1
|
Sun H, Yao Z. Conformal order and Poincaré-Klein mapping underlying electrostatics-driven inhomogeneity in tethered membranes. Phys Rev E 2023; 108:025001. [PMID: 37723772 DOI: 10.1103/physreve.108.025001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2023] [Accepted: 07/17/2023] [Indexed: 09/20/2023]
Abstract
Understanding the organization of matter under the long-range electrostatic force is a fundamental problem in multiple fields. In this work, based on the electrically charged tethered membrane model, we reveal regular structures underlying the lowest-energy states of inhomogeneously stretched planar lattices by a combination of numerical simulation and analytical geometric analysis. Specifically we show the conformal order characterized by the preserved bond angle in the lattice deformation and reveal the Poincaré-Klein mapping underlying the electrostatics-driven inhomogeneity. The discovery of the Poincaré-Klein mapping, which connects the Poincaré disk and the Klein disk for the hyperbolic plane, implies the connection of long-range electrostatic force and hyperbolic geometry. We also discuss lattices with patterned charges of opposite signs for modulating in-plane inhomogeneity and even creating 3D shapes, which may have a connection to metamaterials design. This work suggests the geometric analysis as a promising approach for elucidating the organization of matter under the long-range force.
Collapse
Affiliation(s)
- Honghui Sun
- School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Zhenwei Yao
- School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
| |
Collapse
|
2
|
Yao Z. Charged elastic rings: deformation and dynamics. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 35:045101. [PMID: 36541481 DOI: 10.1088/1361-648x/aca7f8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2022] [Accepted: 12/01/2022] [Indexed: 06/17/2023]
Abstract
We report the counter-intuitive instability of charged elastic rings, and the persistence of sinusoidal deformations in the lowest-energy configurations by the combination of high-precision numerical simulations and analytical perturbation calculation. We also study the dynamical evolution of the charged ring under random disturbance, and reveal the modulation of the dominant frequencies by the electrostatic force. The purely mechanical analysis of the classical ring system presented in this work yields insights into the subtlety of long-range forces in the organization and dynamics of matter.
Collapse
Affiliation(s)
- Zhenwei Yao
- School of Physics and Astronomy, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China
| |
Collapse
|
3
|
Wan D. Configurational entropy of colloidal particles in a confined space. Phys Rev E 2022; 106:034609. [PMID: 36266835 DOI: 10.1103/physreve.106.034609] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Accepted: 09/05/2022] [Indexed: 06/16/2023]
Abstract
We calculate the configurational entropy of colloidal particles in a confined geometry interacting as hard disks using Monte Carlo integration. In particular, we consider systems with three kinds of boundary conditions: hard, periodic, and spherical. For small to moderate packing fraction ϕ values, we find the entropies per particle for systems with the periodic and the spherical boundary conditions tend to reach the same value with the increase of the particle number N, while that for the system with the hard boundary conditions still has obvious differences compared to them within the studied N range. Surprisingly, despite the small system sizes, the estimated entropies per particle at infinite system size from extrapolations in the periodic and spherical systems are in reasonable agreement with that calculated using thermodynamic integration. Besides, as N increases we find the pair correlation function begins to exhibit similar features as that of a thermally equilibrated hard-disk fluid at the same packing fraction. Our findings may contribute to a better understanding of how the configurational entropy changes with the system size and the influence of boundary conditions, and provide insights relevant to engineering particles in confined spaces.
Collapse
Affiliation(s)
- Duanduan Wan
- School of Physics and Technology, Wuhan University, Wuhan 430072, China
| |
Collapse
|
4
|
Siemens A, Schmelcher P. Tunable order of helically confined charges. Phys Rev E 2020; 102:012147. [PMID: 32795010 DOI: 10.1103/physreve.102.012147] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2020] [Accepted: 07/06/2020] [Indexed: 11/07/2022]
Abstract
We investigate a system of equally charged Coulomb-interacting particles confined to a toroidal helix in the presence of an external electric field. Due to the confinement, the particles experience an effective interaction that oscillates with the particle distance and allows for the existence of stable bound states, despite the purely repulsive character of the Coulomb interaction. We design an order parameter to classify these bound states and use it to identify a structural crossover of the particle order, occurring when the electric field strength is varied. Amorphous particle configurations for a vanishing electric field and crystalline order in the regime of a strong electric field are observed. We study the impact of parameter variations on the particle order and conclude that the crossover occurs for a wide range of parameter values and even holds for different helical systems.
Collapse
Affiliation(s)
- Ansgar Siemens
- Zentrum für Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, 22761 Hamburg, Germany
| | - Peter Schmelcher
- Zentrum für Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, 22761 Hamburg, Germany.,Hamburg Center for Ultrafast Imaging, Universität Hamburg, 22761 Hamburg, Germany
| |
Collapse
|
5
|
Crystal-like order and defects in metazoan epithelia with spherical geometry. Sci Rep 2020; 10:7652. [PMID: 32376904 PMCID: PMC7203251 DOI: 10.1038/s41598-020-64598-w] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2019] [Accepted: 04/20/2020] [Indexed: 01/23/2023] Open
Abstract
Since Robert Hooke studied cork cell patterns in 1665, scientists have been puzzled by why cells form such ordered structures. The laws underlying this type of organization are universal, and we study them comparing the living and non-living two-dimensional systems self-organizing at the spherical surface. Such-type physical systems often possess trigonal order with specific elongated defects, scars and pleats, where the 5-valence and 7-valence vertices alternate. In spite of the fact that the same physical and topological rules are involved in the structural organization of biological systems, such topological defects were never reported in epithelia. We have discovered them in the follicular spherical epithelium of ascidians that are emerging models in developmental biology. Surprisingly, the considered defects appear in the epithelium even when the number of cells in it is significantly less than the previously known threshold value. We explain this result by differences in the cell sizes and check our hypothesis considering the self-assembly of different random size particles on the spherical surface. Scars, pleats and other complex defects found in ascidian samples can play an unexpected and decisive role in the permanent renewal and reorganization of epithelia, which forms or lines many tissues and organs in metazoans.
Collapse
|
6
|
Ridgway WJM, Cheviakov AF. Locally and globally optimal configurations of N particles on the sphere with applications in the narrow escape and narrow capture problems. Phys Rev E 2019; 100:042413. [PMID: 31770902 DOI: 10.1103/physreve.100.042413] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2018] [Indexed: 11/07/2022]
Abstract
Determination of optimal arrangements of N particles on a sphere is a well-known problem in physics. A famous example of such is the Thomson problem of finding equilibrium configurations of electrical charges on a sphere. More recently, however, similar problems involving other potentials and nonspherical domains have arisen in biophysical systems. Many optimal configurations have previously been computed, especially for the Thomson problem; however, few results exist for potentials that correspond to more applied problems. Here we numerically compute optimal configurations corresponding to the narrow escape and narrow capture problems in biophysics. We provide comprehensive tables of global energy minima for N≤120 and local energy minima for N≤65, and we exclude all saddle points. Local minima up to N=120 are available online.
Collapse
Affiliation(s)
- Wesley J M Ridgway
- Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada
| | - Alexei F Cheviakov
- Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada
| |
Collapse
|
7
|
Yao Z. Command of Collective Dynamics by Topological Defects in Spherical Crystals. PHYSICAL REVIEW LETTERS 2019; 122:228002. [PMID: 31283259 DOI: 10.1103/physrevlett.122.228002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2019] [Revised: 04/13/2019] [Indexed: 06/09/2023]
Abstract
Directing individual motions of many constituents to a coherent dynamical state is a fundamental challenge in multiple fields. Here, based on the spherical crystal model, we show that topological defects in particle arrays can be a crucial element in regulating collective dynamics. Specifically, we highlight the defect-driven synchronized breathing modes around disclinations and collective oscillations with strong connection to disruption of crystalline order. This work opens the promising possibility of an organizational principle based on topological defects, and may inspire new strategies for harnessing intriguing collective dynamics in extensive nonequilibrium systems.
Collapse
Affiliation(s)
- Zhenwei Yao
- School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
| |
Collapse
|
8
|
Gilbert J, Cheviakov A. Globally optimal volume-trap arrangements for the narrow-capture problem inside a unit sphere. Phys Rev E 2019; 99:012109. [PMID: 30780370 DOI: 10.1103/physreve.99.012109] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2018] [Indexed: 11/07/2022]
Abstract
The determination of statistical characteristics for particles undergoing Brownian motion in constrained domains has multiple applications in various areas of research. This work presents an attempt to systematically compute globally optimal configurations of traps inside a three-dimensional domain that minimize the average of the mean first passage time (MFPT) for the narrow capture problem, the average time it takes a particle to be captured by any trap. For a given domain, the mean first passage time satisfies a linear Poisson problem with Dirichlet-Neumann boundary conditions. While no closed-form general solution of such problems is known, approximate asymptotic MFPT expressions for small traps in a unit sphere have been found. These solutions explicitly depend on trap parameters, including locations, through a pairwise potential function. After probing the applicability limits of asymptotic formulas through comparisons with numerical and available exact solutions of the narrow capture problem, full three-dimensional global optimization was performed to find optimal trap positions in the unit sphere for 2≤N≤100 identical traps. The interaction energy values and geometrical features of the putative optimal trap arrangements are presented.
Collapse
Affiliation(s)
- Jason Gilbert
- Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon S7N 5E6, Canada
| | - Alexei Cheviakov
- Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon S7N 5E6, Canada
| |
Collapse
|
9
|
Chen J, Xing X, Yao Z. Depletion zones and crystallography on pinched spheres. Phys Rev E 2018; 97:032605. [PMID: 29776116 DOI: 10.1103/physreve.97.032605] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Indexed: 11/07/2022]
Abstract
Understanding the interplay between ordered structures and substrate curvature is an interesting problem with versatile applications, including functionalization of charged supramolecular surfaces and modern microfluidic technologies. In this work, we investigate the two-dimensional packing structures of charged particles confined on a pinched sphere. By continuously pinching the sphere, we observe cleavage of elongated scars into pleats, proliferation of disclinations, and subsequently, emergence of a depletion zone at the negatively curved waist that is completely void of particles. We systematically study the geometrics and energetics of the depletion zone, and reveal its physical origin as a finite size effect, due to the interplay between Coulomb repulsion and concave geometry of the pinched sphere. These results further our understanding of crystallography on curved surfaces, and have implications in design and manipulation of charged, deformable interfaces in various applications.
Collapse
Affiliation(s)
- Jingyuan Chen
- School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China.,Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Xiangjun Xing
- School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China.,Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
| | - Zhenwei Yao
- School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China.,Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
| |
Collapse
|
10
|
Ballard AJ, Das R, Martiniani S, Mehta D, Sagun L, Stevenson JD, Wales DJ. Energy landscapes for machine learning. Phys Chem Chem Phys 2018; 19:12585-12603. [PMID: 28367548 DOI: 10.1039/c7cp01108c] [Citation(s) in RCA: 62] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Abstract
Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions. In particular, we can define quantities analogous to molecular structure, thermodynamics, and kinetics, and relate these emergent properties to the structure of the underlying landscape. This Perspective aims to describe these analogies with examples from recent applications, and suggest avenues for new interdisciplinary research.
Collapse
Affiliation(s)
- Andrew J Ballard
- University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, UK.
| | - Ritankar Das
- University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, UK.
| | - Stefano Martiniani
- University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, UK.
| | - Dhagash Mehta
- Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, IN, USA
| | - Levent Sagun
- Mathematics Department, Courant Institute, New York University, NY, USA
| | | | - David J Wales
- University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, UK.
| |
Collapse
|
11
|
Abstract
Understanding geometric frustration of ordered phases in two-dimensional condensed matter on curved surfaces is closely related to a host of scientific problems in condensed matter physics and materials science. Here, we show how two-dimensional Lennard-Jones crystal clusters confined on a sphere resolve geometric frustration and form pentagonal vacancy structures. These vacancies, originating from the combination of curvature and physical interaction, are found to be topological defects and they can be further classified into dislocational and disclinational types. We analyze the dual role of these crystallographic defects as both vacancies and topological defects, illustrate their formation mechanism, and present the phase diagram. The revealed dual role of the topological vacancies may find applications in the fabrication of robust nanopores. This work also shows the promising potential of exploiting richness in both physical interactions and substrate geometries to create new types of crystallographic defects, which have strong connections with the design of crystalline materials.
Collapse
Affiliation(s)
- Zhenwei Yao
- School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China.
| |
Collapse
|
12
|
Arias E, Florez E, Pérez–Torres JF. Algorithm based on the Thomson problem for determination of equilibrium structures of metal nanoclusters. J Chem Phys 2017; 146:244107. [DOI: 10.1063/1.4984049] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Affiliation(s)
- E. Arias
- Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, Colombia
| | - E. Florez
- Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, Colombia
| | - J. F. Pérez–Torres
- Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, Colombia
- Escuela de Qumica, Universidad Industrial de Santander, Bucaramanga, Colombia
| |
Collapse
|
13
|
Hall DM, Grason GM. How geometric frustration shapes twisted fibres, inside and out: competing morphologies of chiral filament assembly. Interface Focus 2017. [PMID: 28630675 DOI: 10.1098/rsfs.2016.0140] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Chirality frustrates and shapes the assembly of flexible filaments in rope-like, twisted bundles and fibres by introducing gradients of both filament shape (i.e. curvature) and packing throughout the structure. Previous models of chiral filament bundle formation have shown that this frustration gives rise to several distinct morphological responses, including self-limiting bundle widths, anisotropic domain (tape-like) formation and topological defects in the lateral inter-filament order. In this paper, we employ a combination of continuum elasticity theory and discrete filament bundle simulations to explore how these distinct morphological responses compete in the broader phase diagram of chiral filament assembly. We show that the most generic model of bundle formation exhibits at least four classes of equilibrium structure-finite-width, twisted bundles with isotropic and anisotropic shapes, with and without topological defects, as well as bulk phases of untwisted, columnar assembly (i.e. 'frustration escape'). These competing equilibrium morphologies are selected by only a relatively small number of parameters describing filament assembly: bundle surface energy, preferred chiral twist and stiffness of chiral filament interactions, and mechanical stiffness of filaments and their lateral interactions. Discrete filament bundle simulations test and verify continuum theory predictions for dependence of bundle structure (shape, size and packing defects of two-dimensional cross section) on these key parameters.
Collapse
Affiliation(s)
- Douglas M Hall
- Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA, USA
| | - Gregory M Grason
- Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA, USA
| |
Collapse
|
14
|
Morgan JWR, Mehta D, Wales DJ. Properties of kinetic transition networks for atomic clusters and glassy solids. Phys Chem Chem Phys 2017; 19:25498-25508. [DOI: 10.1039/c7cp03346j] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Small-world and scale-free properties are analysed for kinetic transition networks of clusters and glassy systems.
Collapse
Affiliation(s)
- John W. R. Morgan
- Department of Chemical Engineering
- University of Michigan
- Ann Arbor
- USA
| | - Dhagash Mehta
- Department of Applied and Computational Mathematics and Statistics
- University of Notre Dame
- Notre Dame
- USA
| | | |
Collapse
|