Polymerization nonlinear

In nonlinear polymerization rate factors depend on chain length or other micro-structural properties, which leads to special problems when utilizing the method of moments [6]. We here address transfer to polymer in radical polymerization leading to long-chain branching (LCB) and random scission. The reaction equation for transfer to polymer in a ID formulation with the rate factor proportional to chain length, ktj,m is Eq. (15). 

The population balance contribution for dead chains is given by  

This illustrates a typical closure problem. In linear polymerization, differential equations of the ath moment contain RHS terms with orth or lower moments, which allows direct solution of the set. In contrast, as shown by Eq. (17), nonlinear polymerization leads to higher moments on the RHS. This implies that the set cannot readily be solved without additional assumptions. Another example is random scission of linear dead chains into two living chains (macroradicals)  

the rate factor is proportional to chain length. The 1D population balance 

Constructing equations for 0th through 4th moments by summing over chain length leads to  

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Gelation and the attendent insolubility mentioned above are encountered in all of the nonlinear polymerizations listed in Table I and in many others likewise. Naturally these characteristics have been attributed to the restraining effects of three-dimensional, or space, network structures of infinite size within the polymer. This is the feature which distinguishes most nonlinear from linear polymers. 

Figure 1. Number fraction of ring structures per molecule (Nr) as a function of extent of reaction (p) for bulk, linear, and nonlinear polyurethane-forming reactions with approximately equimolar concentrations of reactive groups (r = [NCO]J [OH]0 ss 1) (2,3). Conditions O-linear polymerization, HDI + poly(ethyleneglycol) at 70°, [NCOfo — 5.111 mol/kg, [OH], = 5.188 mol/kg number-average number of bonds in chain forming smallest ring structure (v) = 25.2, and nonlinear polymerization, HDI and POP triol at 70°C, [NCO] — 0.9073 mol/kg, [OH]0 = 0.9173 mol/kg v = 115. Reproduced with permission from Ref. 5.

Note 1 A polyelectrolyte network is sometimes called a cross-linked polyelectrolyte. Use of the latter term is not recommended unless the polyelectrolyte network is formed by the cross-linking of existing polyelectrolyte macromolecules rather than by nonlinear polymerization. (See the definition of a crosslink, definition 1.59, ref [4].)... 

The molecular weight distribution in this type of nonlinear polymerization will be much narrower than for a linear polymerization. Molecules of sizes very much different from the average are less likely than in linear polymerization, since this would require having the statistically determined / branches making up a molecule all very long or all very short. The distribution functions for this polymerization have been derived statistically [Peebles, 1971 Schaefgen and Flory, 1948], and the results are given as... 

Flory (1941a,b, 1953) and Stockmayer (1943, 1944) laid out the basic relations for establishing the evolution of structure with conversion in nonlinear polymerizations. Their analysis is based on the following assumptions defining an ideal network ... 

On- and off-lattice Monte Carlo simulations are among the best available methods to analyze complex nonlinear polymerizations, particularly those presenting a high extent of intramolecular cyclization (Somvarsky and Dusek, 1994 Anseth and Bowman, 1994). 

Polymerization rate represents the instantaneous status of reaction locus, but the whole history of polymerization is engraved within the molecular weight distribution (MWD). Recently, a new simulation tool that uses the Monte Carlo (MC) method to estimate the whole reaction history, for both hnear [263-265] and nonlinear polymerization [266-273], has been proposed. So far, this technique has been applied to investigate the kinetic behavior after the nucleation period, where the overall picture of the kinetics is well imderstood. However, the versatility of the MC method could be used to solve the complex problems of nucleation kinetics. 

Knoesen, A., Molau, N. E., Yankelevich, D. R., Mortazavi, M. A., and Dienes, A. Corona-poled nonlinear polymeric films In situ electric field meaairement, characterization and ultrashort-pulse applications. International Journal of Nonlinear Optical Physics, vol. 1, no. 1, 1992, p. 73-102. 

The reaction between similar /-functional molecules is the simplest case of nonlinear polymerization. An example is the etherification of pentaerythritol [27] ... 

The recursive method of Macosko and Miller [27] has been described earlier for calculating molecular weight averages up to the gel point in nonlinear polymerization. A similar recursive method [34] can also be used beyond the gel point, particularly for calculating weight-fraction solubles (so/) and cross-link density. To illustrate the principles, we consider first the simple homopolymerization, that is, reaction between similar /-functional monomers Ay and then a more common stepwise copolymerization, such as reaction of A/ with B2. 

FIGURE 2.44 Structural diagram of nonlinearly polymeric body. (Reprinted from Yu. Potapov, O. Figovsky, Yu. Borisov, S. Pinaev, and D. Beilin, Creep of Polymer Concrete at Compressive Loading, J. Scientific Israel Technological Advantages 5, nos. 1-2 (2003) 1-10. With permission.)... 

Let us consider the nonlinear polymerization of a monomer bearing / functional groups. Ay. The simplest representation of such a concept is shown in Figure 9.6, for a three-functional unit. This unit can hold four reaction states, from 0 to 3, which indicates the number of reacted bonds that link this unit with its neighbor. 

Miller, D.R., Macosko, C.W. Average property relations for nonlinear polymerization with unequal reactivity. Macromolecules 11, 656-662 (1976)... 

Furfural and HMF are readily prepared from various catalytic biomass conversion processes. Both furfural and HMF can be readily converted to a large variety of monomers for polymerizations by chain-growth and/or condensation mechanisms. As the transformation of furfural and HMF to fine chemicals or monomers for polymers has been well documented by several comprehensive reviews [107-113, 130, 131], this chapter has mainly focused on the bio-based furan polymers with self-healing ability through thermally reversible Diels-Alder reactions, which is a recently exploited prosperous research area. In addition, the furan-based DA reaction has also been used in the thermoreversible nonlinear polymerization and dendrimer chemistry. 

A novel methodology for the sjmthesis of complex nonlinear polymeric architectures was proposed, based on the... 

Tumolillo, T. A., Jr., and Ashley, P. R., Fabrication techniques of photopolymer clad waveguides for nonlinear polymeric modulators, SPIE Proc. Photopolym. Device Phys. Chem. Appl. II, 1559 (1991), 65,... 

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