Theory for Undiluted Polymers

From the modification of the Rouse theory for undiluted polymers in Chapter 10, Section A, it is evident also that all contributions to G (equation 1 of Chapter 10) should be proportional to pT and all relaxation times should have the same temperature dependence. In this case the factor ot is expressed from equation 4 of Chapter 10 by 

Composite curve obtained by plotting the data of Fig. 11-1 with reduced variables, representing the behavior over an extended frequency scale at temperature To = 100 C. 

Temperature dependence of the shift factor ar used in plotting Fig. 11 -3. Points, chosen empirically curve, from equation 21. 

Although equation 3 was obtained from the Rouse theory, the application of reduced variables in Fig. 11-3 is based on a much more general hypothesis, namely, that the modulus contributions are proportional. opT and the relaxation times to whether or not the specific spectrum of times predicted by the Rouse theory is applicable. (In fact, the shape of the curve in Fig. 11-3 deviates considerably from the Rouse theory predictions.) This hypothesis is widely fulfilled but must be carefully examined each time it is used. An important criterion is that the shapes of the original curves at different temperatures must match over a substantial range of frequencies other criteria will be discussed in Section B. 

Temperature reduction of the various viscoelastic functions can be obtained by plotting as follows  

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Rouse Theory for Undiluted Polymer of Low Molecular Weight... 

Molecular Theory for Undiluted Amorphous Polymers and Concentrated Solutions Networks and Entanglements... 

For undiluted polymers, unlike dilute solutions, the viscoelastic behavior in stress relaxation and creep is of considerable interest as well as dynamic properties. The functions G t) and J t) as predicted by the Rouse theory have been numerically evaluated by Tschoegl, and are tabulated in reduced dimensionless form in Appendix E. 

For undiluted polymers of low molecular weight to which the Rouse theory is applicable, the X factors are actually determined entirely by fo- Comparison of equations 4, 6, 21, and 23 shows that... 

The steady-state compliance, as expressed by the theories of Graessley or Doi and Edwards for undiluted polymers (equations 52 and 56 of Chapter 10), is proportional to Me/p and independent of M when M MS- For concentrated solutions, p must be replaced by c = pv2, and, if Me — M v as discussed in Section B1 above, the concentration dependence of 7° under conditions of high entanglement becomes... 

For undiluted polymers and concentrated solutions, there are two types of theories (a) the singlechain theories or reptation theories , in which one focuses on the motions of one polymer molecule in the fluid as it moves in some kind of mean field provided by the surrounding polymer molecules and (b) the network theories , in which one visualizes the fluid as a loosely joined network in which the network junctions have a distribution of lifetimes. The chain theories are similar in structure to the dilute solution theories, and one has to make some kinds of assumptions about how the surrounding molecules affect the hydrodynamic drag and the Brownian motion. The network theories are similar in structure to the kinetic theory of rubber elasticity, and one has to make some kinds of assumptions about the junction kinetics. 

In Fiery s theory of the excluded volume (27), the chains in undiluted polymer systems assume their unperturbed dimensions. The expansion factor in solutions is governed by the parameter (J — x)/v, v being the molar volume of solvent and x the segment-solvent interaction (regular solution) parameter. In undiluted polymers, the solvent for any molecule is simply other polymer molecules. If it is assumed that the excluded volume term in the thermodynamic theory of concentrated systems can be applied directly to the determination of coil dimensions, then x is automatically zero but v is very large, reducing the expansion to zero. 

For undiluted poly (propylene) oxide 2025 Baur and STOCKMAYER (13) observed rmaJ (experimental) to be 7.9 x 10 seconds at — 20° C. They found this to be consistent with the RB theory provided that the friction factor is proportional to the viscosity of the undiluted polymer. In going to dilute solution the friction factor changes by a factor of 10s, resulting in an enormous change in the major relaxation time predicted by the RB equation. 

This chapter describes theories based on various models for rigid and flexible molecules and their comparison with experiment. The effects of intermolecular interaction at finite but low concentrations are discussed extensions of theory to higher concentrations, undiluted polymers, and cross-linked systems are treated in Chapter 10. 

It is clear from a combination of equations 22 and 26 that (since a and arc independent of M and P is proportional to A/, in a series of molecular weights for the same polymer) [i ] should be proportional to M, and this is obviously experimentally incorrect, though it is a well-known feature of the theory of free-draining hydrodynamics. In a 0-solvent, [ ] is proportional to and in good solvents the exponent is not higher than 0.8. This discrepancy is removed by inclusion of hydrodynamic interaction. (However, the Rouse form of the frequency dependence is applicable with certain limitations to concentrated solutions and undiluted polymers, as shown in subsequent chapters.)... 

For the restricted case of low molecular weights and no coupling entanglements, the viscoelastic properties of star-branched undiluted polymers can be described by a special case of the Zimm-Kilb theory o in which there is no hydrodynamic interaction. Calculations were made by Ham i by use of a method which is somewhat different from that of Rouse but yields the same results for unbranched molecules. Stars with arms of unequal length were included. For such a branched molecule, the terminal relaxation time ti, the viscosity r/o, and the steady-state compliance are always smaller than for an unbranched molecule of the same molecular weight the more branches and the more nearly equal their lengths, the... 

A quite different type of observation which leads also to the concept of an entanglement network is the dependence of viscosity on molecular weight in undiluted polymers or at constant concentration in concentrated solutions, as advanced by Bueche. This is illustrated in Fig. 10-10 for fractions of polystyrene. At low molecular weights, rjo increases only slightly more rapidly than directly proportional to Af, and its magnitude is actually predicted by the Rouse theory, in accordance with the principle of Bueche. Thus, from equations 4 and 6, rjo is given by... 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

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