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Field-theoretic simulations are numerical treatments of polymer field theory models that go beyond the mean-field self-consistent field theory level and have successfully captured a range of mesoscopic phenomena. Inherent in molecularly-based field theories is a “sign problem” associated with complex-valued Hamiltonian functionals. One route to field-theoretic simulations utilizes the complex Langevin (CL) method to importance sample complex-valued field configurations to bypass the sign problem. Although CL is exact in principle, it can be difficult to stabilize in strongly fluctuating systems. An alternate approach for blends or block copolymers with two segment species is to make a “partial saddle point approximation” (PSPA) in which the stiff pressure-like field is constrained to its mean-field value, eliminating the sign problem in the remaining field theory, allowing for traditional (real) sampling methods. The consequences of the PSPA are relatively unknown, and direct comparisons between the two methods are limited. Here, we quantitatively compare thermodynamic observables, order-disorder transitions, and periodic domain sizes predicted by the two approaches for a weakly compressible model of AB diblock copolymers. Using Gaussian fluctuation analysis, we validate our simulation observations, finding that the PSPA incorrectly captures trends in fluctuation corrections to certain thermodynamic observables, microdomain spacing, and location of order-disorder transitions. For incompressible models with contact interactions, we find similar discrepancies between the predictions of CL and PSPA, but these can be minimized by regularization procedures such as Morse calibration. These findings mandate caution in applying the PSPA to broader classes of soft-matter models and systems.
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Complex Implications of Translational Invariance in Polymer Field Theory
The behavior of complex-Langevin field-theoretic simulations (CL-FTSs) of polymer liquids is sensitive to the nature of saddle-point field configurations, which are solutions of self-consistent field theory (SCFT). Recent work [Kang et al. Macromolecules 2024, 57, 3850] has shown that SCFT saddle-points with real fields are generally not isolated solutions but rather members of a low-dimensional family of continuously-connected complex-valued saddle-points sharing the same Hamiltonian value. We show that this behavior is a natural consequence of the analyticity and translational invariance of the Hamiltonian, which together demand its invariance under generalized translations by displacements with complex components. We also present a numerical algorithm that minimizes the deleterious effects of this generalized symmetry on the stability of CL-FTSs.
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- Award ID(s):
- 2103627
- PAR ID:
- 10629285
- Publisher / Repository:
- American Chemical Society
- Date Published:
- Journal Name:
- Macromolecules
- Volume:
- 57
- Issue:
- 20
- ISSN:
- 0024-9297
- Page Range / eLocation ID:
- 9900-9910
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1021/acs.macromol.4c01617
