The General Dynamic Theory of m-Component Copolymerization

The first important question we need to answer is how the monomer feed composition x(p) will be changed with conversion at various initial values 5° and the parameters of kinetic copolymerization model. When such a trajectory x(p) is known, on the base of the formulae (5.1), (5.3), and (5.7) one can find the main statistical copolymer characteristics at any number of its components within the framework of the chosen kinetic model. 

Each trajectory x(p) is regarded to be a solution of the universal set of dynamic equations (5.2), the form of the right-hand parts of which is determined by the selected copolymerization model. 

Any trajectory can end when p - I at a stationary point (SP), in which all the right-hand parts of equations (5.2) equal zero. In the case of the terminal model (2.8) all such SPs are those solutions of the non-linear set of the algebraic equations (4.13) which have a physical meaning. Inside m-simplex one can find no more than one SP, the location of which is determined by the solution of the linear equations (4.14). In addition to such an inner azeotrope of the m-simplex, azeotropes can also exist on its boundaries which are n-simplexes (2 S n m - 1). For each of these boundary azeotropes (m — n) components of vector X are equal to zero, so it is found to be an inner azeotrope in the system of the rest n monomers. Moreover, the equations (4.13) always have m solutions x( = 8is (where 8js is the Cronecker Delta-symbol which is equal to 1 when i = s and to 0 when i =(= s) corresponding to each of the homopolymers of the monomers Ms (s = 1,. ..,m). Such solutions together with all azeotropes both inside m-simplex and on its boundaries form a complete set of SPs of the dynamic system (5.2). 

In the case of the binary copolymerization described by the terminal model (2.1) there are the following types of phase portraits  

In the systems (I) and (III) 2-simplex consists of a sole cell, all the trajectories inside which approach SP corresponding to homopolymer Ms where rs 1. The systems (I) and (III) topologically are equivalent, since they differ from each other only by the inversion of the monomer indexes therefore their phase portraits are of the same type, too. In the systems (II) and (IV) the azeotropic point separates the simplex into the two cells. However, the system (IV), in which both parameters r, and r2 exceed unity practically is non-realizable [20-24]. That is why the stable binary azeotropes are excluded from the consideration, and the dynamics of the copolymerization of two monomers is exhaustively characterized by only two types (I) and (II) of phase portraits. 

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