1
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Mesarec L, Kralj S, Iglič A. Biaxial Structures of Localized Deformations and Line-like Distortions in Effectively 2D Nematic Films. NANOMATERIALS (BASEL, SWITZERLAND) 2024; 14:246. [PMID: 38334517 PMCID: PMC10856884 DOI: 10.3390/nano14030246] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2023] [Revised: 01/17/2024] [Accepted: 01/20/2024] [Indexed: 02/10/2024]
Abstract
We numerically studied localized elastic distortions in curved, effectively two-dimensional nematic shells. We used a mesoscopic Landau-de Gennes-type approach, in which the orientational order is theoretically considered by introducing the appropriate tensor nematic order parameter, while the three-dimensional shell shape is described by the curvature tensor. We limited our theoretical consideration to axially symmetric shapes of nematic shells. It was shown that in the surface regions of stomatocyte-class nematic shell shapes with large enough magnitudes of extrinsic (deviatoric) curvature, the direction of the in-plane orientational ordering can be mutually perpendicular above and below the narrow neck region. We demonstrate that such line-like nematic distortion configurations may run along the parallels (i.e., along the circular lines of constant latitude) located in the narrow neck regions of stomatocyte-like nematic shells. It was shown that nematic distortions are enabled by the order reconstruction mechanism. We propose that the regions of nematic shells that are strongly elastically deformed, i.e., topological defects and line-like distortions, may attract appropriately surface-decorated nanoparticles (NPs), which could potentially be useful for the controlled assembly of NPs.
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Affiliation(s)
- Luka Mesarec
- Laboratory of Physics, Faculty of Electrical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia;
| | - Samo Kralj
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, 2000 Maribor, Slovenia;
- Condensed Matter Physics Department, Jožef Stefan Institute, 1000 Ljubljana, Slovenia
| | - Aleš Iglič
- Laboratory of Physics, Faculty of Electrical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia;
- Laboratory of Clinical Biophysics, Faculty of Medicine, University of Ljubljana, 1000 Ljubljana, Slovenia
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2
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Ledesma-Durán A, Juárez-Valencia LH. Diffusion coefficients and MSD measurements on curved membranes and porous media. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2023; 46:70. [PMID: 37578670 DOI: 10.1140/epje/s10189-023-00329-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2023] [Accepted: 07/28/2023] [Indexed: 08/15/2023]
Abstract
We study some geometric aspects that influence the transport properties of particles that diffuse on curved surfaces. We compare different approaches to surface diffusion based on the Laplace-Beltrami operator adapted to predict concentration along entire membranes, confined subdomains along surfaces, or within porous media. Our goal is to summarize, firstly, how diffusion in these systems results in different types of diffusion coefficients and mean square displacement measurements, and secondly, how these two factors are affected by the concavity of the surface, the shape of the possible barriers or obstacles that form the available domains, the sinuosity, tortuosity, and constrictions of the trajectories and even how the observation plane affects the measurements of the diffusion. In addition to presenting a critical and organized comparison between different notions of MSD, in this review, we test the correspondence between theoretical predictions and numerical simulations by performing finite element simulations and illustrate some situations where diffusion theory can be applied. We briefly reviewed computational schemes for understanding surface diffusion and finally, discussed how this work contributes to understanding the role of surface diffusion transport properties in porous media and their relationship to other transport processes.
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Affiliation(s)
- Aldo Ledesma-Durán
- Departmento de Matemáticas, Universidad Autónoma Metropolitana, CDMX, Mexico
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3
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Mesarec L, Góźdź W, Kralj-Iglič V, Kralj S, Iglič A. Coupling of nematic in-plane orientational ordering and equilibrium shapes of closed flexible nematic shells. Sci Rep 2023; 13:10663. [PMID: 37393271 DOI: 10.1038/s41598-023-37664-2] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2023] [Accepted: 06/25/2023] [Indexed: 07/03/2023] Open
Abstract
The impact of the intrinsic curvature of in-plane orientationally ordered curved flexible nematic molecules attached to closed 3D flexible shells was studied numerically. A Helfrich-Landau-de Gennes-type mesoscopic approach was adopted where the flexible shell's curvature field and in-plane nematic field are coupled and concomitantly determined in the process of free energy minimisation. We demonstrate that this coupling has the potential to generate a rich diversity of qualitatively new shapes of closed 3D nematic shells and the corresponding specific in-plane orientational ordering textures, which strongly depend on the shell's volume-to-surface area ratio, so far not predicted in mesoscopic-type numerical studies of 3D shapes of closed flexible nematic shells.
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Affiliation(s)
- Luka Mesarec
- Laboratory of Physics, Faculty of Electrical Engineering, University of Ljubljana, Tržaška Cesta 25, 1000, Ljubljana, Slovenia.
| | - Wojciech Góźdź
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224, Warsaw, Poland
| | - Veronika Kralj-Iglič
- Laboratory of Clinical Biophysics, Faculty of Health Sciences, University of Ljubljana, Zdravstvena 5, 1000, Ljubljana, Slovenia
| | - Samo Kralj
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška Cesta 160, 2000, Maribor, Slovenia
- Condensed Matter Physics Department, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana, Slovenia
| | - Aleš Iglič
- Laboratory of Physics, Faculty of Electrical Engineering, University of Ljubljana, Tržaška Cesta 25, 1000, Ljubljana, Slovenia
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4
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Hölbl A, Mesarec L, Polanšek J, Iglič A, Kralj S. Stable Assemblies of Topological Defects in Nematic Orientational Order. ACS OMEGA 2023; 8:169-179. [PMID: 36643572 PMCID: PMC9835183 DOI: 10.1021/acsomega.2c07174] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/07/2022] [Accepted: 12/07/2022] [Indexed: 06/17/2023]
Abstract
We considered general mechanisms enabling the stabilization of localized assemblies of topological defects (TDs). There is growing evidence that physical fields represent fundamental natural entities, and therefore these features are of interest to all branches of physics. In general, cores of TDs are energetically costly, and consequently, assemblies of TDs are unfavorable. Owing to the richness of universalities in the physics of TDs, it is of interest to identify systems where they are easily experimentally accessible, enabling detailed and well-controlled analysis of their universal behavior, and cross-fertilizing knowledge in different areas of physics. In this respect, thermotropic nematic liquid crystals (NLCs) represent an ideal experiment testbed for such studies. In addition, TDs in NLCs could be exploited in several applications. We present examples that emphasize the importance of curvature imposed on the phase component of the relevant order parameter field. In NLCs, it is represented by the nematic tensor order parameter. Using a simple Landau-type approach, we show how the coupling between chirality and saddle splay elasticity, which can be expressed as a Gaussian curvature contribution, can stabilize Meron TDs. The latter have numerous analogs in other branches of physics. TDs in 2D curved manifolds reveal that the Gaussian curvature dominantly impacts the assembling and stabilization of TDs. Furthermore, a strong enough curvature that serves as an attractor for TDs is a respective field that could be imposed in a fast enough phase transition. Assemblies of created TDs created in such a disordered environment could be stabilized by appropriate impurities.
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Affiliation(s)
- Arbresha Hölbl
- Faculty of Natural
Sciences and Mathematics, University of
Maribor, Koroška
160, 2000 Maribor, Slovenia
| | - Luka Mesarec
- Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia
| | - Juš Polanšek
- Faculty of Natural
Sciences and Mathematics, University of
Maribor, Koroška
160, 2000 Maribor, Slovenia
| | - Aleš Iglič
- Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia
| | - Samo Kralj
- Faculty of Natural
Sciences and Mathematics, University of
Maribor, Koroška
160, 2000 Maribor, Slovenia
- Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
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5
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Polanšek J, Holbl A, Starzonek S, Drozd-Rzoska A, Rzoska SJ, Kralj S. History-dependent phase transition character. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2022; 45:70. [PMID: 35997865 PMCID: PMC9399213 DOI: 10.1140/epje/s10189-022-00221-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/23/2022] [Accepted: 07/26/2022] [Indexed: 06/15/2023]
Abstract
We consider history-dependent behavior in domain-type configurations in orientational order that are formed in configurations reached via continuous symmetry-breaking phase transitions. In equilibrium, these systems exhibit in absence of impurities a spatially homogeneous order. We focus on cases where domains are formed via (i) Kibble-Zurek mechanism in fast enough quenches or by (ii) Kibble mechanism in strongly supercooled phases. In both cases, domains could be arrested due to pinned topological defects that are formed at domain walls. In systems exhibiting polar or quadrupolar order, point and line defects (disclinations) dominate, respectively. In particular, the disclinations could form complex entangled structures and are more efficient in stabilizing domains. Domain patterns formed by fast quenches could be arrested by impurities imposing a strong enough random-field type disorder, as suggested by the Imry-Ma theorem. On the other hand, domains formed in supercooled systems could be also formed if large enough energy barriers arresting domains are established due to large enough systems' stiffness. The resulting effective interactions in established domain-type patterns could be described by random matrices. The resulting eigenvectors reveal expected structural excitations formed in such structures. The most important role is commonly played by the random matrix largest eigenvector. Qualitatively different behavior is expected if this eigenvector exhibits a localized or extended character. In the former case, one expects a gradual, non-critical-type transition into a glass-type structure. However, in the latter case, a critical-like phase behavior could be observed.
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Affiliation(s)
- Juš Polanšek
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska 160, 2000, Maribor, Slovenia
| | - Arbresha Holbl
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska 160, 2000, Maribor, Slovenia
| | - Szymon Starzonek
- Institute of High Pressure Physics Polish Academy of Sciences, ul. Sokołowska 29/37, 01-142, Warsaw, Poland.
| | - Aleksandra Drozd-Rzoska
- Institute of High Pressure Physics Polish Academy of Sciences, ul. Sokołowska 29/37, 01-142, Warsaw, Poland
| | - Sylwester J Rzoska
- Institute of High Pressure Physics Polish Academy of Sciences, ul. Sokołowska 29/37, 01-142, Warsaw, Poland
| | - Samo Kralj
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska 160, 2000, Maribor, Slovenia
- Condensed Matter Physics Department, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana, Slovenia
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6
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Zhang GH, Nelson DR. Fractional defect charges in liquid crystals with p-fold rotational symmetry on cones. Phys Rev E 2022; 105:054703. [PMID: 35706319 DOI: 10.1103/physreve.105.054703] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2022] [Accepted: 03/30/2022] [Indexed: 06/15/2023]
Abstract
Conical surfaces, with a δ function of Gaussian curvature at the apex, are perhaps the simplest example of geometric frustration. We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones. For free boundary conditions at the base, we find both the ground state(s) and a discrete ladder of metastable states as a function of both the cone angle and the liquid crystal symmetry p. We find that these states are characterized by a set of fractional defect charges at the apex and that the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction of the cone. We check our predictions for the ground-state energies numerically for a set of commensurate cone angles (corresponding to a set of commensurate Gaussian curvatures concentrated at the cone apex), whose surfaces can be polygonized as a perfect triangular or square mesh, and find excellent agreement with our theoretical predictions.
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Affiliation(s)
- Grace H Zhang
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - David R Nelson
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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7
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Yao X, Zhang L, Chen JZY. Defect patterns of two-dimensional nematic liquid crystals in confinement. Phys Rev E 2022; 105:044704. [PMID: 35590543 DOI: 10.1103/physreve.105.044704] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Accepted: 03/23/2022] [Indexed: 06/15/2023]
Abstract
A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a surface-confined system. When such a system is further confined by external line boundaries or excluded from internal line boundaries, the nematic directors form a deformed texture that may display defect points or defect lines, for which winding numbers can be clearly defined. Here, a particular attention is paid to the case when the liquid crystal molecules prefer to form a boundary nematic texture in parallel to the wall surface (i.e., following the homogeneous boundary condition). A general theory, based on geometric argument, is presented for the relationship between the sum of all winding numbers in the system (the total winding number) and the type of confinement angles and curved segments. The conclusion is validated by comparing the theoretical defect rule with existing nematic textures observed experimentally and theoretically in recent years.
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Affiliation(s)
- Xiaomei Yao
- Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
| | - Lei Zhang
- Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
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8
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Mesarec L, Drab M, Penič S, Kralj-Iglič V, Iglič A. On the Role of Curved Membrane Nanodomains, and Passive and Active Skeleton Forces in the Determination of Cell Shape and Membrane Budding. Int J Mol Sci 2021; 22:2348. [PMID: 33652934 PMCID: PMC7956631 DOI: 10.3390/ijms22052348] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2021] [Revised: 02/18/2021] [Accepted: 02/20/2021] [Indexed: 02/03/2023] Open
Abstract
Biological membranes are composed of isotropic and anisotropic curved nanodomains. Anisotropic membrane components, such as Bin/Amphiphysin/Rvs (BAR) superfamily protein domains, could trigger/facilitate the growth of membrane tubular protrusions, while isotropic curved nanodomains may induce undulated (necklace-like) membrane protrusions. We review the role of isotropic and anisotropic membrane nanodomains in stability of tubular and undulated membrane structures generated or stabilized by cyto- or membrane-skeleton. We also describe the theory of spontaneous self-assembly of isotropic curved membrane nanodomains and derive the critical concentration above which the spontaneous necklace-like membrane protrusion growth is favorable. We show that the actin cytoskeleton growth inside the vesicle or cell can change its equilibrium shape, induce higher degree of segregation of membrane nanodomains or even alter the average orientation angle of anisotropic nanodomains such as BAR domains. These effects may indicate whether the actin cytoskeleton role is only to stabilize membrane protrusions or to generate them by stretching the vesicle membrane. Furthermore, we demonstrate that by taking into account the in-plane orientational ordering of anisotropic membrane nanodomains, direct interactions between them and the extrinsic (deviatoric) curvature elasticity, it is possible to explain the experimentally observed stability of oblate (discocyte) shapes of red blood cells in a broad interval of cell reduced volume. Finally, we present results of numerical calculations and Monte-Carlo simulations which indicate that the active forces of membrane skeleton and cytoskeleton applied to plasma membrane may considerably influence cell shape and membrane budding.
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Affiliation(s)
- Luka Mesarec
- Faculty of Electrical Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia; (L.M.); (M.D.); (S.P.)
| | - Mitja Drab
- Faculty of Electrical Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia; (L.M.); (M.D.); (S.P.)
| | - Samo Penič
- Faculty of Electrical Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia; (L.M.); (M.D.); (S.P.)
| | - Veronika Kralj-Iglič
- Faculty of Health Sciences, University of Ljubljana, SI-1000 Ljubljana, Slovenia;
- Institute of Biosciences and Bioresources, National Research Council, 80131 Napoli, Italy
| | - Aleš Iglič
- Faculty of Electrical Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia; (L.M.); (M.D.); (S.P.)
- Institute of Biosciences and Bioresources, National Research Council, 80131 Napoli, Italy
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9
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Wand CR, Bates MA. Chiral nematic liquid crystals in torus-shaped and cylindrical cavities. Phys Rev E 2019; 100:052702. [PMID: 31869937 DOI: 10.1103/physreve.100.052702] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Indexed: 11/07/2022]
Abstract
We present a Monte Carlo simulation study of chiral nematic liquid crystals confined in torus-shaped and cylindrical cavities. For an achiral nematic with planar degenerate anchoring confined to a toroidal or cylindrical cavity, the ground state is defect free, with an untwisted director field. As chirality is introduced, the ground state remains defect free but the director field becomes twisted within the cavity. For homeotropic anchoring, the ground state for an achiral nematic within a toroidal cavity consists of two disclination rings, one large and one small, that follow the major circumference of the torus. As chirality is introduced and increased, this ground state becomes unstable with respect to twisted configurations. The closed nature of the toroidal cavity requires that only a half integer number of twists can be formed and this leads to the ground state being either a single disclination line that encircles the torus twice or a pair of intertwined disclination rings forming stable, knotted defect structures.
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Affiliation(s)
- Charlie R Wand
- Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom
| | - Martin A Bates
- Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom
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10
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Mesarec L, Góźdź W, Iglič A, Kralj-Iglič V, Virga EG, Kralj S. Normal red blood cells' shape stabilized by membrane's in-plane ordering. Sci Rep 2019; 9:19742. [PMID: 31875042 PMCID: PMC6930264 DOI: 10.1038/s41598-019-56128-0] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2019] [Accepted: 12/04/2019] [Indexed: 11/19/2022] Open
Abstract
Red blood cells (RBCs) are present in almost all vertebrates and their main function is to transport oxygen to the body tissues. RBCs' shape plays a significant role in their functionality. In almost all mammals in normal conditions, RBCs adopt a disk-like (discocyte) shape, which optimizes their flow properties in vessels and capillaries. Experimentally measured values of the reduced volume (v) of stable discocyte shapes range in a relatively broad window between v ~ 0.58 and 0.8. However, these observations are not supported by existing theoretical membrane-shape models, which predict that discocytic RBC shape is stable only in a very narrow interval of v values, ranging between v ~ 0.59 and 0.65. In this study, we demonstrate that this interval is broadened if a membrane's in-plane ordering is taken into account. We model RBC structures by using a hybrid Helfrich-Landau mesoscopic approach. We show that an extrinsic (deviatoric) curvature free energy term stabilizes the RBC discocyte shapes. In particular, we show on symmetry grounds that the role of extrinsic curvature is anomalously increased just below the nematic in-plane order-disorder phase transition temperature.
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Affiliation(s)
- L Mesarec
- Laboratory of Biophysics, Faculty of Electrical Engineering, University of Ljubljana, 1000, Ljubljana, Slovenia
| | - W Góźdź
- Institute of Physical Chemistry, Polish Academy of Sciences, 01-224, Warsaw, Poland
| | - A Iglič
- Laboratory of Biophysics, Faculty of Electrical Engineering, University of Ljubljana, 1000, Ljubljana, Slovenia
- Laboratory of Mass Spectrometry and Proteomics, Institute of Biosciences and BioResources, National Research Council of Italy, Napoli, 80132, Italy
| | - V Kralj-Iglič
- Laboratory of Mass Spectrometry and Proteomics, Institute of Biosciences and BioResources, National Research Council of Italy, Napoli, 80132, Italy
- Laboratory of Clinical Biophysics, Faculty of Health Sciences, University of Ljubljana, 1000, Ljubljana, Slovenia
- Laboratory of Clinical Biophysics, Faculty of Medicine, University of Ljubljana, 1000, Ljubljana, Slovenia
| | - E G Virga
- Department of Mathematics, University of Pavia, Via Ferrata 5, 27100, Pavia, Italy
| | - S Kralj
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, 2000, Maribor, Slovenia.
- Condensed Matter Physics Department, Jožef Stefan Institute, 1000, Ljubljana, Slovenia.
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11
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Mirantsev LV, Sonnet AM, Virga EG. Lifting ordered surfaces: Ellipsoidal nematic shells. Phys Rev E 2018; 98:012701. [PMID: 30110838 DOI: 10.1103/physreve.98.012701] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2018] [Indexed: 11/07/2022]
Abstract
When a material surface is functionalized so as to acquire some type of order, functionalization of which soft condensed matter systems have recently provided many interesting examples, the modeler faces an alternative. Either the order is described on the curved, physical surface where it belongs, or it is described on a flat surface that is unrolled as preimage of the physical surface under a suitable height function. This paper applies a general method that pursues the latter avenue by lifting whatever order tensor is deemed appropriate from a flat to a curved surface. We specialize this method to nematic shells, for which it also provides a simple but perhaps convincing interpretation of the outcomes of some molecular dynamics experiments on ellipsoidal shells.
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Affiliation(s)
- Leonid V Mirantsev
- Institute of the Problems of Mechanical Engineering, Academy of Sciences of Russia, St. Petersburg 199178, Russia
| | - André M Sonnet
- Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland, United Kingdom
| | - Epifanio G Virga
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
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12
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Nitschke I, Nestler M, Praetorius S, Löwen H, Voigt A. Nematic liquid crystals on curved surfaces: a thin film limit. Proc Math Phys Eng Sci 2018; 474:20170686. [PMID: 29977124 DOI: 10.1098/rspa.2017.0686] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2017] [Accepted: 05/16/2018] [Indexed: 11/12/2022] Open
Abstract
We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process, we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. The main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.
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Affiliation(s)
- Ingo Nitschke
- Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany
| | - Michael Nestler
- Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany
| | - Simon Praetorius
- Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany
| | - Hartmut Löwen
- Institut für Theoretische Physik II - Soft Matter, Heinrich-Heine- Universität Düsseldorf, 40225 Düsseldorf, Germany
| | - Axel Voigt
- Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany.,Dresden Center for Computational Materials Science (DCMS), 01062 Dresden, Germany
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13
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Abstract
When nanoparticle self-assembly takes place in an anisotropic liquid crystal environment, fascinating new effects can arise. The presence of elastic anisotropy and topological defects can direct spatial organization. An important goal in nanoscience is to direct the assembly of nanoparticles over large length scales to produce macroscopic composite materials; however, limitations on spatial ordering exist due to the inherent disorder of fluid-based methods. In this paper we demonstrate the formation of quantum dot clusters and spherical capsules suspended within spherical liquid crystal droplets as a method to position nanoparticle clusters at defined locations. Our experiments demonstrate that particle sorting at the isotropic–nematic phase front can dominate over topological defect-based assembly. Notably, we find that assembly at the nematic phase front can force nanoparticle clustering at energetically unfavorable locations in the droplets to form stable hollow capsules and fractal clusters at the droplet centers.
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14
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Gómez LR, García NA, Vega DA, Lorenzana J. Thermal properties of vortices on curved surfaces. Phys Rev E 2018; 97:012117. [PMID: 29448486 DOI: 10.1103/physreve.97.012117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2017] [Indexed: 01/08/2023]
Abstract
We use Monte Carlo simulations to study the finite temperature behavior of vortices in the XY model for tangent vector order on curved backgrounds. Contrary to naive expectations, we show that the underlying geometry does not affect the proliferation of vortices with temperature respect to what is observed on a flat surface. Long-range order in these systems is analyzed by using two-point correlation functions. As expected, in the case of slightly curved substrates these correlations behave similarly to the plane. However, for high curvatures, the presence of geometry-induced unbounded vortices at low temperatures produces the rapid decay of correlations and an apparent lack of long-range order. Our results shed light on the finite-temperature physics of soft-matter systems and anisotropic magnets deposited on curved substrates.
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Affiliation(s)
- Leopoldo R Gómez
- Instituto de Física del Sur (IFISUR), Departamento de Física, Universidad Nacional del Sur (UNS), CONICET, Av. L. N. Alem 1253, B8000CPB-Bahía Blanca, Argentina
| | - Nicolás A García
- Institut Laue-Langevin, 71 Avenue des Martyrs, 3842 Grenoble, France
| | - Daniel A Vega
- Instituto de Física del Sur (IFISUR), Departamento de Física, Universidad Nacional del Sur (UNS), CONICET, Av. L. N. Alem 1253, B8000CPB-Bahía Blanca, Argentina
| | - José Lorenzana
- ISC-CNR, Dipartimento di Fisica, Università di Roma "La Sapienza", Piazzale Aldo Moro 2, 00185 Roma, Italy
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15
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Sonnet AM, Virga EG. Bistable curvature potential at hyperbolic points of nematic shells. SOFT MATTER 2017; 13:6792-6802. [PMID: 28828443 DOI: 10.1039/c7sm01216k] [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
Nematic shells are colloidal particles coated with nematic liquid crystal molecules which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. We describe the nematic order on a shell by a unit director field on an orientable surface. Equilibrium fields can then be found by minimising the elastic energy, which in general is a function of the surface gradient of the director field. We learn how to extract systematically out of this energy a fossil component, related only to the surface and its curvatures, which expresses a curvature potential for the molecular torque. At hyperbolic points on the colloid's surface, and only there, the alignment preferred by the curvature potential may fail to be a direction of principal curvature. There the fossil energy becomes bistable.
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Affiliation(s)
- André M Sonnet
- Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland, UK.
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Travesset A. Topological structure prediction in binary nanoparticle superlattices. SOFT MATTER 2016; 13:147-157. [PMID: 27156535 DOI: 10.1039/c6sm00713a] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
Systems of spherical nanoparticles with capping ligands have been shown to self-assemble into beautiful superlattices of fascinating structure and complexity. In this paper, I show that the spherical geometry of the nanoparticle imposes constraints on the nature of the topological defects associated with the capping ligand and that such topological defects control the structure and stability of the superlattices that can be assembled. All these considerations form the basis for the orbifold topological model (OTM) described in this paper. The model quantitatively predicts the structure of super-lattices where capping ligands are hydrocarbon chains in excellent agreement with experimental results, explains the appearance of low packing fraction lattices as equilibrium, why certain similar structures are more stable (bccAB6vs. CaB6, AuCu vs. CsCl, etc.) and many other experimental observations.
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Affiliation(s)
- A Travesset
- Iowa State University and Ames lab, Department of Physics and Astronomy, Ames, USA.
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Ye S, Zhang P, Chen JZY. Nematic ordering of semiflexible polymers confined on a toroidal surface. SOFT MATTER 2016; 12:5438-5449. [PMID: 27248049 DOI: 10.1039/c6sm01098a] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We study the isotropic-like and nematic states of wormlike liquid-crystal polymers embedded on the surface of a torus. The role played by surface curvature, which couples to the molecular rigidity, is reported as the main reason that causes the weak nematic ordering in an otherwise ordinary isotropic phase. The same coupling has a profound effect on the nematic states as well, which are stabilized by the Onsager excluded-volume interaction; the latter has been frequently used to study lyotropic liquid crystal polymers and is used here as an example of the physical mechanisms that drive the system to make orientational ordering. We identify important parameters in the system which are used as axes of the four-dimensional phase diagram. The numerical study demonstrates a strong correlation between the liquid-crystal defect-free and defect structures and the geometry of the liquid-crystal embedded surface.
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Affiliation(s)
- Shiwei Ye
- LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People's Republic of China.
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Mesarec L, Góźdź W, Iglič A, Kralj S. Effective Topological Charge Cancelation Mechanism. Sci Rep 2016; 6:27117. [PMID: 27250777 PMCID: PMC4890433 DOI: 10.1038/srep27117] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2016] [Accepted: 05/13/2016] [Indexed: 11/18/2022] Open
Abstract
Topological defects (TDs) appear almost unavoidably in continuous symmetry breaking phase transitions. The topological origin makes their key features independent of systems’ microscopic details; therefore TDs display many universalities. Because of their strong impact on numerous material properties and their significant role in several technological applications it is of strong interest to find simple and robust mechanisms controlling the positioning and local number of TDs. We present a numerical study of TDs within effectively two dimensional closed soft films exhibiting in-plane orientational ordering. Popular examples of such class of systems are liquid crystalline shells and various biological membranes. We introduce the Effective Topological Charge Cancellation mechanism controlling localised positional assembling tendency of TDs and the formation of pairs {defect, antidefect} on curved surfaces and/or presence of relevant “impurities” (e.g. nanoparticles). For this purpose, we define an effective topological charge Δmeff consisting of real, virtual and smeared curvature topological charges within a surface patch Δς identified by the typical spatially averaged local Gaussian curvature K. We demonstrate a strong tendency enforcing Δmeff → 0 on surfaces composed of Δς exhibiting significantly different values of spatially averaged K. For Δmeff ≠ 0 we estimate a critical depinning threshold to form pairs {defect, antidefect} using the electrostatic analogy.
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Affiliation(s)
- Luka Mesarec
- Laboratory of Biophysics, Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia
| | - Wojciech Góźdź
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland
| | - Aleš Iglič
- Laboratory of Biophysics, Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia
| | - Samo Kralj
- Condensed Matter Physics Department, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia.,Jožef Stefan International Postgraduate School, Jamova 39, 1000 Ljubljana, Slovenia.,Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, 2000 Maribor, Slovenia
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de Oliveira EJL, de Oliveira IN, Lyra ML, Mirantsev LV. Tunable topological valence in nematic shells on spherocylindrical colloidal particles. Phys Rev E 2016; 93:012703. [PMID: 26871131 DOI: 10.1103/physreve.93.012703] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2015] [Indexed: 06/05/2023]
Abstract
We perform molecular dynamics simulations of the orientational ordering on nematic shells delimited by spherocylindrical nanoscopic colloidal particles. We show that under conditions of degenerate planar anchoring, the equilibrium director field structure in these shells exhibits pairs of +1/2 topological defects at the poles of spherical cups in the absence of an external electric field. In addition, a certain number of pairs of ±1/2 defects occurs on the spherical cups far from the poles, thus resulting in a total of eight valence spots. A strong field applied along the main spherocylindrical axis removes the ±1/2 defect pairs while it coalesces the polar ones into a single +1 topological defect. A strong transverse field destroys all defects on the spherical cups but generates four +1/2 defects in the cylindrical part. Therefore, an external field can be used to control the number of valence centers in spherocylindrical nematic shells, thus unveiling their capability of acting as multivalent building blocks for nanophotonic devices.
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Affiliation(s)
- E J L de Oliveira
- Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió-Alagoas, Brazil
| | - I N de Oliveira
- Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió-Alagoas, Brazil
| | - M L Lyra
- Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió-Alagoas, Brazil
| | - L V Mirantsev
- Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió-Alagoas, Brazil
- Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia
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Jesenek D, Kralj S, Rosso R, Virga EG. Defect unbinding on a toroidal nematic shell. SOFT MATTER 2015; 11:2434-2444. [PMID: 25662487 DOI: 10.1039/c4sm02540g] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We study nematic liquid crystal textures exhibiting topological defects (TDs) on a two-dimensional (2D) toroidal shell. For the toroidal topology the total topological charge of TDs is equal to zero. We use a mesoscopic Landau-de Gennes approach which features a 2D nematic order tensor Q. We show that fat tori unbind TDs. If no extrinsic free energy couples Q with the Weingarten tensor of the torus, then defects and antidefects are assembled along the innermost and the outermost circles of the torus, respectively. In this case, we estimate the critical condition for the onset of TDs using an electrostatic analogy. If, on the other hand, an extrinsic free energy is present, then defects are repelled from these regions.
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Affiliation(s)
- Dalija Jesenek
- Condensed Matter Physics Department, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia.
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Liang Q, Ye S, Zhang P, Chen JZY. Rigid linear particles confined on a spherical surface: Phase diagram of nematic defect states. J Chem Phys 2014; 141:244901. [DOI: 10.1063/1.4903995] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Segatti A, Snarski M, Veneroni M. Equilibrium configurations of nematic liquid crystals on a torus. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012501. [PMID: 25122318 DOI: 10.1103/physreve.90.012501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2014] [Indexed: 06/03/2023]
Abstract
The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy and how new configurations emerge. In particular, our analysis predicts the existence of stable equilibria with complex windings.
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Affiliation(s)
- Antonio Segatti
- Dipartimento di Matematica "F. Casorati", Università di Pavia, Pavia, Italy
| | | | - Marco Veneroni
- Dipartimento di Matematica "F. Casorati", Università di Pavia, Pavia, Italy
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Koning V, van Zuiden BC, Kamien RD, Vitelli V. Saddle-splay screening and chiral symmetry breaking in toroidal nematics. SOFT MATTER 2014; 10:4192-4198. [PMID: 24780941 DOI: 10.1039/c4sm00076e] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We present a theoretical study of director fields in toroidal geometries with degenerate planar boundary conditions. We find spontaneous chirality: despite the achiral nature of nematics the director configuration shows a handedness if the toroid is thick enough. In the chiral state the director field displays a double twist, whereas in the achiral state there is only bend deformation. The critical thickness increases as the difference between the twist and saddle-splay moduli grows. A positive saddle-splay modulus prefers alignment along the meridian of the bounding torus, and hence promotes a chiral configuration. The chiral-achiral transition mimics the order-disorder transition of the mean-field Ising model. The role of the magnetisation in the Ising model is played by the degree of twist. The role of the temperature is played by the aspect ratio of the torus. Remarkably, an external field does not break the chiral symmetry explicitly, but shifts the transition. In the case of toroidal cholesterics, we do find a preference for one chirality over the other - the molecular chirality acts as a field in the Ising analogy.
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Affiliation(s)
- Vinzenz Koning
- Instituut-Lorentz, Universiteit Leiden, Postbus 9506, 2300 RA Leiden, The Netherlands.
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Mbanga BL, Voorhes KK, Atherton TJ. Simulating defect textures on relaxing nematic shells. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052504. [PMID: 25353812 DOI: 10.1103/physreve.89.052504] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/06/2014] [Indexed: 06/04/2023]
Abstract
Two nematic shells brought in contact coalesce in order to reduce their combined interfacial tension, and, following this topological transition, relax to an equilibrium state. In this work, we study the defect textures as the combined shell shape evolves. By varying the sizes of the shells, we perform a quasistatic investigation of the director field and the defect valence on the doublet. Regimes are found where positive and negative defects exist due to the large negative Gaussian curvature at the neck. Using large-scale computer simulations, we determine how annihilating defect pairs on coalescing shells are selected and the stage of coalescence at which annihilation occurs.
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Affiliation(s)
- Badel L Mbanga
- Department of Physics and Astronomy, Center for Nanoscopic Physics, Tufts University, 4 Colby St., Medford, Massachusetts 02155, USA
| | - Kate K Voorhes
- Department of Physics and Astronomy, Center for Nanoscopic Physics, Tufts University, 4 Colby St., Medford, Massachusetts 02155, USA
| | - Timothy J Atherton
- Department of Physics and Astronomy, Center for Nanoscopic Physics, Tufts University, 4 Colby St., Medford, Massachusetts 02155, USA
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Li Y, Miao H, Ma H, Chen JZY. Defect-free states and disclinations in toroidal nematics. RSC Adv 2014. [DOI: 10.1039/c4ra04441j] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
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Morphology transition in lipid vesicles due to in-plane order and topological defects. Proc Natl Acad Sci U S A 2013; 110:3242-7. [PMID: 23401499 DOI: 10.1073/pnas.1213994110] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Complex morphologies in lipid membranes typically arise due to chemical heterogeneity, but in the tilted gel phase, complex shapes can form spontaneously even in a membrane containing only a single lipid component. We explore this phenomenon via experiments and coarse-grained simulations on giant unilamellar vesicles of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine. When cooled from the untilted L(α) liquid-crystalline phase into the tilted gel phase, vesicles deform from smooth spheres to disordered, highly crumpled shapes. We propose that this shape evolution is driven by nucleation of complex membrane microstructure with topological defects in the tilt orientation that induce nonuniform membrane curvature. Coarse-grained simulations demonstrate this mechanism and show that kinetic competition between curvature change and defect motion can trap vesicles in deeply metastable, defect-rich structures.
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Mirantsev LV, Sonnet AM, Virga EG. Geodesic defect anchoring on nematic shells. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:020703. [PMID: 23005713 DOI: 10.1103/physreve.86.020703] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2012] [Revised: 06/23/2012] [Indexed: 06/01/2023]
Abstract
Nematic shells are colloidal particles coated with nematic liquid crystal molecules, which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. Molecular dynamics simulations on a nanoscopic spherical shell indicate that under appropriate adhesion conditions for the molecules on the equator, the equilibrium nematic texture exhibits at each pole a pair of +1/2 defects so close to one another to be treated as one +1 defect. Spirals connect the polar defects, though the continuum limit of the interaction potential would not feature any elastic anisotropy. A molecular averaging justifies an anchoring defect energy that feels the geodesics emanating from the defect. All our observations are explained by such a geodesic anchoring, which vanishes on flat manifolds.
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Affiliation(s)
- Leonid V Mirantsev
- Institute of the Problems of Mechanical Engineering, Academy of Sciences of Russia, St Petersburg 199178, Russia
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Dhakal S, Solis FJ, Olvera de la Cruz M. Nematic liquid crystals on spherical surfaces: control of defect configurations by temperature, density, and rod shape. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:011709. [PMID: 23005439 DOI: 10.1103/physreve.86.011709] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2011] [Revised: 05/30/2012] [Indexed: 06/01/2023]
Abstract
Recent experiments have shown that defect conformations in spherical nematic liquid crystals can be controlled through variations of temperature, shell thickness, and other environmental parameters. These modifications can be understood as a result of the induced changes in the effective elastic constants of the system. To characterize the relation between defect conformations and elastic anisotropy, we carry out Monte Carlo simulations of a nematic on a spherical surface. As the anisotropy is increased, the defects flow from a tetrahedral arrangement to two coalescing pairs and then to a great circle configuration. We also analyze this flow using a variational method based on harmonic configurations.
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Affiliation(s)
- Subas Dhakal
- Department of Materials Science, Northwestern University, Evanston, Illinois 60208, USA
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