Nonlinear chains

Polymers without configurational regularity are called atactic. Configurationally regular polymers can fonn crystalline stmctures, while atactic polymers are almost always amorjihous. Many polymers consist of linear molecules, however, nonlinear chain architectures are also important (figure C2.1.2). 

Figure C2.1.2. Polymers witli linear and nonlinear chain architectures. The nonlinear polymers can have branched chains. Short chains of oligomers can be grafted to tire main chain. The chains may fonn a. stor-like stmcture. The chains can be cross-linked and fonn a network.

However, it should be noted that such dependence in the case of nonlinear alkyl chains will be different. The effective -CH2- increase in the case of nonlinear chains will be lesser (ca. 50%) than in the case of a linear alkyl chain. The tertiary -CH2-group effect would be even lesser. In general, though, one will expect that the change in y per mole substance will increase with any increase in the hydrocarbon group of the amphiphile. 

Miller and Flory (43) calculated of a-helices by taking bond-angle fluctuations into account. This approach gives the molecular rod a flexibility similar to that of wormlike chains but retains the a-helical conformation of the chain. Thus, in a sense, it may be regarded as a refinement of the treatment of Moha et al. The calculated results gave a nonlinear chain-length dependence of experimental results shown in Fig. 21. 

With the above-mentioned calculated results alone, it is impossible to distinguish which of the two mechanisms, interrupted helical structure or bond angle fluctuation, is mainly responsible for the observed nonlinear chain-length... 

To address polymer network formation from nonlinear chain-growth polymerization (or copolymerization), kinetic methods are more appropriate [23, 75-83], Some of the most successful kinetic models to address this type of system are based on the method of moments [23, 75-77, 79, 80, 82, 84], Some divergence problems at the vicinity of the gelation point are common with the method of moments, although there are practical ways to avoid this situation [80], A more refined kinetic method to address the issue of modeling the dynamics of gelation in... 

The modeling of homopolymerization as well as linear and nonlinear (chain transfer to polymer and crosslinking) copolymerizations can be represented with the same polymerization scheme shown in Figure 12.1. The kinetic... 

For lignin polymers with an obviously nonlinear chain topology, formulation of the concept of a homologous polymer may present a quite difficult task. At the current state... 

The assumption of Gaussian chains in the affine network theory can be removed by using nonlinear chains, such as the RF model (Langevin chain), stiff chain model (KP chain), etc. These models show enhanced stress in the high-stretching region. The effect of nonlinear stretching will be detailed in Section 4.6. 

Figure 9.6 shows the calculated modulus of the nonlinear chain with A = 10 with a fixed coupling constant g = 0.2. Although we are studying linear viscoelasticity, the... 

Rg. 9l6 Dynamic mechanical moduli of the nonlinear chain A = 10 with tension-dissociation coupling g =0.2. The dimensionless diffusion parameter e/j is changed from 0 (affine network, broken fine) to 0.20 (solid lines). (Reprinted with permission from Ref. [23].)... 

Gaussian chains (A = 0) show no hardening. For a nonlinear chain with A = 10, for instance, the critical value obtained in this way is Yc = 6.3. 

Amongst the alkali metal derivatives of the type MM (OR)s, the formation of a stable volatile dimeric KZr(OBu )s 2 derivative has been confirmed. In the case of titanium, LiTi(OPr )5 2 has been shown to be dimeric in solution as well as in the solid state, whereas the analogous Na and K derivatives are polymeric, with linear and nonlinear chains respectively as revealed by their X-ray structures. 

Relationship between Nonlinear Rheology and the Extent of Polymerization during the Growth of Nonlinear Chains... 

Note that Rp is defined for linear chains only, but can be defined for any chain architecture including nonlinear chains such as branched chains. In this sense, R gives a more universal measure for the chain dimension. 

The best quantity to characterize the overall dimension of the branched chain is the radius of gyration, R The end-to-end distance cannot be well defined for nonlinear chains. The branching parameter is defined as... 

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