Directed assembly of block polymers is rapidly becoming a viable strategy for lithographic patterning of nanoscopic features. One of the key attributes of directed assembly is that an underlying chemical or topographic substrate pattern used to direc
t assembly need not exhibit a direct correspondence with the sought after block polymer morphology, and past work has largely relied on trial-and-error approaches to design appropriate patterns. In this work, a computational evolutionary strategy is proposed to solve this optimization problem. By combining the Cahn-Hilliard equation, which is used to find the equilibrium morphology, and the covariance-matrix evolutionary strategy, which is used to optimize the combined outcome of particular substrate-copolymer combinations, we arrive at an efficient method for design of substrates leading to non-trivial, desirable outcomes.
Composition fluctuations in disordered melts of symmetric diblock copolymers are studied by Monte Carlo simulation over a range of chain lengths and interaction strengths. Results are used to test three theories: (1) the random phase approximation (R
PA), (2) the Fredrickson-Helfand (FH) theory, which was designed to describe large fluctuations near an order-disorder transition (ODT), and (3) a more recent renormalized one-loop (ROL) theory, which reduces to FH theory near the ODT, but which is found to be accurate over a much wider range of parameters.
A renormalized one-loop theory (ROL) is used to calculate corrections to the random phase approximation (RPA) for the structure factor $Sc(q)$ in disordered diblock copolymer melts. Predictions are given for the peak intensity $S(q^{star})$, peak pos
ition $q^{star}$, and single-chain statistics for symmetric and asymmetric copolymers as functions of $chi N$, where $chi$ is the Flory-Huggins interaction parameter and $N$ is the degree of polymerization. The ROL and Fredrickson-Helfand (FH) theories are found to yield asymptotically equivalent results for the dependence of the peak intensity $S(q^{star})$ upon $chi N$ for symmetric diblock copolymers in the limit of strong scattering, or large $chi N$, but yield qualitatively different predictions for symmetric copolymers far from the ODT and for asymmetric copolymers. The ROL theory predicts a suppression of $S(q^star)$ and a decrease of $q^{star}$ for large values of $chi N$, relative to the RPA predictions, but an enhancement of $S(q^{star})$ and an increase in $q^{star}$ for small $chi N$ ($chi N < 5$). By separating intra- and inter-molecular contributions to $S^{-1}(q)$, we show that the decrease in $q^{star}$ near the ODT is caused by the $q$ dependence of the intermolecular direct correlation function, and is unrelated to any change in single-chain statistics, but that the increase in $q^{star}$ at small values of $chi N$ is a result of non-Gaussian single-chain statistics.
