Copolymerization equation multicomponent

The multicomponent copolymerization equations given above yield the initial copoljnner composition. If the conversion is high, integration of the copoijmerization equation is necessary as in the case of binary copolymerization 30). Walling and Briggs (2) have proposed an... 

Copolymer composition can be predicted for copolymerizations with two or more components, such as those employing acrylonitrile plus a neutral monomer and an ionic dye receptor. These equations are derived by assuming that the component reactions involve only the terminal monomer unit of the chain radical. The theory of multicomponent polymerization kinetics has been treated (35,36). 

An example of multicomponent system that can be dealt with by using kinetics equations of the Smoluchowski type is provided by a step growth alternating copolymerization of two bifunctional monomers [16]. This system requires fit-tie more laborious, but quite straightforward algebra. 

The investigation of the copolymerization dynamics for multicomponent systems in contrast to binary ones becomes a rather complicated problem since the set of the kinetic equations describing the drift of the monomer feed composition with conversion in the latter case has no analytical solution. As for the numerical solutions in the case of the copolymerization of more than three monomers one can speak only about a few particular results [7,8] based on the simplified equations. A simple constructive algorithm [9] was proposed based on the methods of the theory of graphs, free of the above mentioned shortcomings. 

The mostly well-known discussion concerning multicomponent copolymerization theory is probably connected with so-called simplified equations put forward in papers [143-147] for the description of S(x) dependence. These equations can be obtained from the general Walling-Briggs equations (3.8), (4.10) via substituting in them expression a /a for Bs. The derivation of these simplified equations is based on the assumption that the rate of the monomer Mj addition to the radical R, is equal to the rate of the monomer addition to the radical Rj ... 

The conventional [Eq. (7.77)] and simplified [eq. (7.81)] terpolymeriza-tion equations can be used to predict the composition of a terpolymer from the reactivity ratios in the two-component systems M1/M2, M1/M3, and Ms/Ms- The compositions calculated by either of the terpolymerization equations show good agreement with the experimentally observed compositions. Neither equation is found superior to the other in predicting terpolymer compositions. Both equations have been successfully extended to multicomponent copolymerizations of four or more monomers [30,31]. 

The terminal model for copolymerization can be naturally extended to multicomponent systems involving three or more monomers. Multicomponent copolymerizations And practical application in many commercial processes that involve three to five monomers to impart different properties to the final polymer (e.g., chemical resistance or a certain degree of crosslinking) [134]. There is a classical mathematical development for the terpolymerization or three-monomer case, the Alfrey-Goldfinger equation (Eq. 6.43)... 

The second approach implies that the MWD equations for a multicomponent monomer system are treated as those of a homopolymer. These ideas appeared in the literature under slightly different names but around the same period, when more researchers started studying copolymerization systems (e.g., Ballard et al. [20]). Hamielec s group formalized this approach under the name of pseudokinetic rate constants method (PKRCM) and illustrated its use for linear, branched, and crosslinked copolymerization systems [21], as well as for batch, semibatch, and continuous reactors [22], Ray s group also made use of what they referred to as apparent rate constants [23]. The group of Morbidelli used a similar idea that they termed as pseudohomopolymer approach [24]. 

Tip 13 (related to Tip 12) Copolymerization, copolymer composition, composition drift, azeotropy, semibatch reactor, and copolymer composition control. Most batch copolymerizations exhibit considerable drift in monomer composition because of different reactivities (reactivity ratios) of the two monomers (same ideas apply to ter-polymerizations and multicomponent cases). This leads to copolymers with broad chemical composition distribution. The magnirnde of the composition drift can be appreciated by the vertical distance between two items on the plot of the instantaneous copolymer composition (ICC) or Mayo-Lewis (model) equation item 1, the ICC curve (ICC or mole fraction of Mj incorporated in the copolymer chains, F, vs mole fraction of unreacted Mi,/j) and item 2, the 45° line in the plot of versus/j. 

Copolymer composition can be predicted for copolymerizations with two or more components, such as those employing acrylonitrile plus a neutral monomer and an ionic dye receptor. These equations are derived by assuming that the component reactions involve only the terminal monomer unit of the chain radical. This leads to a collection of N x N component reactions and x 1) binary reactivity ratios, where N is the number of components used. The equation for copolymer composition for a specific monomer composition was derived by Mayo and Lewis [74], using the set of binary reactions, rate constants, and reactivity ratios described in Equation 12.13 through Equation 12.18. The drift in monomer composition, for bicomponent systems was described by Skeist [75] and Meyer and coworkers [76,77]. The theory of multicomponent polymerization kinetics has been treated by Ham [78] and Valvassori and Sartori [79]. Comprehensive reviews of copolymerization kinetics have been published by Alfrey et al. [80] and Ham [81,82], while the more specific subject of acrylonitrile copolymerization has been reviewed by Peebles [83]. The general subject of the reactivity of polymer radicals has been treated in depth by Jenkins and Ledwith [84]. 

The rate matrix for multicomponent copolymerization can thus also be expressed in terms of the homopolymerization propagation rate constants of the n monomers, and reactivity ratios as defined in a similar manner to Equation (10.7). The set of copolymerization rate equations given by Equation (10.30) can be given in matrix form as follows ... 

Equation (10.46) is a reasonable assumption for multicomponent copolymerization for the general case of n monomers as well. This implies that the rate of copolymerization propagation of equals the formation of M, radical. When this is applied to all possible pairs of monomers in n comonomers, the QSSA holds as the net production of radicals equals the consumption during the copolymerization propagation itself. Nothing about the termination reactions is brought forward into this analysis. Thus, for n comonomers ... 

---