Chain expansion

In the classical theories of polyelectrolytes, the chain expansion is characterized by the electrostatic excluded volume parameter, zel with ... 

The Manning theory(16,17) predicts a critical value c above which counterions are condensated on polyion. In the case of monovalent ions, =1 and for > c, the chain expansion... 

There are indications which fall short of definite proof, that a certain amount of chain rearrangement takes place, thus reducing the real chain expansion to values less than expected from elementary models. 

In network BI in which chain expansion was the greatest, the measured results show more chain swelling than a network with f=3 but less than a network with f=4. Chain swelling was less than that of the phantom network model for the other two networks, and in one case, the chains coiled to a size slightly less than that of the unperturbed molecule. 

Properties which depend on the size of the chain, for example, viscosity and angular dependence of light scattering, get strongly affected by chain expansion. The viscosity may even increase markedly as polymer concentration decreases, the increase in viscosity results in an increase in the degree of... 

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...

The numerical efficiency of the viscosity lower bound method has allowed calculations on considerably longer chains. The long chain Hmit results for 12-arm stars without intramolecular interactions and with EV (up to 325 beads) and in the theta region (up to 145 beads) [164] are close to the previous estimates with shorter chains (the extrapolated ratio g obtained in this study is also included in Table 4). The lower bound method has also served to characterize globule-coil transitions of 12-arm star chains from intrinsic viscosity calculations [143], though finite size effects are considerably more important than in the characterization of this transition from the radius of gyration data (see Fig. 17). This is due to the noticeable increase in the solvent permeabiHty associated with the chain expansion in better solvent conditions. However, the permeability effects are smaller in the more compact star chains than in their linear counterparts. 

With the assumption of uniform chain expansion due to potential interactions, we... 

Angular scattering functions, PIijI, are computed for subchains located in the middle and at the end of a PE chain. The RIS model developed by Flory et at, (S 004 - S 006) is used for the unperturbed chain. Chain expansion is introduced using a matrix treatment which satisfactorily reproduces several configuration-dependent properties of macromolecules perturbed by long-range interactions. 

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

The influence of the chain expansion produced by excluded volume on the mean-square optical anisotropy is studied in six types of polymers (PE, PVC, PVB, PS, polylp-chlorostyrene), polylp-bromostyrenel. RIS models are used for the configuration statistics of the unperturbed chains. The mean-square optical anisotropy of PE is found to be insensitive to excluded volume. The mean-square optical anisotropy of the five other polymers, on the other hand, is sensitive to the imposition of the excluded volume if the stereochemical composition is exclusively racemic. Much smaller effects are seen in meso chains and in chains with Bernoullian statistics and an equal probability for meso and racemic diads. 

Generator matrix methods are used to compute g, 0, and cp2 > 0 for PVB, PVC, and PS chains as function of the stereochemical composition. Simulations that permit introduction of excluded volume show that for all three chains Is insensitive to /0 unless the stereochemical composition is predominantly racemic. The response of to chain expansion is more dramatic in racemic PVC than in the other two polymers. 

This difference is probably associated with the chain expansion and/or contraction relative to the interface between the blocks within the particular morphology. In analogy to solid inorganic materials whose solubility increases under external pressure and to colloid systems where the capillary pressure leads to an increase in the solubility (Ostwald ripening effect), one would intuitively expect larger swelling of... 

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... 

The viscosity of a linear polyelectrolyte solution depends on the conformation of the molecules, which in turn is affected by intramolecular electrostatic interactions between charged segments located along the polymer backbone, but the interactions in systems of charged polyelectrolytes are still far from being understood. The study on the solution property of cyclic polyelectrolyte is of interest, since the chain expansion of a cyclic... 

The Coulomb interaction causes a chain expansion in dilute solution. With increasing density, the electrostatic interaction is screened and the chain size decreases monotonically [49, 50]. Within our model, the density of polymers are related to the inverse screening length according to K2=4w/lBp. 

The chain expansion varies with %, hence temperature, and there is a... 

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 ... 

The degree of chain expansion also depends on %, hence temperature (Equation 11-58). Clearly, as % approaches values of 1/2, a tends to zero and the chain becomes ideal. Flory reorganized his expression for the interactions in terms of a parameter, 6, and the temperature at which deviations from ideal chain dimensions occur is thus called... 

FIGURE 12-10 Schematic diagram depicting the reduction in chain expansion with decreasing temperature. 

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. ... 

Fluorescent hydrophobes (naphthyl and pyrenyl groups) incorporated into the polysulfobetaines are a powerful tool for studying the formation of intra-and interpolymer aggregates in aqueous and aqueous salt solutions [85,229-231]. Intermacromolecular hydrophobic association is observed as an increase in the excimer emission relative to that of the monomer emission, where h/Iu is the ratio of intensities of excimer and monomer fluorescence which reflects the extent of inter/intra macromolecular interactions. Intramolecular micellization is easily monitored by the quenching efficiency of the thallium ions. The decrease of h/Iu reflects the breaking of the intra- and interchain associations in aqueous salt solutions, leading to chain expansion. 

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