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Martinsons M, Hielscher J, Kapfer SC, Schmiedeberg M. Event-chain Monte Carlo simulations of the liquid to solid transition of two-dimensional decagonal colloidal quasicrystals. J Phys Condens Matter 2019; 31:475103. [PMID: 31342938 DOI: 10.1088/1361-648x/ab3519] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
In event-chain Monte Carlo simulations, we model colloidal particles in two dimensions that interact according to an isotropic short-ranged pair potential which supports the two typical length scales present in decagonal quasicrystals. We investigate the assembled structures as we vary the density and temperature. Our special interest is related to the transition from quasicrystal to liquid. In contrast to the KTHNY melting theory for quasicrystals which predicts an intermediate pentahedratic phase, we find a one-step first-order melting transition. However, we discover that the slow relaxation of phasonic flips, i.e. rearrangements of the particles due to additional degrees of freedom in quasicrystals, changes the positional correlation functions, to the extent that structures with long-range orientational correlations, but exponentially decaying positional correlations, are observed.
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Affiliation(s)
- Juliane U. Klamser
- Laboratoire de Physique Statistique, Département de physique de l’ENS, Ecole Normale Supérieure, PSL Research University, Université Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06, CNRS, 75005 Paris, France
| | - Sebastian C. Kapfer
- Theoretische Physik 1, FAU Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
| | - Werner Krauth
- Laboratoire de Physique Statistique, Département de physique de l’ENS, Ecole Normale Supérieure, PSL Research University, Université Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06, CNRS, 75005 Paris, France
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Klatt MA, Lovrić J, Chen D, Kapfer SC, Schaller FM, Schönhöfer PWA, Gardiner BS, Smith AS, Schröder-Turk GE, Torquato S. Universal hidden order in amorphous cellular geometries. Nat Commun 2019; 10:811. [PMID: 30778054 PMCID: PMC6379405 DOI: 10.1038/s41467-019-08360-5] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2018] [Accepted: 01/03/2019] [Indexed: 12/04/2022] Open
Abstract
Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties. Disordered hyperuniformity implies a hidden order on length scales that can be found in various amorphous materials. Klatt et al. analyse the evolution of random point patterns using Llyod’s algorithm and show that they converge to an effectively hyperuniform state regardless of the initial conditions.
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Affiliation(s)
- Michael A Klatt
- Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76131, Karlsruhe, Germany.,Department of Physics, Princeton University, Princeton, NJ, 08544, USA
| | - Jakov Lovrić
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10 000 Zagreb, Croatia.,School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,PULS Group, Department of Physics and Interdisciplinary Center for Nanostructured Films, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany
| | - Duyu Chen
- Department of Chemistry, Princeton University, Princeton, NJ, 08544, USA
| | - Sebastian C Kapfer
- Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany
| | - Fabian M Schaller
- Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76131, Karlsruhe, Germany.,Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany
| | - Philipp W A Schönhöfer
- School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany
| | - Bruce S Gardiner
- School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,School of Computer Science and Software Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia
| | - Ana-Sunčana Smith
- Division of Physical Chemistry, Ruđer Bošković Institute, Bijenička 54, 10 000 Zagreb, Croatia.,PULS Group, Department of Physics and Interdisciplinary Center for Nanostructured Films, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany
| | - Gerd E Schröder-Turk
- School of Engineering and Information Technology, Murdoch University, 90 South St, Murdoch, WA, 6150, Australia.,Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058, Erlangen, Germany.,Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, ACT, 0200, Australia
| | - Salvatore Torquato
- Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA.
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Schindler T, Kapfer SC. Nonequilibrium steady states, coexistence, and criticality in driven quasi-two-dimensional granular matter. Phys Rev E 2019; 99:022902. [PMID: 30934354 DOI: 10.1103/physreve.99.022902] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2018] [Indexed: 06/09/2023]
Abstract
Nonequilibrium steady states of vibrated inelastic frictionless spheres are investigated in quasi-two-dimensional confinement via molecular dynamics simulations. The phase diagram in the density-amplitude plane exhibits a fluidlike disordered and an ordered phase with threefold symmetry, as well as phase coexistence between the two. A dynamical mechanism exists that brings about metastable traveling clusters and at the same time stable clusters with anisotropic shapes at low vibration amplitude. Moreover, there is a square bilayer state which is connected to the fluid by BKTHNY-type two-step melting with an intermediate tetratic phase. The critical behavior of the two continuous transitions is studied in detail. For the fluid-tetratic transition, critical exponents of γ[over ̃]=1.73, η_{4}≈1/4, and z=2.05 are obtained.
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Affiliation(s)
- Thomas Schindler
- Theoretische Physik 1, FAU Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany
| | - Sebastian C Kapfer
- Theoretische Physik 1, FAU Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany
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Fey S, Kapfer SC, Schmidt KP. Quantum Criticality of Two-Dimensional Quantum Magnets with Long-Range Interactions. Phys Rev Lett 2019; 122:017203. [PMID: 31012713 DOI: 10.1103/physrevlett.122.017203] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2018] [Indexed: 06/09/2023]
Abstract
We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary transformations with classical Monte Carlo simulations to extract high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean-field to nearest-neighbor universality with continuously varying critical exponents. In the frustrated case on the square lattice the system remains in the universality class of the nearest-neighbor model independent of the long-range nature of the interaction, while we argue that the quantum criticality for the triangular lattice is terminated by a first-order phase transition line.
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Affiliation(s)
- Sebastian Fey
- Lehrstuhl für Theoretische Physik I, Staudtstraße 7, Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
| | - Sebastian C Kapfer
- Lehrstuhl für Theoretische Physik I, Staudtstraße 7, Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
| | - Kai Phillip Schmidt
- Lehrstuhl für Theoretische Physik I, Staudtstraße 7, Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
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Abstract
We study the continuous one-dimensional hard-sphere model and present irreversible local Markov chains that mix on faster time scales than the reversible heat bath or Metropolis algorithms. The mixing time scales appear to fall into two distinct universality classes, both faster than for reversible local Markov chains. The event-chain algorithm, the infinitesimal limit of one of these Markov chains, belongs to the class presenting the fastest decay. For the lattice-gas limit of the hard-sphere model, reversible local Markov chains correspond to the symmetric simple exclusion process (SEP) with periodic boundary conditions. The two universality classes for irreversible Markov chains are realized by the totally asymmetric SEP (TASEP), and by a faster variant (lifted TASEP) that we propose here. We discuss how our irreversible hard-sphere Markov chains generalize to arbitrary repulsive pair interactions and carry over to higher dimensions through the concept of lifted Markov chains and the recently introduced factorized Metropolis acceptance rule.
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Affiliation(s)
- Sebastian C Kapfer
- Theoretische Physik 1, FAU Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Werner Krauth
- Laboratoire de Physique Statistique, Département de physique de l'ENS, Ecole Normale Supérieure, PSL Research University, Université Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Université Paris 06, CNRS, 75005 Paris, France
- Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, Japan
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Schmiedeberg M, Achim CV, Hielscher J, Kapfer SC, Löwen H. Dislocation-free growth of quasicrystals from two seeds due to additional phasonic degrees of freedom. Phys Rev E 2017; 96:012602. [PMID: 29347123 DOI: 10.1103/physreve.96.012602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2017] [Indexed: 06/07/2023]
Abstract
We explore the growth of two-dimensional quasicrystals, i.e., aperiodic structures that possess long-range order, from two seeds at various distances and with different orientations by using dynamical phase-field crystal calculations. We compare the results to the growth of periodic crystals from two seeds. There, a domain border consisting of dislocations is observed in case of large distances between the seed and large angles between their orientation. Furthermore, a domain border is found if the seeds are placed at a distance that does not fit to the periodic lattice. In the case of the growth of quasicrystals, we only observe domain borders for large distances and different orientations. Note that all distances do inherently not match to a perfect domain wall-free quasicrystalline structure. Nevertheless, we find dislocation-free growth for all seeds at a small enough distance and for all seeds that approximately have the same orientation. In periodic structures, the stress that occurs due to incommensurate distances between the seeds results in phononic strain fields or, in the case of too large stresses, in dislocations. In contrast, in quasicrystals an additional phasonic strain field can occur and suppress dislocations. Phasons are additional degrees of freedom that are unique to quasicrystals. As a consequence, the additional phasonic strain field helps to distribute the stress and facilitates the growth of dislocation-free quasicrystals from multiple seeds. In contrast, in the periodic case the growth from multiple seeds most likely leads to a structure with multiple domains. Our work lays the theoretical foundations for growing perfect quasicrystals from different seeds and is therefore relevant for many applications.
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Affiliation(s)
- M Schmiedeberg
- Institut für Theoretische Physik 1, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
| | - C V Achim
- Water Research Center for Agriculture and Mining (CRHIAM), University of Concepción, 4030000 Concepción, Chile
| | - J Hielscher
- Institut für Theoretische Physik 1, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
| | - S C Kapfer
- Institut für Theoretische Physik 1, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
| | - H Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40204 Düsseldorf, Germany
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Hielscher J, Martinsons M, Schmiedeberg M, Kapfer SC. Detection of phonon and phason modes in intrinsic colloidal quasicrystals by reconstructing their structure in hyperspace. J Phys Condens Matter 2017; 29:094002. [PMID: 28008870 DOI: 10.1088/1361-648x/aa55a5] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Phasons are additional degrees of freedom which occur in quasicrystals alongside the phonons known from conventional periodic crystals. The rearrangements of particles that are associated with a phason mode are hard to interpret in physical space. We reconstruct the quasicrystal structure by an embedding into extended higher-dimensional space, where phasons correspond to displacements perpendicular to the physical space. In dislocation-free decagonal colloidal quasicrystals annealed with Brownian dynamics simulations, we identify thermal phonon and phason modes. Finite phononic strain is pinned by phasonic excitations even after cooling down to zero temperature. For the phasonic displacements underlying the flip pattern, the reconstruction method gives an approximation within the limits of a multi-mode harmonic ansatz, and points to fundamental limitations of a harmonic picture for phasonic excitations in intrinsic colloidal quasicrystals.
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Affiliation(s)
- J Hielscher
- Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
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Abstract
We present a rigorous efficient event-chain Monte Carlo algorithm for long-range interacting particle systems. Using a cell-veto scheme within the factorized Metropolis algorithm, we compute each single-particle move with a fixed number of operations. For slowly decaying potentials such as Coulomb interactions, screening line charges allow us to take into account periodic boundary conditions. We discuss the performance of the cell-veto Monte Carlo algorithm for general inverse-power-law potentials, and illustrate how it provides a new outlook on one of the prominent bottlenecks in large-scale atomistic Monte Carlo simulations.
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Affiliation(s)
- Sebastian C Kapfer
- Theoretische Physik 1, FAU Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Werner Krauth
- Laboratoire de Physique Statistique, Ecole Normale Supérieure/PSL Research University, UPMC, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France
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Spanner M, Höfling F, Kapfer SC, Mecke KR, Schröder-Turk GE, Franosch T. Splitting of the Universality Class of Anomalous Transport in Crowded Media. Phys Rev Lett 2016; 116:060601. [PMID: 26918973 DOI: 10.1103/physrevlett.116.060601] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2015] [Indexed: 06/05/2023]
Abstract
We investigate the emergence of subdiffusive transport by obstruction in continuum models for molecular crowding. While the underlying percolation transition for the accessible space displays universal behavior, the dynamic properties depend in a subtle nonuniversal way on the transport through narrow channels. At the same time, the different universality classes are robust with respect to introducing correlations in the obstacle matrix as we demonstrate for quenched hard-sphere liquids as underlying structures. Our results confirm that the microscopic dynamics can dominate the relaxational behavior even at long times, in striking contrast to glassy dynamics.
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Affiliation(s)
- Markus Spanner
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Felix Höfling
- Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
- Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany, and IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Sebastian C Kapfer
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Klaus R Mecke
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Gerd E Schröder-Turk
- Murdoch University, School of Engineering and IT, Mathematics and Statistics, Murdoch, Western Australia 6150, Australia
| | - Thomas Franosch
- Institut für Theoretische Physik, Leopold-Franzens-Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria
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Abstract
The phase diagram of two-dimensional continuous particle systems is studied using the event-chain Monte Carlo algorithm. For soft disks with repulsive power-law interactions ∝r^{-n} with n≳6, the recently established hard-disk melting scenario (n→∞) holds: a first-order liquid-hexatic and a continuous hexatic-solid transition are identified. Close to n=6, the coexisting liquid exhibits very long orientational correlations, and positional correlations in the hexatic are extremely short. For n≲6, the liquid-hexatic transition is continuous, with correlations consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario. To illustrate the generality of these results, we demonstrate that Yukawa particles likewise may follow either the KTHNY or the hard-disk melting scenario, depending on the Debye-Hückel screening length as well as on the temperature.
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Affiliation(s)
- Sebastian C Kapfer
- Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France
| | - Werner Krauth
- Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France
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Michel M, Kapfer SC, Krauth W. Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps. J Chem Phys 2014; 140:054116. [DOI: 10.1063/1.4863991] [Citation(s) in RCA: 71] [Impact Index Per Article: 7.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Mickel W, Kapfer SC, Schröder-Turk GE, Mecke K. Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter. J Chem Phys 2013; 138:044501. [DOI: 10.1063/1.4774084] [Citation(s) in RCA: 182] [Impact Index Per Article: 16.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
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Schröder-Turk GE, de Campo L, Evans ME, Saba M, Kapfer SC, Varslot T, Grosse-Brauckmann K, Ramsden S, Hyde ST. Polycontinuous geometries for inverse lipid phases with more than two aqueous network domains. Faraday Discuss 2013; 161:215-47; discussion 273-303. [DOI: 10.1039/c2fd20112g] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Kapfer SC, Mickel W, Mecke K, Schröder-Turk GE. Jammed spheres: Minkowski tensors reveal onset of local crystallinity. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 85:030301. [PMID: 22587029 DOI: 10.1103/physreve.85.030301] [Citation(s) in RCA: 39] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2011] [Revised: 01/26/2012] [Indexed: 05/31/2023]
Abstract
The local structure of disordered jammed packings of monodisperse spheres without friction, generated by the Lubachevsky-Stillinger algorithm, is studied for packing fractions above and below 64%. The structural similarity of the particle environments to fcc or hcp crystalline packings (local crystallinity) is quantified by order metrics based on rank-four Minkowski tensors. We find a critical packing fraction φ(c)≈0.649, distinctly higher than previously reported values for the contested random close packing limit. At φ(c), the probability of finding local crystalline configurations first becomes finite and, for larger packing fractions, increases by several orders of magnitude. This provides quantitative evidence of an abrupt onset of local crystallinity at φ(c). We demonstrate that the identification of local crystallinity by the frequently used local bond-orientational order metric q(6) produces false positives and thus conceals the abrupt onset of local crystallinity. Since the critical packing fraction is significantly above results from mean-field analysis of the mechanical contacts for frictionless spheres, it is suggested that dynamic arrest due to isostaticity and the alleged geometric phase transition in the Edwards framework may be disconnected phenomena.
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Affiliation(s)
- Sebastian C Kapfer
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany.
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Kapfer SC, Hyde ST, Mecke K, Arns CH, Schröder-Turk GE. Minimal surface scaffold designs for tissue engineering. Biomaterials 2011; 32:6875-82. [DOI: 10.1016/j.biomaterials.2011.06.012] [Citation(s) in RCA: 272] [Impact Index Per Article: 20.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2011] [Accepted: 06/07/2011] [Indexed: 10/18/2022]
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Schröder-Turk GE, Varslot T, de Campo L, Kapfer SC, Mickel W. A bicontinuous mesophase geometry with hexagonal symmetry. Langmuir 2011; 27:10475-10483. [PMID: 21728305 DOI: 10.1021/la201718a] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We report that a specific realization of Schwarz's triply periodic hexagonal minimal surface is isotropic with respect to the Doi-Ohta interface tensor and simultaneously has minimal packing and stretching frustration similar to those of the commonly found cubic bicontinuous mesophases. This hexagonal surface, of symmetry P6(3)/mmc with a lattice ratio of c/a = 0.832, is therefore a likely candidate geometry for self-assembled lipid/surfactant or copolymer mesophases. Furthermore, both the peak position ratios in its powder diffraction pattern and the elastic moduli closely resemble those of the cubic bicontinuous phases. We therefore argue that a genuine possibility of experimental misidentification exists.
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Affiliation(s)
- Gerd E Schröder-Turk
- Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany.
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Schröder-Turk GE, Mickel W, Kapfer SC, Klatt MA, Schaller FM, Hoffmann MJF, Kleppmann N, Armstrong P, Inayat A, Hug D, Reichelsdorfer M, Peukert W, Schwieger W, Mecke K. Minkowski tensor shape analysis of cellular, granular and porous structures. Adv Mater 2011; 23:2535-2553. [PMID: 21681830 DOI: 10.1002/adma.201100562] [Citation(s) in RCA: 52] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.
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Affiliation(s)
- G E Schröder-Turk
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany.
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Abstract
The effective linear-elastic moduli of disordered network solids are analyzed by voxel-based finite element calculations. We analyze network solids given by Poisson-Voronoi processes and by the structure of collagen fiber networks imaged by confocal microscopy. The solid volume fraction ϕ is varied by adjusting the fiber radius, while keeping the structural mesh or pore size of the underlying network fixed. For intermediate ϕ, the bulk and shear modulus are approximated by empirical power-laws K(phi)proptophin and G(phi)proptophim with n≈1.4 and m≈1.7. The exponents for the collagen and the Poisson-Voronoi network solids are similar, and are close to the values n=1.22 and m=2.11 found in a previous voxel-based finite element study of Poisson-Voronoi systems with different boundary conditions. However, the exponents of these empirical power-laws are at odds with the analytic values of n=1 and m=2, valid for low-density cellular structures in the limit of thin beams. We propose a functional form for K(ϕ) that models the cross-over from a power-law at low densities to a porous solid at high densities; a fit of the data to this functional form yields the asymptotic exponent n≈1.00, as expected. Further, both the intensity of the Poisson-Voronoi process and the collagen concentration in the samples, both of which alter the typical pore or mesh size, affect the effective moduli only by the resulting change of the solid volume fraction. These findings suggest that a network solid with the structure of the collagen networks can be modeled in quantitative agreement by a Poisson-Voronoi process.
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Affiliation(s)
- Susan Nachtrab
- Institut für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
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