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Designing a minimal Landau model to stabilize desired quasicrystals

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Physical Review Research 2025年第2期7卷 023021-023021页

作者： Wei Si Shifeng Li Pingwen Zhang An-Chang Shi Kai Jiang School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China School of Mathematics and Statistics Wuhan University Wuhan 430072 China LMAM CAPT and School of Mathematical Sciences Peking University Beijing 100871 China Department of Physics and Astronomy McMaster University Hamilton L8S 4M1 Canada

Interparticle interactions with multiple length scales play a pivotal role in the formation and stability of quasicrystals. Choosing a minimal set of length scales to stabilize a given quasicrystal is a challenging problem. To address this challenge, we propose a symmetry-preserving screening method (SPSM) to design a Landau theory with a minimal number of length scales—referred to as the minimal Landau theory—that includes only the essential length scales necessary to stabilize quasicrystals. Based on a generalized multiple-length-scale Landau theory, SPSM first evaluates various spectral configurations of candidate structures under a hard constraint. It then identifies the configuration with the lowest free energy. Using this optimal configuration, SPSM calculates phase diagrams to explore the thermodynamic stability of desired quasicrystals. SPSM can design a minimal Landau theory capable of stabilizing the desired quasicrystals by incrementally increasing the number of length scales. Our application of SPSM has not only confirmed known behaviors in 10- and 12-fold quasicrystals but also led to a significant prediction that quasicrystals with 8-, 14-, 16-, and 18-fold symmetry could be stable within three-length-scale Landau models.

关键词： Quasicrystals

Stability of diverse dodecagonal quasicrystals in T-shaped liquid crystalline molecules

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arXiv 2024年

作者： Wang, Xin Shi, An-Chang Zhang, Pingwen Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Department of Physics and Astronomy McMaster University HamiltonONL8S4M1 Canada School of Mathematics and Statistics Wuhan University Wuhan430072 China

Quasicrystals are intriguing ordered structures characterized by the lack of translational symmetry and the existence of rotational symmetry. The tiling of different geometric units such as triangles and squares in two-dimensional space can result in a great variety of quasicrystals that could be realized by the self-assembly of liquid crystalline molecules. In this study, we introduce three self-similar dodecagonal tilings, including a novel Diamond-Square-Triangle pattern, composed of triangular and quadrangular tiles and examine their thermodynamic stability by using the self-consistent field theory applied to T-shaped liquid crystalline molecules. Specifically, we detail the inflation rules for the construction of these dodecagonal tilings and analyze their self-similarity, and show that these tilings can be viewed as projections of higher-dimensional periodic lattice points with projection windows. Using these dodecagonal tilings as initial configurations of the SCFT results in solutions corresponding to quasicrystals that could form from the T-shaped liquid crystalline molecules. The relative stability of these aperiodic phases is analyzed to obtain design rules that could stabilize quasicrystals. Meanwhile, we provide two criteria for distinguishing three dodecagonal quasicrystals and their approximants by analyzing their diffraction peaks. These findings shed new lighten on the discovery of new quasicrystals in soft materials. Copyright © 2024, The Authors. All rights reserved.

关键词： Quasicrystals

Designing a minimal Landau theory to stabilize desired quasicrystals

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arXiv 2024年

作者： Si, Wei Li, Shifeng Zhang, Pingwen Shi, An-Chang Jiang, Kai School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China LMAM CAPT and School of Mathematical Sciences Peking University Beijing100871 China School of Mathematics and Statistics Wuhan University Wuhan430072 China Department of Physics and Astronomy McMaster University HamiltonL8S 4M1 Canada

Interparticle interactions with multiple length scales play a pivotal role in the formation and stability of quasicrystals. Choosing a minimal set of length scales to stabilize a given quasicrystal is a challenging problem. To address this challenge, we propose an intelligent screening method (ISM) to design a Landau theory with a minimal number of length scales – referred to as the minimal Landau theory – that includes only the essential length scales necessary to stabilize quasicrystals. Based on a generalized multiple-length-scale Landau theory, ISM first evaluates various spectral configurations of candidate structures under a hard constraint. It then identifies the configuration with the lowest free energy. Using this optimal configuration, ISM calculates phase diagrams to explore the thermodynamic stability of desired quasicrystals. ISM can design a minimal Landau theory capable of stabilizing the desired quasicrystals by incrementally increasing the number of length scales. Our application of ISM has not only confirmed known behaviors in 10- and 12-fold quasicrystals but also led to a significant prediction that quasicrystals with 8-, 14-, 16-, and 18-fold symmetry could be stable within three-length-scale Landau models. © 2024, CC BY-NC-ND.

关键词： Free energy

Phase behavior of symmetric diblock copolymers under 3D soft confinement

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arXiv 2024年

作者： He, Zhijuan Huang, Jin Jiang, Kai Shi, An-Chang School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education 411105 China Laboratory of Mathematics and Complex Systems Ministry of Education School of Mathematical Sciences Beijing Normal University Beijing100875 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

The phase behavior of symmetric diblock copolymers under three-dimensional (3D) soft confinement is investigated using the self-consistent field theory. The soft confinement is realized in binary blends composed AB diblock copolymers and C homopolymers, where the copolymers self-assemble to form a droplet embedded in the homopolymer matrix. The phase behavior of the confined block copolymers is regulated by the degree of confinement and the selectivity of the homopolymers, resulting in a rich variety of novel structures. When the C homopolymers are neutral to the A- and B-blocks, stacked lamellae (SL) are formed where the number of layers increases with the droplet volume, resulting in a morphological transition sequence from Janus particle to square SL. When the C homopolymers are strongly selective to the B-blocks, a series of non-lamellar morphologies, including onion-, hamburger-, cross-, ring-, and cookie-like structures, are observed. A detailed free energy analysis reveals a first-order reversible transformation between SL and onion-like (OL) structures when the selectivity of the homopolymers is changed. Our results provide a comprehensive understanding of how various factors, such as the copolymer concentration, homopolymer chain length, degree of confinement, homopolymer selectivity, affect the self-assembled structures of diblock copolymers under soft 3D confinement. Copyright © 2024, The Authors. All rights reserved.

关键词： Free energy

Nucleation and phase transition of decagonal quasicrystals

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arXiv 2024年

作者： Zhou, Tiejun Zhang, Lei Zhang, Pingwen Shi, An-Chang Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Beijing International Center for Mathematical Research Peking University Beijing100871 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China Department of Physics and Astronomy McMaster University HamiltonL8S 4M1 Canada

In this work, we study the nucleation of quasicrystals from liquid or periodic crystals by developing an efficient order-order phase transition algorithm, namely the nullspace-preserving saddle search method. Specifically, we focus on nucleation and phase transitions of the decagonal quasicrystal (DQC) based on the Lifshitz-Petrich model. We present the nucleation path of DQC from the liquid and demonstrate one- and two-stage transition paths between DQC and periodic crystals. We provide a perspective of the group-subgroup phase transition and nucleation rates to understand the nucleation and phase transition mechanisms involving DQC. These results reveal the one-step and stepwise modes of symmetry breaking or recovery in the phase transition from DQC, where the stepwise modes are more probable. Copyright © 2024, The Authors. All rights reserved.

关键词： Quasicrystals

Theory of polygonal phases self-assembled from T-shaped liquid crystalline molecules

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arXiv 2023年

作者： He, Zhijuan Wang, Xin Zhang, Pingwen Shi, An-Chang Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China Department of Physics and Astronomy McMaster University HamiltonONL8S 4M1 Canada

Extensive experimental studies have shown that numerous ordered phases can be formed via the self-assembly of T-shaped liquid crystalline molecules (TLCMs) composed of a rigid backbone, two flexible end chains and a flexible side chain. However, a comprehensive understanding of the stability and formation mechanisms of these intricately nanostructured phases remains incomplete. Here we fill this gap by carrying out a theoretical study of the phase behaviour of TLCMs. Specifically, we construct phase diagrams of TLCMs by computing the free energy of different ordered phases of the system. Our results reveal that the number of polygonal edges increases as the length of side chain or interaction strength increases, consistent with experimental observations. The theoretical study not only reproduces the experimentally observed phases and phase transition sequences, but also systematically analyzes the stability mechanism of the polygonal phases. Copyright © 2023, The Authors. All rights reserved.

关键词： Free energy

A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant

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Communications in computational Physics 2014年第2期15卷 451-469页

作者： Jian-Jun Xu Yunqing Huang Ming-Chih Lai Zhilin Li School of Mathematical and Computational Sciences Xiangtan UniversityHunan 410005China Department of Applied Mathematics National Chiao Tung UniversityHsinchu 30050Taiwan Department of Mathematics North Carolina State UniversityRaleighNC27695USA .4 Hunan Key Lab for Computation and Simulation in Science and Engineering Hunan 410005China School of Mathematical Sciences Nanjing Normal UniversityNanjingChina.

In this paper,a numerical method is presented for simulating the 3D interfacial flows with insoluble *** numerical scheme consists of a 3D immersed interface method(IIM)for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming *** 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the *** assumption is convenient in conjunction with the level-set *** allows standard Lagrangian interpolation for quantities at the projection points on the *** interface jump relations are re-derived accordingly.A novel rotational procedure is given to generate smooth local coordinate systems and make effective *** examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy.A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.

关键词： Multigrid method constrained Cauchy-Born elasticity nanoindentation CauchyBorn rule.

26th Annual computational Neuroscience Meeting (CNS*2017): Part 3 Antwerp, Belgium. 15-20 July 2017 Abstracts

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BMC NEUROscience 2017年第SUPPL 1期18卷 95-176页

作者： [Anonymous] Department of Neuroscience Yale University New Haven CT 06520 USA Department Physiology & Pharmacology SUNY Downstate Brooklyn NY 11203 USA NYU School of Engineering 6 MetroTech Center Brooklyn NY 11201 USA Departament de Matemàtica Aplicada Universitat Politècnica de Catalunya Barcelona 08028 Spain Institut de Neurobiologie de la Méditerrannée (INMED) INSERM UMR901 Aix-Marseille Univ Marseille France Center of Neural Science New York University New York NY USA Aix-Marseille Univ INSERM INS Inst Neurosci Syst Marseille France Laboratoire de Physique Théorique et Modélisation CNRS UMR 8089 Université de Cergy-Pontoise 95300 Cergy-Pontoise Cedex France Department of Mathematics and Computer Science ENSAT Abdelmalek Essaadi’s University Tangier Morocco Laboratory of Natural Computation Department of Information and Electrical Engineering and Applied Mathematics University of Salerno 84084 Fisciano SA Italy Department of Medicine University of Salerno 84083 Lancusi SA Italy Dipartimento di Fisica Università degli Studi Aldo Moro Bari and INFN Sezione Di Bari Italy Data Analysis Department Ghent University Ghent Belgium Coma Science Group University of Liège Liège Belgium Cruces Hospital and Ikerbasque Research Center Bilbao Spain BIOtech Department of Industrial Engineering University of Trento and IRCS-PAT FBK 38010 Trento Italy Department of Data Analysis Ghent University Ghent 9000 Belgium The Wellcome Trust Centre for Neuroimaging University College London London WC1N 3BG UK Department of Electronic Engineering NED University of Engineering and Technology Karachi Pakistan Blue Brain Project École Polytechnique Fédérale de Lausanne Lausanne Switzerland Departement of Mathematics Swansea University Swansea Wales UK Laboratory for Topology and Neuroscience at the Brain Mind Institute École polytechnique fédérale de Lausanne Lausanne Switzerland Institute of Mathematics University of Aberdeen Aberdeen Scotland UK Department of Integrativ

Adaptive mesh refinement and superconvergence for two-dimensional interface problems

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SIAM Journal on Scientific Computing 2014年第4期36卷 A1478-A1499页

作者： Wei, Huayi Chen, Long Huang, Yunqing Zheng, Bin Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Lab. of Intelligent Comp. and Info. Proc. Comp. Min. of Educ. School of Math. and Computational Sci. Xiangtan University Xiangtan Hunan 411105 China Department of Mathematics University of California at Irvine Irvine 92697 CA United States Northwest National Lab Richland 99352 WA United States

Adaptive mesh refinement and the B46;rgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body... 详细信息

Adaptive mesh refinement and the Börgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolation of the exact solution. Based on this superconvergence result, a maximal norm error estimate of order one and half is obtained without using the discrete maximum principle. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients. © 2014 Society for Industrial and Applied Mathematics.

关键词： Adaptive mesh refinement Body-fitted mesh Superconvergence