1
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Willis JD, Matsen MW. Bicontinuous microemulsion in binary blends of complementary diblock copolymers. J Chem Phys 2024; 160:024906. [PMID: 38193556 DOI: 10.1063/5.0185556] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2023] [Accepted: 12/19/2023] [Indexed: 01/10/2024] Open
Abstract
The phase behavior of binary blends of AB diblock copolymers of compositions f and 1 - f is examined using field-theoretic simulations. Highly asymmetric compositions (i.e., f ≈ 0) behave like homopolymer blends macrophase separating into coexisting A- and B-rich phases as the segregation is increased, whereas more symmetric diblocks (i.e., f ≈ 0.5) microphase separate into an ordered lamellar phase. In self-consistent field theory, these behaviors are separated by a Lifshitz critical point at f = 0.2113. However, its lower critical dimension is believed to be four, which implies that the Lifshitz point should be destroyed by fluctuations. Consistent with this, it is found to transform into a tricritical point. Furthermore, the highly swollen lamellar phase near the mean-field Lifshitz point disorders into a bicontinuous microemulsion (BμE), consisting of large interpenetrating A- and B-rich microdomains. BμE has been previously reported in ternary blends of AB diblock copolymer with its parent A- and B-type homopolymers, but in that system the homopolymers have a tendency to macrophase separate. Our alternative system for creating BμE is free of this macrophase separation.
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Affiliation(s)
- J D Willis
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - M W Matsen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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2
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Vigil D, Zhang A, Delaney KT, Fredrickson GH. Phase Separation, Reaction Equilibrium, and Self-Assembly in Binary Telechelic Homopolymer Blends. Macromolecules 2023; 56:9994-10005. [PMID: 38161325 PMCID: PMC10753893 DOI: 10.1021/acs.macromol.3c01653] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2023] [Revised: 10/30/2023] [Accepted: 11/28/2023] [Indexed: 01/03/2024]
Abstract
We study a binary blend of telechelic homopolymers that can form reversible AB-type bonds at the chain ends. Reversibly bonding polymers display novel material properties, including thermal tunability and self-healing, that are not found in conventional covalently bonded polymers. Previous studies of reversibly bonding polymer systems have been limited by the computational demand of accounting for an infinite number of possible reaction products in a spatially inhomogeneous, self-assembled structure. We demonstrate that newly developed theoretical models and numerical methods enable the simultaneous computation of phase equilibrium, reaction equilibrium, and self-assembly via self-consistent field theory. Phase diagrams are computed at a variety of physically relevant conditions and are compared with nonreactive analogues as well as previous experimental studies of telechelic polymer blends.
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Affiliation(s)
- Daniel
L. Vigil
- Department
of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
| | - Amy Zhang
- Department
of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
| | - Kris T. Delaney
- Materials
Research Laboratory, University of California, Santa Barbara, California 93106, United States
| | - Glenn H. Fredrickson
- Department
of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
- Materials
Research Laboratory, University of California, Santa Barbara, California 93106, United States
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3
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Quah T, Delaney KT, Fredrickson GH. Assessment of the partial saddle point approximation in field-theoretic polymer simulations. J Chem Phys 2023; 159:164103. [PMID: 37873956 DOI: 10.1063/5.0173047] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2023] [Accepted: 10/06/2023] [Indexed: 10/25/2023] Open
Abstract
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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Affiliation(s)
- Timothy Quah
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA
| | - Kris T Delaney
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
| | - Glenn H Fredrickson
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
- Materials Department, University of California, Santa Barbara, California 93106, USA
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4
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McGarrigle EC, Delaney KT, Balents L, Fredrickson GH. Emergence of a Spin Microemulsion in Spin-Orbit Coupled Bose-Einstein Condensates. PHYSICAL REVIEW LETTERS 2023; 131:173403. [PMID: 37955470 DOI: 10.1103/physrevlett.131.173403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2023] [Revised: 08/14/2023] [Accepted: 09/01/2023] [Indexed: 11/14/2023]
Abstract
We report the first numerical prediction of a "spin microemulsion"-a phase with undulating spin domains resembling classical bicontinuous oil-water-surfactant emulsions-in two-dimensional systems of spinor Bose-Einstein condensates with isotropic Rashba spin-orbit coupling. Using field-theoretic numerical simulations, we investigated the melting of a low-temperature stripe phase with supersolid character and find that the stripes lose their superfluidity at elevated temperature and undergo a Kosterlitz-Thouless-like transition into a spin microemulsion. Momentum distribution calculations highlight a thermally broadened occupation of the Rashba circle of low-energy states with macroscopic and isotropic occupation around the ring. We provide a finite-temperature phase diagram that positions the emulsion as an intermediate, structured isotropic phase with residual quantum character before transitioning at higher temperature into a structureless normal fluid.
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Affiliation(s)
- Ethan C McGarrigle
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA
| | - Kris T Delaney
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
| | - Leon Balents
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
- Canadian Institute for Advanced Research, Toronto, Ontario M5G 1M1, Canada
| | - Glenn H Fredrickson
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
- Materials Department, University of California, Santa Barbara, California 93106, USA
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5
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Sims MB, Goetze JW, Gorbea GD, Gdowski ZM, Lodge TP, Bates FS. Photocrosslinkable Polymeric Bicontinuous Microemulsions. ACS APPLIED MATERIALS & INTERFACES 2023; 15:10044-10052. [PMID: 36774627 DOI: 10.1021/acsami.2c22927] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
We present an approach to photocrosslink bicontinuous microemulsions derived from ternary blends of poly(methoxyethyl acrylate) (PM, Mn = 4200 g/mol), poly(hexyl methacrylate-co-coumarin methacrylate) (PHC, Mn = 6800 g/mol), and PM-b-PHC diblock polymer (Mn = 19,400 g/mol) in a phase-selective manner, enabling structural characterization at an unprecedented level of detail. This strategy utilizes the [2 + 2] photodimerization reaction of coumarin derivatives to covalently crosslink blends without the use of harsh reagents or disruptive thermal treatment, thus preserving the intricate network structure throughout curing. The resulting crosslinked bicontinuous microemulsions exhibited rubbery behavior at elevated temperatures, achieving an elastic shear modulus of nearly 1 MPa at 70 °C, owing to the presence of the three-dimensional co-continuous network morphology. The dimensional stabilization afforded by crosslinking further allowed the microstructure to be directly imaged by scanning electron microscopy and atomic force microscopy. Contrary to recent theoretical findings, the BμE appears in a wide temperature and compositional window, suggesting that it is a robust feature of these blends. As a proof of concept demonstrating both the utility of bicontinuous microemulsion-derived materials and versatility of this strategy toward broader applications in energy storage and transport, the uncrosslinked portion of a cured blend was extracted by washing and replaced with an ionic liquid; the resultant heterogeneous solid electrolyte exhibited a room-temperature conductivity of 2 mS/cm, approximately one-quarter that of the pure ionic liquid.
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Affiliation(s)
- Michael B Sims
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Joshua W Goetze
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Gabriela Diaz Gorbea
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Zachary M Gdowski
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Timothy P Lodge
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
- Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Frank S Bates
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
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6
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Zhang B, Cui S, Lodge TP, Bates FS. Structure and Phase Behavior of Bottlebrush Diblock Copolymer-Linear Homopolymer Ternary Blends. Macromolecules 2023. [DOI: 10.1021/acs.macromol.2c02434] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
Affiliation(s)
- Bo Zhang
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Shuquan Cui
- Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Timothy P. Lodge
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
- Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Frank S. Bates
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
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7
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Schmid F. Understanding and Modeling Polymers: The Challenge of Multiple Scales. ACS POLYMERS AU 2022. [DOI: 10.1021/acspolymersau.2c00049] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Affiliation(s)
- Friederike Schmid
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128Mainz, Germany
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8
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Fredrickson GH, Xie S, Edmund J, Le ML, Sun D, Grzetic DJ, Vigil DL, Delaney KT, Chabinyc ML, Segalman RA. Ionic Compatibilization of Polymers. ACS POLYMERS AU 2022; 2:299-312. [PMID: 36267546 PMCID: PMC9576261 DOI: 10.1021/acspolymersau.2c00026] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Revised: 07/13/2022] [Accepted: 07/14/2022] [Indexed: 11/29/2022]
Affiliation(s)
- Glenn H. Fredrickson
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States
| | - Shuyi Xie
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States
| | - Jerrick Edmund
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
| | - My Linh Le
- Department of Materials, University of California, Santa Barbara, California 93106, United States
| | - Dan Sun
- Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States
| | - Douglas J. Grzetic
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States
| | - Daniel L. Vigil
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
| | - Kris T. Delaney
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States
| | - Michael L. Chabinyc
- Department of Materials, University of California, Santa Barbara, California 93106, United States
| | - Rachel A. Segalman
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
- Department of Materials, University of California, Santa Barbara, California 93106, United States
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9
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Zheng C, Zhang B, Bates FS, Lodge TP. Self-Assembly of Partially Charged Diblock Copolymer-Homopolymer Ternary Blends. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c00518] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Caini Zheng
- Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Bo Zhang
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Frank S. Bates
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Timothy P. Lodge
- Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, United States
- Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States
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10
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Kulshreshtha A, Hayward RC, Jayaraman A. Impact of Composition and Placement of Hydrogen-Bonding Groups along Polymer Chains on Blend Phase Behavior: Coarse-Grained Molecular Dynamics Simulation Study. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c00055] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Arjita Kulshreshtha
- Department of Chemical and Biomolecular Engineering, Colburn Laboratory, University of Delaware, 150 Academy Street, Newark, Delaware 19716, United States
| | - Ryan C. Hayward
- Department of Chemical and Biological Engineering, University of Colorado, 596 UCB, Boulder, Colorado 80309, United States
| | - Arthi Jayaraman
- Department of Chemical and Biomolecular Engineering, Colburn Laboratory, University of Delaware, 150 Academy Street, Newark, Delaware 19716, United States
- Department of Materials Science and Engineering, University of Delaware, 201 Dupont Hall, Newark, Delaware 19716, United States
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11
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Müller M, Abetz V. Nonequilibrium Processes in Polymer Membrane Formation: Theory and Experiment. Chem Rev 2021; 121:14189-14231. [PMID: 34032399 DOI: 10.1021/acs.chemrev.1c00029] [Citation(s) in RCA: 37] [Impact Index Per Article: 12.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/13/2023]
Abstract
Porous polymer and copolymer membranes are useful for ultrafiltration of functional macromolecules, colloids, and water purification. In particular, block copolymer membranes offer a bottom-up approach to form isoporous membranes. To optimize permeability, selectivity, longevity, and cost, and to rationally design fabrication processes, direct insights into the spatiotemporal structure evolution are necessary. Because of a multitude of nonequilibrium processes in polymer membrane formation, theoretical predictions via continuum models and particle simulations remain a challenge. We compiled experimental observations and theoretical approaches for homo- and block copolymer membranes prepared by nonsolvent-induced phase separation and highlight the interplay of multiple nonequilibrium processes─evaporation, solvent-nonsolvent exchange, diffusion, hydrodynamic flow, viscoelasticity, macro- and microphase separation, and dynamic arrest─that dictates the complex structure of the membrane on different scales.
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Affiliation(s)
- Marcus Müller
- Georg-August Universität, Institut für Theoretische Physik, 37073 Göttingen, Germany
| | - Volker Abetz
- Helmholtz-Zentrum Hereon, Institut für Membranforschung, 21502 Geesthacht, Germany.,Universität Hamburg, Institut für Physikalische Chemie, 20146 Hamburg, Germany
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12
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Classical Critical Behaviors and Ginzburg Criteria for Polymer Mixtures. Macromol Res 2021. [DOI: 10.1007/s13233-021-2083-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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13
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Vigil DL, Delaney KT, Fredrickson GH. Quantitative Comparison of Field-Update Algorithms for Polymer SCFT and FTS. Macromolecules 2021. [DOI: 10.1021/acs.macromol.1c01804] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Daniel L. Vigil
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
| | - Kris T. Delaney
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States
| | - Glenn H. Fredrickson
- Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States
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14
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Dhamankar S, Webb MA. Chemically specific coarse‐graining of polymers: Methods and prospects. JOURNAL OF POLYMER SCIENCE 2021. [DOI: 10.1002/pol.20210555] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Satyen Dhamankar
- Department of Chemical and Biological Engineering Princeton University Princeton New Jersey USA
| | - Michael A. Webb
- Department of Chemical and Biological Engineering Princeton University Princeton New Jersey USA
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15
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Matsen MW, Beardsley TM. Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation. Polymers (Basel) 2021; 13:2437. [PMID: 34372040 PMCID: PMC8347900 DOI: 10.3390/polym13152437] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2021] [Revised: 07/16/2021] [Accepted: 07/20/2021] [Indexed: 11/16/2022] Open
Abstract
Field-theoretic simulations (FTS) provide an efficient technique for investigating fluctuation effects in block copolymer melts with numerous advantages over traditional particle-based simulations. For systems involving two components (i.e., A and B), the field-based Hamiltonian, Hf[W-,W+], depends on a composition field, W-(r), that controls the segregation of the unlike components and a pressure field, W+(r), that enforces incompressibility. This review introduces researchers to a promising variant of FTS, in which W-(r) fluctuates while W+(r) tracks its mean-field value. The method is described in detail for melts of AB diblock copolymer, covering its theoretical foundation through to its numerical implementation. We then illustrate its application for neat AB diblock copolymer melts, as well as ternary blends of AB diblock copolymer with its A- and B-type parent homopolymers. The review concludes by discussing the future outlook. To help researchers adopt the method, open-source code is provided that can be run on either central processing units (CPUs) or graphics processing units (GPUs).
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Affiliation(s)
- Mark W. Matsen
- Department of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada
- Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada
- Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, ON N2L 3G1, Canada;
| | - Thomas M. Beardsley
- Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, ON N2L 3G1, Canada;
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16
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Xie S, Zhang B, Bates FS, Lodge TP. Phase Behavior of Salt-Doped A/B/AB Ternary Polymer Blends: The Role of Homopolymer Distribution. Macromolecules 2021. [DOI: 10.1021/acs.macromol.1c00928] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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17
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Weyman A, Mavrantzas VG, Öttinger HC. Field-theoretic simulations beyond δ-interactions: Overcoming the inverse potential problem in auxiliary field models. J Chem Phys 2021; 155:024106. [PMID: 34266260 DOI: 10.1063/5.0055255] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Modern field-theoretic simulations of complex fluids and polymers are constructed around a particle-to-field transformation that brings an inverse potential u-1 in the model equations. This has restricted the application of the framework to systems characterized by relatively simple pairwise interatomic interactions; for example, excluded volume effects are treated through the use of δ-function interactions. In this study, we first review available nonbonded pair interactions in field-theoretic models and propose a classification. Then, we outline the inverse potential problem and present an alternative approach on the basis of a saddle-point approximation, enabling the use of a richer set of pair interaction functions. We test our approach by using as an example the Morse potential, which finds extensive applications in particle-based simulations, and we calibrate u-1 with results from a molecular dynamics simulation. The u-1 thus obtained is consistent with the field-theoretic model equations, and when used in stand-alone self-consistent field simulations, it produces the correct fluid structure starting from a random initial state of the density field.
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Affiliation(s)
- Alexander Weyman
- Polymer Physics, Department of Materials, ETH Zurich, CH 8093 Zurich, Switzerland
| | - Vlasis G Mavrantzas
- Particle Technology Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, CH 8092 Zurich, Switzerland
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