1
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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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2
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Teng Y, Andersen NT, Chen JZY. Statistical Properties of a Slit-Confined Wormlike Chain of Finite Length. Macromolecules 2021. [DOI: 10.1021/acs.macromol.1c00759] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023]
Affiliation(s)
- Yue Teng
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Nigel T. Andersen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Jeff Z. Y. Chen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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3
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Niu Y, Bu X, Zhang X. Single Chain Mean-Field Theory Study on Responsive Behavior of Semiflexible Polymer Brush. MATERIALS 2021; 14:ma14040778. [PMID: 33562209 PMCID: PMC7914892 DOI: 10.3390/ma14040778] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/14/2020] [Revised: 01/28/2021] [Accepted: 02/04/2021] [Indexed: 11/21/2022]
Abstract
The application of single chain mean-field theory (SCMFT) on semiflexible chain brushes is reviewed. The worm-like chain (WLC) model is the best mode of semiflexible chain that can continuously recover to the rigid rod model and Gaussian chain (GC) model in rigid and flexible limits, respectively. Compared with the commonly used GC model, SCMFT is more applicable to the WLC model because the algorithmic complexity of the WLC model is much higher than that of the GC model in self-consistent field theory (SCFT). On the contrary, the algorithmic complexity of both models in SCMFT are comparable. In SCMFT, the ensemble average of quantities is obtained by sampling the conformations of a single chain or multi-chains in the external auxiliary field instead of solving the modified diffuse equation (MDE) in SCFT. The precision of this calculation is controlled by the number of bonds Nm used to discretize the chain contour length L and the number of conformations M used in the ensemble average. The latter factor can be well controlled by metropolis Monte Carlo simulation. This approach can be easily generalized to solve problems with complex boundary conditions or in high-dimensional systems, which were once nightmares when solving MDEs in SCFT. Moreover, the calculations in SCMFT mainly relate to the assemble averages of chain conformations, for which a portion of conformations can be performed parallel on different computing cores using a message-passing interface (MPI).
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4
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Huang J, Li S, Zhang X, Huang G. Neural network model for structure factor of polymer systems. J Chem Phys 2020; 153:124902. [PMID: 33003754 DOI: 10.1063/5.0022464] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Abstract
As an important physical quantity to understand the internal structure of polymer chains, the structure factor is being studied both in theory and experiment. Theoretically, the structure factor of Gaussian chains has been solved analytically, but for wormlike chains, numerical approaches are often used, such as Monte Carlo simulations, solving the modified diffusion equation. In these works, the structure factor needs to be calculated differently for different regions of the wave vector and chain rigidity, and some calculation processes are resource consuming. In this work, by training a deep neural network, we obtained an efficient model to calculate the structure factor of polymer chains, without considering different regions of wavenumber and chain rigidity. Furthermore, based on the trained neural network model, we predicted the contour and Kuhn lengths of some polymer chains by using scattering experimental data, and we found that our model can get pretty reasonable predictions. This work provides a method to obtain the structure factor for polymer chains, which is as good as previous and more computationally efficient. It also provides a potential way for the experimental researchers to measure the contour and Kuhn lengths of polymer chains.
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Affiliation(s)
- Jie Huang
- Department of Physics, Wenzhou University, Wenzhou, Zhejiang 325035, China
| | - Shiben Li
- Department of Physics, Wenzhou University, Wenzhou, Zhejiang 325035, China
| | - Xinghua Zhang
- School of Science, Beijing Jiaotong University, Beijing 100044, China
| | - Gang Huang
- Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
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5
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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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6
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Numerical Simulation of the Distribution Function and Free Energy of a Single Wormlike Polymer Confined between Hard Walls. CHINESE JOURNAL OF POLYMER SCIENCE 2019. [DOI: 10.1007/s10118-019-2322-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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7
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Liang Q, Jiang Y, Chen JZY. Orientationally ordered states of a wormlike chain in spherical confinement. Phys Rev E 2019; 100:032502. [PMID: 31640076 DOI: 10.1103/physreve.100.032502] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2019] [Indexed: 06/10/2023]
Abstract
One of the basic characteristics of a linear dsDNA molecule is its persistence length, typically of order 50 nm. The DNA chain inflicts a large energy penalty if it is bent sharply at that length scale. Viruses of bacteria, known as bacteriophages, typically have a dimension of a few tens of nanometers. Yet, it is known that a bacteriophage actively packages viral DNA inside the capsid and ejects it afterwards. Here, adopting a commonly used polymer model known as the wormlike chain, we answer an idealized question: Placing a linear DNA molecule inside a spherical cavity, what ordered states can we derive from known tools in statistical physics? Solving the model in a rigorous field-theory framework, we report a universal phase diagram for four orientationally ordered and disordered states, in terms of two relevant physical parameters.
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Affiliation(s)
- Qin Liang
- Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China
- School of Chemistry, Beihang University, Beijing 100191, China
| | - Ying Jiang
- School of Chemistry, Beihang University, Beijing 100191, China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
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8
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Hannon AF, Joseph Kline R, DeLongchamp D. Advancing the computational methodology of rigid rod and semiflexible polymer systems: A new solution to the wormlike chain model with rod‐coil copolymer calculations. ACTA ACUST UNITED AC 2018. [DOI: 10.1002/polb.24745] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
Affiliation(s)
- Adam F. Hannon
- Materials Science and Engineering Division National Institute of Standards and Technology Gaithersburg Maryland 100 Bureau Drive 20899
| | - R. Joseph Kline
- Materials Science and Engineering Division National Institute of Standards and Technology Gaithersburg Maryland 100 Bureau Drive 20899
| | - Dean DeLongchamp
- Materials Science and Engineering Division National Institute of Standards and Technology Gaithersburg Maryland 100 Bureau Drive 20899
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9
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Chen JZY. Conformational Properties of a Back-Folding Wormlike Chain Confined in a Cylindrical Tube. PHYSICAL REVIEW LETTERS 2017; 118:247802. [PMID: 28665664 DOI: 10.1103/physrevlett.118.247802] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2017] [Indexed: 06/07/2023]
Abstract
When a semiflexible chain is confined in a narrow cylindrical tube, the formation of a polymer hairpin is a geometrical conformation that accompanies an exponentially large local free energy and, hence, is a relatively rare event. Numerical solutions of the hairpin distribution functions for persistence-length-to-tube-radius ratios over a wide range are obtained in high precision, by using the Green's function approach for the wormlike-chain model. The crossover region between the narrow and moderately narrow tubes is critically investigated in terms of the hairpin free energy, global persistence length, mean hairpin-tip distance from the tube axis, and hairpin-plane orientational properties. Accurate representations of the solutions by simple interpolation formulae are suggested.
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Affiliation(s)
- Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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10
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Li S, Jiang Y, Chen JZY. Complex liquid-crystal nanostructures in semiflexible ABC linear triblock copolymers: A self-consistent field theory. J Chem Phys 2016; 145:184902. [PMID: 27846676 DOI: 10.1063/1.4967423] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We show that two series of ABC linear triblock copolymers possess sequences of order-to-order phase transitions between microphase-separated states, as the degree of flexibility of the semiflexible middle B-blocks varies. The spatial and orientational symmetries of these phases, some of them containing liquid-crystal ordering, are analysed in comparison with related structures previously determined experimentally and theoretically. A theoretical framework based on the self-consistent field treatment of the wormlike-chain model, which incorporates the Flory-Huggins and Maier-Saupe interactions in the free energy, is used here as a basic foundation for numerical calculations. We suggest that tuning the flexibility parameter, which reduces to the concept of degree of polymerization in the coil-like limit and characterizes the chain-persistency in the rod-like limit, provides a promising approach that can be used to design the resulting microphase-separated structures in semiflexible copolymer melts.
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Affiliation(s)
- Shiben Li
- Department of Physics, Wenzhou University, Wenzhou, Zhejiang 325035, China
| | - Ying Jiang
- School of Chemistry and Environment and Center of Soft Matter Physics and its Applications, Beihang University, Beijing 100191, China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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11
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Jiang Y, Li S, Chen JZY. Perspective: parameters in a self-consistent field theory of multicomponent wormlike-copolymer melts. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2016; 39:91. [PMID: 27696016 DOI: 10.1140/epje/i2016-16091-8] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2016] [Accepted: 09/13/2016] [Indexed: 06/06/2023]
Abstract
We review a formalism that can be used to calculate the microphase-separated crystallographic structures of multi-component wormlike polymer melts. The approach is based on a self-consistent field theory of wormlike polymers where the persistence length of each component is an important parameter. We emphasize on an analysis of the number of independent parameters required to specify a problem in general, for a system that includes Flory-Huggins and Maier-Saupe energies. Examples of recent applications are also briefly demonstrated: AB homopolymer interface, AB diblock copolymers, and rod-coil copolymers.
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Affiliation(s)
- Ying Jiang
- School of Chemistry and Environment, Center of Soft Matter Physics and its Applications, Beihang University, 100191, Beijing, China
| | - Shiben Li
- Department of Physics, Wenzhou University, 325035, Wenzhou, Zhejiang, China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada.
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12
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Ye S, Zhang P, Chen JZY. Nematic ordering of semiflexible polymers confined on a toroidal surface. SOFT MATTER 2016; 12:5438-5449. [PMID: 27248049 DOI: 10.1039/c6sm01098a] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We study the isotropic-like and nematic states of wormlike liquid-crystal polymers embedded on the surface of a torus. The role played by surface curvature, which couples to the molecular rigidity, is reported as the main reason that causes the weak nematic ordering in an otherwise ordinary isotropic phase. The same coupling has a profound effect on the nematic states as well, which are stabilized by the Onsager excluded-volume interaction; the latter has been frequently used to study lyotropic liquid crystal polymers and is used here as an example of the physical mechanisms that drive the system to make orientational ordering. We identify important parameters in the system which are used as axes of the four-dimensional phase diagram. The numerical study demonstrates a strong correlation between the liquid-crystal defect-free and defect structures and the geometry of the liquid-crystal embedded surface.
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Affiliation(s)
- Shiwei Ye
- LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People's Republic of China.
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13
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14
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Jiang Y, Zhang X, Miao B, Yan D, Chen JZY. Microphase separation of short wormlike diblock copolymers with a finite interaction range. SOFT MATTER 2016; 12:2481-2490. [PMID: 26822622 DOI: 10.1039/c5sm02865e] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We investigate several structural properties of low-molecular weight AB diblock copolymer melts, focusing on a number of features that substantially deviate from those of high-molecular weight copolymer melts. The study is based on the wormlike chain formalism aided by random phase approximation and self-consistent field theory. We examine the effects that stemmed from both the finite molecular weight and the finite interaction range between unlike AB monomers. The latter yields profound effects on systems consisting of short wormlike block copolymers. The noticeable shift of the order-disorder transition point is discussed. Attention is also paid to the strong-segregation regime, where low molecular weight polymers are subject to finite stretchability.
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Affiliation(s)
- Ying Jiang
- School of Chemistry and Environment, Center of Soft Matter Physics and its Applications, Beihang University, Beijing 100191, China
| | - Xinghua Zhang
- School of Science, Beijing Jiaotong University, Beijing 100044, China.
| | - Bing Miao
- College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China.
| | - Dadong Yan
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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15
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Tang J, Jiang Y, Zhang X, Yan D, Chen JZY. Phase Diagram of Rod–Coil Diblock Copolymer Melts. Macromolecules 2015. [DOI: 10.1021/acs.macromol.5b02235] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
Affiliation(s)
- Jiuzhou Tang
- Beijing
National Laboratory For Molecular Sciences (BNLMS), Laboratory of
Polymer Physics and Chemistry, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China
| | - Ying Jiang
- School
of Chemistry and Environment, Beihang University, Beijing 100083, China
| | - Xinghua Zhang
- School
of Science, Beijing Jiaotong University, Beijing 100044, China
| | - Dadong Yan
- Department
of Physics, Beijing Normal University, Beijing 100875, China
| | - Jeff Z. Y. Chen
- Department
of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
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16
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Klotz AR, Duong L, Mamaev M, de Haan HW, Chen JZY, Reisner WW. Measuring the Confinement Free Energy and Effective Width of Single Polymer Chains via Single-Molecule Tetris. Macromolecules 2015. [DOI: 10.1021/acs.macromol.5b00977] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Affiliation(s)
| | - Lyndon Duong
- Department
of Physics, McGill University, Montreal, QC, Canada H3A 2T8
| | - Mikhail Mamaev
- Department
of Physics, McGill University, Montreal, QC, Canada H3A 2T8
| | - Hendrick W. de Haan
- Faculty
of Science, University of Ontario Institute of Technology, Oshawa, ON, Canada L1H 7K4
| | - Jeff Z. Y. Chen
- Department
of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada N2L 3G1
| | - Walter W. Reisner
- Department
of Physics, McGill University, Montreal, QC, Canada H3A 2T8
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17
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Liang Q, Ye S, Zhang P, Chen JZY. Rigid linear particles confined on a spherical surface: Phase diagram of nematic defect states. J Chem Phys 2014; 141:244901. [DOI: 10.1063/1.4903995] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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18
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Li S, Jiang Y, Chen JZY. Phase transitions in semiflexible-rod diblock copolymers: a self-consistent field theory. SOFT MATTER 2014; 10:8932-8944. [PMID: 25299162 DOI: 10.1039/c4sm01884b] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We investigate the phase behavior of semiflexible-rod diblock copolymers in a parameter range where the system displays columnar and lamella structures, using a self-consistent field theory based on the wormlike-chain model. Both Flory-Huggins and Maier-Saupe orientational interactions are incorporated in the formalism, which allows us to explore microphase separation and liquid-crystal ordering simultaneously. Order-to-order phase transitions induced by chain rigidity and orientational interaction are both reported and analyzed. Coupled orientational ordering and spatial inhomogeneity of the four microphase-separated states are discussed in this work: hexagonal column, ellipse column, smectic-A, and smectic-C.
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Affiliation(s)
- Shiben Li
- Department of Physics, Wenzhou University, Wenzhou, Zhejiang 325035, China
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19
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Zhang X, Jiang Y, Miao B, Chen Y, Yan D, Chen JZY. The structure factor of a wormlike chain and the random-phase-approximation solution for the spinodal line of a diblock copolymer melt. SOFT MATTER 2014; 10:5405-5416. [PMID: 24932521 DOI: 10.1039/c4sm00374h] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
An efficient and convenient numerical approach to calculate the structure factor of a wormlike chain model is proposed by directly dealing with a formal solution of the Green's function. A precise numerical representation of the structure factor of the wormlike chain model is then obtained, for arbitrary chain rigidity. On one hand, in the flexible limit, the numerical results recover the well-known Debye function of the structure factor of a Gaussian chain and furthermore predict the correct large-k behavior that a Gaussian model fails to capture; on the other hand, in the rigid limit, the numerical results recover the well-known Neugebauer function of the structure factor of a rigid rod. Based on the calculated structure factor, the random phase approximation is employed to study the physical properties of the order-disorder transition for asymmetric wormlike diblock copolymers; particularly, the spinodal line of the disordered phase is calculated. For the case of symmetric diblock copolymer microphase separation, the present calculation reproduces the phase boundary previously determined by self-consistent field theories and yields the entire picture crossing over from the flexible-chain limit to the rigid-chain limit.
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Affiliation(s)
- Xinghua Zhang
- School of Science, Beijing Jiaotong University, Beijing 100044, China
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20
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Gao J, Tang P, Yang Y, Chen JZY. Free energy of a long semiflexible polymer confined in a spherical cavity. SOFT MATTER 2014; 10:4674-4685. [PMID: 24839199 DOI: 10.1039/c4sm00605d] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
The free energy and conformational properties of a wormlike chain confined inside a spherical surface are investigated. We show that in the weak-confinement limit, the wormlike chain model exactly reproduces the confinement properties of a Gaussian chain; in such a case the confinement entropy dominates the free energy; in the strong-confinement limit, the free energy is dominated by the bending energy of the chain, which is forced to wrap around the confining surface. We also present a numerical solution within the crossover region between the two limits, solving the differential equation that the probability distribution function satisfies.
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Affiliation(s)
- Jie Gao
- Department of Macromolecular Science, Fudan University, Shanghai 200433, China.
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21
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Li J, Zhang H, Qiu F. Self-consistent field theory of block copolymers on a general curved surface. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2014; 37:18. [PMID: 24664410 DOI: 10.1140/epje/i2014-14018-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2013] [Revised: 12/17/2013] [Accepted: 03/17/2014] [Indexed: 06/03/2023]
Abstract
In this work, we propose a theoretical framework based on the self-consistent field theory (SCFT) for the study of self-assembling block copolymers on a general curved surface. Relevant numerical algorithms are also developed. To demonstrate the power of the approach, we calculate the self-assembled patterns of diblock copolymers on three distinct curved surfaces with different genus. We specially study the geometrical effects of curved surfaces on the conformation of polymer chains as well as on the pattern formation of block copolymers. By carefully examining the diffusion equation of the propagator on curved surfaces, it is predicted that Gaussian chains are completely unaware of the extrinsic curvature but that they will respond to the intrinsic curvature of the surface. This theoretical assertion is consistent with our SCFT simulations of block copolymers on general curved surfaces.
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Affiliation(s)
- Jianfeng Li
- The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, 200433, Shanghai, China,
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22
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Affiliation(s)
- Jeff Z. Y. Chen
- Department
of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
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23
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Jiang Y, Chen JZY. Self-consistent field theory and numerical scheme for calculating the phase diagram of wormlike diblock copolymers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042603. [PMID: 24229202 DOI: 10.1103/physreve.88.042603] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2013] [Revised: 09/09/2013] [Indexed: 06/02/2023]
Abstract
This paper concerns establishing a theoretical basis and numerical scheme for studying the phase behavior of AB diblock copolymers made of wormlike chains. The general idea of a self-consistent field theory is the combination of the mean-field approach together with a statistical weight that describes the configurational properties of a polymer chain. In recent years, this approach has been extensively used for structural prediction of block copolymers, based on the Gaussian-model description of a polymer chain. The wormlike-chain model has played an important role in the description of polymer systems, covering the semiflexible-to-rod crossover of the polymer properties and the highly stretching regime, which the Gaussian-chain model has difficulties to describe. Although the idea of developing a self-consistent field theory for wormlike chains could be traced back to early development in polymer physics, the solution of such a theory has been limited due to technical difficulties. In particular, a challenge has been to develop a numerical algorithm enabling the calculation of the phase diagram containing three-dimensional structures for wormlike AB diblock copolymers. This paper describes a computational algorithm that combines a number of numerical tricks, which can be used for such a calculation. A phase diagram covering major parameter areas was constructed for the wormlike-chain system and reported by us, where the ratio between the total length and the persistence length of a constituent polymer is suggested as another tuning parameter for the microphase-separated structures; all detailed technical issues are carefully addressed in the current paper.
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Affiliation(s)
- Ying Jiang
- Guelph-Waterloo Physics Institute and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
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