1
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Greenall MJ, Derry MJ. Temperature dependence of micelle shape transitions in copolymer solutions: the role of inter-block incompatibility. SOFT MATTER 2024; 20:3628-3634. [PMID: 38619553 DOI: 10.1039/d4sm00331d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/16/2024]
Abstract
The nature of the transition between worm-like and spherical micelles in block copolymer dispersions varies between systems. In some formulations, heating drives a transition from worms to spheres, while in other systems the same transition is induced by cooling. In addition, a sphere-worm interconversion can be accompanied either by an increase or a decrease in the core solvation, even if the direction of the temperature dependence is the same. Here, self-consistent field theory is used to provide a potential explanation of this range of behaviour. Specifically, we show that, within this model, the dependence of the transition on the incompatibility χBS of the solvophobic block B and the solvent S (the parameter most closely related to the temperature) is strongly influenced by the incompatibility χAB between B and the solvophilic block A. When χAB is small (χAB < 0.1), it is found that increasing χBS produces a transition from worm-like micelles to spheres (or, more generally, from less curved to more curved structures). When χAB is above 0.1, increasing χBS drives the system from spheres to worm-like micelles. Whether a transition is observed within a realistic range of χBS is also found to depend on the fraction of solvophilic material in the copolymer. The relevance of our calculations to experiments is discussed, and we suggest that the direction of the temperature dependence may be controlled not only by the solution behaviour of the solvophobic block (upper critical solution temperature-like versus lower critical solution temperature-like) but also by χAB.
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Affiliation(s)
- M J Greenall
- School of Mathematics and Physics, University of Lincoln, Brayford Pool, Lincoln, LN6 7TS, UK.
| | - M J Derry
- Aston Institute for Membrane Excellence, Aston University, Birmingham, B4 7ET, UK
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2
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Shi J, Huang X, Li W. The impact of intramolecular polydispersity on the self-assembly of AB n miktoarm star copolymers. Phys Chem Chem Phys 2023; 25:20032-20041. [PMID: 37462012 DOI: 10.1039/d3cp00994g] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 07/27/2023]
Abstract
The self-assembly behaviors of ABn miktoarm star copolymers as one typical type of asymmetric architecture have been studied well in the past few decades due to their deflected phase boundaries. In particular, recently, they have attracted renewed theoretical interest due to their expanded spherical phase region that stabilizes complex Frank-Kasper spherical phases. However, previous theoretical studies have never considered ABn copolymers with unequal arm lengths, which is more or less the case for synthesized copolymers. In this work, we investigate the self-assembly behaviors of ABn miktoarm star copolymers with unequal B-arms using self-consistent field theory. We propose an intramolecular polydispersity index (iĐ) to quantify the distribution of unequal B-blocks. Accordingly, we further propose a simple quantity of an effective arm number nequ = n/iĐ for quantitatively comparing the phase boundaries between various ABn copolymer samples with different arm numbers or different distributions of B-blocks. Our results indicate that different ABn copolymers with equal nequ exhibit similar phase diagrams. On the other hand, we also found that the phase boundaries of two different samples with same nequ are not exactly overlapped. We speculate that the effect of spontaneous curvature may be mainly controlled by nequ, but the packing frustration of B-blocks may also be dependent on the other quantities that are closely related to the shape of the distribution of B-arms, such as higher order polydispersity indexes.
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Affiliation(s)
- Jiahao Shi
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China.
| | - Xianbo Huang
- National-certified Enterprise Technology Center, Kingfa Science and Technology Co., Ltd, Science City, Guangzhou 510663, China.
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China.
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3
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Zhao B, Dong Q, Yang W, Xu Y. Theoretical Study of Phase Behaviors of Symmetric Linear B 1A 1B 2A 2B 3 Pentablock Copolymer. Molecules 2023; 28:molecules28083536. [PMID: 37110770 PMCID: PMC10146716 DOI: 10.3390/molecules28083536] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2023] [Revised: 04/08/2023] [Accepted: 04/16/2023] [Indexed: 04/29/2023] Open
Abstract
The nanostructures that are self-assembled from block copolymer systems have attracted interest. Generally, it is believed that the dominating stable spherical phase is body-centered cubic (BCC) in linear AB-type block copolymer systems. The question of how to obtain spherical phases with other arrangements, such as the face-centered cubic (FCC) phase, has become a very interesting scientific problem. In this work, the phase behaviors of a symmetric linear B1A1B2A2B3 (fA1 = fA2, fB1 = fB3) pentablock copolymer are studied using the self-consistent field theory (SCFT), from which the influence of the relative length of the bridging B2-block on the formation of ordered nanostructures is revealed. By calculating the free energy of the candidate ordered phases, we determine that the stability regime of the BCC phase can be replaced by the FCC phase completely by tuning the length ratio of the middle bridging B2-block, demonstrating the key role of B2-block in stabilizing the spherical packing phase. More interestingly, the unusual phase transitions between the BCC and FCC spherical phases, i.e., BCC → FCC → BCC → FCC → BCC, are observed as the length of the bridging B2-block increases. Even though the topology of the phase diagrams is less affected, the phase windows of the several ordered nanostructures are dramatically changed. Specifically, the changing of the bridging B2-block can significantly adjust the asymmetrical phase regime of the Fddd network phase.
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Affiliation(s)
- Bin Zhao
- Department of Physics, Taizhou University, Taizhou 318000, China
| | - Qingshu Dong
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Wei Yang
- Department of Physics, Taizhou University, Taizhou 318000, China
| | - Yuci Xu
- Faculty of Materials Science and Chemical Engineering, Ningbo University, Ningbo 315211, China
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4
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Vorselaars B. Efficient Langevin and Monte Carlo sampling algorithms: The case of field-theoretic simulations. J Chem Phys 2023; 158:114117. [PMID: 36948836 DOI: 10.1063/5.0131183] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/24/2023] Open
Abstract
We introduce Langevin sampling algorithms to field-theoretic simulations (FTSs) of polymers that, for the same accuracy, are ∼10× more efficient than a previously used Brownian dynamics algorithm that used predictor corrector for such simulations, over 10× more efficient than the smart Monte Carlo (SMC) algorithm, and typically over 1000× more efficient than a simple Monte Carlo (MC) algorithm. These algorithms are known as the Leimkuhler-Matthews (the BAOAB-limited) method and the BAOAB method. Furthermore, the FTS allows for an improved MC algorithm based on the Ornstein-Uhlenbeck process (OU MC), which is 2× more efficient than SMC. The system-size dependence of the efficiency for the sampling algorithms is presented, and it is shown that the aforementioned MC algorithms do not scale well with system sizes. Hence, for larger sizes, the efficiency difference between the Langevin and MC algorithms is even greater, although, for SMC and OU MC, the scaling is less unfavorable than for the simple MC.
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Affiliation(s)
- Bart Vorselaars
- School of Mathematics and Physics, University of Lincoln, Brayford Pool, Lincoln LN6 7TS, United Kingdom
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5
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He W, Wang F, Qiang Y, Pan Y, Li W, Liu M. Asymmetric Binary Spherical Phases Self-Assembled by Mixing AB Diblock/ABC Triblock Copolymers. Macromolecules 2023. [DOI: 10.1021/acs.macromol.2c02217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/13/2023]
Affiliation(s)
- Wangping He
- Department of Chemistry, Key Laboratory of Surface & Interface Science of Polymer Materials of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China
| | - Feng Wang
- Department of Chemistry, Key Laboratory of Surface & Interface Science of Polymer Materials of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China
| | - Yicheng Qiang
- Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Yuchao Pan
- Department of Chemistry, Key Laboratory of Surface & Interface Science of Polymer Materials of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Meijiao Liu
- Department of Chemistry, Key Laboratory of Surface & Interface Science of Polymer Materials of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310018, China
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6
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Zhao F, Dong Q, Li Q, Liu M, Li W. Emergence and Stability of Exotic “Binary” HCP-Type Spherical Phase in Binary AB/AB Blends. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c01957] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Affiliation(s)
- Fengmei Zhao
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Qingshu Dong
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Qingyun Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Meijiao Liu
- Department of Chemistry, Zhejiang Sci-Tech University, Hangzhou 310018, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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7
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Li L, Xu Z, Li W. Emergence of Connected Binary Spherical Structures from the Self-assembly of an AB 2C Four-Arm Star Terpolymer. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c01919] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Luyang Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Zhanwen Xu
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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8
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Zhang L, Yang J, Li W. Emergence of Multi-strand Helices from the Self-Assembly of AB-Type Multiblock Copolymer under Cylindrical Confinement. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c01651] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Lixun Zhang
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Junying Yang
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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9
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Xie O, Olsen BD. A Self-Consistent Field Theory Formalism for Sequence-Defined Polymers. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c00721] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Oliver Xie
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Bradley D. Olsen
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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10
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Yong D, Kim JU. Accelerating Langevin Field-Theoretic Simulation of Polymers with Deep Learning. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c00705] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Daeseong Yong
- Department of Physics, Ulsan National Institute of Science and Technology, 50 UNIST-gil, Ulsan 44919, Korea
| | - Jaeup U. Kim
- Department of Physics, Ulsan National Institute of Science and Technology, 50 UNIST-gil, Ulsan 44919, Korea
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11
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Li Q, Woo D, Kim JK, Li W. Truly “Inverted” Cylinders and Spheres Formed in the A(AB) 3/AC Blends of B/C Hydrogen Bonding Interactions. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c01084] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Qingyun Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200438, P. R. China
| | - Dokyung Woo
- National Creative Research Initiative Center for Smart Block Copolymers, Department of Chemical Engineering, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea
| | - Jin-Kon Kim
- National Creative Research Initiative Center for Hybrid Nano Materials by High-level Architectural Design of Block Copolymer, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200438, P. R. China
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12
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Lin D, Yu HY. Deep learning and inverse discovery of polymer self-consistent field theory inspired by physics-informed neural networks. Phys Rev E 2022; 106:014503. [PMID: 35974507 DOI: 10.1103/physreve.106.014503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2022] [Accepted: 07/06/2022] [Indexed: 06/15/2023]
Abstract
We devise a deep learning solver inspired by physics-informed neural networks (PINNs) to tackle the polymer self-consistent field theory (SCFT) equations for one-dimensional AB-diblock copolymers. The PINNs framework comprises two parallel feedforward neural networks that separately represent the segmental partition functions and self-consistent chemical potential fields. The two networks are coupled through a loss function incorporating the governing equation, initial and boundary conditions, and the incompressibility constraint. To avoid the metastable homogeneous solution, the network parameters are initialized based on known self-consistent fields obtained from the numerical pseudospectral method. For copolymers of length N at a given volume fraction of A block (f) and the reduced Flory-Huggins interaction parameter (χN), the minimization of the loss function leads to the converged network parameters that successfully capture the stable lamellar phase. The periodicity of the lamellar structure is correctly reproduced for the explored sets of [f,χN], irrespective of the presumed computational domain size for initialization. Moreover, the proposed PINNs are applicable to the inverse discovery of the interaction parameter and the embedded chemical potential fields for an observed structure. This capability of solving the inverse SCFT problem demonstrates the potential of using PINNs to accelerate the exploration of new polymeric materials.
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Affiliation(s)
- Danny Lin
- Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan
| | - Hsiu-Yu Yu
- Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan
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13
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Dong L, Wylie K, Nabae Y, Hayakawa T. An experimental and theoretical investigation into the self-assembly of a chemically modified high- χ coil-rod diblock copolymer. RSC Adv 2022; 12:17950-17958. [PMID: 35765337 PMCID: PMC9204556 DOI: 10.1039/d2ra02536a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2022] [Accepted: 06/02/2022] [Indexed: 11/23/2022] Open
Abstract
A precursor diblock copolymer with a silicon backbone, polystyrene-block-poly(methylvinylsiloxane), was synthesized, and 1H,1H,2H,2H-perfluorodecanethiol was quantitatively introduced into the backbone via a thiol-ene reaction to yield a novel coil–rod diblock copolymer, poly(styrene-block-poly(2-((3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,10-heptadecafluorodecyl)thio)ethyl)methylsiloxane). The ultra-hydrophobicity of the introduced perfluoroalkyl side chain enhanced the segregation between counter-blocks and significantly increased the χ value, which is essential for minimizing the size of self-assembled domains for lithographic applications. Thus, self-assembled domains with a minimal spacing of approximately 10 nm were formed. A hexagonally packed array with significant anisotropy was observed in the self-assembled morphology by small-angle X-ray scattering and transmission electron microscopy. Such an array was precisely reproduced by modified self-consistent field theory (SCFT) calculation developed for the coil–rod structure. Furthermore, the phase diagram was estimated, and the morphological dependence on the relative scale of the rod unit was investigated by SCFT prediction. A high-χ coil-rod diblock copolymer capable of forming unique anisotropic hexagonally packed cylindrical domains evidenced by experimental and theoretical study.![]()
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Affiliation(s)
- Lei Dong
- School of Materials Science and Engineering, Southeast University Nanjing 211189 China
| | - Kevin Wylie
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology S8-813, 2-12-1-S8-36, Ookayama, Meguro-ku Tokyo 152-8552 Japan
| | - Yuta Nabae
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology S8-813, 2-12-1-S8-36, Ookayama, Meguro-ku Tokyo 152-8552 Japan
| | - Teruaki Hayakawa
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology S8-813, 2-12-1-S8-36, Ookayama, Meguro-ku Tokyo 152-8552 Japan
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14
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Cai Y. Tilt Modulus of Bilayer Membranes Self-Assembled from Rod-Coil Diblock Copolymers. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2022; 38:5820-5828. [PMID: 35437996 DOI: 10.1021/acs.langmuir.2c00427] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
Quantitatively understanding membrane fission and fusion requires a mathematical model taking their underlying elastic degrees of freedom, such as the molecule's tilt, into account. Hamm-Kozlov's model is such a framework that includes a tilt modulus along with the bending modulus and Gaussian modulus. This paper investigates the tilt modulus of liquid-crystalline bilayer membranes by applying self-consistent field theory. Unlike the widely used method in molecular dynamics simulation which extracts the tilt modulus by simulating bilayer buckles with various single modes, we introduce a tilt constrain term in the free energy to stabilize bilayers with various tilt angles. Fitting the energy curve as a function of the tilt angle to Hamm-Kozlov's elastic energy allows us to extract the tilt modulus directly. Based on this novel scheme and focused on the bilayers self-assembled from rod-coil diblock copolymers, we carry out a systematic study of the dependence of the tensionless A-phase bilayer's tilt modulus on the microscopic parameters.
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Affiliation(s)
- Yongqiang Cai
- School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, 100875 Beijing, China
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15
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Spencer RKW, Ha BY, Saeidi N. Interplay between nematic and cholesteric interactions in self-consistent field theory. Phys Rev E 2022; 105:054501. [PMID: 35706232 DOI: 10.1103/physreve.105.054501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2021] [Accepted: 03/29/2022] [Indexed: 06/15/2023]
Abstract
Chirality is a design feature of a number of biomolecules (e.g., collagen). In these molecules, cholesteric (chiral-nematic) behavior emerges from a combination of the tendency for the biopolymers to align (nematic interactions) and for the alignment direction to change with position, rotating around an axis normal to the alignment direction. This paper presents self-consistent field theory (SCFT) of chiral-nematic polymers, which takes into account polymer flexibility and the orientational degrees of freedom of polymer segments. Using the resulting SCFT, we construct a phase diagram showing regions of stability for isotropic, nematic, and cholesteric phases. Furthermore, we find that nematic interactions can stabilize the cholesteric phase, pushing the isotropic-cholesteric phase transition to lower cholesteric interaction strength, until the isotropic-nematic-cholesteric triple point is reached.
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Affiliation(s)
- Russell K W Spencer
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
| | - Bae-Yeun Ha
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
| | - Nima Saeidi
- Department of Surgery, The Center for Engineering in Medicine, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts 02114, USA
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16
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Zhang X, Li W. Periodic Patchy Spheres Self-Assembled by A mBCA n' Multiblock Terpolymers. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2022; 38:4407-4414. [PMID: 35352945 DOI: 10.1021/acs.langmuir.2c00139] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
We have designed AmBCAn' multiblock terpolymers and studied their self-assembly using self-consistent field theory, aiming to generate the periodically arranged patchy spheres and thus to clarify the regulation mechanism of the number of patches. A number of two-dimensional phase diagrams are constructed for three typical architectures A2BCA2', A2BCA3', and A3BCA2'. Four kinds of stable patchy spheres with the number of patches as 2 (S2), 4 (S4), 5 (S5), and 6 (S6) are obtained. These phases follow a common transition sequence of S2 → S4 → S5 → S6 along with the increasing of the volume fraction of C-block (fC), which forms the core sphere patched with B-domains. Moreover, the S6 phase exhibits the widest stability window, while S5 has the narrowest one. The increased arms of A'-blocks in A2BCA3' architecture deflect the phase boundaries toward large fC and accordingly expand the regions of these patchy spheres due to the amplified effect of spontaneous curvature. In contrast, the increased arms of A-blocks in A3BCA2' remarkably expands the window of S6 but narrows those of the other patchy spheres, which is mainly caused by increased packing frustration resulting from the reduced extension of the more divided A-blocks. The widest window of the S6 phase reaches ΔfC ∼ 0.13, which is readily accessed by experiment. Our work not only demonstrates a self-assembly strategy to engineer the patchy spheres, but also sheds light on the regulation mechanism of the patchy number.
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Affiliation(s)
- Xiaohui Zhang
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai, 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai, 200433, China
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17
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Spencer RKW, Ha BY, Saeidi N. Self-consistent field theory of chiral nematic worm-like chains. J Chem Phys 2022; 156:114902. [PMID: 35317576 PMCID: PMC8934192 DOI: 10.1063/5.0078937] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
Many macromolecules of biological and technological interest are both chiral and semi-flexible. DNA and collagen are good examples. Such molecules often form chiral nematic (or cholesteric) phases, as is well-documented in collagen and chitin. This work presents a method for studying cholesteric phases in the highly successful self-consistent field theory of worm-like chains, offering a new way of studying many biologically relevant molecules. The method involves an effective Hamiltonian with a chiral term inspired by the Oseen-Frank (OF) model of liquid crystals. This method is then used to examine the formation of cholesteric phases in chiral-nematic worm-like chains as a function of polymer flexibility, as well as the optimal cholesteric pitch and distribution of polymer segment orientations. Our approach not only allows for the determination of the isotropic-cholesteric transition and segment distributions, beyond what the OF model promises, but also explicitly incorporates polymer flexibility into the study of the cholesteric phase, offering a more complete understanding of the behavior of semiflexible chiral-nematic polymers.
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Affiliation(s)
- Russell K. W. Spencer
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Bae-Yeun Ha
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Authors to whom correspondence should be addressed: and
| | - Nima Saeidi
- Department of Surgery, The Center for Engineering in Medicine (CEM), Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts 02114, USA
- Authors to whom correspondence should be addressed: and
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18
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Yang J, Dong Q, Liu M, Li W. Universality and Specificity in the Self-Assembly of Cylinder-Forming Block Copolymers under Cylindrical Confinement. Macromolecules 2022. [DOI: 10.1021/acs.macromol.1c02504] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Junying Yang
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Qingshu Dong
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Meijiao Liu
- Department of Chemistry, Zhejiang Sci-Tech University, Hangzhou 310018, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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19
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Sillaste S, Thompson RB. Molecular Bonding in an Orbital-Free-Related Density Functional Theory. J Phys Chem A 2022; 126:325-332. [PMID: 34994568 DOI: 10.1021/acs.jpca.1c07128] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
A density functional theory based on polymer self-consistent field theory is applied to systems of two atoms in order to show that this approach is capable of predicted molecular bonding. Periodic table elements from hydrogen up to neon are examined and homonuclear diatomic molecules are found to form for H2, N2, O2, and F2, in agreement with known results. The heteronuclear molecules CO and HF, which are known to exist under ambient conditions, are also found to be stable. Bond lengths for most of these molecules agree with experimental results to within less than 8%, with the exception of O2 and F2 which deviate more significantly. The bonding energy for H2 is given and is within 16% of the known value, but fundamental vibrational frequencies do not agree well with experiment. The main approximations of the theory are very simple and include a Fermi-Amaldi correction to the electron-electron interaction to account for self-interactions and a basic expression for the Pauli potential to account for the exclusion principle. The self-consistent equations are solved in terms of basis functions that encode the cylindrical symmetry of diatomic molecules. Since orbitals are not used, the approach is related to orbital-free density functional theory.
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Affiliation(s)
- Spencer Sillaste
- Department of Physics & Astronomy and Waterloo Institute for Nanotechnology, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
| | - Russell B Thompson
- Department of Physics & Astronomy and Waterloo Institute for Nanotechnology, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
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20
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Zhao F, Xu Z, Li W. Self-Assembly of Asymmetric Diblock Copolymers under the Spherical Confinement. Macromolecules 2021. [DOI: 10.1021/acs.macromol.1c02250] [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)
- Fengmei Zhao
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Zhanwen Xu
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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21
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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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22
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Chen G, Zhang H, Lu T, Jiang Y. The stress deformation response influenced by the chain rigidity for mesostructures in diblock copolymers. Phys Chem Chem Phys 2021; 23:22992-23004. [PMID: 34611676 DOI: 10.1039/d1cp03159g] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A self-consistent field theory formalism based on the wormlike chain model is developed to investigate the stress-strain relation for mesostructures in diblock copolymers under the influence of chain rigidity, involving the adjustable simulation cell in the non-orthogonal coordinates by means of optimization of free energy. We elucidate the effect of the chain persistency broadly spanning from the Gaussian chain to the rigid rodlike chain on the elastic response of mesophases that deviate from the initial equilibrium structures. We analytically and numerically demonstrate that our current approach in the long chain limit recovers to the Gaussian-chain-based theory. Being ascribed to the distinct conformational behaviors for flexible chains and rigid rodlike chains, the tensile and compressive stresses applied to lamellae exhibit asymmetric deformation behaviors and the shear stress applied to the initial equilibrium hexagonal cylinders results in noticeable deviations in the shape and spatial arrangement of cylindroids for various chain rigidity values. For the zero stress, in addition, our approach can be straightforwardly utilized to explore the optimal size and shape of the simulation cell in order to achieve a stress free configuration of systems.
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Affiliation(s)
- Gaohang Chen
- School of Chemistry and Key Laboratory of Bio-Inspired Smart Interfacial Science and Technology of Ministry of Education and Center of Soft Matter Physics and Its Applications, Beihang University, Beijing 100191, China. .,School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
| | - Hui Zhang
- School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
| | - Teng Lu
- Computer Network Information Center of the Chinese Academy of Sciences, Beijing 100190, China
| | - Ying Jiang
- School of Chemistry and Key Laboratory of Bio-Inspired Smart Interfacial Science and Technology of Ministry of Education and Center of Soft Matter Physics and Its Applications, Beihang University, Beijing 100191, China. .,Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing 100191, China
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23
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Magruder BR, Dorfman KD. The C36 Laves phase in diblock polymer melts. SOFT MATTER 2021; 17:8950-8959. [PMID: 34550147 DOI: 10.1039/d1sm01063h] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
The C14 and C15 Laves phases form as micelle packing structures in many types of soft matter, but the related C36 phase, which consists of alternating C14-type and C15-type layers, has not been observed in any such system. To understand this absence in the context of diblock polymers, we used self-consistent field theory to relate the morphology and energetics of C36 to other known mesophases. Two case studies were conducted: blends of AB diblock polymers with A homopolymers (where A forms the micelle core), in which C14 and C15 have stability windows, and neat AB diblock melts, in which Laves phases are metastable. Laves phases exhibit nearly identical micelle morphologies and nearly degenerate free energies, with the free energy of C36 being a near-perfect bisector of the C14 and C15 free energies in all cases, revealing an intrinsic symmetry in free energy that is attributed solely to the structural relationship between the phases in which the packing of C36 is intermediate between C14 and C15. Based on this connection between structure and free energy, C36 is thus not expected to form in flexible diblock polymers, since C14 and C15 can always form instead via facile mass transfer.
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Affiliation(s)
- Benjamin R Magruder
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE, Minneapolis, MN 55455, USA.
| | - Kevin D Dorfman
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE, Minneapolis, MN 55455, USA.
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24
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Qiang Y, Li W. Accelerated Method of Self-Consistent Field Theory for the Study of Gaussian Ring-Type Block Copolymers. Macromolecules 2021. [DOI: 10.1021/acs.macromol.1c01683] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Yicheng Qiang
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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25
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Partition and selectivity of electrolytes in cylindrical nanopores with heterogeneous surface charge. J Mol Liq 2021. [DOI: 10.1016/j.molliq.2021.116839] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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26
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Li Q, Li W. Expanding Alternating Spherical and Cylindrical Regions by Tailoring Binary Symmetric ABC/ABC Blends. Macromolecules 2021. [DOI: 10.1021/acs.macromol.1c01559] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Qingyun Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200438, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200438, China
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27
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Case LJ, Delaney KT, Fredrickson GH, Bates FS, Dorfman KD. Open-source platform for block polymer formulation design using particle swarm optimization. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2021; 44:115. [PMID: 34532757 DOI: 10.1140/epje/s10189-021-00123-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2021] [Accepted: 09/06/2021] [Indexed: 06/13/2023]
Abstract
Facile exploration of large design spaces is critical to the development of new functional soft materials, including self-assembling block polymers, and computational inverse design methodologies are a promising route to initialize this task. We present here an open-source software package coupling particle swarm optimization (PSO) with an existing open-source self-consistent field theory (SCFT) software for the inverse design of self-assembling block polymers to target bulk morphologies. To lower the barrier to use of the software and facilitate exploration of novel design spaces, the underlying SCFT calculations are seeded with algorithmically generated initial fields for four typical morphologies: lamellae, network phases, cylindrical phases, and spherical phases. In addition to its utility within PSO, the initial guess tool also finds generic applicability for stand-alone SCFT calculations. The robustness of the software is demonstrated with two searches for classical phases in the conformationally symmetric diblock system, as well as one search for the Frank-Kasper [Formula: see text] phase in conformationally asymmetric diblocks. The source code for both the initial guess generation and the PSO wrapper is publicly available.
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Affiliation(s)
- Logan J Case
- Department of Chemical Engineering and Materials Science, University of Minnesota - Twin Cities, 421 Washington Avenue SE, Minneapolis, MN, 55455, USA
| | - Kris T Delaney
- Department of Chemical Engineering and Materials Research Laboratory, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA
| | - Glenn H Fredrickson
- Department of Chemical Engineering and Materials Research Laboratory, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA
| | - Frank S Bates
- Department of Chemical Engineering and Materials Science, University of Minnesota - Twin Cities, 421 Washington Avenue SE, Minneapolis, MN, 55455, USA
| | - Kevin D Dorfman
- Department of Chemical Engineering and Materials Science, University of Minnesota - Twin Cities, 421 Washington Avenue SE, Minneapolis, MN, 55455, USA.
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28
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Xie J, Li Y, Shi A. Binary Blends of Diblock Copolymers: An Efficient Route to Complex Spherical Packing Phases. MACROMOL THEOR SIMUL 2021. [DOI: 10.1002/mats.202100053] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023]
Affiliation(s)
- Jiayu Xie
- Department of Physics and Astronomy McMaster University 1280 Main Street West Hamilton Ontario L8S 4M1 Canada
| | - Yu Li
- Department of Physics and Astronomy McMaster University 1280 Main Street West Hamilton Ontario L8S 4M1 Canada
| | - An‐Chang Shi
- Department of Physics and Astronomy McMaster University 1280 Main Street West Hamilton Ontario L8S 4M1 Canada
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29
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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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30
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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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31
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Zhao B, Liu MJ, Wang C, Chen YC, Xu YC. Enlarged Stable Phase Region of Hybrid Lamella-sphere Phase Enabled by A1B1A2(B2)m Branched Tetrablock Copolymer. CHINESE JOURNAL OF POLYMER SCIENCE 2021. [DOI: 10.1007/s10118-021-2554-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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32
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Wang X, Li S, Cai Y. Analytical Calculation of the Elastic Moduli of Self-Assembled Liquid-Crystalline Bilayer Membranes. J Phys Chem B 2021; 125:5309-5320. [PMID: 33989496 DOI: 10.1021/acs.jpcb.1c01116] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Liquid-crystalline orders are ubiquitous in membranes and could significantly affect the elastic properties of the self-assembled bilayers. Calculating the free energy of bilayer membranes with different geometries and fitting them to their theoretical expressions allow us to extract the elastic moduli, such as the bending modulus and Gaussian modulus. However, this procedure is time-consuming for liquid-crystalline bilayers. In paper reports a novel method to calculate the elastic moduli of the self-assembled liquid-crystalline bilayers within the self-consistent field theory framework. Based on the asymptotic expansion method, we derive the analytical expression of the elastic moduli, which reduces the computational cost significantly. Numerical simulations illustrate the validity and efficiency of the proposed method.
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Affiliation(s)
- Xiaoyuan Wang
- School of Mathematics and Statistics, Guizhou University, Huaxi District, 550025 Guiyang, China
| | - Sirui Li
- School of Mathematics and Statistics, Guizhou University, Huaxi District, 550025 Guiyang, China.,School of Mathematical Sciences, Zhejiang University, 886 Yuhang Road, Xihu District, 310027 Hangzhou, China
| | - Yongqiang Cai
- School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, 100875 Beijing, China
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33
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Zhao B, Wang C, Chen Y, Liu M. Frank-Kasper Phases Self-Assembled from a Linear A 1B 1A 2B 2 Tetrablock Copolymer. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2021; 37:5642-5650. [PMID: 33929198 DOI: 10.1021/acs.langmuir.1c00503] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The Frank-Kasper (FK) phases self-assembled from block copolymer systems have attracted abiding interest. In this work, the formation mechanism of the complex FK phases from the self-assembly of simple A1B1A2B2 tetrablock copolymers is investigated using self-consistent field theory (SCFT). For a typical set of parameter spaces, we utilize SCFT to construct a number of phase diagrams. In these phase diagrams, the FK phases exhibit a notable stability region. The stable region of the FK phases reveals that the distribution of A1 and A2 blocks can be precisely regulated by tuning the ratio of the A1/A2 block, wherein the long A1 blocks can aggregate within the "core" while the short A2 blocks can form the "shell" of a spherical domain in the FK phases, respectively, to accommodate the sizes and shapes of the spherical domains in the complex spherical packing phases. Besides, we also demonstrate that the existence of the B2 block plays a crucial factor to stabilize the FK phases.
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Affiliation(s)
- Bin Zhao
- School of Big Data Science, Taizhou University, Taizhou 318000, China
- Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Chao Wang
- School of Big Data Science, Taizhou University, Taizhou 318000, China
| | - Yingcai Chen
- School of Big Data Science, Taizhou University, Taizhou 318000, China
| | - Meijiao Liu
- Department of Chemistry, Zhejiang Sci-Tech University, Hangzhou 310018, China
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34
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Lai CT, Shi A. Binary Blends of Diblock Copolymers: An Effective Route to Novel Bicontinuous Phases. MACROMOL THEOR SIMUL 2021. [DOI: 10.1002/mats.202100019] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Affiliation(s)
- Chi To Lai
- Department of Physics & Astronomy McMaster University 1280 Main St. W, Hamilton Ontario L8S 4M1 Canada
| | - An‐Chang Shi
- Department of Physics & Astronomy McMaster University 1280 Main St. W, Hamilton Ontario L8S 4M1 Canada
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35
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Beardsley TM, Matsen MW. Fluctuation correction for the order-disorder transition of diblock copolymer melts. J Chem Phys 2021; 154:124902. [PMID: 33810684 DOI: 10.1063/5.0046167] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023] Open
Abstract
The order-disorder transition (ODT) of diblock copolymer melts is evaluated for an invariant polymerization index of N¯=104, using field-theoretic simulations (FTS) supplemented by a partial saddle-point approximation for incompressibility. For computational efficiency, the FTS are performed using the discrete Gaussian-chain model, and results are then mapped onto the continuous model using a linear approximation for the Flory-Huggins χ parameter. Particular attention is paid to the complex phase window. Results are found to be consistent with the well-established understanding that the gyroid phase extends down to the ODT. Furthermore, our simulations are the first to predict that the Fddd phase survives fluctuation effects, consistent with experiments.
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Affiliation(s)
- T M Beardsley
- Department of Chemical Engineering, Department of Physics & Astronomy, and Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - M W Matsen
- Department of Chemical Engineering, Department of Physics & Astronomy, and Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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36
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Spencer RKW, Ha BY. How a Polymer Brush Interacts with Inclusions and Alters Their Interaction. Macromolecules 2021. [DOI: 10.1021/acs.macromol.0c02179] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Affiliation(s)
- Russell K. W. Spencer
- Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Bae-Yeun Ha
- Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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37
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Seo Y, Park SY, Lee J, Kim JK, Duan C, Li W. Inverted Cylindrical Microdomains by Blending Star-Shaped and Linear Block Copolymers. Macromolecules 2021. [DOI: 10.1021/acs.macromol.0c02037] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Yeseong Seo
- National Creative Research Initiative Center for Smart Block Copolymers, Department of Chemical Engineering, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea
| | - So Yeong Park
- National Creative Research Initiative Center for Smart Block Copolymers, Department of Chemical Engineering, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea
| | - Jaeyong Lee
- National Creative Research Initiative Center for Smart Block Copolymers, Department of Chemical Engineering, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea
| | - Jin Kon Kim
- National Creative Research Initiative Center for Smart Block Copolymers, Department of Chemical Engineering, Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Republic of Korea
| | - Chao Duan
- Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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38
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Li C, Dong Q, Li W. Largely Tunable Asymmetry of Phase Diagrams of A(AB) n Miktoarm Star Copolymer. Macromolecules 2020. [DOI: 10.1021/acs.macromol.0c02272] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Congcong Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200438, China
| | - Qingshu Dong
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200438, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200438, China
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39
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Qiang Y, Li W. Accelerated Pseudo-Spectral Method of Self-Consistent Field Theory via Crystallographic Fast Fourier Transform. Macromolecules 2020. [DOI: 10.1021/acs.macromol.0c01974] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022]
Affiliation(s)
- Yicheng Qiang
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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40
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Gu X, Li W. Impact of Thin-Film Confinement on the Packing of Low-Coordinate Spheres in Bulk. Macromolecules 2020. [DOI: 10.1021/acs.macromol.0c01635] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Xueying Gu
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 2004338, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 2004338, China
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41
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Xie Q, Qiang Y, Zhang G, Li W. Emergence and Stability of Janus-Like Superstructures in an ABCA Linear Tetrablock Copolymer. Macromolecules 2020. [DOI: 10.1021/acs.macromol.0c01328] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023]
Affiliation(s)
- Qiong Xie
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Yicheng Qiang
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Guojie Zhang
- Department of Chemical Engineering, School of Chemistry and Chemical Engineering, Guangzhou University, Guangzhou 510006, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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42
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Cai Y, Li S, Shi AC. Elastic properties of self-assembled bilayer membranes: Analytic expressions via asymptotic expansion. J Chem Phys 2020; 152:244121. [PMID: 32610980 DOI: 10.1063/5.0009734] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Abstract
Bilayer membranes self-assembled from amphiphilic molecules are ubiquitous in biological and soft matter systems. The elastic properties of bilayer membranes are essential in determining the shape and structure of bilayers. A novel method to calculate the elastic moduli of the self-assembled bilayers within the framework of the self-consistent field theory is developed based on an asymptotic expansion of the order parameters in terms of the bilayer curvature. In particular, the asymptotic expansion method is used to derive analytic expressions of the elastic moduli, which allows us to design more efficient numerical schemes. The efficiency of the proposed method is illustrated by a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents.
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Affiliation(s)
- Yongqiang Cai
- Department of Mathematics, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 119077, Singapore
| | - Sirui Li
- School of Mathematics and Statistics, Guizhou University, Huaxi District, Guiyang 550025, People's Republic of China
| | - An-Chang Shi
- Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada
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Jiang J, Chen X, Yang S, Chen EQ. The size and affinity effect of counterions on self-assembly of charged block copolymers. J Chem Phys 2020; 152:124901. [PMID: 32241155 DOI: 10.1063/5.0002896] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023] Open
Abstract
The effect of counterions' size and affinity on the microphase separated morphologies of neutral-charged diblock copolymers is investigated systematically using a random phase approximation (RPA) and self-consistent field theory (SCFT). The phase diagrams as a function of χAB and fA at different counterion sizes and different affinities to neutral blocks are constructed, respectively. Stability limits calculated using the RPA are in good agreement with the disorder-body-centered cubic phase boundaries from SCFT calculations. It was found that increasing the size of counterions causes the phase diagram to shift upward and leftward, which is attributed to electrostatic interactions and the intrinsic volume of counterions. The domain size of the ordered phase shows an unexpected tendency that it decreases with increasing counterions' size. The counterions' distributions in H and G phases demonstrate that it is electrostatic interaction, instead of packing frustration, that plays a leading role in such systems. For finite size counterions, with the increase in affinity between counterions and neutral blocks, the phase diagram shifts upward, indicating the improved compatibility between different blocks. Furthermore, the affinity effect between counterions and neutral blocks can be mapped into an effective Flory parameter χAB ' = χAB + 0.27χBC.
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Affiliation(s)
- Jiadi Jiang
- Beijing National Laboratory for Molecular Sciences, Department of Polymer Science and Engineering and Key Laboratory of Polymer Chemistry and Physics of Ministry of Education, College of Chemistry, Peking University, Beijing 100871, People's Republic of China
| | - Xu Chen
- Beijing National Laboratory for Molecular Sciences, Department of Polymer Science and Engineering and Key Laboratory of Polymer Chemistry and Physics of Ministry of Education, College of Chemistry, Peking University, Beijing 100871, People's Republic of China
| | - Shuang Yang
- Beijing National Laboratory for Molecular Sciences, Department of Polymer Science and Engineering and Key Laboratory of Polymer Chemistry and Physics of Ministry of Education, College of Chemistry, Peking University, Beijing 100871, People's Republic of China
| | - Er-Qiang Chen
- Beijing National Laboratory for Molecular Sciences, Department of Polymer Science and Engineering and Key Laboratory of Polymer Chemistry and Physics of Ministry of Education, College of Chemistry, Peking University, Beijing 100871, People's Republic of China
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Abstract
This perspective addresses the development of polymer field theory for predicting the equilibrium phase behavior of block polymer melts. The approach is tailored to the high-molecular-weight limit, where universality reduces all systems to the standard Gaussian chain model, an incompressible melt of elastic threads interacting by contact forces. Using mathematical identities, this particle-based version of the model is converted to an equivalent field-based version that depends on fields rather than particle coordinates. The statistical mechanics of the field-based model is typically solved using the saddle-point approximation of self-consistent field theory (SCFT), which equates to mean field theory, but it can also be evaluated using field theoretic simulations (FTS). While SCFT has matured into one of the most successful theories in soft condensed matter, FTS are still in its infancy. The two main obstacles of FTS are the high computational cost and the occurrence of an ultraviolet divergence, but fortunately there has been recent groundbreaking progress on both fronts. As such, FTS are now well poised to become the method of choice for predicting fluctuation corrections to mean field theory.
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Affiliation(s)
- M W Matsen
- Department of Chemical Engineering, Department of Physics and Astronomy, and Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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Cheong GK, Chawla A, Morse DC, Dorfman KD. Open-source code for self-consistent field theory calculations of block polymer phase behavior on graphics processing units. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2020; 43:15. [PMID: 32086593 DOI: 10.1140/epje/i2020-11938-y] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2019] [Accepted: 02/11/2020] [Indexed: 06/10/2023]
Abstract
Self-consistent field theory (SCFT) is a powerful approach for computing the phase behavior of block polymers. We describe a fast version of the open-source Polymer Self-Consistent Field (PSCF) code that takes advantage of the massive parallelization provided by a graphical processing unit (GPU). Benchmarking double-precision calculations indicate up to 30× reduction in time to converge SCFT calculations of various diblock copolymer phases when compared to the Fortran CPU version of PSCF using the same algorithms, with the speed-up increasing with increasing unit cell size for the diblock polymer problems examined here. Where double-precision accuracy is not needed, single-precision calculations can provide speed-up of up to 60× in convergence time. These improvements in speed within an open-source format open up new vistas for SCFT-driven block polymer materials discovery by the community at large.
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Affiliation(s)
- Guo Kang Cheong
- Department of Chemical Engineering and Materials Science, University of Minnesota - Twin Cities, 421 Washington Avenue SE, 55455, Minneapolis, MN, USA
| | - Anshul Chawla
- Department of Chemical Engineering and Materials Science, University of Minnesota - Twin Cities, 421 Washington Avenue SE, 55455, Minneapolis, MN, USA
| | - David C Morse
- Department of Chemical Engineering and Materials Science, University of Minnesota - Twin Cities, 421 Washington Avenue SE, 55455, Minneapolis, MN, USA
| | - Kevin D Dorfman
- Department of Chemical Engineering and Materials Science, University of Minnesota - Twin Cities, 421 Washington Avenue SE, 55455, Minneapolis, MN, USA.
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Wylie K, Dong L, Chandra A, Nabae Y, Hayakawa T. Modifying the Interaction Parameters of a Linear ABC Triblock Terpolymer by Functionalizing the Short, Reactive Middle Block To Induce Morphological Change. Macromolecules 2020. [DOI: 10.1021/acs.macromol.9b02567] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Affiliation(s)
- Kevin Wylie
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology, 2-12-1-S8-36 Ookayama, Meguro-ku, Tokyo 152-8552, Japan
| | - Lei Dong
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology, 2-12-1-S8-36 Ookayama, Meguro-ku, Tokyo 152-8552, Japan
| | - Alvin Chandra
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology, 2-12-1-S8-36 Ookayama, Meguro-ku, Tokyo 152-8552, Japan
| | - Yuta Nabae
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology, 2-12-1-S8-36 Ookayama, Meguro-ku, Tokyo 152-8552, Japan
| | - Teruaki Hayakawa
- Department of Materials Science and Engineering, School of Materials and Chemical Technology, Tokyo Institute of Technology, 2-12-1-S8-36 Ookayama, Meguro-ku, Tokyo 152-8552, Japan
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Cai Y, Zhang P, Shi AC. Elastic properties of liquid-crystalline bilayers self-assembled from semiflexible-flexible diblock copolymers. SOFT MATTER 2019; 15:9215-9223. [PMID: 31642464 DOI: 10.1039/c9sm01844a] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
The mechanical response and shape of self-assembled bilayer membranes depend crucially on their elastic properties. Most of the studies focused on the elastic properties of fluid membranes, despite the ubiquitous presence of membranes with liquid-crystalline order. Here the elastic properties of liquid-crystalline bilayers self-assembled from diblock copolymers composed of a semiflexible block are studied theoretically. Specifically, the self-consistent field theory (SCFT) is applied to a model system composed of semiflexible-flexible diblock copolymers dissolved in flexible homopolymers that act as solvents. The free energy of self-assembled tensionless bilayer membranes in three different geometries, i.e. planar, cylindrical and spherical, is obtained by solving the SCFT equations using a hybrid method, in which the orientation-dependent functions are treated using the spherical harmonics, whereas the position-dependent operators are treated using the compact difference schemes. The bending modulus κM and Gaussian modulus κG of the bilayer are extracted from the free energies. The effects of the molecular parameters of the system, such as the chain rigidity and the orientational interaction, are systematically examined.
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Affiliation(s)
- Yongqiang Cai
- Department of Mathematics, National University of Singapore, Singapore 119076, Singapore.
| | - Pingwen Zhang
- LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China.
| | - An-Chang Shi
- Department of Physics and Astronomy, McMaster University, Hamilton L8S 4M1, Ontario, Canada.
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Beardsley TM, Spencer RKW, Matsen MW. Computationally Efficient Field-Theoretic Simulations for Block Copolymer Melts. Macromolecules 2019. [DOI: 10.1021/acs.macromol.9b01904] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
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49
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Zhang Q, Qiang Y, Duan C, Li W. Single Helix Self-Assembled by Frustrated ABC 2 Branched Terpolymers. Macromolecules 2019. [DOI: 10.1021/acs.macromol.9b00110] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Qi Zhang
- State Key Laboratory of Molecular Engineering of Polymers, Collaborative Innovation Center of Polymers and Polymer Composite Materials, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Yicheng Qiang
- State Key Laboratory of Molecular Engineering of Polymers, Collaborative Innovation Center of Polymers and Polymer Composite Materials, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Chao Duan
- State Key Laboratory of Molecular Engineering of Polymers, Collaborative Innovation Center of Polymers and Polymer Composite Materials, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Collaborative Innovation Center of Polymers and Polymer Composite Materials, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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50
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Zhao M, Li W. Laves Phases Formed in the Binary Blend of AB4 Miktoarm Star Copolymer and A-Homopolymer. Macromolecules 2019. [DOI: 10.1021/acs.macromol.8b02407] [Citation(s) in RCA: 44] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Affiliation(s)
- Mingtian Zhao
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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