Block copolymers and polymer mixtures in dilute solution: General crossover analysis and comparison with Monte Carlo calculations

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Abstract
Gell‐Mann–Low style renormalization group is applied to the two‐parameter type model of block copolymers and mixtures. These systems have multiple excluded volume interaction variables between like and unlike chemical species in addition to variables characterizing the lengths of the substituent blocks. This leads to a five parameter model for monodisperse binary polymer mixtures and diblock copolymers. We derive the full crossover dependence of dilute solutionblock copolymer and mixture properties on all five parameters by calculating effective exponents in the crossover region to second order in ε=4−d and the prefactors to order ε. The multiparameter renormalization group equation is solved to first order in ε, and a simplifying approximation is introduced to derive closed analytic forms charcterizing average polymer dimensions. Specific radial observables of block copolymers, such as the radius of gyration and the mean square end‐to‐end vector distance of di‐ and triblock copolymers as a whole a n d of the individual blocks, are evaluated in order to compare with Monte Carlo calculations in the avoiding block limit. An important component of this comparison involves the use of a physical reference state to replace the nonexistent theta state for blocks and mixtures. Good agreement is found between the Monte Carlo data and the renormalization group predictions. Moreover, the renormalization group calculations cover a much larger domain of the parameter space of variable excluded volume interaction and composition than explored in previous Monte Carlo studies. Our calculations also provide the basic input parameters for a renormalization group description of semidilute solutions of block copolymers and mixtures.