Calibration of the Flory-Huggins interaction parameter in field-theoretic simulations

Abstract

Field-theoretic simulations (FTS) offer a versatile method of dealing with complicated block copolymer systems, but unfortunately they struggle to cope with the level of fluctuations typical of experiments. Although the main obstacle, an ultraviolet divergence, can be removed by renormalizing the Flory-Huggins χ parameter, this only works for unrealistically large invariant polymerization indexes, N¯. Here, we circumvent the problem by applying the Morse calibration, where a nonlinear relationship between the bare χb used in FTS and the effective χ corresponding to the standard Gaussian-chain model is obtained by matching the disordered-state structure function, S(k), of symmetric diblock copolymers to renormalized one-loop predictions. This calibration brings the order-disorder transition obtained from FTS into agreement with the universal results of particle-based simulations for values of N¯ characteristic of the experiment. In the limit of weak interactions, the calibration reduces to a linear approximation, χ ≈ z∞χb, consistent with the previous renormalization of χ for large N¯.

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

“…For instance, the transition from CPUs to GPUs has allowed the system size to be scaled up considerably [ 83 , 84 ]. A similar improvement was also achieved by switching from Monte Carlo dynamics used in the early studies [ 13 , 38 , 39 , 40 , 50 , 51 , 80 ] to the Langevin dynamics in Equation ( 36 ) [ 48 ].…”

“…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).…”

“…Consequently, they can all be mapped onto the standard Gaussian chain model (GCM) [ 5 ], which is a minimal model that treats block copolymer systems as incompressible melts of thin elastic threads that interact by pairwise contact forces. At present, the most accurate method of performing the mapping is the Morse calibration [ 4 , 6 ], which has been well demonstrated for experiments [ 7 ] and particle-based simulations [ 8 , 9 , 10 , 11 , 12 ], as well as field-theoretic simulations [ 13 ].…”

Computationally Efficient Field-Theoretic Simulations for Block Copolymer Melts

“…As it turns out, w + (r) is real valued, which then allows for the use of conventional simulation methods. Most studies apply Monte Carlo dynamics (MC-FTS), [15][16][17][18][19] but a couple have implemented standard Langevin dynamics (L-FTS). 20,21 The simulations are typically performed in a cubic simulation box of volume V = L 3 with periodic boundaries.…”