Simple Calibration\nof Block Copolymer Melt Models

According to the universality hypothesis, the phase behavior\nof\ndifferent block copolymer melt models having a fixed chain architecture\ndepends solely on two parameters: the invariant chain length <i>N̅</i> and the effective interaction parameter <i>χN</i>. If models behave universally, they can be compared\nto each other and can predict experiments quantitatively. The majority\nof simulations, however, do not use the Flory–Huggins parameter\nχ directly but instead operate with the exchange energy α\n= ϵ<sub><i>AB</i></sub>– (ϵ<sub><i>AA</i></sub>+ ϵ<sub><i>BB</i></sub>)/2, where\nϵ<sub><i>xy</i></sub> is the interaction energy between\nmonomers of type <i>x</i> and <i>y</i>. The simplest\nFlory-like definition of χ is a standard linear relation: χ\n∝ α; however, previous studies have shown that the universality\nhypothesis does not hold for all models using this approximation.\nHere, we analyzed the behavior of more than 30 coarse-grained symmetric\ndiblock copolymer melt models. We discovered that the phase behavior\nof a wide <i>family</i> of models depends only on two parameters, <i>N̅</i> and <i>χN</i>, if the simple linear\ndefinition of χ is imposed. This family is comprised of the\nmodels in which the monomer interaction potential energy <i>z</i> has a near-symmetric distribution around its mean. This phenomenon\ncould be explained by the coincidence of normalized peak scattering\nintensity at the same <i>χN</i> and <i>N̅</i>, which was observed for selected models with symmetric <i>z</i>-distribution when the χ ∝ α expression was used.\nOur results indicate that the main parameter controlling the symmetry\nof the <i>z</i>-distribution is the monomer density ρ.\nAbove certain ρ, models have symmetric <i>z</i>-distributions,\nand their order–disorder transition points follow a master\ncurve similar to the one predicted by Fredrickson–Helfand theory\nin the experimentally relevant <i>N̅</i> > 10<sup>2</sup> range. On the other hand, low-ρ models exhibit skewed <i>z</i>-distributions, and the simple χ ∝ α\nformula is no longer universally applicable to them. We provided a\ntable containing the linear χ(α) mapping; if that mapping\nis used, all models with a symmetric <i>z</i>-distribution\nhave the phase behavior depending solely on <i>N̅</i> and <i>χN</i>. Our results will enable a simple\ncomparison between a broad family of coarse-grained models and experiments\nand can guide block copolymer model building. This, in turn, will\nfacilitate the screening of new block copolymer morphologies and support\nmaterial design.