pubs.acs.org/Langmuir © 2009 American Chemical Society
Design of 2D Binary Colloidal Crystals in a Nematic Liquid Crystal ‡ U. Ognysta,† A. Nych,† V. Nazarenko,† M. Skarabot, and I. Musevic*,‡ †
Institute of Physics, 46 Nauky avenue, Kyiv 680028, Ukraine, and ‡J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Received May 14, 2009. Revised Manuscript Received July 22, 2009
In this paper, we examine directed self-assembly in a 2D binary system of dipolar and quadrupolar colloidal particles with normal surface boundary conditions, dispersed in the nematic liquid crystal. Using the laser tweezers, we assembled a large variety of stable 2D colloidal crystal structures. In all analyzed structures, the particles, their surface treatment and the cell conditions were the same, which gives us the ability to systematically follow the evolution of colloidal assembly when many particles are present. We present an analogy between molecular self-assembly and organization of colloidal microspheres in liquid crystalline medium to extend the strategy for designing colloidal crystalline structures of different level of complexity.
Introduction Colloids have long been used in applications such as paints, ink, coatings, foods and many manufacturing processes.1,2 In all these cases, the dispersed material is the important agent, and the colloidal dispersion simply facilitates its application. More recently, colloidal particles have been deliberately used to modify rheological or optical properties of their carrier fluids.3 For these applications, the material from which the colloidal particles are made is less important than their dispersal influence on the carrier medium. For example, spatial distribution of colloidal particles is in many cases an important issue.4 Many efforts have been devoted to assemble spatially periodic structures of dielectric microspheres from or within water colloidal dispersions using self-assembly or directed assembly. The reason for this interest is that regularly ordered arrays of colloidal dielectric particles can affect the propagation of light in a similar way as a semiconductor crystal affects the propagation of electrons.5 This could give rise to some spectacular optical phenomena and could be used to manufacture photonic bandgap materials for novel photonic devices. Similar to the band gap structure for electron propagation in semiconductor crystals, forbidden gaps are formed in photonic crystals for electromagnetic waves. In a photonic bandgap material, light of a selected frequency cannot propagate within the crystal in any direction regardless of its polarization, because there are no available electromagnetic eigenmodes. In other words, light of a frequency within the gap will be Bragg-reflected, such that electromagnetic standing waves are formed in all directions for all polarizations. Thus, a photonic band gap is the optical analogue of an electronic band gap in semiconductors. However, since Maxwell’s equations are scale-invariant, a structure can be designed to have a band gap at any desired wavelength. For example, if the gap is tuned to the wavelength of 1.3 or *To whom correspondence should be addressed. E-mail: igor.musevic@ ijs.si. (1) Russel, W.; Saville, D.; Schowalter, W. Colloidal Dispersions; Cambridge University Press:Cambridge, U.K., 1991. (2) Drzaic, P. Liquid Crystal Dispersions; World Scientific: 1995. (3) Grier, D. MRS Bull. 1998, 23, 21. (4) Murray, C. MRS Bull. 1998, 23, 33. (5) Joannopoulos, J.; Johnson, S.; Winn, J. Photonic Crystals: Molding the Flow of Light; Princeton University Press: 2008.
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1.5 μm, photonic crystals could be used in photonic-crystal fibers for long-distance telecommunication. In these fibers, total internal reflection which is used for ordinary light guiding fibers, is replaced by Bragg-reflection from the photonic-crystal core of the fiber, which can be air or even vacuum. It has been shown in a number of recent publications, that colloidal particles dispersed in nematic liquid crystals (NLCs) demonstrate a novel colloidal system with great advantages compared to water-based colloidal dispersions. The novelty is in the colloidal pair interaction, which is not van der Waals or electrostatic interaction, but is generated from the topological properties of the dispersion and orientational elasticity of the liquid crystalline host.6-23 Instead of electric charges, which are essential for the pair interaction in ordinary, water based colloids, the topological charges, originating from the order in NLCs are important in nematic colloids. The elasticity of the liquid crystalline matrix also leads to strong spatial anisotropy of the pair interaction. It is known that to satisfy global boundary conditions and normal boundary conditions on the surface of a colloidal particle, nematic liquid crystal creates configuration that exhibits either dipolar or quadrupolar symmetry of the NLC orientation (6) Lopatnikov, S. L.; Namiot, V. A. Sov. Phys. JETP 1978, 48, 180. (7) Ramaswamy, S.; Nityananda, R.; Raghunathan, V.; Prost, J. Mol. Cryst. Liq. Cryst. 1996, 288, 175. (8) Kuksenok, O. V.; Ruhwandl, R. W.; Shiyanovskii, S. V.; Terentjev, E. M. Phys. Rev. E 1996, 54, 5198. (9) Poulin, P.; Stark, H.; Lubensky, T. Science 1997, 275, 1770. (10) Poulin, P.; Weitz, D. Phys. Rev. E 1998, 57, 626. (11) Lubensky, T.; Pettey, D.; Currier, N.; Stark, H. Phys. Rev. E 1998, 57, 610. (12) Lev, B. I.; Tomchuk, P. M. Phys. Rev. E 1999, 59, 591. (13) Stark, H. Phys. Rep 2001, 351, 387. (14) Lev, B. I.; Chernyshuk, S. B.; Tomchuk, P. M.; Yokoyama, H. Phys. Rev. E 2002, 65, 021709. (15) Musevic, I.; Skarabot, M.; Tkalec, U.; Ravnik, M.; Zumer, S. Science 2006, 313, 954. (16) Pergamenshchik, V.; Uzunova, V. Eur. Phys. J. E 2007, 23, 161. (17) Yoon, D.; Choi, M.; Kim, Y.; Kim, M.; Lavrentovich, O.; Jung, H. Nat. Mater. 2007, 6, 866. (18) Ognysta, U.; Nych, A.; Nazarenko, V.; Musevic, I.; Skarabot, M.; Ravnik, M.; Zumer, S.; Poberaj, I.; Babic, D. Phys. Rev. Lett. 2008, 100, 217803. (19) Gu, Y.; Abbott, N. Phys. Rev. Lett. 2000, 85, 4719. (20) Stark, H. Phys. Rev. E 2002, 66, 32701. (21) Smalyukh, I.; Chernyshuk, S.; Lev, B.; Nych, A.; Ognysta, U.; Nazarenko, V.; Lavrentovich, O. Phys. Rev. Lett. 2004, 93, 117801. (22) Smalyukh, I.; Lavrentovich, O.; Kuzmin, A.; Kachynski, A.; Prasad, P. Phys. Rev. Lett. 2005, 95, 157801. (23) Nych, A.; Ognysta, U.; Pergamenshchik, V.; Lev, B.; Nazarenko, V.; Musevic, I.; Skarabot, M.; Lavrentovich, O. Phys. Rev. Lett. 2007, 98, 057801.
Published on Web 09/16/2009
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around the colloidal particles. A dipolar colloidal particle is accompanied with hyperbolic hedgehog point defect, thus being reminiscent to an electric dipole. Similarly, a quadrupolar colloidal particle is encircled with a disclination loop, called Saturn ring defect. In thin layers of a NLC, the elastic interaction between two particles is qualitatively similar to that between electrostatic dipoles and quadrupoles. It has recently been shown, that this electrostatic picture is not valid in real 3D environment.24 Since the elastic deformation created by colloidal particle extends on much larger distances compared to the size of the colloidal particle, the colloidal pair interaction is of long-range and extends practically as far as tens of micrometers from the micrometer-sized particle. This opens new and exciting routes to nontrivial self-assembly of colloidal particles in anisotropic and complex solvents,15,18,25-27 which cannot be realized in any known colloidal system. In contrast to the ordinary colloidal systems, where the interaction has the radial symmetry only and the resultant structure has rather cubic or hexagonal ordering, the anisotropic solvent may result in a structure with practically any level of symmetry, that is indispensable for modern photonics applications ranging from diffraction gratings and beam steering devices to sophisticated photonic structures. In this paper, we suggest that the elastic interactions between colloidal inclusions in the liquid crystal matrix can be viewed as colloidal “bonds”, similar to chemical bonds between real atoms in solids. First, every colloidal particle deforms liquid crystal director field in a specific manner, spreading the deformation over long, macroscopic scale. Second, colloidal particles interact or “feel” each other when the surrounding and deformed regions of the nematic start to overlap.7,11,14 Thus, like atomic bonds, which are formed by overlapping electron orbitals, colloidal bonds in our systems are formed by overlapping field of elastic deformation of liquid crystal. Since NLC-mediated elastic interaction is strongly anisotropic, the colloidal bonds have their own characteristic energies together with well-defined angular profiles. The type and amplitude of the elastic deformation around one particle together with the number and type of accompanying topological defects determine the character and strength of the interaction with another inclusions. The same applies to atomic bonds, which are dependent on spatial shape of corresponding electron orbitals: their signs and amplitudes determine their ability to form a bond. Although the two kinds of bonding have different nature, this still allows us to draw an analogy between them, as it was successively used for capillary interaction.28 The purpose of this paper is to examine directed self-assembly in a 2D binary system of dipolar and quadrupolar colloidal particles with normal surface boundary conditions dispersed in the nematic liquid crystal. By means of laser tweezers setup and utilizing analogy to bonds between atoms, we have realized a large variety of different 2D colloidal crystal structures and present a general strategy and design rules of their assembling.
Experimental Section Binary mixtures of dipolar and quadrupolar colloidal particles were prepared following the procedure described in ref 18. We applied a monolayer of N,N-dimethyl-N-octadecyl-3-aminopropyl trimethoxysilyl chloride (DMOAP) to the surface of 4 μm (24) Pergamenshchik, V.; Uzunova, V. Phys. Rev. E 2009, 79, 021704. (25) Nazarenko, V.; Nych, A.; Lev, B. Phys. Rev. Lett. 2001, 87, 075504. (26) Skarabot, M.; Ravnik, M.; Zumer, S.; Tkalec, U.; Poberaj, I.; Babic, D.; Osterman, N.; Musevic, I. Phys. Rev. E 2007, 76, 51406. (27) Skarabot, M.; Ravnik, M.; Zumer, S.; Tkalec, U.; Poberaj, I.; Babic, D.; Osterman, N.; Musevic, I. Phys. Rev. E 2008, 77, 31705. (28) Bowden, N.; Choi, I.; Grzybowski, B.; Whitesides, G. J. Am. Chem. Soc. 1999, 121, 5373.
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colloidal particles (LICRISTAR 40, Merck GmbH) to have strong normal surface anchoring of liquid crystal (LC) molecules. Surface treated colloidal particles were mixed with pentylcyanobiphenyl (5CB) nematic liquid crystal (Merck GmbH). Measuring cells were prepared by using cleaned indium-tin-oxide (ITO) coated glass plates covered with a thin layer of polyimide, which was unidirectionally rubbed to obtain strong planar anchoring of LC molecules on the surfaces. Then the glasses were assembled into small-angle wedge cell and filled with colloidal dispersion. The wedge angle was small, producing ∼0.04 μm thickness change over 100 μm field of view, which allowed us to neglect the effects of the wedge. We have used optical tweezers setup to manipulate colloidal particles and assemble 2D colloidal structures in 5CB.29 The tweezers setup was built around an inverted microscope (Nikon Eclipse, TE2000-U) with Argon laser (Coherent, Innova 90C) at 514 nm as a laser source. A pair of acousto-optic deflectors (AOD), driven by a computerized system (Aresis, Tweez 70) was used for trap manipulation. A beam expander was used to match the laser beam to the AOD apertures and some additional optics was used to image the pivotal point of the AOD onto the entrance pupil of the water immersion microscope objective (Nikon, NIR Apo 60/1.0W). All colloidal structures were prepared in the region with cell thickness ∼5.5-6.5 μm, where the probabilities for a 4 μm DMOAP coated LICRISTAR colloidal particle to adopt either dipolar or quadrupolar director configuration are approximately equal.18,27 In our experiments, it was possible to switch director configuration around single particle from dipolar to quadrupolar or back by means of localized heating using high power laser beam and subsequent quenching, similar to experiments described in ref 27.
Results and Discussion NLC-mediated elastic interaction between pair of colloidal particles can be expanded to multipole series using well-known solutions from the electrostatics.7,11 With analogy to the electrostatics, the individual terms of this expansion are called dipoledipole (DD), dipole-quadrupole (DQ), and quadrupolequadrupole (QQ) interaction, respectively.12,14 It has been shown on pure dipolar and pure quadrupolar nematic colloids, that dipolar and quadrupolar interactions lead to the formation of stable 2D dipolar26 and quadrupolar27 colloidal crystalline lattices, respectively. In our recent experiments on a mixture of dipolar and quadrupolar nematic colloids,18 we have shown that the symmetry of the dipole-quadrupole colloidal interaction leads to a novel variety of 2D nematic “binary” colloidal crystals. The distinguishing feature of these NLC-mediated elastic interactions is that they are highly anisotropic, that is, they depend not only on the distance between pair of colloidal particles but also on their orientation relative to undisturbed NLC director field. Figure 1 presents the angular dependence of the dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole colloidal interactions, which are schematically presented with particles placed along the strongest attraction direction. It should be noted that we are dealing with two-dimensional systems, and Figure 1 represents only a part of all possible scenarios. To obtain threedimensional attraction-repulsion diagram one has to rotate the corresponding 2D diagram around the director orientation, turning all 2D sectors into 3D cones. In our experiments, we were able to create a NLC colloidal system, in which particles demonstrated all three main kinds of elastic interactions, predicted in Figure 1. Experimental observations are shown on microscopic images in Figure 2, where (29) Musevic, I.; Skarabot, M.; Babic, D.; Osterman, N.; Poberaj, I.; Nazarenko, V.; Nych, A. Phys. Rev. Lett. 2004, 93, 187801.
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equilibrium configurations of a small cluster of topologically equal or different colloidal particles are shown: (a,b) pure dipole-dipole,(c-h) mixed dipole-quadrupole, and (i) pure quadrupole-quadrupole clusters. This observation that some colloidal particle can attract and hold another particle at some specific distance and at some specific angle, is a key evidence that justifies our analogy to real atomic systems one can observe in chemistry. Usually, a selected atom, such as carbon, can create finite number of bonds to another atom like, for example, hydrogen. A number of these bonds, their spatial distribution and strength are determined by electron orbitals of the atoms involved. The behavior of colloidal spheres dispersed in NLC system is very much the same and can also be regarded as a formation of intercolloidal bonds. Like bonds between real atoms, intercolloidal bonds also have their specific spatial distribution that changes depending on what kind of particles are involved in the “reaction”. For example, a dipolar particle can attract another dipolar particle with parallel dipolar moment only along the NLC
Figure 1. Schematic representation of angular dependence of colloidal pair interactions, as reconstructed from11,15,18,27 (a) dipole-dipole interaction, (b) dipole-quadrupole, and (c) quadrupole-quadrupole interaction terms. The black dots are the hyperbolic hedgehog defects, and the black rings are “Saturn ring” defects. The central particle is the reference one and remaining particles are placed along the direction of the strongest attraction to the reference particle. In the red-shaded sectors, pairwise interaction is repulsive.
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director field either from right or from left side, as is shown in Figure 2a. When dipolar moment of another particle is antiparallel to that of the reference particle, one observes situation presented in Figure 2b. Pair of dipolar and quadrupolar colloids can form one of the three configurations presented in Figure 2c-e, two of which are equivalent (Figure 2d, e). By adding more particles one can create more complex aggregates Figure 2f-h. Pair of quadrupolar particles in uniform director field is oriented at some specific angle with respect to the director that leads to the symmetrical configuration shown in Figure 2i. These intercolloidal bonds differ not only in symmetry and spatial distribution but also in their strength. Our measurements showed that binding energy of dipole-quadrupole pair (of particle diameter 4 μm) in configurations presented in Figure 2d and e was about ∼3000 kBT, while for configuration depicted in Figure 2c, the binding energy was about ∼6000 kBT.18 Dipole-dipole pair has the largest binding energy of about ∼10 000 kBT15 and quadrupole-quadrupole pair has the weakest binding energy of about ∼800 kBT,27 all measured for 2.7 μm diameter colloidal particles. Colloidal crystals, assembled from mixed dipolar and quadrupolar particles, are considered robust from our experience, if they do not contain quadrupole-quadrupole bonds. Such crystals are stable in cells for a very long time and can also be manipulated as a whole, using laser tweezers. Another similarity between our colloidal systems and real atomic systems lies in their ability to form multiple structures or “substances”. Generally, atoms of some specific kind can form limited number of substances like, for example, oxygen atoms can form gas oxygen and ozone or carbon atoms that can form diamond, graphite, etc. When atoms of other kind are added, it results usually in large variety of chemical substances with different properties. For example, carbohydrates can be treated as a “mixture” of carbon and hydrogen atoms. We have a similar situation in our NLC colloidal systems. Having only dipolar or quadrupolar colloids, one can realize just two 2D colloidal crystalline lattices.15 But when dipolar and quadrupolar colloids are mixed together, we show that they can form a great variety of 2D colloidal crystals. These crystalline colloidal structures can also be considered as structures, formed by repetition of a
Figure 2. Schematic representation of several basic arrangements of 4 μm colloids in planar NLC cell between crossed polarizers (top row) and unpolarized light (second row): (a, b) particles with dipolar director configurations only; (c, d, e, f, g, h) mixed particles with dipolar and quadrupolar configurations; (i) particles with quadrupolar configurations only. Each arrangement is described by a chemical-like formula (see text for details) We assume that dipolar moment of the dipolar colloid is positive, when its accompanying hyperbolic hedgehog defect (black dot) lies in the direction of the X-axis (to the right on all images) with respect to the particle. 12094 DOI: 10.1021/la901719t
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“colloidal molecule”, an elementary building block of the structure. Since these colloidal structures are built using intercolloidal bonds of different strengths, they differ in their stability and mechanical properties. Thus the result of the similarity between the colloidal structures under study and chemistry, we introduce chemical-like formulas that can describe a pair of colloidal particles or aggregates of particles. For notation purposes, we assign positive sign to the dipolar colloidal particle when its accompanying hyperbolic hedgehog defect (black dots in Figure 1a) lies in the direction of the X-axis (to the right on all images) with respect to the particle itself. In these terms, the configuration in Figure 2a can be represented as [D+ 3 ], because it consists of three dipolar particles lying along direction of the nematic director. Configuration in Figure 2b has [D+ 2 D ] formula, because it consists of two “positive” and one “negative” dipolar particle. Configuration in Figure 2c can be represented as [D+Q], since it is formed by dipolar and quadrupolar particles oriented along the director field. Configurations in Figure 2d and e are presen~ Here [Q] ~ implies that in this case quadrupolar ted as [D+Q]. particle has distorted Saturn ring.18 Unlike [D+Q] formula that ~ formula corresponds to only one configuration, the [D+Q] describes two configurations, which can be transformed one into another by π rotation around the X-axis. Configuration presented in Figure 2f is written as [D+QD-]. Configuration presented in Figure 2g represents another situation common to chemistry, when one can find different chemical formulas + + for the same molecule. In this case both [D+ 3 Q] and [D QD2 ] formulas can be applied. Configuration shown in Figure 2h also can be described by more than one formula: [D+ 3 QD ] + + and [D QD2 D ]. Finally, configuration shown in Figure 2i is written as [Q4]. Proposed chemical-like notation can be successfully applied to describe periodic 2D colloidal structures. In this case, one formula describes the elementary building block or a set of particles belonging to the elementary cell of the structure. Structures that are formed solely by dipolar or quadrupolar particles are written in this notation as [D+D-] (or [D-D+]) and [Q], respectively. More complex structures will be represented by more sophisticated formulas. The simplest example of 2D DQ structure is [D+Q] structure, presented in Figure 3. This structure is stabilized solely by DQ interaction and contains all possible kinds of DQ bonds. Since symmetry of DQ interaction differs significantly from the DD and QQ interactions, this structure is not as dense and contains regularly spaced voids of the same shape unlike the dipolar or quadrupolar lattices do.15,26,27 Thus, each particle in this lattice has three nearest neighbors of another kind. Due to the presence of dipolar particles below and above each quadrupolar particle in the lattice, its Saturn-ring is almost undistorted (see Figure 2g). Saturn rings are distorted only around colloidal particles at the edge of the crystal structure, like in Figure 2d and e. This is caused by the uncompensated effect of missing neighboring dipolar particle. Another feature of this lattice is that all dipolar particles have their elastic dipolar moments pointing in the same direction, and the crystal is “ferroelectric” in the topological sense. Hence, the two closest dipolar particles located on the vertical line should repel each other in the absence of other particles. But their attraction through the quadrupolar particle, that lies between them, is sufficient to counterbalance their dipolar repulsion. Two closest dipolar particles that do not lie on the vertical line, are located at an angle close to the border between repulsion and (30) Yada, M.; Yamamoto, J.; Yokoyama, H. Phys. Rev. Lett. 2004, 92, 185501.
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Figure 3. Structure [D+Q] made of 4 μm colloidal particles under crossed polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d).
Figure 4. Structure [D+QD-] under crossed (vertical and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d).
attraction sectors.30 Their repulsion, if any at all, is too weak to destabilize the structure. It should be noted, however, that building block of this structure shown in Figure 3d may be selected in different ways. This is not a feature of this specific 2D DQ structure, but rather common property of all of them, when different building blocks can be found for the same DQ lattice. Although this situation is similar to what one finds in molecular crystals, here we have one major difference: “colloidal bonds” between colloidal particles belonging to one building block are essentially the same as the bonds between different building blocks. This is generally not the case for molecular crystals, where the interatomic bonds inside one molecule are much stronger and of different nature than the intermolecular forces. DOI: 10.1021/la901719t
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We found that voids in the [D+Q] structure can be occupied with dipolar particles with opposite direction of their dipolar moment, making total dipolar moment of the building block zero. It leads to another DQ structure with formula [D+QD-], as shown in Figure 4. In this case there are strong D+-D-, D--Q, ~ and D+-Q bonds in the structure, but there are no true D-Q bonds and each dipolar particle always has two neighboring dipolar particles with opposite dipolar moment. Since the ~ D+-D- attraction is stronger in comparison to the D+-Q, relative positions of dipolar particles are determined mostly by the interaction between them. Quadrupolar particles can make only minor changes, as evident when comparing Figure 2g and h. After addition of the D- particle to the right of the quadrupolar particle, upper and bottom D+ particles move from their previous equilibrium positions closer to the new D- particle because of strong D+-D- interaction. Structure in Figure 4 also shows, how two collinear and antiparallel elastic dipoles can attract each other, if there is another quadrupole in the space between them. These two dipolar particles, will normally repel each other, but in this case repulsive force is compensated by their attraction to other particles through Q-D- and D+-Q bonds. On the other hand, dipolar particles that belong to neighboring building blocks are stabilized by their attraction to the particles above and below through D+-D--Q-D+-D- chains. Alternatively, the [D+QD-] structure may be treated as a system of vertical zigzag chains with antiparallel elastic dipoles (see Figure 4), bonded together by quadrupolar particles. In contrast to the “ferroelectric” lattice presented in Figure 3, the [D+QD-] building block is “antiferroelectric” with zero total elastic dipolar moment, which is also the case with the whole structure in Figure 4. By adding another two dipolar particles to the ends of the building block shown in Figure 4, one obtains [D2+QD2-] lattice, shown in Figure 5. Its structural and mechanical properties are similar to those of [D+QD-]. Such elementary building blocks can be arranged in different ways in 2D structure, depending on how many of their dipolar particles form D+-D- bonds. Since there are two dipolar particles on each end of the building block with total of five colloidal particles, there are four total possibilities to arrange the building blocks. The elementary cell consists of two blocks, which may overlap with either one or two dipolar particles. In turn, the two building blocks forming the elementary cell can also overlap with one or two dipolar particles. Among the four resulting configurations two are essentially the same, thus it is possible to build three different structures with different lattice constants using the [D2+QD2-] building block and, accordingly, different filling factor. In the experiment we were able to stabilize only the structure, shown in Figure 5. Whereas the structures in Figures 4 and 5 have equal number of dipolar colloids on both sides of the quadrupolar particle, in principle they do not have to be equal. Figure 6 shows a [D+QD2-] lattice with an asymmetric building block. Since we have only one dipolar colloidal particle on the left side of the quadrupolar one, we have just one possibility to arrange these building blocks into 2D lattice with periodic voids. Due to its asymmetry, the building block has nonzero total dipolar moment and the total dipolar moment of whole lattice is not zero. Summarizing the above-described lattices, one can see that they belong to more general class of 2D binary DQ lattices, described by [Dm+QDn-] (m, n > 0) general formula. Theoretically, there should be no restrictions on magnitude of m and n, so lattices of this kind with arbitrary m and n may be assembled. But in practice, not all lattices will be stable and robust like [D+QD-] 12096 DOI: 10.1021/la901719t
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Figure 5. Structure [D+QD2-] under crossed (vertical and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d).
Figure 6. Structure [D+QD2-] under crossed (vertical and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d).
(see Figure 4). We also tried to assemble a structure with more asymmetric building block with m=1 and n=4. In this case at early stage of assembling small pieces of 2D structure were not so stable and difficult to handle. These building blocks have long dipolar chains at the right side and only one dipolar particle at the left side, so they can overlap only with one dipolar colloidal pair, leaving three-particle-long dipolar parts with parallel elastic dipolar moments open. These open tails interact and repel each other, which results in substantial bending, especially at the edges of the structure. In general, [Dm+QDn-] structures have one specific feature: they always have periodically placed voids, whose shape depends on m and n. Consider, for example, m=1 and n=0, which is the structure shown in Figure 3. There are voids formed by three Langmuir 2009, 25(20), 12092–12100
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Figure 7. Structure [D+Q2D-] under crossed (vertical and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d).
Figure 8. Structure [D+Q3D-] under crossed (vertical and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d).
dipolar and three quadrupolar particles in this structure. When m and n are equal and the building blocks overlap as much as possible, we have voids formed by two quadrupolar and two dipolar particles, as can be seen from Figures 4 and 5. When m and n differ by unity, we have voids formed by two quadrupolar and four dipolar particles, as one can see in Figure 6. Voids of even more complicated shape can be produced when m and n differ more than unity or m and n are equal and the building blocks do not overlap completely. Adding more dipolar particles to the building block is not the only way to create more sophisticated structures, as quadrupolar particles can be added to the building block as well. In Figure 7, the [D+Q2D-] the structure is presented with two dipolar and two quadruplar particles in the elementary building block. Similar structure with formula [D+Q3D-] is shown in Figure 8, but its building block has three quadrupolar particles. Like the structure [D+QD-] shown in Figure 4, these two structures can be alternatively presented as a set of vertical zigzag chains of antiparallel dipoles, stabilized by two- (see Figure 7) or three-particle wide (see Figure 8) vertical stripes of quadrupolar lattice.27 Generalizing all the structures described above, one can see that they can be described with one general formula [Dm+QpDn-]. By varying m, n, and p, one can realize great variety of stable structures, including those already described. We also noticed in our laser tweezers experiments, that quadrupolar particles in [D+Q2D-] and [D+Q3D-] structures do not form perfect 2D lattice. They tend to form the structure made of quadrupolar chains that is shown in Figure 8. Two main factors are responsible for that. First of all, the vertical spacing of quadrupolar particles in the [D+Q2D-] and [D+Q3D-] structures is determined by the equilibrium vertical spacing of the antiparallel dipolar particles, since the dipolar interaction is much stronger than quadrupolar. Second, a pair of quadrupolar particles is tilted at 73° with respect to the far-field nematic director, whereas in 2D quadrupolar structure this angle changes to 56°, as was shown in ref 27. Therefore, quadrupolar particles can not remain positioned at the same angle with respect to the nematic director like in 2D quadrupolar structure, because this scenario should involve overlapping of neighboring particles which is impossible for hard
sphere colloids. As a result, this angle decreases which also decreases the attraction of the neighboring particles. It has also been shown that in 2D quadrupolar nematic colloidal structures, the particles form straight or kinked chains instead of regular 2D lattice.27 These quadrupolar crystals are quite fragile (see Figure 8 in ref 27 and the discussion therein), as the lateral interaction between quadrupolar chains is quite weak and cannot resist external perturbation, such as a change in temperature and the flow of a liquid crystal. Thus, in the [DmþQpDn-] structures, these two factors will create imperfections in the quadrupolar part of the lattice and this effect may be more important with increasing number of quadrupoles per building bock. As regards the stability of this crystalline class for large m or n, the most stable structures are obtained when their building blocks overlap as much as possible. Structure stability degrades when the building blocks with high m and n do not overlap completely and there is an uncompensated repulsion between nonoverlapped dipolar particles. Because of the presence of strong Dþ-D- and weak Q-Q intercolloidal bonds, these structures are not mechanically strong and can be split into separate parts along the quadrupolar stripes. However, if the separation between the two parts is of the order of the size of the colloidal particle, Q-Q bonds bring the two parts together and may completely restore the lattice. By the addition of more particles to the building block of a DQ structure, much more complex structures can be realized, such as one, shown in Figure 9. It has a general formula [D10þD10-Q4], because its building block consists of 4 quadrupolar and 20 dipolar particles, with equal numbers of dipolar particles with the positive and negative dipolar moments. This structure contains a set of periodically placed voids of the same shape and can also be obtained from [DþQD2-], shown in Figure 6. To do that one has to (1) remove two central particles from the building blocks periodically; 2) symmetrize half of the remaining building blocks horizontally to obtain the [D10þD10-Q4] structure. Due to the presence of the second step, number of positive and negative dipolar particles are equal. In our experiments we see that despite their repulsion, unidirectional dipolar particles may be stabilized in 2D structure by
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Figure 9. Structure [D10þD10-Q4] under crossed (vertical and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d).
Figure 10. Row structure R{DþQ} under crossed (vertical and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d). Diameter of particles is 4 μm.
quadrupolar particles, as presented in Figure 2(g). The inserted quadrupolar particle somehow “neutralizes” dipolar repulsion and this phenomena was used for assembling [DþQ] structure shown in Figure 3. We found that it is also possible to stabilize parallel dipolar chains in the similar manner, and this can be achieved in a specific subclass of 2D DQ structures. The simplest structure of this kind is [R{DþQ}] structure, presented in Figure 10. Rows of dipolar particles, oriented along the director field, which would normally repel each other, are partially stabilized with inserted quadrupolar particles. Hence, we call this subclass of 2D DQ structures “row structures”. Here, [R{...}] means that the building block should be 12098 DOI: 10.1021/la901719t
Ognysta et al.
~ Q}] ~ under crossed (vertical Figure 11. Row structure [R{D2þQ and horizontal) polarizers (a) and unpolarized light (b) together with its schematic representation (c) and the elementary building block (d). Diameter of particles is 4 μm.
repeated in such manner (in this case horizontally) that alternative rows of dipolar and quadrupolar particles are created. Due to strong repulsion between unidirectional dipolar chains at small lateral separations, this arrangement is not very stable. For our experimental conditions it was nearly impossible to build bigger colloidal structure of this kind. It should be noted, however, that this situation may be different for different NLC matrices with different elastic constants and/or colloidal particles made of another material or with another surface treatment, resulting in different surface anchoring energy. Because of the Dþ-Dþ repulsion and the strong Dþ-Q attraction, the quadrupolar particles of the latter structure often rearrange forming quadrupolar doublets that results in increasing ~ Q}] ~ separation between dipolar chains. The resulting [R{D2þQ lattice is shown in Figure 11. In this case the repulsion between quadrupolar particles does not allow quadrupolar doublets to be attached to each dipolar particle. When they are attached to every other dipolar particle, their repulsion is weaker, and the repulsion between dipolar chains is weaker as well because of the increased separation between them. Despite the fact that we have smaller number of quadrupoles between the dipolar chains, attraction þ ~ Q-D ~ is sufficient to maintain the ordering. through the Dþ-QBy further increasing the separation between the dipolar chains, stable colloidal organization can also be realized as it is ~ Q}] ~ structure in Figure 12. Like for the shown for [R{D2þQQ previous structure we were also unable to stabilize dense packed structure because of the Q-Q repulsion. This structure contains quadrupolar particles with distorted and nondistorted Saturn rings. Quadrupolar particles that lie in the middle between dipolar chains have their Saturn rings unperturbed unlike the quadrupoles that have dipolar particles in direct neighborhood. This ~ weakens the attractive force created through the Dþ-Q-Qþ þ þ ~ Q-D chains, but since D -D repulsion is also weaker in ~ Q}] ~ structure, the structure is stabilized. comparison to [R{D2þQ Building block of this structure was deliberately made big in order to show that quadrupolar particles with deformed Saturn rings can be arranged both into “>”- and “
