We develop a simple recursive approach to treat reversible condensation polymerization with cyclization. Based upon a minimum set of balance equations, the law of mass action, Gaussian chain statistics, and the assumption of independent reactions, we derive exact analytical solutions for systems without cyclization, for systems containing only smallest loops, or systems that exclusively form loops. Exact numerical solutions are computed for the general case of a homopolymerization of flexible precursor polymers. All solutions were tested with Monte-Carlo simulations. A generalization for good solvent is discussed and it is shown that this generalization agrees with previous work in the limit of low and high polymer volume fractions. The new aspect of our approach is its flexibility that allows for a rather simple generalization to more complex situations. These include different kinds of reversible linear polymerization, nonlinear polymerization, first shell substitution effect, semiflexibility, or a low-molecular-weight cutoff for cyclization.

In a preceding work [10.1021/acs.macromol.1c00718Macromolecules2021], we have introduced a simple recursive scheme that allows us to treat stepwise linear reversible polymerizations of any kind with cyclization. This approach is used to discuss the polymerization of linear Gaussian strands (LGS) with two different reactive groups A and B on either chain end that participate in two orthogonal reactions and the strictly alternating copolymerization of LGS that carry A reactive groups with LGS equipped with type B reactive groups. The former of these cases has not been discussed theoretically in literature, the latter only regarding some special cases. We provide either analytical expressions or exact numerical solutions for the general cases with and without cyclization. Weight distributions, averages, polydispersity, and the weight fractions of cyclic and linear species are computed. All numerical solutions were tested by Monte-Carlo simulations.

We develop a theoretical approximation for the micro-structure and the properties of reversible networks made of star polymers based upon a set of suitable balance equations. Our model is tested by Monte-Carlo simulations of model systems with a controlled formation of cyclic structures. If only pending loops are considered, both irreversible and reversible networks develop a unique critical concentration $c_{\text{crit}}$ for gelation in the limit of high functionality $f\rightarrow\infty$ of the star polymers. Intra-molecular reactions are preferred in reversible networks as compared to irreversible systems at the same concentration and fraction of bound reactive groups. Odd-even effects develop at small f once pending loops are possible and these regard the critical concentration, network properties like weight fraction of gel, weight fraction of the elastically active material, and modulus. A precise consideration of loops involving $g\ge2$ junctions requires consideration of correlations between connected pairs of stars.