1
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Montero AM, Santos A, Gurin P, Varga S. Ordering properties of anisotropic hard bodies in one-dimensional channels. J Chem Phys 2023; 159:154507. [PMID: 37861120 DOI: 10.1063/5.0169605] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2023] [Accepted: 09/26/2023] [Indexed: 10/21/2023] Open
Abstract
The phase behavior and structural properties of hard anisotropic particles (prisms and dumbbells) are examined in one-dimensional channels using the Parsons-Lee (PL) theory, and the transfer-matrix and neighbor-distribution methods. The particles are allowed to move freely along the channel, while their orientations are constrained such that one particle can occupy only two or three different lengths along the channel. In this confinement setting, hard prisms behave as an additive mixture, while hard dumbbells behave as a non-additive one. We prove that all methods provide exact results for the phase properties of hard prisms, while only the neighbor-distribution and transfer-matrix methods are exact for hard dumbbells. This shows that non-additive effects are incorrectly included into the PL theory, which is a successful theory of the isotropic-nematic phase transition of rod-like particles in higher dimensions. In the one-dimensional channel, the orientational ordering develops continuously with increasing density, i.e., the system is isotropic only at zero density, while it becomes perfectly ordered at the close-packing density. We show that there is no orientational correlation in the hard prism system, while the hard dumbbells are orientationally correlated with diverging correlation length at close packing. On the other hand, positional correlations are present for all the systems, the associated correlation length diverging at close packing.
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Affiliation(s)
- Ana M Montero
- Departamento de Física, Universidad de Extremadura, E-06006 Badajoz, Spain
| | - Andrés Santos
- Departamento de Física, Universidad de Extremadura, E-06006 Badajoz, Spain
- Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06006 Badajoz, Spain
| | - Péter Gurin
- Physics Department, Centre for Natural Sciences, University of Pannonia, P.O. Box 158, Veszprém H-8201, Hungary
| | - Szabolcs Varga
- Physics Department, Centre for Natural Sciences, University of Pannonia, P.O. Box 158, Veszprém H-8201, Hungary
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2
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Wensink HH, Grelet E. Elastic response of colloidal smectic liquid crystals: Insights from microscopic theory. Phys Rev E 2023; 107:054604. [PMID: 37329078 DOI: 10.1103/physreve.107.054604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2023] [Accepted: 05/05/2023] [Indexed: 06/18/2023]
Abstract
Elongated colloidal rods at sufficient packing conditions are known to form stable lamellar or smectic phases. Using a simplified volume-exclusion model, we propose a generic equation of state for hard-rod smectics that is robust against simulation results and is independent of the rod aspect ratio. We then extend our theory by exploring the elastic properties of a hard-rod smectic, including the layer compressibility (B) and bending modulus (K_{1}). By introducing weak backbone flexibility we are able to compare our predictions with experimental results on smectics of filamentous virus rods (fd) and find quantitative agreement between the smectic layer spacing, the out-of-plane fluctuation strength, as well as the smectic penetration length λ=sqrt[K_{1}/B]. We demonstrate that the layer bending modulus is dominated by director splay and depends sensitively on lamellar out-of-plane fluctuations that we account for on the single-rod level. We find that the ratio between the smectic penetration length and the lamellar spacing is about two orders of magnitude smaller than typical values reported for thermotropic smectics. We attribute this to the fact that colloidal smectics are considerably softer in terms of layer compression than their thermotropic counterparts while the cost of layer bending is of comparable magnitude.
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Affiliation(s)
- H H Wensink
- Laboratoire de Physique des Solides-UMR 8502, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - E Grelet
- Centre de Recherche Paul Pascal-UMR 5031, CNRS, Université de Bordeaux, 33600 Pessac, France
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3
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Petrov DA. Liquid-crystal composites of carbon nanotubes in a magnetic field: Bridging continuum theory and a molecular-statistical approach. Phys Rev E 2023; 107:054701. [PMID: 37329002 DOI: 10.1103/physreve.107.054701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2023] [Accepted: 05/06/2023] [Indexed: 06/18/2023]
Abstract
We propose an approach combining the continuum theory and molecular-statistical approach for a suspension of carbon nanotubes based on a negative diamagnetic anisotropy liquid crystal. Using the continuum theory, we show that in the case of an infinite sample in suspension it is possible to observe peculiar magnetic Fréedericksz-like transitions between three nematic phases: planar, angular, and homeotropic with different mutual orientations of liquid-crystal and nanotube directors. The transition fields between these phases are found analytically as functions of material parameters of the continuum theory. To account for the effects associated with temperature changes, we propose a molecular-statistical approach that allows obtaining the equations of orientational state for the orientation angles of the main axes of the nematic order, i.e., the liquid-crystal and carbon-nanotube directors in a similar form as was obtained within the continuum theory. Thus, it is possible to relate the parameters of the continuum theory, such as the surface-energy density of a coupling between molecules and nanotubes, to the parameters of the molecular-statistical model and the order parameters of the liquid crystal and carbon nanotubes. This approach allows determining the temperature dependencies of the threshold fields of transitions between different nematic phases, which is impossible in the framework of the continuum theory. In the framework of the molecular-statistical approach we predict the existence of an additional direct transition between the planar and homeotropic nematic phases of the suspension, which cannot be described based on the continuum theory. As the main results, the magneto-orientational response of the liquid-crystal composite is studied and a possible biaxial orientational ordering of the nanotubes in the magnetic field is shown.
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Affiliation(s)
- Danil A Petrov
- Physics of Phase Transitions Department, Perm State University, Bukirev Street 15, 614990 Perm, Russia
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4
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Aliabadi R, Nasirimoghadam S, Wensink HH. Evidence of T-type structures of hard square boards in capillary confinement. Phys Rev E 2023; 107:054117. [PMID: 37329060 DOI: 10.1103/physreve.107.054117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Accepted: 04/05/2023] [Indexed: 06/18/2023]
Abstract
We employ Onsager's second virial density functional theory combined with the Parsons-Lee theory within the restricted orientation (Zwanzig) approximation to examine the phase structure of hard square boards of dimensions (L×D×D) uniaxially confined in narrow slabs. Depending on the wall-to-wall separation (H), we predict a number of distinctly different capillary nematic phases, including a monolayer uniaxial or biaxial planar nematic, homeotropic with a variable number of layers, and a T-type structure. We determine that the favored phase is homotropic, and we observe first-order transitions from the homeotropic structure with n layers to n+1 layers as well as from homeotropic surface anchoring to a monolayer planar or T-type structure involving both planar and homeotropic anchoring at the pore surface. By increasing the packing fraction, we further demonstrate a reentrant homeotropic-planar-homeotropic phase sequence in a particular range (i.e., H/D=1.1 and 0.25≤L/D<0.26). We find that the T-type structure is more stable when the pore is wide enough with respect to the planar phase. The enhanced stability of the mixed-anchoring T-structure is unique for square boards and becomes manifest at pore width exceeding L+D. More specifically, the biaxial T-type structure emerges directly from the homeotropic state without intervention of a planar layer structure as observed for other convex particle shapes.
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Affiliation(s)
| | | | - Henricus Herman Wensink
- Laboratoire de Physique des Solides - UMR 8502, CNRS, Université Paris-Saclay, 91405 Orsay, France
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5
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Renner J, Schmidt M, de Las Heras D. Reduced-variance orientational distribution functions from torque sampling. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2023; 35:235901. [PMID: 36974000 DOI: 10.1088/1361-648x/acc522] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2022] [Accepted: 03/08/2023] [Indexed: 06/18/2023]
Abstract
We introduce a method to sample the orientational distribution function in computer simulations. The method is based on the exact torque balance equation for classical many-body systems of interacting anisotropic particles in equilibrium. Instead of the traditional counting of events, we reconstruct the orientational distribution function via an orientational integral of the torque acting on the particles. We test the torque sampling method in two- and three-dimensions, using both Langevin dynamics and overdamped Brownian dynamics, and with two interparticle interaction potentials. In all cases the torque sampling method produces profiles of the orientational distribution function with better accuracy than those obtained with the traditional counting method. The accuracy of the torque sampling method is independent of the bin size, and hence it is possible to resolve the orientational distribution function with arbitrarily small angular resolutions.
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Affiliation(s)
- Johannes Renner
- Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - Matthias Schmidt
- Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - Daniel de Las Heras
- Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany
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6
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Revignas D, Ferrarini A. Spontaneous Twisting of Achiral Hard Rod Nematics. PHYSICAL REVIEW LETTERS 2023; 130:028102. [PMID: 36706401 DOI: 10.1103/physrevlett.130.028102] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2022] [Revised: 05/25/2022] [Accepted: 12/09/2022] [Indexed: 06/18/2023]
Abstract
Since Onsager's seminal work, hard rods have been taken as a prototype of nematic liquid crystals, characterized by uniaxial order and a uniform director field as a ground state. Here, using Onsager theory to calculate the free energy in the presence of arbitrary deformations, we find that hard rod nematics have an intrinsic tendency to twist around their ordering axis (double twist), driven by a mechanism in which the orientational fluctuations of particles play a key role. The anisotropic hard core potential used here is arguably the simplest form of interaction able to originate spontaneous breaking of mirror symmetry in a 3D fluid. Our results are discussed in relation to the recent discovery of a double twisted ground state in cylindrically confined lyotropic chromonic liquid crystals.
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Affiliation(s)
- Davide Revignas
- Department of Chemical Sciences, University of Padova Via Marzolo 1, 35131 Padova, Italy
| | - Alberta Ferrarini
- Department of Chemical Sciences, University of Padova Via Marzolo 1, 35131 Padova, Italy
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7
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Opdam J, Peters VFD, Wensink HH, Tuinier R. Multiphase Coexistence in Binary Hard Colloidal Mixtures: Predictions from a Simple Algebraic Theory. J Phys Chem Lett 2023; 14:199-206. [PMID: 36580685 PMCID: PMC9841575 DOI: 10.1021/acs.jpclett.2c03138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
A general theoretical framework is proposed to quantify the thermodynamic properties of multicomponent hard colloidal mixtures. This framework is used to predict the phase behavior of mixtures of rods with spheres and rods with plates taking into account (liquid) crystal phases of both components. We demonstrate a rich and complex range of phase behaviors featuring a large variety of different multiphase coexistence regions, including two five-phase coexistence regions for hard rod/sphere mixtures, and even a six-phase equilibrium for hard rod/plate dispersions. The various multiphase coexistences featured in a particular mixture are in line with a recently proposed generalized phase rule and can be tuned through subtle variations of the particle shape and size ratio. Our approach qualitatively accounts for certain multiphase equilibria observed in rod/plate mixtures of clay colloids and will be a useful guide in tuning the phase behavior of shape-disperse mixtures in general.
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Affiliation(s)
- J. Opdam
- Laboratory
of Physical Chemistry, Department of Chemical Engineering and Chemistry,
and Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, P.O. Box 513, 5600 MBEindhoven, The Netherlands
| | - V. F. D. Peters
- Laboratory
of Physical Chemistry, Department of Chemical Engineering and Chemistry,
and Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, P.O. Box 513, 5600 MBEindhoven, The Netherlands
- Department
of Earth Sciences, Utrecht University, Princetonlaan 8a, 3584CBUtrecht, The Netherlands
| | - H. H. Wensink
- Laboratoire
de Physique des Solides, Université Paris-Saclay and CNRS, 91405Orsay, France
| | - R. Tuinier
- Laboratory
of Physical Chemistry, Department of Chemical Engineering and Chemistry,
and Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, P.O. Box 513, 5600 MBEindhoven, The Netherlands
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8
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Gurin P, Varga S. Anomalous phase behavior of quasi-one-dimensional attractive hard rods. Phys Rev E 2022; 106:044606. [PMID: 36397485 DOI: 10.1103/physreve.106.044606] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2022] [Accepted: 09/28/2022] [Indexed: 06/16/2023]
Abstract
We study a two-state model of attractive hard rods using the transfer matrix method, where the centers of the particles are confined to a straight line, but the orientations of the rods can be parallel or perpendicular to the confining line. The rods are modeled as hard rectangles with length L and width D and decorated with attractive sites at both ends of the rectangles. We find that the particles align parallel to the line and form long chains at low densities, while they turn out of the line and form a Tonks gas at high densities. With increasing the stickiness between the rods, the structural change between parallel and perpendicular states becomes stronger and the pressure vs density curve becomes almost a horizontal line at the transition pressure. We show that such a behavior is reminiscent of the first-order phase transition. This manifests in the validity of the lever rule of the phase transitions for very sticky cases.
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Affiliation(s)
- Péter Gurin
- Physics Department, Centre for Natural Sciences, University of Pannonia, PO Box 158, Veszprém, H-8201 Hungary
| | - Szabolcs Varga
- Physics Department, Centre for Natural Sciences, University of Pannonia, PO Box 158, Veszprém, H-8201 Hungary
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9
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Rafael EM, Tonti L, Daza FAG, Patti A. Active microrheology of colloidal suspensions of hard cuboids. Phys Rev E 2022; 106:034612. [PMID: 36266794 DOI: 10.1103/physreve.106.034612] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2022] [Accepted: 08/31/2022] [Indexed: 06/16/2023]
Abstract
By performing dynamic Monte Carlo simulations, we investigate the microrheology of isotropic suspensions of hard-core colloidal cuboids. In particular, we infer the local viscoelastic behavior of these fluids by studying the dynamics of a probe spherical particle that is incorporated in the host phase and is dragged by an external force. This technique, known as active microrheology, allows one to characterize the microscopic response of soft materials upon application of a constant force, whose intensity spans here three orders of magnitude. By tuning the geometry of cuboids from oblate to prolate as well as the system density, we observe different responses that are quantified by measuring the effective friction perceived by the probe particle. The resulting friction coefficient exhibits a linear regime at forces that are much weaker and larger than the thermal forces, whereas a nonlinear, force-thinning regime is observed at intermediate force intensities.
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Affiliation(s)
- Effran Mirzad Rafael
- Department of Chemical Engineering, The University of Manchester, Manchester, M13 9PL, United Kingdom
| | - Luca Tonti
- Department of Chemical Engineering, The University of Manchester, Manchester, M13 9PL, United Kingdom
| | - Fabián A García Daza
- Department of Chemical Engineering, The University of Manchester, Manchester, M13 9PL, United Kingdom
| | - Alessandro Patti
- Department of Chemical Engineering, The University of Manchester, Manchester, M13 9PL, United Kingdom
- Department of Applied Physics, University of Granada, Avenida Fuente Nueva s/n, 18071 Granada, Spain
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10
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Kubala P, Tomczyk W, Cieśla M. In silico study of liquid crystalline phases formed by bent-shaped molecules with excluded volume type interactions. J Mol Liq 2022. [DOI: 10.1016/j.molliq.2022.120156] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/15/2022]
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11
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Empting E, Bader N, Oettel M. Interplay of orientational order and roughness in simulated thin film growth of anisotropically interacting particles. Phys Rev E 2022; 105:045306. [PMID: 35590594 DOI: 10.1103/physreve.105.045306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2021] [Accepted: 03/29/2022] [Indexed: 06/15/2023]
Abstract
Roughness and orientational order in thin films of anisotropic particles are investigated using kinetic Monte Carlo simulations on a cubic lattice. Anisotropic next-neighbor interactions between the lattice particles were chosen to mimic the effects of shape anisotropy in the interactions of disk- or rodlike molecules with van der Waals attractions. Increasing anisotropy leads first to a preferred orientation in the film (which is close to the corresponding equilibrium transition) while the qualitative mode of roughness evolution (known from isotropic systems) does not change. At strong anisotropies, an effective step-edge (Ehrlich-Schwoebel) barrier appears and a nonequilibrium roughening effect is found, accompanied by reordering in the film which can be interpreted as the nucleation and growth of domains of lying-down disks or rods. The information on order and roughness is combined into a diagram of dynamic growth modes.
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Affiliation(s)
- E Empting
- Institut für Angewandte Physik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
| | - N Bader
- Institut für Angewandte Physik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
| | - M Oettel
- Institut für Angewandte Physik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
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12
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Mazzilli V, Satoh K, Saielli G. Mixtures of discotic and spherical soft particles: de-mixing, liquid crystal behaviour and relative solubility. J Mol Liq 2022. [DOI: 10.1016/j.molliq.2021.117973] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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13
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Ramírez González JP, Cinacchi G. Phase behavior of hard circular arcs. Phys Rev E 2021; 104:054604. [PMID: 34942798 DOI: 10.1103/physreve.104.054604] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Accepted: 10/12/2021] [Indexed: 01/26/2023]
Abstract
By using Monte Carlo numerical simulation, this work investigates the phase behavior of systems of hard infinitesimally thin circular arcs, from an aperture angle θ→0 to an aperture angle θ→2π, in the two-dimensional Euclidean space. Except in the isotropic phase at lower density and in the (quasi)nematic phase, in the other phases that form, including the isotropic phase at higher density, hard infinitesimally thin circular arcs autoassemble to form clusters. These clusters are either filamentous, for smaller values of θ, or roundish, for larger values of θ. Provided the density is sufficiently high, the filaments lengthen, merge, and straighten to finally produce a filamentary phase while the roundels compact and dispose themselves with their centers of mass at the sites of a triangular lattice to finally produce a cluster hexagonal phase.
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Affiliation(s)
- Juan Pedro Ramírez González
- Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
| | - Giorgio Cinacchi
- Departamento de Física Teórica de la Materia Condensada, Instituto de Física de la Materia Condensada (IFIMAC), Instituto de Ciencias de Materiales "Nicolás Cabrera", Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
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14
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Lázaro MT, Aliabadi R, Wensink HH. Second-virial theory for shape-persistent living polymers templated by disks. Phys Rev E 2021; 104:054505. [PMID: 34942807 DOI: 10.1103/physreve.104.054505] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2021] [Accepted: 11/03/2021] [Indexed: 11/07/2022]
Abstract
Living polymers composed of noncovalently bonded building blocks with weak backbone flexibility may self-assemble into thermoresponsive lyotropic liquid crystals. We demonstrate that the reversible polymer assembly and phase behavior can be controlled by the addition of (nonadsorbing) rigid colloidal disks which act as an entropic reorienting "template" onto the supramolecular polymers. Using a particle-based second-virial theory that correlates the various entropies associated with the polymers and disks, we demonstrate that small fractions of discotic additives promote the formation of a polymer nematic phase. At larger disk concentrations, however, the phase is disrupted by collective disk alignment in favor of a discotic nematic fluid in which the polymers are dispersed antinematically. We show that the antinematic arrangement of the polymers generates a nonexponential molecular-weight distribution and stimulates the formation of oligomeric species. At sufficient concentrations the disks facilitate a liquid-liquid phase separation which can be brought into simultaneously coexistence with the two fractionated nematic phases, providing evidence for a four-fluid coexistence in reversible shape-dissimilar hard-core mixtures without cohesive interparticle forces. We stipulate the conditions under which such a phenomenon could be found in experiment.
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Affiliation(s)
- M Torres Lázaro
- Laboratoire de Physique des Solides, UMR 8502, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - R Aliabadi
- Physics Department, Sirjan University of Technology, Sirjan 78137, Iran
| | - H H Wensink
- Laboratoire de Physique des Solides, UMR 8502, CNRS, Université Paris-Saclay, 91405 Orsay, France
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15
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Monderkamp PA, Wittmann R, Cortes LBG, Aarts DGAL, Smallenburg F, Löwen H. Topology of Orientational Defects in Confined Smectic Liquid Crystals. PHYSICAL REVIEW LETTERS 2021; 127:198001. [PMID: 34797147 DOI: 10.1103/physrevlett.127.198001] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2021] [Revised: 06/28/2021] [Accepted: 09/22/2021] [Indexed: 06/13/2023]
Abstract
We propose a general formalism to characterize orientational frustration of smectic liquid crystals in confinement by interpreting the emerging networks of grain boundaries as objects with a topological charge. In a formal idealization, this charge is distributed in pointlike units of quarter-integer magnitude, which we identify with tetratic disclinations located at the end points and nodes. This coexisting nematic and tetratic order is analyzed with the help of extensive Monte Carlo simulations for a broad range of two-dimensional confining geometries as well as colloidal experiments, showing how the observed defect networks can be universally reconstructed from simple building blocks. We further find that the curvature of the confining wall determines the anchoring behavior of grain boundaries, such that the number of nodes in the emerging networks and the location of their end points can be tuned by changing the number and smoothness of corners, respectively.
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Affiliation(s)
- Paul A Monderkamp
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - René Wittmann
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - Louis B G Cortes
- School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA
| | - Dirk G A L Aarts
- Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
| | - Frank Smallenburg
- Laboratoire de Physique des Solides, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
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16
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Peters VD, González García Á, Wensink HH, Vis M, Tuinier R. Multiphase Coexistences in Rod-Polymer Mixtures. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2021; 37:11582-11591. [PMID: 34553593 PMCID: PMC8495896 DOI: 10.1021/acs.langmuir.1c01896] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/16/2021] [Revised: 09/09/2021] [Indexed: 06/13/2023]
Abstract
Using recently derived analytical equations of state for hard rod dispersions, we predict the phase behavior of athermal rod-polymer mixtures with free volume theory. The rods are modeled as hard spherocylinders, while the nonadsorbing polymer chains are described as penetrable hard spheres. It is demonstrated that all of the different types of phase states that are stable for pure colloidal rod dispersions can coexist with any combination of these phases if polymers are added, depending on the concentrations, rod aspect ratio, and polymer-rod size ratio. This includes novel two-, three-, and four-phase coexistences and isostructural coexistences between dilute and concentrated phases of the same kind, even for the more ordered (liquid) crystal phases. This work provides insight into the conditions at which particular multiphase coexistences are expected for well-defined model colloidal rod-polymer mixtures. We provide a quantitative map detailing the various types of isostructural coexistences, which confirms an early qualitative hypothesis by Bolhuis et al. ( J. Chem. Phys. 107, 1997 1551).
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Affiliation(s)
- Vincent
F. D. Peters
- Laboratory
of Physical Chemistry, Department of Chemical Engineering and Chemistry
& Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
| | - Álvaro González García
- Sustainable
Polymer Chemistry Group, Department of Molecules & Materials, MESA + Institute for Nanotechnology, University of
Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
| | - Henricus H. Wensink
- Laboratoire
de Physique des Solides − UMR 8502, CNRS & Université
Paris-Saclay, 91400 Orsay, France
| | - Mark Vis
- Laboratory
of Physical Chemistry, Department of Chemical Engineering and Chemistry
& Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
| | - Remco Tuinier
- Laboratory
of Physical Chemistry, Department of Chemical Engineering and Chemistry
& Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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17
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Behzadi F, Ghazi SM, Aliabadi R. From n-layer planar ordering to the monolayer homeotropic structure of confined hard rods: The effect of shape anisotropy and wall-to-wall separation. Phys Rev E 2021; 103:022702. [PMID: 33735962 DOI: 10.1103/physreve.103.022702] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/24/2020] [Accepted: 01/13/2021] [Indexed: 11/07/2022]
Abstract
Using the Parsons-Lee theory, we examined the effect of shape anisotropy and the wall-to-wall separation (H) on the phase behavior of the hard parallelepiped rods with dimensions L, D, and D (L>D) in such narrow slitlike pores which only one homeotropic layer can form. The phase structures, including biaxiality, planar nematic layering transition as well as planar to homeotropic, were studied for some separations in the range 2.5D≤H≤10.0D for H-D≤L<H.
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Affiliation(s)
- Fahimeh Behzadi
- Department of Physics, Faculty of Science, Fasa University, 74617-81189 Fasa, Iran
| | - Seyed Mohammad Ghazi
- Department of Physics, Faculty of Science, Fasa University, 74617-81189 Fasa, Iran
| | - Roohollah Aliabadi
- Department of Physics, Faculty of Science, Fasa University, 74617-81189 Fasa, Iran
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18
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Martínez-Ratón Y, Velasco E. Orientational ordering in a fluid of hard kites: A density-functional-theory study. Phys Rev E 2020; 102:052128. [PMID: 33327136 DOI: 10.1103/physreve.102.052128] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2020] [Accepted: 11/09/2020] [Indexed: 11/07/2022]
Abstract
Using density-functional theory we theoretically study the orientational properties of uniform phases of hard kites-two isosceles triangles joined by their common base. Two approximations are used: scaled particle theory and a new approach that better approximates third virial coefficients of two-dimensional hard particles. By varying some of their geometrical parameters, kites can be transformed into squares, rhombuses, triangles, and also very elongated particles, even reaching the hard-needle limit. Thus, a fluid of hard kites, depending on the particle shape, can stabilize isotropic, nematic, tetratic, and triatic phases. Different phase diagrams are calculated, including those of rhombuses, and kites with two of their equal interior angles fixed to 90^{∘}, 60^{∘}, and 75^{∘}. Kites with one of their unequal angles fixed to 72^{∘}, which have been recently studied via Monte Carlo simulations, are also considered. We find that rhombuses and kites with two equal right angles and not too large anisometry stabilize the tetratic phase but the latter stabilize it to a much higher degree. By contrast, kites with two equal interior angles fixed to 60^{∘} stabilize the triatic phase to some extent, although it is very sensitive to changes in particle geometry. Kites with the two equal interior angles fixed to 75^{∘} have a phase diagram with both tetratic and triatic phases, but we show the nonexistence of a particle shape for which both phases are stable at different densities. Finally, the success of the new theory in the description of orientational order in kites is shown by comparing with Monte Carlo simulations for the case where one of the unequal angles is fixed to 72^{∘}. These particles also present a phase diagram with stable tetratic and triatic phases.
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Affiliation(s)
- Yuri Martínez-Ratón
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad 30, E-28911, Leganés, Madrid, Spain
| | - Enrique Velasco
- Departamento de Física Teórica de la Materia Condensada, Instituto de Física de la Materia Condensada (IFIMAC) and Instituto de Ciencia de Materiales Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049, Madrid, Spain
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19
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Ramírez González JP, Cinacchi G. Dense packings of hard circular arcs. Phys Rev E 2020; 102:042903. [PMID: 33212650 DOI: 10.1103/physreve.102.042903] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2020] [Accepted: 09/09/2020] [Indexed: 11/07/2022]
Abstract
This work investigates dense packings of congruent hard infinitesimally thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle θ∈(π,2π]. Differently than those denotable as minor whose subtended angle θ∈[0,π], it is impossible for two hard infinitesimally thin circular arcs with θ∈(π,2π] to arbitrarily closely approach once they are arranged in a configuration, e.g., on top of one another, replicable ad infinitum without introducing any overlap. This makes these hard concave particles, in spite of being infinitesimally thin, most densely pack with a finite number density. This raises the question as to what are these densest packings and what is the number density that they achieve. Supported by Monte Carlo numerical simulations, this work shows that one can analytically construct compact closed circular groups of hard major circular arcs in which a specific, θ-dependent, number of them (counter) clockwise intertwine. These compact closed circular groups then arrange on a triangular lattice. These analytically constructed densest-known packings are compared to corresponding results of Monte Carlo numerical simulations to assess whether they can spontaneously turn up.
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Affiliation(s)
- Juan Pedro Ramírez González
- Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, E-28049 Madrid, Spain
| | - Giorgio Cinacchi
- Departamento de Física Teórica de la Materia Condensada, Instituto de Física de la Materia Condensada (IFIMAC), Instituto de Ciencias de Materiales "Nicolás Cabrera", Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, E-28049 Madrid, Spain
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20
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Ngo VT, Nguyen PT, Diep HT. Statistical Physics Approach to Liquid Crystals: Dynamics of Mobile Potts Model Leading to Smectic Phase, Phase Transition by Wang-Landau Method. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1232. [PMID: 33286999 PMCID: PMC7711670 DOI: 10.3390/e22111232] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/16/2020] [Revised: 10/18/2020] [Accepted: 10/26/2020] [Indexed: 11/16/2022]
Abstract
We study the nature of the smectic-isotropic phase transition using a mobile 6-state Potts model. Each Potts state represents a molecular orientation. We show that with the choice of an appropriate microscopic Hamiltonian describing the interaction between individual molecules modeled by a mobile 6-state Potts spins, we observe the smectic phase dynamically formed when we cool the molecules from the isotropic phase to low temperatures (T). In order to elucidate the order of the transition and the low-T properties, we use the high-performance Wang-Landau flat energy-histogram technique. We show that the smectic phase goes to the liquid (isotropic) phase by melting/evaporating layer by layer starting from the film surface with increasing T. At a higher T, the whole remaining layers become orientationally disordered. The melting of each layer is characterized by a peak of the specific heat. Such a succession of partial transitions cannot be seen by the Metropolis algorithm. The successive layer meltings/evaporations at low T are found to have a first-order character by examining the energy histogram. These results are in agreement with experiments performed on some smectic liquid crystals.
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Affiliation(s)
- V. Thanh Ngo
- Center for Informatics and Computing, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10000, Vietnam;
- Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10000, Vietnam;
| | - Phuong-Thuy Nguyen
- Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10000, Vietnam;
- Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Hanoi 10000, Vietnam
| | - Hung T. Diep
- Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université (Formerly, University of Cergy-Pontoise), CNRS, UMR 8089, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France
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21
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Mizani S, Gurin P, Aliabadi R, Salehi H, Varga S. Demixing and tetratic ordering in some binary mixtures of hard superellipses. J Chem Phys 2020; 153:034501. [PMID: 32716200 DOI: 10.1063/5.0009705] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
Abstract
We examine the fluid phase behavior of binary mixtures of hard superellipses using the scaled particle theory. The superellipse is a general two-dimensional convex object that can be tuned between the elliptical and rectangular shapes continuously at a given aspect ratio. We find that the shape of the particle affects strongly the stability of isotropic, nematic, and tetratic phases in the mixture even if the side lengths of both species are fixed. While the isotropic-isotropic demixing transition can be ruled out using the scaled particle theory, the first order isotropic-nematic and the nematic-nematic demixing transition can be stabilized with strong fractionation between the components. It is observed that the demixing tendency is strongest in small rectangle-large ellipse mixtures. Interestingly, it is possible to stabilize the tetratic order at lower densities in the mixture of hard squares and rectangles where the long rectangles form a nematic phase, while the squares stay in the tetratic order.
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Affiliation(s)
- Sakine Mizani
- Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
| | - Péter Gurin
- Institute of Physics and Mechatronics, University of Pannonia, P.O. Box 158, Veszprém H-8201, Hungary
| | - Roohollah Aliabadi
- Department of Physics, Faculty of Science, Fasa University, 74617-81189 Fasa, Iran
| | - Hamdollah Salehi
- Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
| | - Szabolcs Varga
- Institute of Physics and Mechatronics, University of Pannonia, P.O. Box 158, Veszprém H-8201, Hungary
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22
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Peters VFD, Vis M, Wensink HH, Tuinier R. Algebraic equations of state for the liquid crystalline phase behavior of hard rods. Phys Rev E 2020; 101:062707. [PMID: 32688562 DOI: 10.1103/physreve.101.062707] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Accepted: 06/01/2020] [Indexed: 05/24/2023]
Abstract
Based on simplifications of previous numerical calculations [H. Graf and H. Löwen, Phys. Rev. E 59, 1932 (1999)1063-651X10.1103/PhysRevE.59.1932], we propose algebraic free energy expressions for the smectic-A liquid crystal phase and the crystal phases of hard spherocylinders. Quantitative agreement with simulations is found for the resulting equations of state. The free energy expressions can be used to straightforwardly compute the full phase behavior for all aspect ratios and to provide a suitable benchmark for exploring how attractive interrod interactions mediate the phase stability through perturbation approaches such as free-volume or van der Waals theory.
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Affiliation(s)
- V F D Peters
- Laboratory of Physical Chemistry, Department of Chemical Engineering and Chemistry & Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
| | - M Vis
- Laboratory of Physical Chemistry, Department of Chemical Engineering and Chemistry & Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
- Laboratoire de Chimie, École Normale Supérieure de Lyon, 69364 Lyon Cedex 07, France
| | - H H Wensink
- Laboratoire de Physique des Solides - UMR 8502, CNRS & Université Paris-Saclay, Orsay, France
| | - R Tuinier
- Laboratory of Physical Chemistry, Department of Chemical Engineering and Chemistry & Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
- Van t Hoff Laboratory for Physical and Colloid Chemistry, Department of Chemistry & Debye Institute for Nanomaterials Science, Utrecht University, 3584 CH Utrecht, The Netherlands
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23
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Mizani S, Aliabadi R, Salehi H, Varga S. Orientational ordering and layering of hard plates in narrow slitlike pores. Phys Rev E 2019; 100:032704. [PMID: 31639981 DOI: 10.1103/physreve.100.032704] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2019] [Indexed: 11/07/2022]
Abstract
We examine the ordering behavior of hard platelike particles in a very narrow, slitlike pore using the Parsons-Lee density functional theory and the restricted orientation approximation. We observe that the plates are orientationally ordered and align perpendicularly (face-on) to the walls at low densities, a first-order layering transition occurs between uniaxial nematic structures having n and n+1 layers at intermediate densities, and even a phase transition between a monolayer with parallel (edge-on) orientational order and n layers with a perpendicular one can be detected at high densities. In addition to this, the edge-on monolayer is usually biaxial nematic, and a uniaxial-biaxial nematic phase transition can be also seen at very high densities.
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Affiliation(s)
- Sakine Mizani
- Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
| | - Roohollah Aliabadi
- Department of Physics, Faculty of Science, Fasa University, 74617-81189 Fasa, Iran
| | - Hamdollah Salehi
- Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
| | - Szabolcs Varga
- Institute of Physics and Mechatronics, University of Pannonia, P.O. Box 158, Veszprém H-8201, Hungary
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24
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Chiappini M, Drwenski T, van Roij R, Dijkstra M. Biaxial, Twist-bend, and Splay-bend Nematic Phases of Banana-shaped Particles Revealed by Lifting the "Smectic Blanket". PHYSICAL REVIEW LETTERS 2019; 123:068001. [PMID: 31491177 DOI: 10.1103/physrevlett.123.068001] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/11/2019] [Revised: 06/20/2019] [Indexed: 06/10/2023]
Abstract
We perform an extensive computational study on the phase behavior of hard banana-shaped particles, and show that biaxial, twist-bend, and splay-bend nematic phases are metastable with respect to a smectic phase for a system of hard bent spherocylinders. However, if the smectic phase is destabilized-either by polydispersity in the particle length or by curvature in the particle shape-stable biaxial, twist-bend, and splay-bend nematic phases are obtained. This provides a unified and consistent picture on the subtle role of particle shape on the phase behavior of bent rods.
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Affiliation(s)
- Massimiliano Chiappini
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Department of Physics, Utrecht University, Princetonplein 1, Utrecht 3584 CC, The Netherlands
| | - Tara Drwenski
- Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - René van Roij
- Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - Marjolein Dijkstra
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Department of Physics, Utrecht University, Princetonplein 1, Utrecht 3584 CC, The Netherlands
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25
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Geigenfeind T, de Las Heras D. Principal component analysis of the excluded area of two-dimensional hard particles. J Chem Phys 2019; 150:184906. [PMID: 31091902 DOI: 10.1063/1.5092865] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The excluded area between a pair of two-dimensional hard particles with given relative orientation is the region in which one particle cannot be located due to the presence of the other particle. The magnitude of the excluded area as a function of the relative particle orientation plays a major role in the determination of the bulk phase behavior of hard particles. We use principal component analysis (PCA) to identify the different types of excluded areas corresponding to randomly generated two-dimensional hard particles modeled as non-self-intersecting polygons and star lines (line segments radiating from a common origin). Only three principal components are required to have an excellent representation of the value of the excluded area as a function of the relative particle orientation for sufficiently anisotropic particles. Independent of the particle shape, the minimum value of the excluded area is always achieved when the particles are antiparallel to each other. The property that affects the value of the excluded area most strongly is the elongation of the particle shape. PCA identifies four limiting cases of excluded areas with one to four global minima at equispaced relative orientations. We study selected particle shapes using Monte Carlo simulations.
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Affiliation(s)
- Thomas Geigenfeind
- Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - Daniel de Las Heras
- Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany
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26
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Affiliation(s)
- Michael P. Allen
- Department of Physics, University of Warwick, Coventry, UK
- H. H. Wills Physics Laboratory, Royal Fort, Bristol, UK
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27
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Maryshev I, Marenduzzo D, Goryachev AB, Morozov A. Kinetic theory of pattern formation in mixtures of microtubules and molecular motors. Phys Rev E 2018; 97:022412. [PMID: 29548141 DOI: 10.1103/physreve.97.022412] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2017] [Indexed: 11/07/2022]
Abstract
In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)PLEEE81539-375510.1103/PhysRevE.74.031915] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behavior by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behavior than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.
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Affiliation(s)
- Ivan Maryshev
- Centre for Synthetic and Systems Biology, Institute of Cell Biology, School of Biological Sciences, University of Edinburgh, Max Born Crescent, Edinburgh EH9 3BF, United Kingdom
| | - Davide Marenduzzo
- SUPA, School of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
| | - Andrew B Goryachev
- Centre for Synthetic and Systems Biology, Institute of Cell Biology, School of Biological Sciences, University of Edinburgh, Max Born Crescent, Edinburgh EH9 3BF, United Kingdom
| | - Alexander Morozov
- SUPA, School of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
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28
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Drwenski T, van Roij R, van der Schoot P. Connectedness percolation of hard convex polygonal rods and platelets. J Chem Phys 2018; 149:054902. [DOI: 10.1063/1.5040185] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Affiliation(s)
- Tara Drwenski
- Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - René van Roij
- Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - Paul van der Schoot
- Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
- Theory of Polymers and Soft Matter, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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29
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Torquato S. Perspective: Basic understanding of condensed phases of matter via packing models. J Chem Phys 2018; 149:020901. [DOI: 10.1063/1.5036657] [Citation(s) in RCA: 73] [Impact Index Per Article: 12.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Affiliation(s)
- S. Torquato
- Department of Chemistry, Department of Physics, Princeton Center for Theoretical Science, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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30
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Dussi S, Tasios N, Drwenski T, van Roij R, Dijkstra M. Hard Competition: Stabilizing the Elusive Biaxial Nematic Phase in Suspensions of Colloidal Particles with Extreme Lengths. PHYSICAL REVIEW LETTERS 2018; 120:177801. [PMID: 29756829 DOI: 10.1103/physrevlett.120.177801] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2018] [Indexed: 06/08/2023]
Abstract
We use computer simulations to study the existence and stability of a biaxial nematic N_{b} phase in systems of hard polyhedral cuboids, triangular prisms, and rhombic platelets, characterized by a long (L), medium (M), and short (S) particle axis. For all three shape families, we find stable N_{b} states provided the shape is not only close to the so-called dual shape with M=sqrt[LS] but also sufficiently anisotropic with L/S>9,11,14,23 for rhombi, (two types of) triangular prisms, and cuboids, respectively, corresponding to anisotropies not considered before. Surprisingly, a direct isotropic-N_{b} transition does not occur in these systems due to a destabilization of N_{b} by a smectic (for cuboids and prisms) or a columnar (for platelets) phase at small L/S or by an intervening uniaxial nematic phase at large L/S. Our results are confirmed by a density functional theory provided the third virial coefficient is included and a continuous rather than a discrete (Zwanzig) set of particle orientations is taken into account.
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Affiliation(s)
- Simone Dussi
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands
| | - Nikos Tasios
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands
| | - Tara Drwenski
- Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands
| | - René van Roij
- Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands
| | - Marjolein Dijkstra
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands
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31
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Brumby PE, Wensink HH, Haslam AJ, Jackson G. Structure and Interfacial Tension of a Hard-Rod Fluid in Planar Confinement. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2017; 33:11754-11770. [PMID: 28885848 DOI: 10.1021/acs.langmuir.7b02254] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The structural properties and interfacial tension of a fluid of rodlike hard-spherocylinder particles in contact with hard structureless flat walls are studied by means of Monte Carlo simulation. The calculated surface tension between the rod fluid and the substrate is characterized by a nonmonotonic trend as a function of the bulk concentration (density) over the range of isotropic bulk concentrations. As suggested by earlier theoretical studies, a surface-ordering scenario is confirmed by our simulations: the local orientational order close to the wall changes from uniaxial to biaxial nematic when the bulk concentration reaches about 85% of the value at the onset of the isotropic-nematic phase transition. The surface ordering coincides with a wetting transition whereby the hard wall is wetted by a nematic film. Accurate values of the fluid-solid surface tension, the adsorption, and the average particle-wall contact distance are reported (over a broad range of densities into the dense nematic region for the first time), which can serve as a useful benchmark for future theoretical and experimental studies on confined rod fluids. The simulation data are supplemented with predictions from second-virial density functional theory, which are in good qualitative agreement with the simulation results.
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Affiliation(s)
- Paul E Brumby
- Department of Mechanical Engineering, Keio University , 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Henricus H Wensink
- Laboratoire de Physique des Solides, Université Paris Sud & CNRS , 91405 Orsay Cedex, France
| | - Andrew J Haslam
- Department of Chemical Engineering and Qatar Carbonates and Carbon Storage Research Centre, Imperial College London , South Kensington Campus, London SW7 2AZ, United Kingdom
| | - George Jackson
- Department of Chemical Engineering and Qatar Carbonates and Carbon Storage Research Centre, Imperial College London , South Kensington Campus, London SW7 2AZ, United Kingdom
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32
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Nascimento ES, Palffy-Muhoray P, Taylor JM, Virga EG, Zheng X. Density functional theory for dense nematic liquid crystals with steric interactions. Phys Rev E 2017; 96:022704. [PMID: 28950602 DOI: 10.1103/physreve.96.022704] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2017] [Indexed: 06/07/2023]
Abstract
The celebrated work of Onsager on hard particle systems, based on the truncated second order virial expansion, is valid at relatively low volume fractions for large aspect ratio particles. While it predicts the isotropic-nematic phase transition, it does not provide a realistic equation of state in that the pressure remains finite for arbitrarily high densities. In this work, we derive a mean field density functional form of the Helmholtz free energy for nematics with hard core repulsion. In addition to predicting the isotropic-nematic transition, the model provides a more realistic equation of state. The energy landscape is much richer, and the orientational probability distribution function in the nematic phase possesses a unique feature-it vanishes on a nonzero measure set in orientation space.
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Affiliation(s)
| | | | - Jamie M Taylor
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Epifanio G Virga
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Xiaoyu Zheng
- Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240, USA
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33
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Gschwind A, Klopotek M, Ai Y, Oettel M. Isotropic-nematic transition for hard rods on a three-dimensional cubic lattice. Phys Rev E 2017; 96:012104. [PMID: 29347067 DOI: 10.1103/physreve.96.012104] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2017] [Indexed: 06/07/2023]
Abstract
Using grand-canonical Monte Carlo (GCMC) simulations, we investigate the isotropic-nematic phase transition for hard rods of size L×1×1 on a three-dimensional cubic lattice. We observe such a transition for L≥6. For L=6, the nematic state has a negative order parameter, reflecting the co-occurrence of two dominating orientations. For L≥7, the nematic state has a positive order parameter, corresponding to the dominance of one orientation. We investigate rod lengths up to L=25 and find evidence for a very weakly first-order isotropic-nematic transition, while we cannot completely rule out a second-order transition. It was not possible to detect a density jump at the transition, despite using large systems containing several 10^{5} particles. The probability density distributions P(Q) from the GCMC simulations near the transition are very broad, pointing to strong fluctuations. Our results complement earlier results on the demixing (pseudonematic) transition for an equivalent system in two dimensions, which is presumably of Ising type and occurs for L≥7. We compare our results to lattice fundamental measure theory (FMT) and find that FMT strongly overestimates nematic order and consequently predicts a strong first-order transition. The rod packing fraction of the nematic coexisting states, however, agree reasonably well between FMT and GCMC.
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Affiliation(s)
- A Gschwind
- Institut für Angewandte Physik, Eberhard Karls Universität Tübingen, D-72076 Tübingen, Germany
| | - M Klopotek
- Institut für Angewandte Physik, Eberhard Karls Universität Tübingen, D-72076 Tübingen, Germany
| | - Y Ai
- School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - M Oettel
- Institut für Angewandte Physik, Eberhard Karls Universität Tübingen, D-72076 Tübingen, Germany
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34
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Tortora MMC, Doye JPK. Perturbative density functional methods for cholesteric liquid crystals. J Chem Phys 2017. [DOI: 10.1063/1.4982934] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Affiliation(s)
- Maxime M. C. Tortora
- Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
| | - Jonathan P. K. Doye
- Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
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35
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Rickayzen G, Heyes DM. Isotropic-nematic phase transition of uniaxial variable softness prolate and oblate ellipsoids. J Chem Phys 2017; 146:164505. [PMID: 28456193 DOI: 10.1063/1.4981887] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/24/2023] Open
Abstract
Onsager's theory of the isotropic-nematic phase separation of rod shaped particles is generalized to include particle softness and attractions in the anisotropic interparticle force field. The procedure separates a scaled radial component from the angular integral part, the latter being treated in essentially the same way as in the original Onsager formulation. Building on previous treatments of more idealised hard-core particle models, this is a step toward representing more realistic rod-like systems and also allowing temperature (and in principle specific chemical factors) to be included at a coarse grained level in the theory. The focus of the study is on the coexisting concentrations and associated coexistence properties. Prolate and oblate ellipsoids are considered in both the small and very large aspect ratio limits. Approximations to the terms in the angular integrals derived assuming the very large (prolate) and very small (oblate) aspect ratios limits are compared with the formally exact treatment. The approximation for the second virial coefficient matches the exact solution for aspect ratios above about 20 for the prolate ellipsoids and less than ca. 0.05 for the oblate ellipsoids from the numerical evaluation of the angular integrals. The temperature dependence of the coexistence density could be used to help determine the interaction potential of two molecules. The method works at temperatures above a certain threshold temperature where the second virial coefficient is positive.
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Affiliation(s)
- G Rickayzen
- School of Physical Sciences, University of Kent, Canterbury, Kent CT2 7NH, United Kingdom
| | - D M Heyes
- Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 OEX, United Kingdom
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36
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Cortes LBG, Gao Y, Dullens RPA, Aarts DGAL. Colloidal liquid crystals in square confinement: isotropic, nematic and smectic phases. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:064003. [PMID: 28002038 DOI: 10.1088/1361-648x/29/6/064003] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We report on the confinement of colloidal liquid crystals in three dimensional chambers with a square footprint. To this end we use colloidal silica rods and exploit their relatively large density difference with respect to the dispersing solvent to study isotropic, nematic and smectic phases confined into a single chamber. Combining laser scanning confocal microscopy and soft-lithography techniques enables us to characterize the configurations down to the single particle level. We will focus on the smectic phase and compare to recent theories and simulations.
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Affiliation(s)
- Louis B G Cortes
- Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, UK
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37
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Ferreiro-Córdova C, Wensink HH. Spinodal instabilities in polydisperse lyotropic nematics. J Chem Phys 2016; 145:244904. [DOI: 10.1063/1.4972523] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023] Open
Affiliation(s)
- C. Ferreiro-Córdova
- Laboratoire de Physique des Solides - UMR 8502, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - H. H. Wensink
- Laboratoire de Physique des Solides - UMR 8502, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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38
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González-Pinto M, Martínez-Ratón Y, Varga S, Gurin P, Velasco E. Phase behaviour and correlations of parallel hard squares: from highly confined to bulk systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:244002. [PMID: 27115832 DOI: 10.1088/0953-8984/28/24/244002] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We study a fluid of two-dimensional parallel hard squares in bulk and under confinement in channels, with the aim of evaluating the performance of fundamental-measure theory (FMT). To this purpose, we first analyse the phase behaviour of the bulk system using FMT and Percus-Yevick (PY) theory, and compare the results with molecular dynamics and Monte Carlo simulations. In a second step, we study the confined system and check the results against those obtained from the transfer matrix method and from our own Monte Carlo simulations. Squares are confined to channels with parallel walls at angles of 0° or 45° relative to the diagonals of the parallel hard squares, respectively, which allows for an assessment of the effect of the external-potential symmetry on the fluid structural properties. In general FMT overestimates bulk correlations, predicting the existence of a columnar phase (absent in simulations) prior to crystallization. The equation of state predicted by FMT compares well with simulations, although the PY approach with the virial route is better in some range of packing fractions. The FMT is highly accurate for the structure and correlations of the confined fluid due to the dimensional crossover property fulfilled by the theory. Both density profiles and equations of state of the confined system are accurately predicted by the theory. The highly non-uniform pair correlations inside the channel are also very well described by FMT.
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Affiliation(s)
- Miguel González-Pinto
- Departamento de Física Teórica de la Materia Condensada, Facultad de Ciencias, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
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39
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Dussi S, Dijkstra M. Entropy-driven formation of chiral nematic phases by computer simulations. Nat Commun 2016; 7:11175. [PMID: 27067806 PMCID: PMC4832067 DOI: 10.1038/ncomms11175] [Citation(s) in RCA: 58] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2015] [Accepted: 02/29/2016] [Indexed: 12/24/2022] Open
Abstract
Predicting the macroscopic chiral behaviour of liquid crystals from the microscopic chirality of the particles is highly non-trivial, even when the chiral interactions are purely entropic in nature. Here we introduce a novel chiral hard-particle model, namely particles with a twisted polyhedral shape and obtain a stable fully entropy-driven cholesteric phase by computer simulations. By slightly modifying the triangular base of the particle, we are able to switch from a left-handed prolate (calamitic) to a right-handed oblate (discotic) cholesteric phase using the same right-handed twisted particle model. Furthermore, we show that not only prolate and oblate chiral nematic phases, but also other novel entropy-driven phases, namely chiral blue phases, chiral nematic phases featuring both twist and splay deformations, chiral biaxial nematic phases with one of the axes twisted, can be obtained by varying particle biaxiality and chirality. Our results allow to identify general guidelines for the stabilization of these phases.
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Affiliation(s)
- Simone Dussi
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
| | - Marjolein Dijkstra
- Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
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40
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Wöhrle T, Wurzbach I, Kirres J, Kostidou A, Kapernaum N, Litterscheidt J, Haenle JC, Staffeld P, Baro A, Giesselmann F, Laschat S. Discotic Liquid Crystals. Chem Rev 2015; 116:1139-241. [PMID: 26483267 DOI: 10.1021/acs.chemrev.5b00190] [Citation(s) in RCA: 418] [Impact Index Per Article: 46.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Tobias Wöhrle
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Iris Wurzbach
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Jochen Kirres
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Antonia Kostidou
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Nadia Kapernaum
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Juri Litterscheidt
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Johannes Christian Haenle
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Peter Staffeld
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Angelika Baro
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Frank Giesselmann
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Sabine Laschat
- Institut für Organische Chemie, and ‡Institut für Physikalische Chemie, Universität Stuttgart , Pfaffenwaldring 55, 70569 Stuttgart, Germany
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41
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Greco C, Ferrarini A. Entropy-Driven Chiral Order in a System of Achiral Bent Particles. PHYSICAL REVIEW LETTERS 2015; 115:147801. [PMID: 26551824 DOI: 10.1103/physrevlett.115.147801] [Citation(s) in RCA: 43] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2015] [Indexed: 05/16/2023]
Abstract
Why should achiral particles organize into a helical structure? Here, using theory and molecular dynamics simulations we show that at high concentration crescent-shaped particles interacting through a purely repulsive potential form the twist-bend nematic phase, which features helical order of the twofold symmetry axes of particles, with doubly degenerate handedness. Spontaneous breaking of the chiral symmetry is driven by the entropic gain that derives from the decrease in excluded volume in the helical arrangement. Crucial to this purpose is the concave shape of particles. This study is based on a general formulation of the Onsager theory, which includes biaxiality and polarity of phase and particles, in addition to the space modulation of order. Molecular dynamics simulations corroborate the theoretical predictions and provide further insights into the structure of the helical phase.
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Affiliation(s)
- Cristina Greco
- Dipartimento di Scienze Chimiche, Università di Padova, via Marzolo 1, 35131 Padova, Italy
| | - Alberta Ferrarini
- Dipartimento di Scienze Chimiche, Università di Padova, via Marzolo 1, 35131 Padova, Italy
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42
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Piastra M, Virga EG. Explicit excluded volume of cylindrically symmetric convex bodies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062503. [PMID: 26172727 DOI: 10.1103/physreve.91.062503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2015] [Indexed: 06/04/2023]
Abstract
We represent explicitly the excluded volume V(e){B(1),B(2)} of two generic cylindrically symmetric, convex rigid bodies, B(1) and B(2), in terms of a family of shape functionals evaluated separately on B(1) and B(2). We show that V(e){B(1),B(2)} fails systematically to feature a dipolar component, thus making illusory the assignment of any shape dipole to a tapered body in this class. The method proposed here is applied to cones and validated by a shape-reconstruction algorithm. It is further applied to spheroids (ellipsoids of revolution), for which it shows how some analytic estimates already regarded as classics should indeed be emended.
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Affiliation(s)
- Marco Piastra
- Dipartimento di Ingegneria Industriale e dell'Informazione, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy
| | - Epifanio G Virga
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, I-27100 Pavia, Italy
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43
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Wittmann R, Marechal M, Mecke K. Elasticity of nematic phases with fundamental measure theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052501. [PMID: 26066185 DOI: 10.1103/physreve.91.052501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2015] [Indexed: 06/04/2023]
Abstract
In a previous publication [R. Wittmann, M. Marechal, and K. Mecke, Europhys. Lett. 109, 26003 (2015)], we introduced fundamental mixed measure theory (FMMT) for mixtures of anisotropic hard bodies, which shows that earlier results with an empirical parameter are inaccurate. Now we provide a deeper insight into the background of this theory in integral geometry. We study the Frank elastic coefficients in the nematic phase of the hard spherocylinder fluid. The framework of FMMT provides us with the required direct correlation function without additional input of an equation of state. A series representation of the mixed measure gives rise to closed analytical formulas for the elastic constants that only depend on the density, order parameters, and the particle geometry, pointing out a significant advantage of our geometry-based approach compared to other density functionals. Our elastic coefficients are in good agreement with computer simulations and increase with the density and the nematic order parameter. We confirm earlier mean-field predictions in the limits of low orientational order and infinitely long rods.
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Affiliation(s)
- René Wittmann
- Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstraße 7, D-91058 Erlangen, Germany
| | - Matthieu Marechal
- Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstraße 7, D-91058 Erlangen, Germany
| | - Klaus Mecke
- Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstraße 7, D-91058 Erlangen, Germany
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