Since in ionic and pseudoionic dispersion polymerization of cyclic esters all propagating chains are located in microspheres that occupy only a relatively small fraction of the total volume of reacting mixture (after a short induction

FIG. 13 Number of chains per microsphere as a function of time. Calculations based on Eqs. (5) and (12), values of kinetic parameters: k1 = 300 L/mol x s and k2/k1 = 7.36 x 104. Initial concentrations of chains: a, 1.0 x 10-3 mol/L; b, 2.5 x 10-3 mol/L; c, 5.01 x 10-3 mol/L; d, 1.0 x 10-2 mol/L; e, 2.0 x 10-2 mol/L.

FIG. 13 Number of chains per microsphere as a function of time. Calculations based on Eqs. (5) and (12), values of kinetic parameters: k1 = 300 L/mol x s and k2/k1 = 7.36 x 104. Initial concentrations of chains: a, 1.0 x 10-3 mol/L; b, 2.5 x 10-3 mol/L; c, 5.01 x 10-3 mol/L; d, 1.0 x 10-2 mol/L; e, 2.0 x 10-2 mol/L.

period when particles are nucleated), kinetics of this process should differ from kinetics of homogeneous polymerizations in solution and/or in bulk in which active centers are uniformly distributed over the whole volume. Kinetics of ionic and pseudoionic dispersion polymerizations should also be different from the standard radical emulsion and dispersion polymerizations because in radical polymerizations propagating species are produced continuously throughout the polymerization process, often in a liquid phase, and primary radicals enter into the growing particles during polymerization. (In some systems there is a certain probability of their escape from microspheres.)

Formal kinetics of polymerization in solution has been well developed in recent decades and has been presented in many textbooks. The kinetics of radical dispersion polymerizations in heterogeneous and "miniheterogeneous" systems (emulsion, miniemulsion, dispersion polymerizations), albeit more complicated, has been discussed in many textbooks and monographs as well (e.g., [1]). However, the kinetics of ionic and/or pseudoionic polymerizations still awaits thorough analysis. Below is given a basic description of kinetic relations characterizing dispersion ionic and/or pseudoionic processes from the moment when microspheres have been nucleated. This description is based on analysis discussed in Ref. 54.

In dispersion polymerization the total volume of reaction mixture (V) is a sum of the volumes of microspheres (Vm) and of a liquid phase (Vs):

Similarly, the total number of moles of monomer (m) partitioned between the microspheres (mm) and the liquid phase (ms) is derived as follows:

Monomer concentration averaged over the whole volume of polymerizing mixture ([M]) is described by:

In Eq. (15) [Mm] and [Ms] describe monomer concentrations in particles and in the liquid phase, respectively.

The concentration of active centers averaged over the whole volume of polymerizing mixture ([/]0) is derived as follows:

where i and [/m] ([/m] = i/Vm) denote number of moles of propagating species (in case of quantitative initiation with initiator with functionality 1 equal to the initial initiator concentration) and their concentration in microspheres.

Differentiation of Eq. (15) with respect to time (the simplifying assumption has been made that contraction of the volume of polymerizing mixture equals 0 and thus, V is constant) yields:

When all propagating chains are already inside of microspheres, changes of monomer concentration in solution are due only to monomer diffusion from the liquid phase to microspheres and from microspheres to the liquid phase. Thus, for d(Vs[Ms])/dt, one could write a formula:

dt in which Sm denotes total surface of microspheres and Fs,m and Fm,s coefficients determining flux of monomer from liquid phase to microspheres and from mi-crospheres to liquid phase, respectively.

The differential equation describing changes of monomer concentration inside of microspheres could be written as follows:

The first term in the right side of this equation corresponds to monomer consumption due to polymerization inside of microspheres; the second term characterizes flux of monomer from liquid phase into particles; and the third term describes a flux of monomer from microspheres to the liquid phase. In Eq. (19), £ppm denotes apparent propagation rate constant in particles. This constant is called apparent because it is a function of rate constants of propagation involving various physical forms of active centers (e.g., ions, ion pairs, and ionic aggregates in ionic polymerization, and monomeric and aggregated propagating species in the pseudoionic process) and of the equilibrium constant between these species.

Substituting expressions (18) and (19) in Eq. (17) and taking into account Eq. (16) yields:

According to Eq. (20), the rate of monomer consumption averaged over the whole volume of reaction mixture depends on the average concentration of active centers ([/]0) and on the local monomer concentration in microspheres ([Mm]).

At stationary state d(Vm[Mm])/dt = 0 and, thus, from Eq. (19) and formula (16), the following relation follows:

In Eq. (22), a = Vm/V denotes volume fraction occupied by microspheres. Substituting the right-hand side of Eq. (22) in Eq. (20) gives:

d[M] = _Sm Fs,m kppm dt Sm[Fm,s + a(Fs,m - Fm,s)] + (1 - a)kppm[/](

Introducing a parameter kppp that could be called the apparent propagation rate constant in dispersion,

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