Computationally Efficient Field-Theoretic Simulations for Block Copolymer Melts

Abstract

Field-theoretic simulations (FTS) provide fluctuation corrections to self-consistent field theory (SCFT) by simulating its field-theoretic Hamiltonian rather than applying the saddle-point approximation. Although FTS work well for ultrahigh molecular weights, they have struggled with experimentally relevant values. Here, we consider FTS for two-component (i.e., AB-type) melts, where the composition field fluctuates but the saddle-point approximation is still applied to the pressure field that enforces incompressibility. This results in real-valued fields, thereby allowing for conventional simulation methods. We discover that Langevin simulations are one to two orders of magnitude faster than previous Monte Carlo simulations, which permits us to accurately calculate the order-disorder transition of symmetric diblock copolymer melts at realistic molecular weights. This remarkable speedup will, likewise, facilitate FTS for more complicated block copolymer systems, which might otherwise be unfeasible with traditional particle-based simulations.

Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation

“…A Markov sequence of configurations for the partition function in Equation ( 26 ) is generated by evolving the composition field, , using Langevin dynamics, where thermal noise is provided by random numbers, , generated from a normal distribution of zero mean and variance. To improve accuracy, we apply the predictor-corrector algorithm [ 47 , 48 ]. A predicted field at is first obtained using Equation ( 36 ) with and then a corrected field is obtained with evaluated using the predicted field and corresponding composition.…”

“…In the Morse calibration, the nonlinear correction is calculated by fitting the peak of the structure function, , from FTS to ROL predictions for symmetric diblocks. Beardsley and Matsen [ 13 , 48 ] demonstrated the calibration for and , assuming the empirical relationship [ 4 ] between the bare of the FTS and the effective of the ROL theory. The resulting calibration, , , and , is plotted in the inset of Figure 2 .…”

“…The initial applications of FTS have largely focused on two test cases. The first one is neat diblock copolymer melts, which is a natural choice given its relative simplicity and the fact that it is the most thoroughly understood [ 13 , 33 , 34 , 40 , 42 , 48 , 50 ]. As mentioned, the greatest impact of fluctuations is on the disordered phase, and thus particular attention is paid to the order-disorder transition (ODT).…”

“…Beardsley and Matsen [ 48 ] located the ODT of symmetric diblocks using a slight variation of the technique where the ensemble average and integration are performed simultaneously [ 66 ]. Figure 3 shows the resulting free energy difference between the lamellar and disordered phases, , for diblocks of and in a cubic simulation box of size .…”

Instability of the Microemulsion Channel in Block Copolymer-Homopolymer Blends

“…By adapting the FTS to a variety of different ensembles, they found evidence for A þ B þ BμE coexistence, but the high computational costs limited the size of their simulations boxes and resulted in significant metastability issues. Since then, however, Beardsley et al [46] found a speedup of about two orders of magnitude by switching to conventional Langevin simulations. With this improvement in hand, we revisit the ternary blends.…”