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On the Stability of Archimedean Tilings Formed by Patchy Particles

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Home Search Collections Journals About Contact us My IOPscience On the stability of Archimedean tilings formed by patchy particles This content has been downloaded from IOPscience. Please scroll down to see the full text. 2011 J. Phys.: Condens. Matter 23 404206 (http://iopscience.iop.org/0953-8984/23/40/404206) View the table of contents for this issue, or go to the journal homepage for more Download details: This content was downloaded by: kukitxu IP Address: 128.131.48.66 This content was downloaded on 28/01/2014 at 16:48 Please note that terms and conditions apply. IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 23 (2011) 404206 (8pp) doi:10.1088/0953-8984/23/40/404206 On the stability of Archimedean tilings formed by patchy particles Moritz Antlanger1,Gunther¨ Doppelbauer1 and Gerhard Kahl Institut fur¨ Theoretische Physik and Centre for Computational Materials Science (CMS), Technische Universitat¨ Wien, Wien, Austria E-mail: [email protected] Received 28 July 2011, in final form 30 August 2011 Published 19 September 2011 Online at stacks.iop.org/JPhysCM/23/404206 Abstract We have investigated the possibility of decorating, using a bottom-up strategy, patchy particles in such a way that they self-assemble in (two-dimensional) Archimedean tilings. Except for the trihexagonal tiling, we have identified conditions under which this is indeed possible. The more compact tilings, i.e., the elongated triangular and the snub square tilings (which are built up by triangles and squares only) are found to be stable up to intermediate pressure values in the vertex representation, i.e., where the tiling is decorated with particles at its vertices. The other tilings, which are built up by rather large hexagons, octagons and dodecagons, are stable over a relatively large pressure range in the centre representation where the particles occupy the centres of the polygonal units. S Online supplementary data available from stacks.iop.org/JPhysCM/23/404206/mmedia (Some figures in this article are in colour only in the electronic version) 1. Introduction establish bonds with other particles in a highly selective fashion. As a consequence of their strongly anisotropic Is it possible to design colloidal particles so that interactions and their selectivity in the formation of bonds, they self-assemble into a desired target structure? Recent patchy particles are considered to be very promising building experiments [1,2] have impressively demonstrated that such blocks for larger entities on a mesoscopic scale [5]. a bottom-up strategy can indeed be successfully realized. To Overviews of recent progress in experimental and theoretical be more specific, Chen et al designed and fabricated so-called investigations on patchy particles can be found in [6] and [7], patchy colloidal particles, carrying hydrophobic caps on the respectively. two opposite poles, as reported in [1]. Under suitable external In the present contribution we deal with the question conditions, the colloids did indeed self-organize in the of whether—via a bottom-up strategy similar to the one targeted two-dimensional kagome lattice. Simulations [3,4] mentioned above—patchy particles can be designed in such complemented these experimental observations and provided a way as to self-assemble in even more complex ordered in addition exhaustive information about the phase behaviour two-dimensional particle arrangements. As target structures of the system in terms of a temperature versus density phase we have chosen Archimedean tilings. The eight Archimedean diagram. (or semiregular) tilings are presumably known and have The term ‘patchy particles’ stands for a particular class been used since antiquity. They are characterized by an of colloidal particles whose surfaces have been treated with edge-to-edge tiling of the plane with at least two regular suitable physical or chemical methods, creating thereby polygons such that all vertices are of the same type [8]. The regions which differ in their interaction behaviour from polygons involved are the equilateral triangle, the square, the untreated, naked surface of the particle. Via those the regular hexagon, the regular octagon, and the regular well-defined regions (‘patches’) the particles are able to dodecagon. For completeness, we have also included the three related edge-to-edge tilings of the plane formed by a 1 Authors who contributed equally to this work. single regular polygon in our investigations; these tilings are C08$33.00 10953-8984/11/404206 c 2011 IOP Publishing Ltd Printed in the UK & the USA J. Phys.: Condens. Matter 23 (2011) 404206 M Antlanger et al Table 1. List of the three Platonic and eight Archimedean tilings experimental realization. However, this representation has (first column) and their vertex specifications (second column). In the serious problems in minimizing the volume contribution to two other columns parameters are collected that specify patchy the thermodynamic potential. In particular one encounters particles for the vertex representation (see the text): N stands for the number of particles required per unit cell (cf figure1). The last problems for those tilings that are built up by polygonal column specifies the patch positions along the circumference via the units with six vertices or more. The centre representation, indicated angles. Note that in the vertex representation all particles on the other hand, requires considerably more complex have the same size; only for the truncated trihexagonal tiling are two unit cells and at least two particle species, characterized particle species of equal size but different chirality in their by a large size disparity and a relatively large number of decoration required. patches. These issues might represent a particular challenge Vertex representation in an experimental realization. This representation allows one, Tiling Vertices N Patch angles (deg) however, to stabilize tilings that are built up by hexagons, octagons, and dodecagons. 6 Triangular 3 1 f0; 60; 120; 180; 240; 300g Based on these considerations we have addressed the 4 f g Square 4 1 0; 90; 180; 270 following two questions. (i) Can patchy particles be suitably Hexagonal 63 2 f0; 120; 240g decorated to self-assemble in Archimedean tilings? (ii) How Elongated triangular 33:42 2 f0; 60; 120; 180; 270g Snub square 32:4:3:4 4 f0; 60; 120; 210; 270g stable are these self-assembly scenarios as the pressure Snub hexagonal 34:6 6 f0; 60; 120; 180; 240g increases? Working in the NPT ensemble, by construction Trihexagonal 3:6:3:6 3 f0; 60; 180; 240g the desired target structures are stable at vanishing pressure. Rhombitrihexagonal 3:4:6:4 6 f0; 60; 150; 270g However, as the system is exposed even to a tiny pressure, the Truncated square 4:82 4 f0; 90; 225g competition between the energy and the volume contributions 2 Truncated hexagonal 3:12 6 f0; 60; 210g to the thermodynamic potential sets in, leading possibly to f g Truncated trihexagonal 4:6:12 6 0; 90; 240 structures other than the desired target structure. As will be 4:12:6 6 f0; 90; 210g discussed in detail in the body of the paper (in particular in section3), all tilings, except for the trihexagonal tiling, can be stabilized at low, sometimes even at intermediate also known as regular or Platonic tilings [8]. In table1 we pressure values via either of the two representations. We have listed the resulting 11 tilings, introducing the commonly emphasize at this point that we only consider the case used nomenclature and specification via their vertices; further of vanishing temperature, T D 0. Thus from the results information compiled in this table will be discussed below. In presented in this contribution no conclusions can be drawn as figure1 we have depicted these planar tilings; their decoration regards which of the structures identified will survive at finite with particles will be addressed later and can therefore be temperature. ignored for the moment. The paper is organized as follows. In the next section At this point one might argue that the formation of we briefly summarize the characteristic features of our Archimedean tilings by patchy particles represents a rather model for patchy particles and of our optimization strategy academic problem. At least two examples can be put forward (referring the reader to the literature for more details to refute these objections: (i) experiments on colloids, exposed in both cases). Section3 is dedicated to a thorough to a force field induced by interfering laser beams, give discussion of the results. In the final section the results are evidence [9] of a self-assembly scenario of those particles into summarized. an Archimedean-like tiling; (ii) theoretical investigations [10] indicate that deformable spheres form Archimedean tilings in 2. The model and theoretical tools coexistence regions of the phase diagram. Finally, some of the more open Archimedean tilings might show some interesting 2.1. The model features in their phonon spectra. In an effort to demonstrate a successful bottom-up In the present contribution we have used a model for patchy strategy for forming the desired target structures via suitably particles proposed by Doye et al [11], suitably adapted to the decorated patchy particles, Platonic and Archimedean tilings two-dimensional case. In this model the interaction between are ideal candidates: all vertices are of the same type and all two particles is given by a pair potential V.rij; piα; pjβ /, the distances between two vertices are the same. These two discussed below; here rij D ri − rj is the vector between features make an appropriate ‘guess’ for the decoration of the particles i and j and the patch vector, piα, specifies the patch particles with patches rather easy. α of particle i. The explicit expression for V.rij; piα; pjβ / In realizing our bottom-up strategy there are essentially consists of a spherically symmetric Lennard-Jones potential two options for positioning the particles on the tilings: either modulated by an angular factor taking into account the relative at the vertices or at the centres of the polygons (to be orientation of the two interacting particles.
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