1
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Melio J, Henkes SE, Kraft DJ. Soft and Stiff Normal Modes in Floppy Colloidal Square Lattices. PHYSICAL REVIEW LETTERS 2024; 132:078202. [PMID: 38427878 DOI: 10.1103/physrevlett.132.078202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2023] [Revised: 11/08/2023] [Accepted: 01/03/2024] [Indexed: 03/03/2024]
Abstract
Floppy microscale spring networks are widely studied in theory and simulations, but no well-controlled experimental system currently exists. Here, we show that square lattices consisting of colloid-supported lipid bilayers functionalized with DNA linkers act as microscale floppy spring networks. We extract their normal modes by inverting the particle displacement correlation matrix, showing the emergence of a spectrum of soft modes with low effective stiffness in addition to stiff modes that derive from linker interactions. Evaluation of the softest mode, a uniform shear mode, reveals that shear stiffness decreases with lattice size. Experiments match well with Brownian particle simulations, and we develop a theoretical description based on mapping interactions onto a linear response model to describe the modes. Our results reveal the importance of entropic steric effects and can be used for developing reconfigurable materials at the colloidal length scale.
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Affiliation(s)
- Julio Melio
- Huygens-Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
| | - Silke E Henkes
- Lorentz Institute, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands
| | - Daniela J Kraft
- Huygens-Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
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2
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Caprini L, Marini Bettolo Marconi U, Löwen H. Entropy production and collective excitations of crystals out of equilibrium: The concept of entropons. Phys Rev E 2023; 108:044603. [PMID: 37978682 DOI: 10.1103/physreve.108.044603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2023] [Accepted: 09/21/2023] [Indexed: 11/19/2023]
Abstract
We study the collective vibrational excitations of crystals under out-of-equilibrium steady conditions that give rise to entropy production. Their excitation spectrum comprises equilibriumlike phonons of thermal origin and additional collective excitations called entropons because each of them represents a mode of spectral entropy production. Entropons coexist with phonons and dominate them when the system is far from equilibrium while they are negligible in near-equilibrium regimes. The concept of entropons has been recently introduced and verified in a special case of crystals formed by self-propelled particles. Here we show that entropons exist in a broader class of active crystals that are intrinsically out of equilibrium and characterized by the lack of detailed balance. After a general derivation, several explicit examples are discussed, including crystals consisting of particles with alignment interactions and frictional contact forces.
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Affiliation(s)
- L Caprini
- Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik II: Weiche Materie, Universitätsstrasse, 40225 Düsseldorf, Germany
| | - U Marini Bettolo Marconi
- Physics Department, Scuola di Scienze e Tecnologie, Università di Camerino - via Madonna delle Carceri, 62032 Camerino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via A. Pascoli, 06123 Perugia, Italy
| | - H Löwen
- Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik II: Weiche Materie, Universitätsstrasse, 40225 Düsseldorf, Germany
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3
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van Baalen C, Vialetto J, Isa L. Tuning Electrostatic Interactions of Colloidal Particles at Oil-Water Interfaces with Organic Salts. PHYSICAL REVIEW LETTERS 2023; 131:128202. [PMID: 37802948 DOI: 10.1103/physrevlett.131.128202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2023] [Accepted: 08/23/2023] [Indexed: 10/08/2023]
Abstract
Monolayers of colloidal particles at oil-water interfaces readily crystallize owing to electrostatic repulsion, which is often mediated through the oil. However, little attempts exist to control it using oil-soluble electrolytes. We probe the interactions among charged hydrophobic microspheres confined at a water-hexadecane interface and show that repulsion can be continuously tuned over orders of magnitude upon introducing nanomolar amounts of an organic salt into the oil. Our results are compatible with an associative discharging mechanism of surface groups at the particle-oil interface, similar to the charge regulation observed for charged colloids in nonpolar solvents.
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Affiliation(s)
- Carolina van Baalen
- Laboratory for Soft Materials and Interfaces, Department of Materials, ETH Zürich, Vladimir-Prelog-Weg 5, 8093 Zürich, Switzerland
| | - Jacopo Vialetto
- Laboratory for Soft Materials and Interfaces, Department of Materials, ETH Zürich, Vladimir-Prelog-Weg 5, 8093 Zürich, Switzerland
| | - Lucio Isa
- Laboratory for Soft Materials and Interfaces, Department of Materials, ETH Zürich, Vladimir-Prelog-Weg 5, 8093 Zürich, Switzerland
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4
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Qian D, Olvera de la Cruz M. Field-driven cluster formation in two-dimensional colloidal binary mixtures. Phys Rev E 2023; 107:044605. [PMID: 37198853 DOI: 10.1103/physreve.107.044605] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2022] [Accepted: 03/22/2023] [Indexed: 05/19/2023]
Abstract
We study size- and charge-asymmetric oppositely charged colloids driven by an external electric field. The large particles are connected by harmonic springs, forming a hexagonal-lattice network, while the small particles are free of bonds and exhibit fluidlike motion. We show that this model exhibits a cluster formation pattern when the external driving force exceeds a critical value. The clustering is accompanied with stable wave packets in vibrational motions of the large particles.
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Affiliation(s)
- Dingwen Qian
- Applied Physics Program, Northwestern University, Evanston, Illinois 60208, USA
| | - Monica Olvera de la Cruz
- Applied Physics Program, Department of Materials Science and Engineering, Department of Chemistry, and Department of Physic and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
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5
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The Stability of a Hydrodynamic Bravais Lattice. Symmetry (Basel) 2022. [DOI: 10.3390/sym14081524] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Abstract
We present the results of a theoretical investigation of the stability and collective vibrations of a two-dimensional hydrodynamic lattice comprised of millimetric droplets bouncing on the surface of a vibrating liquid bath. We derive the linearized equations of motion describing the dynamics of a generic Bravais lattice, as encompasses all possible tilings of parallelograms in an infinite plane-filling array. Focusing on square and triangular lattice geometries, we demonstrate that for relatively low driving accelerations of the bath, only a subset of inter-drop spacings exist for which stable lattices may be achieved. The range of stable spacings is prescribed by the structure of the underlying wavefield. As the driving acceleration is increased progressively, the initially stationary lattices destabilize into coherent oscillatory motion. Our analysis yields both the instability threshold and the wavevector and polarization of the most unstable vibrational mode. The non-Markovian nature of the droplet dynamics renders the stability analysis of the hydrodynamic lattice more rich and subtle than that of its solid state counterpart.
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6
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Gasser U, Zhou B. Accurate detection of spherical objects in a complex background. OPTICS EXPRESS 2021; 29:37048-37065. [PMID: 34808784 DOI: 10.1364/oe.434652] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2021] [Accepted: 10/02/2021] [Indexed: 06/13/2023]
Abstract
The automated detection of particles in microscopy images has become a routinely used method for quantitative image analysis in biology, physics, and other research fields. While the majority of particle detection algorithms have been developed for bulk materials, the detection of particles in a heterogenous environment due to surfaces or other objects in the studied material is of great interest. However, particle detection is hindered by a complex background due to the diffraction of light resulting in a decreased contrast and image noise. We present a new heuristic method for the reliable detection of spherical particles that suppresses false detections due to a heterogenous background without additional background measurements. Further, we discuss methods to obtain particle coordinates with improved accuracy and compare with other methods, in particular with that of Crocker and Grier.
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7
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Tlusty T. Exceptional topology in ordinary soft matter. Phys Rev E 2021; 104:025002. [PMID: 34525622 DOI: 10.1103/physreve.104.025002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2021] [Accepted: 07/14/2021] [Indexed: 01/22/2023]
Abstract
Hydrodynamics is shown to induce non-Hermitian topological phenomena in ordinary, passive soft matter. This is demonstrated by subjecting a two-dimensional elastic lattice to a low-Reynolds viscous flow. The interplay of hydrodynamics and elasticity splits Dirac cones into bulk Fermi arcs, pairing exceptional points with opposite half-integer topological charges. The bulk Fermi arc is a generic hallmark of the system exhibited in all lattice and flow symmetries. An analytic model and simulations explain how the emergent singularities shape the spectral bands and give rise to a web of van Hove singularity lines in the density of states. The present findings suggest that non-Hermitian physics can be explored in a broad class of ordinary soft matter, living and artificial alike, opening avenues for topology-based technology in this regime.
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Affiliation(s)
- Tsvi Tlusty
- Center for Soft and Living Matter, Institute for Basic Science, and Physics and Chemistry Departments, Ulsan National Institute of Science and Technology, Ulsan 44919, Korea
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8
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Ma X, Mishra CK, Habdas P, Yodh AG. Structural and short-time vibrational properties of colloidal glasses and supercooled liquids in the vicinity of the re-entrant glass transition. J Chem Phys 2021; 155:074902. [PMID: 34418931 DOI: 10.1063/5.0059084] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
Abstract
We investigate the short-time vibrational properties and structure of two-dimensional, bidisperse, colloidal glasses and supercooled liquids in the vicinity of the re-entrant glass transition, as a function of interparticle depletion attraction strength. The long-time spatiotemporal dynamics of the samples are measured to be non-monotonic, confirming that the suspensions evolve from repulsive glass to supercooled liquid to attractive glass with increasing depletion attraction. Here, we search for vibrational signatures of the re-entrant behavior in the short-time spatiotemporal dynamics, i.e., dynamics associated with particle motion inside its nearest-neighbor cage. Interestingly, we observe that the anharmonicity of these in-cage vibrations varies non-monotonically with increasing attraction strength, consistent with the non-monotonic long-time structural relaxation dynamics of the re-entrant glass. We also extract effective spring constants between neighboring particles; we find that spring stiffness involving small particles also varies non-monotonically with increasing attraction strength, while stiffness between large particles increases monotonically. Last, from study of depletion-dependent local structure and vibration participation fractions, we gain microscopic insight into the particle-size-dependent contributions to short-time vibrational modes in the glass and supercooled liquid states.
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Affiliation(s)
- Xiaoguang Ma
- Center for Complex Flows and Soft Matter Research, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
| | - Chandan K Mishra
- Discipline of Physics, Indian Institute of Technology (IIT) Gandhinagar Palaj, Gandhinagar, Gujarat 382355, India
| | - P Habdas
- Department of Physics, Saint Joseph's University, Philadelphia, Pennsylvania 19131, USA
| | - A G Yodh
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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9
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Kryuchkov NP, Yurchenko SO. Collective excitations in active fluids: Microflows and breakdown in spectral equipartition of kinetic energy. J Chem Phys 2021; 155:024902. [PMID: 34266286 DOI: 10.1063/5.0054854] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The effect of particle activity on collective excitations in active fluids of microflyers is studied. With an in silico study, we observed an oscillating breakdown of equipartition (uniform spectral distribution) of kinetic energy in reciprocal space. The phenomenon is related to short-range velocity-velocity correlations that were realized without forming of long-lived mesoscale vortices in the system. This stands in contrast to well-known mesoscale turbulence operating in active nematic systems (bacterial or artificial) and reveals the features of collective dynamics in active fluids, which should be important for structural transitions and glassy dynamics in active matter.
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Affiliation(s)
- Nikita P Kryuchkov
- Bauman Moscow State Technical University, 2nd Baumanskaya str. 5, 105005 Moscow, Russia
| | - Stanislav O Yurchenko
- Bauman Moscow State Technical University, 2nd Baumanskaya str. 5, 105005 Moscow, Russia
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10
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Chen Y, Tan X, Wang H, Zhang Z, Kosterlitz JM, Ling XS. 2D Colloidal Crystals with Anisotropic Impurities. PHYSICAL REVIEW LETTERS 2021; 127:018004. [PMID: 34270301 DOI: 10.1103/physrevlett.127.018004] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2021] [Revised: 04/13/2021] [Accepted: 06/04/2021] [Indexed: 06/13/2023]
Abstract
We report a study of 2D colloidal crystals with anisotropic ellipsoid impurities using video microscopy. It is found that at low impurity densities, the impurity particles behave like floating disorder with which the quasi-long-range orientational order survives and the elasticity of the system is actually enhanced. There is a critical impurity density above which the 2D crystal loses the quasi-long-range orientational order. At high impurity densities, the 2D crystal breaks into polycrystalline domains separated by grain boundaries where the impurity particles aggregate. This transition is accompanied by a decrease in the elastic moduli, and it is associated with strong heterogeneous dynamics in the system. The correlation length vs impurity density in the disordered phase exhibits an essential singularity at the critical impurity density.
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Affiliation(s)
- Ya Chen
- Institute for Advanced Study, Center for Soft Condensed Matter Physics and Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Xinlan Tan
- Institute for Advanced Study, Center for Soft Condensed Matter Physics and Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Huaguang Wang
- Institute for Advanced Study, Center for Soft Condensed Matter Physics and Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, China
- College of Chemistry, Chemical Engineering and Materials Science, Soochow University, Suzhou 215123, China
| | - Zexin Zhang
- Institute for Advanced Study, Center for Soft Condensed Matter Physics and Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, China
- College of Chemistry, Chemical Engineering and Materials Science, Soochow University, Suzhou 215123, China
| | - J M Kosterlitz
- Department of Physics, Brown University, Providence, Rhode Island 02912, USA
| | - Xinsheng Sean Ling
- Department of Physics, Brown University, Providence, Rhode Island 02912, USA
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11
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Guzmán E, Abelenda-Núñez I, Maestro A, Ortega F, Santamaria A, Rubio RG. Particle-laden fluid/fluid interfaces: physico-chemical foundations. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:333001. [PMID: 34102618 DOI: 10.1088/1361-648x/ac0938] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 06/08/2021] [Indexed: 06/12/2023]
Abstract
Particle-laden fluid/fluid interfaces are ubiquitous in academia and industry, which has fostered extensive research efforts trying to disentangle the physico-chemical bases underlying the trapping of particles to fluid/fluid interfaces as well as the properties of the obtained layers. The understanding of such aspects is essential for exploiting the ability of particles on the stabilization of fluid/fluid interface for the fabrication of novel interface-dominated devices, ranging from traditional Pickering emulsions to more advanced reconfigurable devices. This review tries to provide a general perspective of the physico-chemical aspects associated with the stabilization of interfaces by colloidal particles, mainly chemical isotropic spherical colloids. Furthermore, some aspects related to the exploitation of particle-laden fluid/fluid interfaces on the stabilization of emulsions and foams will be also highlighted. It is expected that this review can be used for researchers and technologist as an initial approach to the study of particle-laden fluid layers.
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Affiliation(s)
- Eduardo Guzmán
- Departamento de Química Física, Universidad Complutense de Madrid, Madrid, Spain
- Instituto Pluridisciplinar, Universidad Complutense de Madrid, Madrid, Spain
| | - Irene Abelenda-Núñez
- Departamento de Química Física, Universidad Complutense de Madrid, Madrid, Spain
| | | | - Francisco Ortega
- Departamento de Química Física, Universidad Complutense de Madrid, Madrid, Spain
- Instituto Pluridisciplinar, Universidad Complutense de Madrid, Madrid, Spain
| | - Andreas Santamaria
- Instituto Pluridisciplinar, Universidad Complutense de Madrid, Madrid, Spain
- Institut Laue-Langevin, Grenoble, France
| | - Ramón G Rubio
- Departamento de Química Física, Universidad Complutense de Madrid, Madrid, Spain
- Instituto Pluridisciplinar, Universidad Complutense de Madrid, Madrid, Spain
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12
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13
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Abstract
Over the last two decades, understanding of the attachment of colloids to fluid interfaces has attracted the interest of researchers from different fields. This is explained by considering the ubiquity of colloidal and interfacial systems in nature and technology. However, to date, the control and tuning of the assembly of colloids at fluid interfaces remain a challenge. This review discusses some of the most fundamental aspects governing the organization of colloidal objects at fluid interfaces, paying special attention to spherical particles. This requires a description of different physicochemical aspects, from the driving force involved in the assembly to its thermodynamic description, and from the interactions involved in the assembly to the dynamics and rheological behavior of particle-laden interfaces.
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14
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Goh S, Wittmann R, Menzel AM, Löwen H. Classical density functional theory for a two-dimensional isotropic ferrogel model with labeled particles. Phys Rev E 2019; 100:012605. [PMID: 31499838 DOI: 10.1103/physreve.100.012605] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2019] [Indexed: 01/30/2023]
Abstract
In this study, we formulate a density functional theory (DFT) for systems of labeled particles, considering a two-dimensional bead-spring lattice with a magnetic dipole on every bead as a model for ferrogels. On the one hand, DFT has been widely studied to investigate fluidlike states of materials, in which constituent particles are not labeled as they can exchange their positions without energy cost. On the other hand, in ferrogels consisting of magnetic particles embedded in elastic polymer matrices, the particles are labeled by their positions as their neighbors do not change over time. We resolve such an issue of particle labeling, introducing a mapping of the elastic interaction mediated by springs onto a pairwise additive interaction (pseudosprings) between unlabeled particles. We further investigate magnetostriction and changes in the elastic constants under altered magnetic interactions employing the pseudospring potential. It is revealed that there are two different response scenarios in the mechanical properties of the dipole-spring systems: While systems at low packing fractions are hardened as the magnetic moments increase in magnitude, at high packing fractions softening due to diminishing effects from the steric force, associated with increases in the volume, is observed. The validity of the theory is also verified by Monte Carlo simulations with both real springs and pseudosprings. We expect that our DFT approach may promote our understanding of materials with particle inclusions.
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Affiliation(s)
- Segun Goh
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
| | - René Wittmann
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
| | - Andreas M Menzel
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
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15
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Emamy H, Gang O, Starr FW. The Stability of a Nanoparticle Diamond Lattice Linked by DNA. NANOMATERIALS (BASEL, SWITZERLAND) 2019; 9:E661. [PMID: 31035462 PMCID: PMC6567282 DOI: 10.3390/nano9050661] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/21/2019] [Revised: 04/12/2019] [Accepted: 04/18/2019] [Indexed: 11/30/2022]
Abstract
The functionalization of nanoparticles (NPs) with DNA has proven to be an effective strategy for self-assembly of NPs into superlattices with a broad range of lattice symmetries. By combining this strategy with the DNA origami approach, the possible lattice structures have been expanded to include the cubic diamond lattice. This symmetry is of particular interest, both due to the inherent synthesis challenges, as well as the potential valuable optical properties, including a complete band-gap. Using these lattices in functional devices requires a robust and stable lattice. Here, we use molecular simulations to investigate how NP size and DNA stiffness affect the structure, stability, and crystallite shape of NP superlattices with diamond symmetry. We use the Wulff construction method to predict the equilibrium crystallite shape of the cubic diamond lattice. We find that, due to reorientation of surface particles, it is possible to create bonds at the surface with dangling DNA links on the interior, thereby reducing surface energy. Consequently, the crystallite shape depends on the degree to which such surface reorientation is possible, which is sensitive to DNA stiffness. Further, we determine dependence of the lattice stability on NP size and DNA stiffness by evaluating relative Gibbs free energy. We find that the free energy is dominated by the entropic component. Increasing NP size or DNA stiffness increases free energy, and thus decreases the relative stability of lattices. On the other hand, increasing DNA stiffness results in a more precisely defined lattice structure. Thus, there is a trade off between structure and stability of the lattice. Our findings should assist experimental design for controlling lattice stability and crystallite shape.
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Affiliation(s)
- Hamed Emamy
- Department of Physics, Wesleyan University, Middletown, CT 06459, USA.
- Department of Chemical Engineering, and Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA.
| | - Oleg Gang
- Department of Chemical Engineering, and Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA.
- Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA.
| | - Francis W Starr
- Department of Physics, Wesleyan University, Middletown, CT 06459, USA.
- Department of Molecular Biology & Biochemistry, Wesleyan University, Middletown, CT 06459, USA.
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16
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Maestro A, Santini E, Guzmán E. Physico-chemical foundations of particle-laden fluid interfaces. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2018; 41:97. [PMID: 30141087 DOI: 10.1140/epje/i2018-11708-6] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/31/2018] [Accepted: 07/27/2018] [Indexed: 06/08/2023]
Abstract
Particle-laden interfaces are ubiquitous nowadays. The understanding of their properties and structure is essential for solving different problems of technological and industrial relevance; e.g. stabilization of foams, emulsions and thin films. These rely on the response of the interface to mechanical perturbations. The complex mechanical response appearing in particle-laden interfaces requires deepening on the understanding of physico-chemical mechanisms underlying the assembly of particles at interface which plays a central role in the distribution of particles at the interface, and in the complex interfacial dynamics appearing in these systems. Therefore, the study of particle-laden interfaces deserves attention to provide a comprehensive explanation on the complex relaxation mechanisms involved in the stabilization of fluid interfaces.
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Affiliation(s)
- Armando Maestro
- Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, 38042, Grenoble, Cedex 9, France
| | - Eva Santini
- Istituto di Chimica della Materia Condensata e di Tecnologia per l'Energia (ICMATE), U.O.S. Genova-Consiglio Nazionale delle Ricerche (CNR), Via De Marini 6, 16149, Genova, Italy
| | - Eduardo Guzmán
- Departamento de Química Física I, Universidad Complutense de Madrid, Ciudad Universitaria s/n, 28040, Madrid, Spain.
- Instituto Pluridisciplinar, Universidad Complutense de Madrid, Paseo Juan XXIII, 1, 28040, Madrid, Spain.
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17
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Gibaud T, Constantin D. Direct Liquid to Crystal Transition in a Quasi-Two-Dimensional Colloidal Membrane. J Phys Chem Lett 2018; 9:4302-4307. [PMID: 30004230 DOI: 10.1021/acs.jpclett.8b01524] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Using synchrotron-based small-angle X-ray scattering, we study rigid fd viruses assembled into isolated monolayers from mixtures with a nonabsorbing polymer, which acts as an osmotic agent. As the polymer concentration increases, we observe a direct liquid to crystal transition, without an intermediate hexatic phase, in contrast with many other similar systems, such as concentrated DNA phases or packings of surfactant micelles. We tentatively attribute this effect to the difference in stiffness. The liquid phase can be well described by a hard-disk fluid, while we model the crystalline one as a hexagonal harmonic lattice and we evaluate its elastic constants.
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Affiliation(s)
- Thomas Gibaud
- Univ. Lyon, Ens de Lyon, Univ. Claude Bernard, CNRS , Laboratoire de Physique , F-69342 Lyon , France
| | - Doru Constantin
- Laboratoire de Physique des Solides , CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France
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18
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Shiba H, Keim P, Kawasaki T. Isolating long-wavelength fluctuation from structural relaxation in two-dimensional glass: cage-relative displacement. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:094004. [PMID: 29345245 DOI: 10.1088/1361-648x/aaa8b8] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
It has recently been revealed that long-wavelength fluctuation exists in two-dimensional (2D) glassy systems, having the same origin as that given by the Mermin-Wagner theorem for 2D crystalline solids. In this paper, we discuss how to characterise quantitatively the long-wavelength fluctuation in a molecular dynamics simulation of a lightly supercooled liquid. We employ the cage-relative mean-square displacement (MSD), defined on relative displacement to its cage, to quantitatively separate the long-wavelength fluctuation from the original MSD. For increasing system size the amplitude of acoustic long wavelength fluctuations not only increases but shifts to later times causing a crossover with structural relaxation of caging particles. We further analyse the dynamic correlation length using the cage-relative quantities. It grows as the structural relaxation becomes slower with decreasing temperature, uncovering an overestimation by the four-point correlation function due to the long-wavelength fluctuation. These findings motivate the usage of cage-relative MSD as a starting point for analysis of 2D glassy dynamics.
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Affiliation(s)
- Hayato Shiba
- Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
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19
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Leahy BD, Lin NY, Cohen I. Quantitative light microscopy of dense suspensions: Colloid science at the next decimal place. Curr Opin Colloid Interface Sci 2018. [DOI: 10.1016/j.cocis.2018.03.002] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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20
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Abstract
The normal modes and relaxation rates of weak colloidal gels are investigated in calculations using different models of the hydrodynamic interactions between suspended particles. The relaxation spectrum is computed for freely draining, Rotne-Prager-Yamakawa, and accelerated Stokesian dynamics approximations of the hydrodynamic mobility in a normal mode analysis of a harmonic network representing several colloidal gels. We find that the density of states and spatial structure of the normal modes are fundamentally altered by long-ranged hydrodynamic coupling among the particles. Short-ranged coupling due to hydrodynamic lubrication affects only the relaxation rates of short-wavelength modes. Hydrodynamic models accounting for long-ranged coupling exhibit a microscopic relaxation rate for each normal mode, λ that scales as l^{-2}, where l is the spatial correlation length of the normal mode. For the freely draining approximation, which neglects long-ranged coupling, the microscopic relaxation rate scales as l^{-γ}, where γ varies between three and two with increasing particle volume fraction. A simple phenomenological model of the internal elastic response to normal mode fluctuations is developed, which shows that long-ranged hydrodynamic interactions play a central role in the viscoelasticity of the gel network. Dynamic simulations of hard spheres that gel in response to short-ranged depletion attractions are used to test the applicability of the density of states predictions. For particle concentrations up to 30% by volume, the power law decay of the relaxation modulus in simulations accounting for long-ranged hydrodynamic interactions agrees with predictions generated by the density of states of the corresponding harmonic networks as well as experimental measurements. For higher volume fractions, excluded volume interactions dominate the stress response, and the prediction from the harmonic network density of states fails. Analogous to the Zimm model in polymer physics, our results indicate that long-ranged hydrodynamic interactions play a crucial role in determining the microscopic dynamics and macroscopic properties of weak colloidal gels.
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Affiliation(s)
- Zsigmond Varga
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - James W Swan
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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21
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Khrapak SA, Kryuchkov NP, Yurchenko SO. Thermodynamics and dynamics of two-dimensional systems with dipolelike repulsive interactions. Phys Rev E 2018; 97:022616. [PMID: 29548185 DOI: 10.1103/physreve.97.022616] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2017] [Indexed: 06/08/2023]
Abstract
Thermodynamics and dynamics of a classical two-dimensional system with dipolelike isotropic repulsive interactions are studied systematically using extensive molecular dynamics (MD) simulations supplemented by appropriate theoretical approximations. This interaction potential, which decays as an inverse cube of the interparticle distance, belongs to the class of very soft long-ranged interactions. As a result, the investigated system exhibits certain universal properties that are also shared by other related soft-interacting particle systems (like, for instance, the one-component plasma and weakly screened Coulomb systems). These universalities are explored in this article to construct a simple and reliable description of the system thermodynamics. In particular, Helmholtz free energies of the fluid and solid phases are derived, from which the location of the fluid-solid coexistence is determined. The quasicrystalline approximation is applied to the description of collective modes in dipole fluids. Its simplification, previously validated on strongly coupled plasma fluids, is used to derive explicit analytic dispersion relations for the longitudinal and transverse wave modes, which compare satisfactory with those obtained from direct MD simulations in the long-wavelength regime. Sound velocities of the dipole fluids and solids are derived and analyzed.
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Affiliation(s)
- Sergey A Khrapak
- Aix Marseille University, CNRS, PIIM, 13397 Marseille, France
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 82234 Weßling, Germany
- Joint Institute for High Temperatures, Russian Academy of Sciences, 125412 Moscow, Russia
| | - Nikita P Kryuchkov
- Bauman Moscow State Technical University, 2nd Baumanskaya street 5, 105005 Moscow, Russia
| | - Stanislav O Yurchenko
- Bauman Moscow State Technical University, 2nd Baumanskaya street 5, 105005 Moscow, Russia
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22
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Fritschi S, Fuchs M. Elastic moduli of a Brownian colloidal glass former. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:024003. [PMID: 29182519 DOI: 10.1088/1361-648x/aa9de4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The static, dynamic and flow-dependent shear moduli of a binary mixture of Brownian hard disks are studied by an event-driven molecular dynamics simulation. Thereby, the emergence of rigidity close to the glass transition encoded in the static shear modulus [Formula: see text] is accessed by three methods. Results from shear stress auto-correlation functions, elastic dispersion relations, and the elastic response to strain deformations upon the start-up of shear flow are compared. This enables one to sample the time-dependent shear modulus [Formula: see text] consistently over several decades in time. By that a very precise specification of the glass transition point and of [Formula: see text] is feasible. Predictions by mode coupling theory of a finite shear modulus at the glass transition, of α-scaling in fluid states close to the transition, and of shear induced decay in yielding glass states are tested and broadly verified.
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Affiliation(s)
- S Fritschi
- Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany
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23
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Popli P, Ganguly S, Sengupta S. Translationally invariant colloidal crystal templates. SOFT MATTER 2017; 14:104-111. [PMID: 29210439 DOI: 10.1039/c7sm01877k] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We show that dynamic, feed-back controlled optical traps, whose positions depend on the instantaneous local configuration of particles in a pre-determined way, can stabilise colloidal particles in finite lattices of any given symmetry. Unlike in a static template, the crystal so formed is invariant under uniform translations and retains all possible zero energy modes. We demonstrate this in silico by stabilising the unstable two-dimensional square lattice in a model soft solid with isotropic interactions.
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Affiliation(s)
- Pankaj Popli
- TIFR Centre for Interdisciplinary Sciences, 36/P Gopanapally, Hyderabad 500107, India.
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24
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Abstract
In a recent commentary, J. M. Kosterlitz described how D. Thouless and he got motivated to investigate melting and suprafluidity in two dimensions [Kosterlitz JM (2016) J Phys Condens Matter 28:481001]. It was due to the lack of broken translational symmetry in two dimensions-doubting the existence of 2D crystals-and the first computer simulations foretelling 2D crystals (at least in tiny systems). The lack of broken symmetries proposed by D. Mermin and H. Wagner is caused by long wavelength density fluctuations. Those fluctuations do not only have structural impact, but additionally a dynamical one: They cause the Lindemann criterion to fail in 2D in the sense that the mean squared displacement of atoms is not limited. Comparing experimental data from 3D and 2D amorphous solids with 2D crystals, we disentangle Mermin-Wagner fluctuations from glassy structural relaxations. Furthermore, we demonstrate with computer simulations the logarithmic increase of displacements with system size: Periodicity is not a requirement for Mermin-Wagner fluctuations, which conserve the homogeneity of space on long scales.
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25
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Illing B, Fritschi S, Hajnal D, Klix C, Keim P, Fuchs M. Strain Pattern in Supercooled Liquids. PHYSICAL REVIEW LETTERS 2016; 117:208002. [PMID: 27886484 DOI: 10.1103/physrevlett.117.208002] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2016] [Indexed: 06/06/2023]
Abstract
Investigations of strain correlations at the glass transition reveal unexpected phenomena. The shear strain fluctuations show an Eshelby-strain pattern [∼cos(4θ)/r^{2}], characteristic of elastic response, even in liquids, at long times. We address this using a mode-coupling theory for the strain fluctuations in supercooled liquids and data from both video microscopy of a two-dimensional colloidal glass former and simulations of Brownian hard disks. We show that the long-ranged and long-lived strain signatures follow a scaling law valid close to the glass transition. For large enough viscosities, the Eshelby-strain pattern is visible even on time scales longer than the structural relaxation time τ and after the shear modulus has relaxed to zero.
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Affiliation(s)
- Bernd Illing
- University of Konstanz, D-78457 Konstanz, Germany
| | | | - David Hajnal
- Johannes Gutenberg-University Mainz, D-55099 Mainz, Germany
| | | | - Peter Keim
- University of Konstanz, D-78457 Konstanz, Germany
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26
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Wei WS, Gharbi MA, Lohr MA, Still T, Gratale MD, Lubensky TC, Stebe KJ, Yodh AG. Dynamics of ordered colloidal particle monolayers at nematic liquid crystal interfaces. SOFT MATTER 2016; 12:4715-4724. [PMID: 27109759 DOI: 10.1039/c6sm00295a] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We prepare two-dimensional crystalline packings of colloidal particles on surfaces of the nematic liquid crystal (NLC) 5CB, and we investigate the diffusion and vibrational phonon modes of these particles using video microscopy. Short-time particle diffusion at the air-NLC interface is well described by a Stokes-Einstein model with viscosity similar to that of 5CB. Crystal phonon modes, measured by particle displacement covariance techniques, are demonstrated to depend on the elastic constants of 5CB through interparticle forces produced by LC defects that extend from the interface into the underlying bulk material. The displacement correlations permit characterization of transverse and longitudinal sound velocities of the crystal packings, as well as the particle interactions produced by the LC defects. All behaviors are studied in the nematic phase as a function of increasing temperature up to the nematic-isotropic transition.
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Affiliation(s)
- Wei-Shao Wei
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
| | - Mohamed Amine Gharbi
- Department of Physics, McGill University, Montreal, Quebec, Canada and Department of Chemistry, McGill University, Montreal, Quebec, Canada
| | - Matthew A Lohr
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
| | - Tim Still
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
| | - Matthew D Gratale
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
| | - T C Lubensky
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
| | - Kathleen J Stebe
- Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - A G Yodh
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
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27
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Kelleher CP, Wang A, Guerrero-García GI, Hollingsworth AD, Guerra RE, Krishnatreya BJ, Grier DG, Manoharan VN, Chaikin PM. Charged hydrophobic colloids at an oil-aqueous phase interface. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062306. [PMID: 26764691 DOI: 10.1103/physreve.92.062306] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2015] [Indexed: 06/05/2023]
Abstract
Hydrophobic poly(methyl methacrylate) (PMMA) colloidal particles, when dispersed in oil with a relatively high dielectric constant, can become highly charged. In the presence of an interface with a conducting aqueous phase, image-charge effects lead to strong binding of colloidal particles to the interface, even though the particles are wetted very little by the aqueous phase. We study both the behavior of individual colloidal particles as they approach the interface and the interactions between particles that are already interfacially bound. We demonstrate that using particles which are minimally wetted by the aqueous phase allows us to isolate and study those interactions which are due solely to charging of the particle surface in oil. Finally, we show that these interactions can be understood by a simple image-charge model in which the particle charge q is the sole fitting parameter.
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Affiliation(s)
- Colm P Kelleher
- Department of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, New York 10003, USA
| | - Anna Wang
- Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Guillermo Iván Guerrero-García
- Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA and Instituto de Fisica, Universidad Autonoma de San Luis Potosi, San Luis Potosi, Mexico
| | - Andrew D Hollingsworth
- Department of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, New York 10003, USA
| | - Rodrigo E Guerra
- Department of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, New York 10003, USA
| | - Bhaskar Jyoti Krishnatreya
- Department of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, New York 10003, USA
| | - David G Grier
- Department of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, New York 10003, USA
| | - Vinothan N Manoharan
- Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA and Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Paul M Chaikin
- Department of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, New York 10003, USA
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28
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Yurchenko SO, Kryuchkov NP, Ivlev AV. Pair correlations in classical crystals: The shortest-graph method. J Chem Phys 2015. [DOI: 10.1063/1.4926945] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Affiliation(s)
| | - Nikita P. Kryuchkov
- Bauman Moscow State Technical University, 2nd Baumanskaya str. 5, 105005 Moscow, Russia
| | - Alexei V. Ivlev
- Max-Planck-Institut für extraterrestrische Physik, Giessenbachstr. 1, 85748 Garching, Germany
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29
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Ganguly S, Sengupta S, Sollich P. Statistics of non-affine defect precursors: tailoring defect densities in colloidal crystals using external fields. SOFT MATTER 2015; 11:4517-4526. [PMID: 25953064 DOI: 10.1039/c5sm00301f] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Coarse-graining atomic displacements in a solid produces both local affine strains and "non-affine" fluctuations. Here we study the equilibrium dynamics of these coarse grained quantities to obtain space-time dependent correlation functions. We show how a subset of these thermally excited, non-affine fluctuations act as precursors for the nucleation of lattice defects and suggest how defect probabilities may be altered by an experimentally realisable "external" field conjugate to the global non-affinity parameter. Our results are amenable to verification in experiments on colloidal crystals using commonly available holographic laser tweezer and video microscopy techniques, and may lead to simple ways of controlling the defect density of a colloidal solid.
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Affiliation(s)
- Saswati Ganguly
- Indian Association for the Cultivation of Science, 2A&2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700032, India.
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30
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Parolini L, Law AD, Maestro A, Buzza DMA, Cicuta P. Interaction between colloidal particles on an oil-water interface in dilute and dense phases. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:194119. [PMID: 25924056 DOI: 10.1088/0953-8984/27/19/194119] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The interaction between micron-sized charged colloidal particles at polar/non-polar liquid interfaces remains surprisingly poorly understood for a relatively simple physical chemistry system. By measuring the pair correlation function g(r) for different densities of polystyrene particles at the decane-water interface, and using a powerful predictor-corrector inversion scheme, effective pair-interaction potentials can be obtained up to fairly high densities, and these reproduce the experimental g(r) in forward simulations, so are self consistent. While at low densities these potentials agree with published dipole-dipole repulsion, measured by various methods, an apparent density dependence and long range attraction are obtained when the density is higher. This condition is thus explored in an alternative fashion, measuring the local mobility of colloids when confined by their neighbors. This method of extracting interaction potentials gives results that are consistent with dipolar repulsion throughout the concentration range, with the same magnitude as in the dilute limit. We are unable to rule out the density dependence based on the experimental accuracy of our data, but we show that incomplete equilibration of the experimental system, which would be possible despite long waiting times due to the very strong repulsions, is a possible cause of artefacts in the inverted potentials. We conclude that to within the precision of these measurements, the dilute pair potential remains valid at high density in this system.
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Affiliation(s)
- Lucia Parolini
- Cavendish Laboratory, University of Cambridge, The Old Schools, Trinity Ln, Cambridge CB2 1TN,UK
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31
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Schella A, Melzer A, July C, Bechinger C. Effect of confinement on the mode dynamics of dipole clusters. SOFT MATTER 2015; 11:1197-1207. [PMID: 25563898 DOI: 10.1039/c4sm02333a] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Dynamical properties of colloidal clusters composed of paramagnetic beads are presented. The clusters were trapped either in a parabolic trough or in a hard-wall confinement. In order to access the dynamics of the ensembles, the instantaneous normal mode (INM) approach is utilized, which uses cluster configurations as an input. The peaks in the mode spectra weaken when the system size is increased and when the coupling strength is lowered. The short-time diffusive properties of the clusters are deduced using the INM technique. It is found that angular diffusion is always larger than radial diffusion regardless of the shape of the external trap. Further, short-time diffusion seems to be almost independent of the coupling strength in the solid regime, but decreases with increasing packing fraction and size of the ensembles. In general, it is found that diffusion is larger for parabolically confined than for hard-wall trapped clusters.
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Affiliation(s)
- André Schella
- Institute of Physics, Ernst-Moritz-Arndt-University Greifwald, Felix Hausdorff Str. 6, 17489 Greifswald, Germany.
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32
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Ramos IRO, Ferreira WP, Munarin FF, Peeters FM. Dynamical properties and melting of binary two-dimensional colloidal alloys. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062311. [PMID: 25615097 DOI: 10.1103/physreve.90.062311] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/11/2013] [Indexed: 06/04/2023]
Abstract
A two-dimensional (2D) binary colloidal system consisting of interacting dipoles is investigated using an analytical approach. Within the harmonic approximation we obtain the phonon spectrum of the system as a function of the composition, dipole-moment ratio, and mass ratio between the small and big particles. Through a systematic analysis of the phonon spectra we are able to determine the stability region of the different lattice structures of the colloidal alloys. The gaps in the phonon frequency spectrum, the optical frequencies in the long-wavelength limit, and the sound velocity are discussed as well. Using the modified Lindemann criterion and within the harmonic approximation we estimate the melting temperature of the sublattice generated by the big particles.
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Affiliation(s)
- I R O Ramos
- Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil and Universidade Federal Rural do Semi-árido, Campus Pau dos Ferros, Pau dos Ferros, Rio Grande do Norte, Brazil
| | - W P Ferreira
- Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil
| | - F F Munarin
- Centro de Tecnologia, Bloco 710, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil
| | - F M Peeters
- Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil and Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
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33
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Deutschländer S, Puertas AM, Maret G, Keim P. Specific heat in two-dimensional melting. PHYSICAL REVIEW LETTERS 2014; 113:127801. [PMID: 25279643 DOI: 10.1103/physrevlett.113.127801] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2013] [Indexed: 06/03/2023]
Abstract
We report the specific heat cN around the melting transition(s) of micrometer-sized superparamagnetic particles confined in two dimensions, calculated from fluctuations of positions and internal energy, and corresponding Monte Carlo simulations. Since colloidal systems provide single particle resolution, they offer the unique possibility to compare the experimental temperatures of the peak position of cN(T) and symmetry breaking, respectively. While order parameter correlation functions confirm the Kosterlitz-Thouless-Halperin-Nelson-Young melting scenario where translational and orientational order symmetries are broken at different temperatures with an intermediate so called hexatic phase, we observe a single peak of the specific heat within the hexatic phase, with excellent agreement between experiment and simulation. Thus, the peak is not associated with broken symmetries but can be explained with the total defect density, which correlates with the maximum increase of isolated dislocations. The absence of a latent heat strongly supports the continuous character of both transitions.
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Affiliation(s)
| | - Antonio M Puertas
- Department of Applied Physics, University of Almeria, 04120 Almeria, Spain
| | - Georg Maret
- Physics Department, University of Konstanz, 78464 Konstanz, Germany
| | - Peter Keim
- Physics Department, University of Konstanz, 78464 Konstanz, Germany
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34
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Particle laden fluid interfaces: dynamics and interfacial rheology. Adv Colloid Interface Sci 2014; 206:303-19. [PMID: 24200090 DOI: 10.1016/j.cis.2013.10.010] [Citation(s) in RCA: 122] [Impact Index Per Article: 12.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2013] [Revised: 10/09/2013] [Accepted: 10/10/2013] [Indexed: 12/30/2022]
Abstract
We review the dynamics of particle laden interfaces, both particle monolayers and particle+surfactant monolayers. We also discuss the use of the Brownian motion of microparticles trapped at fluid interfaces for measuring the shear rheology of surfactant and polymer monolayers. We describe the basic concepts of interfacial rheology and the different experimental methods for measuring both dilational and shear surface complex moduli over a broad range of frequencies, with emphasis in the micro-rheology methods. In the case of particles trapped at interfaces the calculation of the diffusion coefficient from the Brownian trajectories of the particles is calculated as a function of particle surface concentration. We describe in detail the calculation in the case of subdiffusive particle dynamics. A comprehensive review of dilational and shear rheology of particle monolayers and particle+surfactant monolayers is presented. Finally the advantages and current open problems of the use of the Brownian motion of microparticles for calculating the shear complex modulus of monolayers are described in detail.
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35
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Still T, Goodrich CP, Chen K, Yunker PJ, Schoenholz S, Liu AJ, Yodh AG. Phonon dispersion and elastic moduli of two-dimensional disordered colloidal packings of soft particles with frictional interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012301. [PMID: 24580221 DOI: 10.1103/physreve.89.012301] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2013] [Indexed: 06/03/2023]
Abstract
Particle tracking and displacement covariance matrix techniques are employed to investigate the phonon dispersion relations of two-dimensional colloidal glasses composed of soft, thermoresponsive microgel particles whose temperature-sensitive size permits in situ variation of particle packing fraction. Bulk, B, and shear, G, moduli of the colloidal glasses are extracted from the dispersion relations as a function of packing fraction, and variation of the ratio G/B with packing fraction is found to agree quantitatively with predictions for jammed packings of frictional soft particles. In addition, G and B individually agree with numerical predictions for frictional particles. This remarkable level of agreement enabled us to extract an energy scale for the interparticle interaction from the individual elastic constants and to derive an approximate estimate for the interparticle friction coefficient.
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Affiliation(s)
- Tim Still
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA and Complex Assemblies of Soft Matter, CNRS-Rhodia-UPenn UMI 3254, Bristol, Pennsylvania 19007, USA
| | - Carl P Goodrich
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Ke Chen
- Beijing National Laboratory for Condensed Matter Physics and Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Peter J Yunker
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA and School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Samuel Schoenholz
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Andrea J Liu
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - A G Yodh
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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36
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Straube AV, Dullens RPA, Schimansky-Geier L, Louis AA. Zigzag transitions and nonequilibrium pattern formation in colloidal chains. J Chem Phys 2013; 139:134908. [DOI: 10.1063/1.4823501] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023] Open
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37
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Chen K, Still T, Schoenholz S, Aptowicz KB, Schindler M, Maggs AC, Liu AJ, Yodh AG. Phonons in two-dimensional soft colloidal crystals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022315. [PMID: 24032840 DOI: 10.1103/physreve.88.022315] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2012] [Revised: 05/08/2013] [Indexed: 06/02/2023]
Abstract
The vibrational modes of pristine and polycrystalline monolayer colloidal crystals composed of thermosensitive microgel particles are measured using video microscopy and covariance matrix analysis. At low frequencies, the Debye relation for two-dimensional harmonic crystals is observed in both crystal types; at higher frequencies, evidence for van Hove singularities in the phonon density of states is significantly smeared out by experimental noise and measurement statistics. The effects of these errors are analyzed using numerical simulations. We introduce methods to correct for these limitations, which can be applied to disordered systems as well as crystalline ones, and we show that application of the error correction procedure to the experimental data leads to more pronounced van Hove singularities in the pristine crystal. Finally, quasilocalized low-frequency modes in polycrystalline two-dimensional colloidal crystals are identified and demonstrated to correlate with structural defects such as dislocations, suggesting that quasilocalized low-frequency phonon modes may be used to identify local regions vulnerable to rearrangements in crystalline as well as amorphous solids.
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Affiliation(s)
- Ke Chen
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA and Beijing National Laboratory for Condensed Matter Physics and Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
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38
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Ganguly S, Sengupta S, Sollich P, Rao M. Nonaffine displacements in crystalline solids in the harmonic limit. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042801. [PMID: 23679467 DOI: 10.1103/physreve.87.042801] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/26/2012] [Indexed: 06/02/2023]
Abstract
A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor D as well as a positive definite scalar χ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of χ and D and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution P(χ,D) and the two-point spatial correlation functions for χ and D. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume Ω and dimensionality. As a corollary to our work, we identify the field h(χ) conjugate to χ and show that this field may be tuned to produce a transition to a state where the ensemble average <χ> and the correlation length of χ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.
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Affiliation(s)
- Saswati Ganguly
- Indian Association for the Cultivation of Science, 2A & 2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700032, India
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Mao X, Chen Q, Granick S. Entropy favours open colloidal lattices. NATURE MATERIALS 2013; 12:217-222. [PMID: 23314105 DOI: 10.1038/nmat3496] [Citation(s) in RCA: 68] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2012] [Accepted: 10/22/2012] [Indexed: 06/01/2023]
Abstract
Burgeoning experimental and simulation activity seeks to understand the existence of self-assembled colloidal structures that are not close-packed. Here we describe an analytical theory based on lattice dynamics and supported by experiments that reveals the fundamental role entropy can play in stabilizing open lattices. The entropy we consider is associated with the rotational and vibrational modes unique to colloids interacting through extended attractive patches. The theory makes predictions of the implied temperature, pressure and patch-size dependence of the phase diagram of open and close-packed structures. More generally, it provides guidance for the conditions at which targeted patchy colloidal assemblies in two and three dimensions are stable, thus overcoming the difficulty in exploring by experiment or simulation the full range of conceivable parameters.
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Affiliation(s)
- Xiaoming Mao
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Philadelphia 19104, USA.
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Klix CL, Ebert F, Weysser F, Fuchs M, Maret G, Keim P. Glass elasticity from particle trajectories. PHYSICAL REVIEW LETTERS 2012; 109:178301. [PMID: 23215226 DOI: 10.1103/physrevlett.109.178301] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2011] [Indexed: 06/01/2023]
Abstract
Using positional data from video microscopy of a two-dimensional colloidal system and from simulations of hard disks, we determine the wave-vector-dependent elastic dispersion relations in glass. The emergence of rigidity based on the existence of a well defined displacement field in amorphous solids is demonstrated. Continuum elastic theory is recovered in the limit of long wavelengths which provides the glass elastic shear and bulk modulus as a function of temperature. The onset of a finite static shear modulus upon cooling marks the fluid-glass transition in an intuitive and unique way.
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41
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Schindler M, Maggs AC. Anisotropic elasticity in confocal studies of colloidal crystals. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2011; 34:115. [PMID: 22042209 DOI: 10.1140/epje/i2011-11115-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2011] [Accepted: 09/12/2011] [Indexed: 05/31/2023]
Abstract
We consider the theory of fluctuations of a colloidal solid observed in a confocal slice. For a cubic crystal we study the evolution of the projected elastic properties as a function of the anisotropy of the crystal using numerical methods based on the fast Fourier transform. In certain situations of high symmetry we find exact analytic results for the projected fluctuations.
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Affiliation(s)
- M Schindler
- Laboratoire PCT, UMR Gulliver CNRS-ESPCI 7083, 10 rue Vauquelin, 75231 Paris Cedex 05, France.
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42
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Kaya D, Green NL, Maloney CE, Islam MF. Normal Modes and Density of States of Disordered Colloidal Solids. Science 2010; 329:656-8. [DOI: 10.1126/science.1187988] [Citation(s) in RCA: 87] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
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Franzrahe K, Nielaba P, Sengupta S. Coarse-graining microscopic strains in a harmonic, two-dimensional solid: elasticity, nonlocal susceptibilities, and nonaffine noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:016112. [PMID: 20866694 DOI: 10.1103/physreve.82.016112] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2009] [Revised: 06/16/2010] [Indexed: 05/29/2023]
Abstract
In soft matter systems the local displacement field can be accessed directly by video microscopy enabling one to compute local strain fields and hence the elastic moduli in these systems using a coarse-graining procedure. We study this process in detail for a simple triangular, harmonic lattice in two dimensions. Coarse-graining local strains obtained from particle configurations in a Monte Carlo simulation generates nontrivial, nonlocal strain correlations (susceptibilities). These may be understood within a generalized, Landau-type elastic Hamiltonian containing up to quartic terms in strain gradients [K. Franzrahe, Phys. Rev. E 78, 026106 (2008)10.1103/PhysRevE.78.026106]. In order to demonstrate the versatility of the analysis of these correlations and to make our calculations directly relevant for experiments on colloidal solids, we systematically study various parameters such as the choice of statistical ensemble, presence of external pressure and boundary conditions. Crucially, we show that special care needs to be taken for an accurate application of our results to actual experiments, where the analyzed area is embedded within a larger system, to which it is mechanically coupled. Apart from the smooth, affine strain fields, the coarse-graining procedure also gives rise to a noise field (χ) made up of nonaffine displacements. Several properties of χ may be rationalized for the harmonic solid using a simple "cell model" calculation. Furthermore the scaling behavior of the probability distribution of the noise field (χ) is studied. We find that for any inverse temperature β, spring constant f, density ρ and coarse-graining length Λ the probability distribution can be obtained from a master curve of the scaling variable X=χβf/ρΛ(2).
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Affiliation(s)
- K Franzrahe
- Fachbereich Physik, Universität Konstanz, Konstanz, Germany.
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Ghosh A, Chikkadi VK, Schall P, Kurchan J, Bonn D. Density of states of colloidal glasses. PHYSICAL REVIEW LETTERS 2010; 104:248305. [PMID: 20867344 DOI: 10.1103/physrevlett.104.248305] [Citation(s) in RCA: 75] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2009] [Revised: 02/20/2010] [Indexed: 05/29/2023]
Abstract
Glasses are structurally liquidlike, but mechanically solidlike. Most attempts to understand glasses start from liquid state theory. Here we take the opposite point of view, and use concepts from solid state physics. We determine the vibrational modes of a colloidal glass experimentally, and find soft low-frequency modes that are very different in nature from the usual acoustic vibrations of ordinary solids. These modes extend over surprisingly large length scales.
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Affiliation(s)
- Antina Ghosh
- Van der Waals Zeeman Institute, University of Amsterdam, Valckenierstraat 67, 1018 XE Amsterdam The Netherlands
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Fornleitner J, Kahl G, Likos CN. Tailoring the phonon band structure in binary colloidal mixtures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:060401. [PMID: 20866368 DOI: 10.1103/physreve.81.060401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2010] [Indexed: 05/29/2023]
Abstract
We analyze the phonon spectra of periodic structures formed by two-dimensional mixtures of dipolar colloidal particles. These mixtures display an enormous variety of complex ordered configurations [J. Fornleitner, Soft Matter 4, 480 (2008)], allowing for the systematic investigation of the ensuing phonon spectra and the control of phononic gaps. We show how the shape of the phonon bands and the number and width of the phonon gaps can be controlled by changing the susceptibility ratio, the composition, and the mass ratio between the two components.
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Affiliation(s)
- Julia Fornleitner
- Center for Computational Materials Science and Institut für Theoretische Physik, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
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Henseler P, Erbe A, Köppl M, Leiderer P, Nielaba P. Density reduction and diffusion in driven two-dimensional colloidal systems through microchannels. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041402. [PMID: 20481722 DOI: 10.1103/physreve.81.041402] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2010] [Indexed: 05/29/2023]
Abstract
The behavior of particles driven through a narrow constriction is investigated in experiment and simulation. The system of particles adapts to the confining potentials and the interaction energies by a self-consistent arrangement of the particles. It results in the formation of layers throughout the channel and of a density gradient along the channel. The particles accommodate to the density gradient by reducing the number of layers one by one when it is energetically favorable. The position of the layer reduction zone fluctuates with time while the particles continuously pass this zone. The flow behavior of the particles is studied in detail. The velocities of the particles and their diffusion behavior reflect the influence of the self-organized order of the system.
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Affiliation(s)
- P Henseler
- Universität Konstanz, Fachbereich für Physik, 78457 Konstanz, Germany.
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Löwen H. Twenty years of confined colloids: from confinement-induced freezing to giant breathing. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2009; 21:474203. [PMID: 21832482 DOI: 10.1088/0953-8984/21/47/474203] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
The physics of colloidal suspensions confined in slits and cavities has significantly advanced during the last twenty years. In particular, freezing transitions in confinement have been addressed by theory and simulations and experimental realizations were proposed to confine colloidal particles to two dimensions. After reviewing this progress, we discuss the generalization to time-dependent confinement which leads to nonequilibrium situations. This is elaborated further for unstable situations where the particles can leave the confinement. In particular, the completely overdamped Brownian motion of a colloidal particle in a time-dependent harmonic trap is considered. The analytically soluble model of a time-dependent quadratic potential is used to extract the dynamical properties of the potential if the potential undergoes periodic switching from a confining harmonic potential to an unstable one. The amplitudes of the oscillating particle response can strongly grow in time, which we refer to as 'giant breathing'. This giant breathing process occurs also in anharmonic potentials and is verifiable in real-space experiments of colloids in laser-optical fields.
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Affiliation(s)
- H Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany
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Chremos A, Likos CN. Crystal Structures of Two-Dimensional Binary Mixtures of Dipolar Colloids in Tilted External Magnetic Fields. J Phys Chem B 2009; 113:12316-25. [DOI: 10.1021/jp903820d] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Alexandros Chremos
- School of Chemistry, University of Edinburgh, West Mains Road, Edinburgh EH3 9JJ, United Kingdom, and Institute for Theoretical Physics II: Soft Matter, Heinrich-Heine-Universität Düsseldorf, Universitätsstrasse 1, D-40225 Düsseldorf, Germany
| | - Christos N. Likos
- School of Chemistry, University of Edinburgh, West Mains Road, Edinburgh EH3 9JJ, United Kingdom, and Institute for Theoretical Physics II: Soft Matter, Heinrich-Heine-Universität Düsseldorf, Universitätsstrasse 1, D-40225 Düsseldorf, Germany
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Deb D, von Grünberg HH. Colloidal model system for island formation. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2009; 21:245102. [PMID: 21693935 DOI: 10.1088/0953-8984/21/24/245102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
We present model calculations to explore the possibility of colloidal island formation over strained surfaces. Colloids, aggregating due to attractive depletion interactions, are deposited onto a colloidal surface whose lattice constant and geometry can be varied by optical forces. This allows precise control of the strain between the substrate and the colloidal adsorbate. Three different strain fields are considered: fields with either an unidirectional or a hexagonal variation of strain, and fields with a combination of both variations. We find that the unidirectional field induces the formation of infinitely extended ridges, while hexagonal strain fields lead to regular pyramidal island structures which can be distorted in a controlled way by adding the unidirectional strain component. We furthermore study the dependence of island size on strain strength for the hexagonal strain pattern and find that the area occupied by an island is a constant fraction of the strain field's repeat unit.
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Affiliation(s)
- D Deb
- Institut für Chemie, Karl-Franzens Universität Graz, A-8010 Graz, Austria
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