  Directed self-assembly (DSA) of block-copolymers (BCPs) is one of the most
promising developments in the cost-effective production of nanoscale devices.
The process makes use of the natural tendency for BCP mixtures to form
nanoscale structures upon phase separation. The phase separation can be
directed through the use of chemically patterned substrates to promote the
formation of morphologies that are essential to the production of semiconductor
devices. Moreover, the design of substrate pattern can formulated as an
optimization problem for which we seek optimal substrate designs that
effectively produce given target morphologies.
  In this paper, we adopt a phase field model given by a nonlocal
Cahn--Hilliard partial differential equation (PDE) based on the minimization of
the Ohta--Kawasaki free energy, and present an efficient PDE-constrained
optimization framework for the optimal design problem. The design variables are
the locations of circular- or strip-shaped guiding posts that are used to model
the substrate chemical pattern. To solve the ensuing optimization problem, we
propose a variant of an inexact Newton conjugate gradient algorithm tailored to
this problem. We demonstrate the effectiveness of our computational strategy on
numerical examples that span a range of target morphologies. Owing to our
second-order optimizer and fast state solver, the numerical results demonstrate
five orders of magnitude reduction in computational cost over previous work.
The efficiency of our framework and the fast convergence of our optimization
algorithm enable us to rapidly solve the optimal design problem in not only
two, but also three spatial dimensions.
