Effect of Orientational Specificity of Complexation on the Behavior of Supramolecular Polymers:  Theory and Simulation

Abstract

Using Monte Carlo simulations we study the association of flexible oligomers terminated by a donor and an acceptor group capable of orientationally specific reversible bonding. On the basis of simulation results, we have obtained equilibrium constants for chain growth and ring closure. These constants were employed in an analytical model, which reproduces the large-scale simulation results very well. We also propose an analytical approach which can be used to analyze experimental data or make predictions of molecular weight, chain/ring distributions, etc., which are hard to obtain experimentally. Our simulation and analytical results show that an increase of orientational specificity of reversible bonding decreases the degree of association and molecular weight and leads to the suppression of small rings. As a result the ring-chain crossover concentration (i.e., the concentration at which the number of reversible bonds in chains and rings coincide) decreases and exhibits a maximum as a function of oligomer length N. With a decrease in the energy of reversible association or increase in temperature the ring-chain crossover shifts to lower concentrations and molecular weight either systematically decreases if the system is in the chain-dominated regime (high concentrations) or increases and exhibit a maximum if the system is in ring-dominated regime (low concentrations). Higher orientational specificity of association in combination with a short spacer length ensures a larger value of the molecular weight at its maximum which is reached at lower temperatures and higher oligomer concentrations. These results are supported by recent experimental observations and can be explained based on the oligomer redistribution between chains and rings near the ring-chain crossover.

Simple and General Approach for Reversible Condensation Polymerization with Cyclization

“…It is worthwhile to mention that conversion p must not be a monotonic function of concentration in the cross-over from the ring-dominated to the chain-dominated regime, which is in agreement with estimates based upon Ercolani’s work. Our numerical results fully support this observation, but on top of treating intramolecular reactions properly, our approach also contains a proper discussion of the effect of contacts with other chains.…”

“…In Figure , we compare the Monte-Carlo simulation data (see section Monte-Carlo Simulations of the SI for simulation details) with the numerical solutions for parameters that are typical for the high conversion limit, since p ≈ 0.95 for all concentrations. Regarding conversion, we observe a characteristic dip at the transition from the loop-dominated regime at small c e toward the chain-dominated regime at large c e that agrees with previous work. , Note that ω is equivalent to the fraction of bonds among all possible bonds that are in loops for case 1. For comparison, we have also included an estimate for the smallest loop model in case of irreversible reactions.…”

“…Regarding conversion, we observe a characteristic dip at the transition from the loop-dominated regime at small c e toward the chain-dominated regime at large c e that agrees with previous work. 38,84 Note that ω is equivalent to the fraction of bonds among all possible bonds that are in loops for case 1. For comparison, we have also included an estimate for the smallest loop model in case of irreversible reactions.…”

Reversible association and network formation in 3 : 1 ligand–metal polymer solutions

“…) is the equilibrium constant for the formation of mono (i ¼ 1), bis (i ¼ 2) and tris (i ¼ 3) ligandmetal complexes; DE i is the energy of binding and DS i is the entropic penalty for the formation of the corresponding complexes, m i is the fraction of the corresponding complexes, V is the total volume of the system and v is the reference volume [in the BFM the reference volume reflects the number of available sites for nearest-neighbors (v is about 54a 3 )]. 52 Eqn 1 expresses the equilibrium (with equilibrium constant K i ) between the fraction of m i and m i-1 complexes and concentration of free oligomers (expression in the square brackets multiplied by v/V).…”

“…For instance, increasing the spacer length and rigidity would decrease the probability of (monomolecular) ring formation. 32,52 As a result one can expect preferential formation of branched molecules of larger molecular weight and network formation at lower oligomer concentrations. These factors may also influence the metalto-oligomer ratio corresponding to the maximum of molecular weight, elastic plateau modulus or network fraction.…”