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Jager PM, Briels WJ, Slot JJM. General weak segregation theory with an application to monodisperse semi-flexible diblock copolymers. J Chem Phys 2023; 158:104901. [PMID: 36922130 DOI: 10.1063/5.0138244] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/16/2023] Open
Abstract
A general theory has been developed for a polydisperse semi-flexible multi-block copolymer melt. Using the Bawendi-Freed approach to model semi-flexible chains, an expression for the Landau free energy is derived in the weak segregation regime, which includes density and orientation order-parameters. The orientation order-parameter is described in the smectic phase and in more complicated structures, such as the hexagonal phase. The Landau free energy contains contributions of two kinds of interactions. The first kind is the Flory-Huggins interaction, which describes the incompatibility of chemically different blocks and may induce microphase separation. The second kind is the Maier-Saupe interaction, which may induce nematic ordering. In the framework of the weak segregation limit, the Landau theory allows us to predict phase structures in the melt as a function of the composition, persistence length, and the strength of the Flory-Huggins and Maier-Saupe interaction. The general theory is applied to a simple system of monodisperse semi-flexible diblock copolymers. In several phase diagrams, a number of possible phase structures are predicted, such as the bcc, hexagonal, smectic-A, smectic-C, and nematic phase. The influence of the Maier-Saupe interaction on the microphase structure is thoroughly discussed.
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Affiliation(s)
- P M Jager
- EY, Centre for Tax and Legal Knowledge, Boompjes 258, P.O. Box 2295, 3011 XZ Rotterdam, The Netherlands
| | - W J Briels
- MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
| | - J J M Slot
- Department of Mathematics and Computer Science, CASA/Applied Analysis, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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Tunable Thin Film Periodicities by Controlling the Orientation of Cylindrical Domains in Side Chain Liquid Crystalline Block Copolymers. INT J POLYM SCI 2022. [DOI: 10.1155/2022/8286518] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
A facile approach to block copolymer (BCP) domain orientation control in thin films has been demonstrated by employing a BCP with liquid crystalline semifluorinated side chains by tuning the composition of the copolymers of the bottom surface layer (BSL). 1H,1H,2H,2H-Perfluorodecanethiol was attached to a precursor polymer, polystyrene-block-poly(glycidyl methacrylate) (PS-b-PGMA), to obtain a novel BCP with a C8F17-containing liquid crystal (LC) side chain (PS-b-P8FMA). Anisotropic hexagonally packed cylinder domains in a bulk state were first characterized by transmission electron microscopy (TEM) and small angle X-ray scattering (SAXS). The observed morphology transition of BCPs with different fluorinated side chain lengths of –CF3, –C4F9, and –C6F13 suggested the decisive effects of LC side chain ordering on the anisotropic nanostructures. In the thin film study, poly(methyl methacrylate-random-2,2,2-trifluoroethyl methacrylate-random-methacrylic acid) (PMMA-ran-PTFEMA-ran-PMAA) solution was used as BSLs for tuning the desired periodicities. The surface free energy (SFE) of BSL was simply tailored by changing the composition of comonomers. In atomic force microscopy (AFM) characterization, long-range ordered perpendicularly oriented BCP domains in a hexagonally packed array or parallel oriented BCP domains as striation patterns were easily fabricated on non-preferential or preferential BSL, respectively. The study presents a novel approach to tunable thin film periodicities without changing or modifying BCPs, which is desired in next-generation BCP lithography.
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Han XG, Sun ZH, Liang N, Zhang H. Aggregate behavior in amphiphilic coil/rod block copolymer solutions. Chem Phys 2022. [DOI: 10.1016/j.chemphys.2022.111663] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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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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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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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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Shao J, Jiang N, Zhang H, Yang Y, Tang P. Target-Directed Design of Phase Transition Path for Complex Structures of Rod-Coil Block Copolymers. ACS OMEGA 2019; 4:20367-20380. [PMID: 31815241 PMCID: PMC6894153 DOI: 10.1021/acsomega.9b02984] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/13/2019] [Accepted: 10/24/2019] [Indexed: 06/10/2023]
Abstract
We apply the string method to the self-consistent mean-field theory framework of the rod-coil block copolymer system to calculate the minimum energy pathways in the rearrangement transitions of lamellae and cylinders with different orientations under certain epitaxial growth relationship. Metastable phases appearing in the reordering transition pathway tend to form the structure at low χN side of the order-order transition boundary compared with the initial phase. In particular, for complex network, metastable phases, such as single gyroid and perforated lamellae, need to select a rearrangement transition between lamellae or cylinders near the order-disorder transition boundary with the same epitaxial growth relationship but different orientations. It is confirmed that this strategy for obtaining complex metastable phases by rational design of rearrangement transition between specific phases in the phase diagram can be applied to a wide range of χN as well as the coil-coil block copolymer system. We further investigate the rearrangement transition behavior combining with the analysis of contribution from the free energy, entropy, degree of mixing between different blocks, and the average orientation degree of rods during the phase transitions. Based on this mechanism, we have developed a target-directed design strategy for constructing self-assembled metastable structures of rod-coil block copolymers.
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Affiliation(s)
- Jingyu Shao
- State Key Laboratory of Molecular
Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Nuofei Jiang
- State Key Laboratory of Molecular
Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Hongdong Zhang
- State Key Laboratory of Molecular
Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Yuliang Yang
- State Key Laboratory of Molecular
Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Ping Tang
- State Key Laboratory of Molecular
Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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Magnetic Field Effects on Chemical Reaction of Power-Law Fluid over an Axisymmetric Stretched Sheet. MAGNETOCHEMISTRY 2019. [DOI: 10.3390/magnetochemistry5040057] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Numerical investigation of the effects of magnetic field strength, thermal radiation, Joule heating, and viscous heating on a forced convective flow of a non-Newtonian, incompressible power-law fluid in an axisymmetric stretching sheet with variable temperature wall is accomplished. The power-law shear-thinning viscosity-shear rate model for the anisotropic solutions and the Rosseland approximation for the thermal radiation through a highly absorbing medium is considered. The temperature-dependent heat sources, Joule heating, and viscous heating are considered to be the source terms in the energy balance. The non-dimensional boundary-layer equations are solved numerically in terms of similarity variable. A parameter study on the boundary value of chemical reaction and Nusselt number is performed as a function of thermal radiation parameter, Brinkman number, Prandtl number, Hartmann number, power-law index, heat source coefficient, Brownian parameter, thermophoresis parameter, and the chemical reaction parameter. The results show that the chemical reaction parameter has an increasing effect on the chemical reaction boundary while the magnetic, thermophoresis, and Brownian effects decrease the rate of the chemical reaction at the boundary.
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Kriksin YA, Potemkin II, Khalatur PG. Chirality in Self-Assembling Rod-Coil Copolymers: Macroscopic Homochirality Versus Local Chirality. POLYMER SCIENCE SERIES C 2018. [DOI: 10.1134/s1811238218020133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Liu F, Tang P, Zhang H, Yang Y. Archimedean Tiling Patterns Self-Assembled from X-Shaped Rod–Coil Copolymers with Hydrogen Bonds. Macromolecules 2018. [DOI: 10.1021/acs.macromol.8b01078] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Faqiang Liu
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Ping Tang
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Hongdong Zhang
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Yuliang Yang
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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Sun T, Liu F, Tang P, Qiu F, Yang Y. Construction of Rod-Forming Single Network Mesophases in Rod–Coil Diblock Copolymers via Inversely Designed Phase Transition Pathways. Macromolecules 2018. [DOI: 10.1021/acs.macromol.8b00773] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Tongjie Sun
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Faqiang Liu
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Ping Tang
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Feng Qiu
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Yuliang Yang
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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Liu F, Sun T, Tang P, Zhang H, Qiu F. Understanding chain folding morphology of semicrystalline polymers based on a rod-coil multiblock model. SOFT MATTER 2017; 13:8250-8263. [PMID: 29071322 DOI: 10.1039/c7sm01892d] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We employ a rod-coil multiblock molecular chain model to investigate chain folding behavior, which is a significant characteristic in semicrystalline polymers, by using the method of self-consistent field theory (SCFT). Polymer chains with different conformations in crystalline and amorphous regions are described by rigid rod chains and flexible Gaussian chains, respectively. At present, we concentrate on the thermodynamic behaviors of polymer semi-crystals after the formation of the initial lamellar crystals. A new mechanism for lamellar thickening is proposed to realize that the end of lamellar thickening depends on the crystallinity degree. In other words, it is impossible for lamellae to develop into extended-chain crystals by means of lamellar thickening if crystallinity is limited to a certain degree. We further discuss the competition between crystalline and amorphous regions and its influence on crystallization behaviors, such as the formation of double lamellae, chain tilt, the anomalies and adjacent re-entry. The synergistic influences of the driving force of crystallization, interfacial energy and crystallinity degree on chain folding behavior are also investigated when the density anomalies in amorphous regions are excluded. Our model demonstrates advantages in accurately describing the mesoscopic layered structures of semicrystalline polymers based upon a microscopic chain model and provides at least a semi-quantitative thermodynamic picture for chain folding.
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Affiliation(s)
- Faqiang Liu
- 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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Cai Y, Zhang P, Shi AC. Liquid crystalline bilayers self-assembled from rod-coil diblock copolymers. SOFT MATTER 2017; 13:4607-4615. [PMID: 28604893 DOI: 10.1039/c7sm00354d] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The structure and phase behaviour of bilayer membranes self-assembled from rod-coil diblock copolymers are studied using the self-consistent field theory, focusing on the occurrence and relative stability of liquid crystalline phases induced by the geometric shape and orientational interaction of the rod-blocks. A variety of liquid crystalline bilayers, corresponding to the smectic phases in bulk systems, are predicted to occur as equilibrium phases of the system. The ordered morphologies and phase behaviour of the system are analyzed. Phase diagrams of the self-assembled bilayers are constructed. The theoretical results provide an understanding of the formation mechanisms of these intricate phases.
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Affiliation(s)
- Yongqiang Cai
- School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China.
| | - 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, Ontario L8S 4M1, Canada.
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Numerical Study of the Magnetic Field Effects on the Heat Transfer and Entropy Generation Aspects of a Power Law Fluid over an Axisymmetric Stretching Plate Structure. ENTROPY 2017. [DOI: 10.3390/e19030094] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
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