Chain length distributions in linear polyaddition proceeding in nano-scale small volumes without mass transfer

Computer simulations (Monte Carlo and numerical integration of differential equations) and theoretical analysis show that the statistical nature of polyaddition, both irreversible and reversible one, affects the way the macromolecules of different lengths are distributed among the small volume nano-reactors (droplets in this study) at any reaction time. The corresponding droplet distributions in respect to the number of reacting chains as well as the chain length distributions depend, for the given reaction time, on rate constants of polyaddition k_p and depolymerization k_d (reversible process), and the initial conditions: monomer concentration and the number of its molecules in a droplet. As a model reaction, a simple polyaddition process $(M)1+(M)1⟶⟵(M)2$⁠, $(M)i+(M)j⟶⟵(M)i+j$ was chosen, enabling to observe both kinetic and thermodynamic (apparent equilibrium constant) effects of a small number of reactant molecules in a droplet. The average rate constant of polymerization is lower than in a macroscopic system, depending on the average number of reactant molecules in a droplet. The apparent equilibrium constants of polymerization $Kij=[(M)i+j]¯/([(M)i]¯[(M)j]¯)$ appear to depend on oligomer/polymer sizes as well as on the initial number of monomer molecules in a droplet. The corresponding equations, enabling prediction of the equilibrium conditions, were derived. All the analyzed effects are observed not only for ideally dispersed systems, i.e. with all droplets containing initially the same number of monomer (M)₁ molecules, but also when initially the numbers of monomer molecules conform the Poisson distribution, expected for dispersions of reaction mixtures.