Chain expansion factor

The ratio of the root mean square lengths is called the chain expansion factor ... 

Thus if we know [tj] and [rj]e as a function of molecular weight we can plot the chain expansion factor as a function of concentration. A plot for polybutadiene from the work of Graessley is shown in Figure 5.21 and uses Equation (5.81) to describe the relationship between concentration and intrinsic viscosity. 

Figure 5.21 A plot for the chain expansion factor for polybutadiene from the work of...

Investigation of the multivariate Gaussian distribution and the dipole moments of perturbed chains expansion factors for perturbed chains. 

In a series of papers [216,217], Nakata and Nakagawa have studied the coil-globule transition by static light scattering measurements on poly(methyl methacrylate) in a selective solvent. They have found that the chain expansion factor, a2 = R2/R20, plotted against the reduced temperature, r = 1 - 0/T, first decreases with decreasing r, as it should be, but then begins to increase (see, e.g., Fig. 2 presented in [217]) In the authors opinion, the increase of... 

Flory modeled this by considering the seg-1 ments to be like a swarm of particles tKs- tributed about the center of gravity of the coil in a Gaussian fashion, then considered the balance between the free energy of mixing I and the free energy associated with the elas-1 tic deformation of the chain. He then found an expression for a chain expansion factor, a, that minimized the free energy, which was expressed in terms of a series in the volume fraction of polymer, 0p. The result is shown in Equation 11-58 ... 

You may recall that the temperature where % 2is what Floiy called the theta tern--perature and can now be seen to describe the situation where the second virial coefficient becomes zero (Figure 12-10). This means that at this point pair-wise interactions cancel and the chain becomes nearly ideal, as we discussed in the section on dilute solutions (Chapter 11), where we referred to the Floiy excluded volume model in which the chain expansion factor is given by Equation 12-18 ... 

For a polymer in dilute solution we have seen that 0-5 is proportional to M°Ja, where a is the chain expansion factor (see Chapter 11 note that previously we related °-S to the number of segments, but this is obviously equal to the molecular weight of the chain, M, divided by the molecular weight of segment M. ... 

Now in some cases a direct study of a polymer under 0 conditions is not feasible or, more frequently, not desired. The need to work under 0 conditions when determining unperturbed chain dimensions might be circumvented if one could rely on theories connecting accurately measurable quantities such as intrinsic viscosity, second virial coefficient etc. obtained in good solvents, with the chain expansion factor. 

These data were obtained by estimating the chain expansion factor a from viscosity and osmotic second virial coefficient measurements, through the combined use of equations (10) and (11). These results tend to confirm the main predictions of the theory presented by the above mentioned authors both as regards the steep decrease in with... 

Solubility parameters can also be estimated from intrinsic viscosity. Flory [101] related intrinsic viscosity to polymer molecular weight and the chain-expansion factor. The chain-expansion factor can, in turn, be related to the polymer-solvent interaction parameter using the Flory-Hug-gins theory. A variety of models can be used to relate the interaction parameter to solubility parameters [87,102,103] these equations have the form... 

Chain expansion factor n. The ratio of the length of a polymer chain in any given solvent to the length of the polymer chain in a theta solvent. 

The chain-expansion factor can be calculated by evaluating the mean-squared radius of gyration using Equation 5.5. Flory obtained an equation that can be written as ... 

The phantom-chain quantity in the denominator is proportional to When the chain-expansion factor is large, the squared expansion factor is proportional to This is what is observed for long chains in good solvents. 

Figure6.11 (a) Effective charge/ and (b) chain expansion factor I] versus Ig at salt-free conditions with only monovalent counterions for N = 1000, Q - 0.0005, iv = 0, iV] =0, and IV2 = 0, Cs) = 0 = 2- S = 1 (dashed),...

As in the case of the chain expansion factors, terms of order Zj become insignificant in the perturbation expansion of the -second virial coefficient when P>P3- Hence the two-parameter approximation becomes acceptable and only the I and II terms are needed in equation (73). 

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