Unperturbed Gaussian chain

In 0 solvents, the radius of gyration of unperturbed Gaussian chain enters the following relationship ... 

The RPA is a mean field approximation that neglects contributions from thermal composition fluctuations and that assumes the chain conformations to be unperturbed Gaussian chains. The last assumption becomes visible from the Debye form factor in the first two terms, which for Vp, = are in accordance with Eq. 7, while the third term involves the FH interaction parameter. 

For poly(methylene), an exclusion distance (hard sphere diameter) of 2.00 A was used to prevent overlap of methylene residues. The calculation reproduced the accepted theoretical and experimental characteristic ratios (mean square unperturbed end-to-end distance relative to that for a freely jointed gaussian chain with the same number of segments) of 5.9. This wps for zero angular bias and a trans/gauche energy separation of 2.09 kJ mol". ... 

Monte-Carlo simulations are applied to estimate the characteristic ratios and p parameters from the RIS models for PE, POM, polybutadiene, and polyisoprene. Here the p parameter is defined as the ratio of the radius of gyration to the hydrodynamic radius. The p parameters of these real chains in the unperturbed state show only a slight dependence on the microconformation in the limit of large molecular weights and are found close to 1.504, which is the value for an idealized Gaussian chain. The estimated p parameters of the real chains appear to be correlated to the chain stiffness and increase with the characteristic ratios. 

To derive the results quoted in Sect. 3.1 for our Gaussian chain model, we here calculate the unperturbed (since there is no interaction) Greensfunction defined as... 

Variations of the Equilibrium Distance r0 between Crosslinks. Thermodynamical calculations start from the hypothesis that the (gaussian) chain is unperturbed and lead to... 

For indefmitly large M/(qML), an unperturbed polymeric chain may be regarded as Gaussian. For this chain, q is related to the unperturbed chain dimension Aw at infinite Mw by... 

The Gaussian chain expanded in this way is then the best Gaussian approximation to the real chain. This equation shows that as du tends to zero, a tends to 1, so that the chain retains its unperturbed value of /"nns i-S-the best Gaussian approximation is then the unperturbed Gaussian. On the other hand, if du becomes large, a becomes proportional to and /"nns becomes proportional to n, in agreement with equation (3.12). A solvent at a temperature T = 0 is called a theta-solvent. 

For an unperturbed (Gaussian) spring-bead chain the mean square and mean reciprocal rij ) of the distance rij between its beads i and j are equal to a i — j and 6/na Y i —These are well-known facts and are cited here without proof (see Ref. [2]). How are these averages expressed when the chain is perturbed by intrachain excluded-volume interactions This problem was first brought us by Peterlin [56] in 1955. Solution of it needs information about Wij(R), but, as mentioned above, work on this function is as yet in the process of development. 

The Gaussian chain provides an exceQent description of conformations in unperturbed polymer liquids and in the glasses obtained cooling them. 

For a Gaussian chain p controls the mean-square end-to-end distance cross-links (eqn 3.2). It follows, therefore, from eqns 3.2 and 3.24 that... 

Region Ifo appears for semifiexibie chains (p > 1) only at p > 1, the overlap concentration c for Gaussian chains is small as a result, the individual chain conformation remains unperturbed by interactions with other drains in spite of significant overlap in the region IIq. 

Unperturbed Dimenaons.—The most fundamental model of the statistics of a polymer in solution is the linear Gaussian chain, which exhibits the Markovian property that the mean square end-to-end distance o is directly proportional to the number of chain segments i.e. ounperturbed dimensions , and b is the (Kuhn) statistical segment length. 

The Gaussian chain provides an excellent description of conformations in unperturbed polymer liquids and in the glasses obtained by cooling them. In real polymer products, however, the molecules often do not have the random shape of a Gaussian chain. Forces exerted on the liquid during forming cause... 

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