressibility measurements provide a con-venient means ot measuring 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 ... 

The hydrodynamic screening length in semidilute solutions is expected to be proportional to the static correlation length ... 

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 correlation length in semidilute solution can be experimentally determined by measuring the diffusion coefficient of very dilute colloidal spheres of various sizes, provided that the spheres do not interact with the polymers. Consider diffusion of a non-interacting sphere in a semidilute unentangled solution. 

Data for different molar masses of the same polymer species combine into a single plot in good solvent [Fig. 9.10(a)] if specific viscosity = ( 7 — r/sZ/ ls is plotted as a function of reduced concentration This simple data collapse works in a good solvent because the correlation length and the tube diameter are proportional to each other, with the same concentration exponents. The line in Fig. 9.10(a) has the slope of 3.9 expected by Eq. (9.46) for semidilute entangled solutions. 

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)...

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