Fluctuation effects in the theory of microphase separation in block copolymers

TLDR

The Flory interaction parameter χ and the chain length N (number of statistical segments) are key variables in block copolymer theory. The study investigates how composition fluctuations affect the microphase separation transition in diblock copolymers. The authors map the block copolymer Hamiltonian onto Brazovskii’s model to analyze fluctuation effects. Fluctuation corrections, neglected in Leibler’s mean‑field theory, shift the microphase separation transition to a higher χN, predict a first‑order transition at (χN)t = 10.495 + 41.022 N⁻¹/³ with a peak scattering of 0.12328 N¹/³, reveal finite composition windows for transitions to lamellar, hexagonal, and bcc phases, and recover Leibler’s results in the infinite‑molecular‑weight limit.

Abstract

The effect of composition fluctuations on the microphase separation transition in diblock copolymers is investigated. Such fluctuation corrections, which were neglected in the mean field treatment of Leibler, are found to be significant for the molecular weights usually encountered. The analysis is facilitated by reducing the block copolymer Hamiltonian to a form previously studied by Brazovskii. Our principal results are the following: (i) A symmetric diblock copolymer is predicted to undergo a first order phase transition at a larger value of χN than the second order transition found by Leibler. The Flory interaction parameter is denoted χ and N is the number of statistical segments per chain. The location of the transition is predicted to be at (χN)t=10.495+41.022 N−1/3, where the peak in the scattering function attains its maximum value of 0.12328 N1/3. (ii) We find windows in composition, with finite width, through which it is possible to pass from the disordered phase to each of the ordered microphases (lamellar, hexagonal, and body-centered-cubic) by changing temperature. (iii) All the results of Leibler are recovered in the limit of block copolymers with infinite molecular weight.