Overlap volume fraction

For small chains in solution the translational diffusion significantly contributes to the overall decay of Schain(Q>0- Therefore precise knowledge of the centre of mass diffusion is essential. Combing dynamic light scattering (DLS) and NSE revealed effective collective diffusion coefficients. Measurements at different concentrations showed that up to a polymer volume fraction of 10% no concentration dependence could be detected. All data are well below the overlap volume fraction of (p =0.23. Since no -dependence was seen, the data may be directly compared with the Zimm prediction [6] for dilute solutions ... 

If the volume fraction cf) of the polymer solution is equal to the overlap volume fraction pervaded volumes of macromolecules densely fill space and chains are just at overlap (0 = i i ) (see Fig. 1.15). ... 

If the polymer volume fraction in solution is below the overlap volume fraction 0, the solution is called dilute (p<4> ). The average distance between chains in dilute solutions is larger than their size. Therefore, polymer coils in dilute solutions are far from each other swimming happily in surrounding solvent. Most properties of dilute solutions are very similar to pure solvent with slight modifications due to the presence of the polymer. 

In the definition of the overlap volume fraction 0, the pervaded volume V is taken at, since polymer size and hence its pervaded volume may change with concentration. 

At the overlap volume fraction (for 4> = < > ) P=, and as the concentration of linear chains is increased P steadily grows, reflecting the presence of additional chains inside the pervaded volume of each molecule. Notice that the overlap parameter counts the number of whole chains that share the pervaded volume. In reality, small parts of numerous chains are within... 

A macromolecule can adopt many conformations, defined by relative locations of its monomers in space. Polymer conformations are often self-similar (fractal) with pervaded volume Fmuch larger than their occupied volume Avmoti where Vmon is the monomer volume. The overlap volume fraction... 

Calculate the overlap volume fraction of a polymer with degree of... 

A mean-field estimate of this probability can be made for the general case of an ideal chain in cf-dimensional space by replacing a chain with an ideal gas of N monomers in the pervaded volume of a coil R. The probability of a given monomer to contact any other monomer within this mean-field approximation is simply the overlap volume fraction (j>, of a chain inside its pervaded volume, determined as the product of the monomer volume and the number density of monomers in the pervaded volume of the coil NjR ... 

Ideal chains obey Gaussian statistics in any dimension with R — bN, leading to the overlap volume fraction ... 

Using the results of Problem 3.6, calculate the overlap volume fraction for the three cases ... 

K 01 0 4 lhody Tp S, t) X Xb Xc overlap volume iraction. laimensioniessi. d. overlap volume fraction for /9-solvents, [dimensionless], p. 172 semidilute-concentrated crossover volume fraction, [dimensionless], p. 180 crossover volume fraction in mean-field theory, [dimensionless], p. 181 probability for segment s to still be part of the tube at time t, [dimensionless], p. 405 Flory interaction parameter, [dimensionless], p. 142 Flory interaction parameter for a binodal, [dimensionless], p. 150 critical interaction parameter, [dimensionless], p. 152 Florv interaction oarameter for a soinodal. 

These suggestions are supported by the observation that the equilibrium swelling degree corresponds to the overlap volume fraction v of uncrosslinked chams with the molecular mass From this correspondence it can be concluded that... 

Figure 19.24. There is a close analogy between solutions of long micelles and polymer solutions, including transient networks. The figure illustrates the transition from dilute to semi-dilute solutions and the overlap concentration 0 is the overlap volume fraction...

Likewise the virial expansion of Il/Ilideai in terms of allows us to find the overlap volume fraction 4> as 4> = [N( /2-x) - This result is, however, wrong. We know that 4> should rather be or for real chains in a good sol-... 

In a good solvent, c whereas in a theta solvent, c It is often more convenient to use the polymer volume fraction, 0, rather than concentration, because then for a polymer in the absence of solvent, 0 = 1. Using 0, two concentration regimes can be distinguished from the dilute regime. If the coils are overlapped, i.e. 0 > 0, where 0 is the overlap volume fraction, but there is still not much polymer in solution, 0 < 1, then the solution is said to be semi-dilute. However, if 0 > 0 and 0 1, the solution is concentrated. Polymer chains in dilute, semi-dilute and concentrated solutions are sketched in Fig. 2.13. 

The overlap volume fraction is typically <() = 3.5 x 10"3. The inverse wave numbers defining the onset of total reflection are respectively... 

We note that Eqs.(32) and (33) imply A = 0.037, which is an order of magnitude greater than the overlap volume fraction... 

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