Non-Gaussian chain

This transformation applies equally well to a non-Gaussian chain, for example in a good solvent where v = 0.5 ... 

There is an alternative and very direct way to generalize the Rouse-Zimm model for non-Gaussian chains. This approach takes advantage of the expression given by the original theory for the chain elastic potential energy in terms of normal coordinates ... 

The approximation consists of assuming that the same expression applies in non-Gaussian chains [21,89], using for the calculation of < i >dL general formula in terms of the averages (Rj is the position vector of unit i) and the transformation matrix that diagonalizes HA. This approach is consistent with the general relationship... 

Models which also describe the molecular weight between crosslinks for neutral polymer networks but use a non-Gaussian chain distribution have also been derived. These models would be useful in cases of highly crosslinked polymer networks. Examples of these types of models include those of Peppas and Lucht [7], Kovac [8], and Galli and Brummage [9]. 

Smith, K. J., A. Ciferri, and J. J. Hermans Anisotropic elasticity of composite molecular networks formed from non-Gaussian chains. J. Polymer Sci., Pt. A, 2, 1025 (1964). 

Figure 15.8 2H NMR spectra obtained in a randomly crosslinked, deuterated PB network. Precursor chain molecular weight 115700 g.mol"1, 1.2% crosslink agent, the average molecular weight between junctions is 27400 g.mol 1 (as determined by swelling experiments) or 11600 g.mol"1 (elastic measurements). The smooth curves are fits with a most probable distribution of chain lengths with number average molecular weight Mc = 11600 and with non-Gaussian chain statistics...

To calculate k) theoretically we must determine rj(r) by solving the equations of motion for the chain. Akcasu and Gurol [41] in 1976 attacked this on the basis of the Kirkwood diffusion equation [42], and Akcasu et al. [43] presented a more general theory in a review article of 1980. In what follows, without going to mathematical details, we summarize some important results on for Gaussian chains. Benmouna and Akcasu [44] and Akcasu et al. [43] extended the calculation of to non-Gaussian chains by invoking the Weill-des Cloizeaux approximation, eq 1.4. However, as mentioned in Section 1, this approximation seems too crude to explore excluded-volume effects on Q, quantitatively. 

Benoit H., Doty P., Light scattering from non-gaussian chains, J. Phys. Chem., 57, 1953, 958—963. 

Using Fade approximation of the inverse Langevin function (4.13) the elastic free energy of a non-Gaussian chain with the chain extension h/Na expresses by the following closed formula. 

Kloczkowski, A. Erman, B. Mark, J. E., Effect of Non-Gaussian Chains on Eluctuations of Junctions in Bimodal Networks. Polymer 2002,43,... 

Direct measurement of the cross-link density of thermoset polymers including those from epoxy resins remains one of the most difficult analytical challenges in the field. A far too common approach simply relates the rubbery modulus (Gr), the thermoset modulus above Tg, to the molecular weight between cross-links (Me) using the theory of rubbery elasticity (133,134). Unfortunately thermoset networks have much more complex features than do true elastomers, including non-Gaussian chain behavior, interchain interactions, and entanglements (172). 

Other possible explanations include non-Gaussian chain or network statistics (see Section 9.10.6) and internal energy effects (42). The latter, bearing on the front factor, will be treated in Section 9.10. 

Although Equation 7.44 is derived for a Gaussian chain with hydrodynamic interaction, the proportionality off with Rg is valid for non-Gaussian chains as well. In fact, with the assumption of uniform chain expansion due to excluded volume interactions, the result of Equation 7.44 can be shown to be valid even in good solutions. Thus, generally. 

Mean Field Theory for Non-Gaussian Chain Architectures 5.3.2.1 Partial Enumeration Schemes... 

In the case of a highly crosslinked network, the assumption that the chain lengths between junctions follows a Gaussian distribution may be incorrect. Therefore, the Peppas and Lucht [12] model (equation (17)) was developed to describe the equilibrium swelling behavior for a polymeric gel where the network was formed in the solid state, but with a non-Gaussian chain length distribution. 

Recent work by Peppas etal. [9] presents a model for the equilibrium swelling where the network is crosslinked in the presence of solvent and a non-Gaussian chain length distribution is assumed (equation (18)). 

Equation (21) is applicable to a hydrogel which is crosslinked in the presence of a solvent and exhibits a non-Gaussian chain length distribution. Here N may be expressed as ... 

Therefore, for polymers crosslinked with solvent present and exhibiting a non-Gaussian chain length distribution, the complete equilibrium expression is equation (41). 2... 

Tables 3 through 6 present the full swelling models for a number of cases. Tables 3 and 4 are applicable for anionic and cationic polymers where the concentration of ions outside the polymer is much less than the concentration of ions inside the gel. Tables 5 and 6 similarly describe the swelling behavior of anionic and cationic polymer gels but here the Ion concentration in the fluid surrounding the polymer is comparable in magnitude to that of the ion species within the polymer itself. In all these tables, equations A and B are valid for polymers with a Gaussian chain length distribution and equations C and D should be used with polymers exhibiting a non-Gaussian chain length distribution. For polymer crosslinked In the solid state, equations A and C are appropriate. Equations B and C should be used with polymers crosslinked in the presence of solvent.

Comparison of Figures 1 and 2 shows that for swelling degrees greater than about ten, finite chain extensibility must be accounted for through non-Gaussian chain statistics. While such theoretical calculations have been proven to be qualitatively... 

The intent of this paper is to present a theory (more correctly, a formalism) that provides a rational, self-consistent scheme for obtaining the macroscopic behavior of a three-dimensional crosslinked elastomeric network in terms of the response of its constituent elements, the macromolecules. The formalism is capable of application to Gaussian and non-Gaussian chain behavior, and to chain behavior that encompasses both energic and entropic force contributions upon change in conformation. 

Extension of the Classical Models to Non-Gaussian Chains. The affine and the phantom network models are based on the Gaussian distribution of end-to-end distances of the network chains. This distribution, however, is not suitable for very short chains, or for any chains stretched to near the limits of their extensibility (2,13,34,91,198,199). In these cases, the modulus shows a marked upturn at high elongations, a result that may be of practical as well as fundamental importance. 

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