Chainwise Polymerizations

As stated in Sec. 3.1, only ideal systems will be considered in this section. This definition implies that there is no intramolecular reaction, a condition which is satisfied in practice for very low concentrations of Af monomers (f 2), in the A2 + Af chainwise polymerization. To take into account intramolecular reactions it would be necessary to introduce more advanced methods to describe network formation, such as dynamic Monte Carlo simulations. 

We will first analyze the case of the pure A2 homopolymerization that leads to a linear polymer e.g., a vinyl polymerization. This will enable the reader to get acquainted with the usual parameters that are necessary to describe a chainwise polymerization. Then, we will consider the general Af + A2 copolymerization, leading to the formation of a polymer network. 

We consider styrene homopolymerization by a free-radical mechanism. Styrene, like any other vinyl monomer, is bifunctional, because the double bond opens two arms in the polymerization processs (one of the carbons is attacked by a free radical, activating the other carbon atom, which may continue to propagate the chain). 

Once initiated, a chain propagates at a fast rate (rp = propagation rate), until termination takes place by one of the following events  

In a particular system, one, two, or the three termination mechanisms may be simultaneously present. But we assume that a termination step is always present (the treatment of living polymerizations is beyond our scope). The first two termination events have the same consequence the dead chain that is formed keeps the same length that it had at the time the termination event took place. In this sense, these termination mechanisms are statistically equivalent. On the contrary, the combination mechanism leads to a dead chain that has a length equal to the sum of the lengths of the two chains that were combined. 

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Both the stepwise and chainwise polymerizations produce polymers that are polydisperse with respect to chain lengths. 

A chainwise polymerization proceeds exclusively by monomer + macromolecule reactions. When the propagation step is fast compared to the initiation step, long chains are already formed at the beginning of the reaction. The main parameters controlling the polymer structure are the functionalities of the monomers and the ratios between the initiation and propagation rates and between initiator and monomer concentrations. 

Figure 2.2 Evolution of the degree of polymerization, DPn, of primary chains for (i) free-radical chainwise polymerization and (ii) anionic (living) polymerization.

In several types of chainwise polymerizations, species with high molar mass are generated from the beginning of the reaction. This is depicted in Fig. 3.2 for an A2 (one double bond per molecule) +A4 (two double bonds per molecule) free-radical polymerization - e.g., a vinyl-divinyl system. 

Living polymerizations (2.3.1, Fig. 2.2) exhibit a different type of network growth. They are classified among the chainwise polymerizations because it is always the monomer that reacts, adding to growing chains. But the growth of primary chains occurs smoothly, as in the case of stepwise polymerizations. 

The mathematical model of network formation in the pregel stage will focus on the prediction of the gel conversion and the evolution of number-and mass-average molar masses, Mn and Mw, respectively. For chainwise polymerizations, calculations will be restricted to the limit of a very low concentration of the polyfunctional monomer (A4 in the previous example). Thus, homogeneous systems will always be considered. 

A simple mathematical description of the postgel stage will be presented for stepwise and free-radical chainwise polymerizations (in this case, the description will be limited to the range of low concentrations of the polyfunctional monomer leading to a homogeneous system). Calculations will be restricted to the evolution of sol and gel fractions, the mass fractions of pendant and elastic chains, and the concentration of crosslinks and EANC as a function of conversion. 

In the rest of the chapter we will consider separately stepwise and chainwise polymerizations. A small separate section will be devoted to the hydrolytic condensation of alkoxysilanes. This system exhibits such a large departure from the usual assumptions involved in the description of network formation that it merits particular consideration. 

For the limit of x — 1, the polydispersity is (almost) the same as the one obtained for chainwise polymerizations, provided that the termination mechanism is chain transfer or disproportionation and primary chains are long enough so that q — 1. However, in the former case polydispersity increases continuously with conversion, while in the latter it gets a value close to 2 from the very beginning of reaction. 

Temperature may have a very significant effect in the case of chainwise polymerizations, because it may affect the values of both q and J , thus modifying the molar-mass distribution. 

An example of this case is a vinyl (A2 ) - divinyl (A4) polymerization. The assumption of an ideal polymerization means that we consider equal initial reactivities, absence of substitution effects, no intramolecular cycles in finite species, and no phase separation in polymer- and monomer-rich phases. These restrictions are so strong that it is almost impossible to give an actual example of a system exhibiting an ideal behavior. An A2 + A4 copolymerization with a very low concentration of A4 may exhibit a behavior that is close to the ideal one. But, in any case, the example developed in this section will show some of the characteristic features of network formation by a chainwise polymerization. 

As in most chainwise polymerizations, q -> 1, the first observation arising from Eq. (3.180) is the very low value of the gel conversion. In actual systems, intramolecular cyclization and microgel formation produce an increase in the gel conversion. But reported values of xgei for the free-radical polymerization of systems containing multifunctional monomers are usually below 0.10. This is the case for the crosslinking of unsaturated polyesters (Af) with styrene (A2 ). 

Equation (3.180) constitutes a particular case of a general equation derived by Stockmayer (1944) to predict gelation in chainwise polymerization. This equation may be written as... 

Let us consider an A2 + A4 chainwise polymerization e.g., a vinyl - divinyl system with af = 0.01 (a very small concentration of the crosslinker in order to keep ideal conditions), and q = 0.999. Figure 3.22 shows the fraction of tetrafunctional crosslinks, nc 4/[A4o], as a function of conversion, for two limiting values of ,. Termination by combination (E, = 1) increases the crosslink concentration with respect to termination by chain transfer or disproportionation (2, = 0). At full conversion, termination by combination leads... 

The first aim of this chapter is to analyze the significant implications associated with the mere statement of a rate equation. Limitations of phenomenological kinetic equations are discussed and more rigorous analysis based on the reaction pathway, for both stepwise and chainwise polymerizations, is presented. The effect of vitrification on polymerization rate is... 

In this section, constitutive equations describing the polymerization kinetics when the system is in the liquid or rubbery state are analyzed. The influence of vitrification on reaction rate is considered in a subsequent section. First, phenomenological kinetic equations are analyzed then, the use of a set of kinetic equations based on a reaction model is discussed in separate subsections for stepwise and chainwise polymerizations. 

Kinetic Equations Based on a Reaction Model Chainwise Polymerizations... 

The kinetic description of chainwise polymerizations requires a set of rate equations accounting for inhibition, initiation, propagation, termination, and transfer steps. The polymerization rate is usually given by the propagation steps (one or more), because they consume functional groups much more frequently than any other step involving them e.g. initiation or transfer steps. However, it is always necessary to consider the overall set of kinetic equations to determine the evolution of the concentration of active species (free radicals, ions, etc.) that participate in the propagation step. 

Therefore, for particular values of the conversion of functional groups and temperature, the rate of chainwise polymerizations depends on the concentration of active species which, in turn, depends on the particular thermal history. Thus, phenomenological equations derived from Eq. (5.1), or isoconversional methods of kinetic analysis, should not be applied for this case. 

When vitrification sets in, there is an overall diffusion control that affects the rate of both stepwise and chainwise polymerizations, because segmental motions are considerably slowed down. 

For chainwise polymerizations, the analysis of model systems implies consideration of the homopolymerization or copolymerization of bifunctional monomers. Kinetic results cannot be directly extrapolated to the case of networks, because very important features such as intramolecular cycliza-tion reactions are not present in the case of linear polymers. However, the nature of initiation and termination reactions may be assessed. For example, using electron spin resonance (ESR), Brown and Sandreczki (1990) identified different types of radicals produced during the homopolymerization of a monomaleimide (a model compound of bismaleimides). 

The presence of inhomogeneities in some polymer networks, particularly those formed by a free-radical chainwise polymerization, has long been recognized (Labana et al., 1971 Dusek, 1971) however, this does not mean that all thermosets must be inhomogeneous. 

Poly(ethylene oxide) can be made using a catalyzed chainwise polymerization of ethylene oxide, or through stepwise condensation polymerization of ethylene glycol. 

Hydrocarbon polymers can be made by the typical chainwise polymerization from ethylene and by the stepwise polymerization from 1,8- dibromooctane. 

Chainwise Polymerizations A typical example of a thermoset produced by a chainwise polymerizahon is the case of the cure of unsaturated polyesters with styrene by a free-radical mechanism. Styrene is a bifiinctional monomer, A2, characterized by the presence of one C=C group that is transformed into a -C-C- bond in the polymerization reaction. The unsaturated polyester is a multifunctional monomer, Aj, characterized by the presence of if 12) C=C groups in its chemical structure. The molar fraction of C=C groups belonging to the multifunctional monomer is given by ... 

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