Accelerated Pseudo-Spectral Method of Self-Consistent Field Theory via Crystallographic Fast Fourier Transform

Self-consistent field theory (SCFT)
has been proven as one of the
most successful methods for studying the phase behavior of block copolymers. In
the past decades, a number of numerical methods have been developed
for solving SCFT equations. Recently, the pseudo-spectral method based
on fast Fourier transform (FFT) has become one of the most frequently
used methods due to its versatility and high efficiency. However,
the computational cost is still rather high, especially for some complex
structures or in the strong-segregation case. To accelerate the calculation,
we introduce crystallographic FFT into the pseudo-spectral method,
which utilizes the symmetry of ordered phases. Thus, a general algorithm
is developed by making partial use of symmetry operations commonly
contained by many different space groups, leading to a speed-up of
about six times for most of the three-dimensional ordered morphologies
observed in AB-type block copolymers, including BCC, FCC, HCP, G,
D, O70, and PL phases as well as complex Frank–Kasper
phases (σ, A15, C14, C15, and Z). In addition, we demonstrate
that more efficient algorithms can be specifically designed by fully
considering symmetry operations for some complex structures. For instance,
a very large speed-up of about 30 times is achieved with a specific
algorithm for the complex Frank–Kasper σ phase with the P42/mnm space group. Besides
acceleration, the memory used by the pseudo-spectral method with crystallographic
FFT is concomitantly saved by many times.