Supplementary Information Assembly and Control of 3D Nematic Colloidal Crystals

Transcription

1 Supplementary Information Assembly and Control of 3D Nematic Colloidal Crystals A. Nych 1,2, U. Ognysta 1,2, M. Škarabot1, M. Ravnik 3, S. Žumer3,1, and I. Muševič 1,3 1 J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia 2 Institute of Physics, prospect Nauky, 46, Kyiv , Ukraine 3 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (Dated: September 4, 2012) 1

2 A.- DETERMINATION OF DEFECT LOCATION IN ISOLATED TOPOLOGICAL DIPOLES AND QUASI-3D CHECKERBOARD STRUCTURE Understanding the orientation of topological dipoles formed by colloidal particles in the nematic LC is crucial for the topological assembly of 3D nematic colloidal crystals. To this aim, we have used high resolution optical microscopy and fluorescent confocal polarizing microscopy (FCPM) [2] to discriminate the positions of topological defects, allocated to individual colloidal particles in the nematic liquid crystal. A polarizing inverted optical microscope Nikon TE-2000 with 60 x objective with N. A.=1.0 was used in the transmission mode to determine the position of the topological defect, accompanying individual colloidal microsphere in the nematic liquid crystal. Left panel of Fig. S1 presents the position of the experimental cell with respect to microscope objective with the topological defects of the colloids located either above (up, left particle in left panel) or below the microsphere (down, right particle on the left panel). Right panel shows a sequence of images of two colloidal particles, positioned close to each other, but with different orientations of their topological dipoles, as we vary the z-position of the focal plane of the objective. The -z direction refers to the situation, where the sample is approaching to the microscope objective (and correspondingly smaller separation), while the +z direction refers to the sample going away from the microscope objective. From the sequence of images taken at different heights, we can clearly see, that these two particles have their topological dipoles antiparallel and the topological defects are located on different sides of the particles. A careful inspection of the sequence of images reveals that the topological defect of the left particle is located further away from the objective, whereas the topological defect of the particle on the right is located at closer separation to the microscope objective. This explains why the particles with up/down orientation of their topological dipoles appear differently in the checkerboard structure, shown in Figure 1 of the main text. We have used fluorescent confocal polarizing microscopy to determine the director field distribution within the checkerboard structure, formed by dense packing of topological dipoles in a homeotropic nematic liquid crystal cell, accommodating only a single layer of colloidal particles. Schematic drawing of the director field in the vertical cross-section of the checkerboard structure (shown in Fig.1,d of the main paper), is presented in Fig. S2. It contains both up- and down dipolar particles, with antiparallel orientation of the neighbor- 2

3 ing topological dipoles. The drawing was obtained by overlaying several ansatz functions for the director field of the dipolar particle with proper up/down orientation and the vertical shift [3]. The voids indicate the positions of the colloidal particles with perpendicular orientation of LC molecules at their surfaces. This drawing clearly indicates the locations of hyperbolic hedgehog defects accompanying each particle. However, when one calculates the corresponding FCPM image intensity I cos 4 (α) [4], where α is the angle between the local LC director and the polarization direction which is along the x (horizontal) axis, the calculated cross-section appears artificially sharper compared to the experimental one (Fig.1,d of the paper). This is specially evident near the hyperbolic hedgehog defects in Fig. S3. However, a fair comparison between the calculated and the observed confocal crosssection has to take into account finite spatial resolution of the confocal microscope. This means that we have to apply a 2D point spread function (PSF) of the confocal microscope by convolving the calculated FCPM cross-section with PSF kernel of the form K P SF = exp [ {(x/w x ) 2 + (z/w z ) 2 }]. Here w x and w z are the lateral and the axial dimensions of the kernel. In our calculations w x and w z were chosen so that their ratio was w z /w x = 2.5, which is in qualitative agreement with the PFS of a real confocal microscope. The resulting convolved image is shown in Fig. S4. It is clear that the FCPM image obtained by using approximate PSF kernel, is in qualitative agreement with the experimental one, shown in Fig. 1,d of the paper. B.- ASSEMBLY OF 3D NEMATIC COLLOIDAL CRYSTALS IN PLANAR CELLS A single colloidal particle in bulk nematic liquid crystal can appear either in a form of a topological dipole or a topological quadrupole [5, 6]. The topological state depends on the strength of the surface anchoring of LC molecules, the diameter of the colloidal microparticle 2R and the ratio of the thickness of the nematic layer h to the diameter of colloidal microspheres. Using DMOAP or polyimide surface-treated materials results in very strong surface anchoring of LC molecules. In thick planar cells, h >> 2R colloidal particles appear in a form of topological dipoles. If the thickness of planar nematic layer is decreased, dipolar colloids will be transformed into quadrupoles under a certain threshold thickness [1]. 3

4 For 2.3 µm silica colloids with DMOAP surface treatment, the critical thickness for the dipolar-to-quadrupolar topological transition is h 3.5µm [7]. In this way it is possible to control the topological state of nematic colloids. We have also studied the topological protocols and rules of assembling 3D nematic colloidal crystals from nematic topological dipoles in cells with planar orientation of the nematic liquid crystal. It is known that dipolar nematic colloids in planar cells of intermediate thickness form a variety of very stable 2D colloidal structures, such as dipolar and quadrupolar nematic colloidal crystals [1] and binary colloidal crystals in a mixture of dipolar and quadrupolar colloids [8]. In the case of dipolar colloidal crystals, the thickness of the cell is 40 50% larger than the colloidal diameter. The two surfaces of the experimental cells with planar orientation of the nematic LC therefore stabilize 2D nematic colloidal structures. It has been demonstrated, for example, that dipolar nematic colloids of 2.32µm diameter in planar cells of thickness 3 4µm form stable structures consisting of antiparallel chains of nematic colloidal dipoles. An odd-even effect was also observed, where a system of an even number of chains was tilted for several degrees with respect to the overall orientation of the planar cell, while an odd number of dipolar chains was oriented exactly along the overall orientation of the nematic LC. However, when the thickness of the experimental cells with planar orientation of the nematic LC is increased beyond the value of several colloidal diameters to accommodate 3D colloidal structures, an instability of the intermediate nematic colloidal structures is observed, which is presented in Fig. S5. Here, the thickness of the planar cell is 8µm and the diameter of colloidal particles is 2.32µm. The colloidal particles appear in a form of topological dipoles, which is recognized by dark spots-hyperbolic point defects, that accompany the particles and form topological (i.e. elastic) dipoles. On the top panel of Fig. S5 one can see on the left side of the image a pair of dipolar chains with antiparallel orientation of their topological dipoles. The pair of dipolar chains is tilted with respect to the orientation of the LC, which is an odd-even effect, mentioned before [7]. When the third dipolar chain is brought closer to the pair of chains, with the orientation of its topological dipoles antiparallel to the nearest neighbor dipolar chain, topological frustration obviously takes place. Instead of direct attraction of three antiparallel dipolar chains, observed in thin planar cells, the chains form a disordered, topologically frustrated trio, shown in the lowest panel of Fig. S5. The nature of topological frustration of a trio of colloidal chains is even better presented 4

5 in thicker planar cells. Fig. S6 shows a sequence of images of three colloidal chains of 4.32 µm silica microspheres, treated with DMOAP, in a planar cell of thickness h = 14 µm, filled with ZLI The images were taken at different positions of the focal plane of the microscope. One can clearly see from these images that the three chains are strongly tilted in an otherwise planar cell, and, what is even more striking, they are not lying in the same plane. This outof-plane tilted colloidal trio of dipolar chains is a striking evidence of the degree of topological frustration of three topological dipoles in the nematic LC. To overcome this topological frustration of three dipolar colloidal chains in the NLC, we have prepared a specially designed LC cell with two planar compartments on top of each other, with both compartments merging into a single and larger compartment, as illustrated in Fig. S7a. The whole cell was filled with a dispersion of 2.32µm silica microspheres coated with DMOAP as described before. Using the laser tweezers, we have prepared a small colloidal platelet in each compartment, consisting of three dipolar colloidal chains. As the compartments are thin, the colloidal chains in each compartment assembled into well defined colloidal platelets of microspheres each, as shown in the far left panel of Fig. S7b. The two colloidal platelets were then grabbed with the laser tweezers and pulled across the edge of the mica sheet, where there was a much thicker, single compartment of a planar nematic LC. In this compartment, (far right panel of Fig. S7b), four of the colloidal chains assembled into a 2 x 2 x 4 colloidal crystallite and the two remaining chains were only loosely attached to this crystallite. This result clearly shows that a major problem in self-assembly of 3D nematic colloidal crystal is the topological frustration of a trio of topological dipolar colloidal chains. An even number of dipolar chains does not show this topological frustration and can be assembled into 3D crystals of tetragonal symmetry. C.- LANDAU-DE GENNES MODELING OF 3D NEMATIC COLLOIDS After checking several example papers from the Nature Communications I moved details on the numerical simulations to Materials and Methods section of the paper. Now I m not sure how the remaining paragraph fits into support information material to the 3D dipolar paper. As for me I do not mind to leave it here, but it might need a kind of introductory paragraph before to link it to the paper. /Andriy/ 5

6 The true reach of numerically modeling three-dimensional nematic colloids can be revealed by performing calculations also in the regime of colloidal particles as elastic quadrupoles [9], which are much more difficult to achieve experimentally. Stable 3D quadrupolar nematic colloidal crystal where topological defects in the form of rings around the waists of the particles -the elastic quadrupoles- determine the ordering of the colloidal crystal are predicted [10]. The equilibrium 3D quadrupolar crystal unit cell is found to be tetragonal with basis, quite similar to the dipolar nematic colloids. Specifically, the 3D quadrupolar crystal exhibits motifs of two-dimensional quadrupolar colloidal lattices -which are commonly used as seeds in the experimental assembly- in diagonal planes, which indicates that a full 3D assembly method may be required to build such crystals and not only a generalization of 2D assembly. Indeed, as show by the 3D dipolar colloidal crystals and these two further examples of 3D structures and this quadrupolar example, 3D liquid crystal colloids can exhibit a wide richness of structures, varying in lattice, symmetry, and topology. REFERENCES [1] Škarabot, M., Ravnik, M., Žumer, S., Tkalec, U., Poberaj, I., Babič, D., Osterman, N. and Muševič, I. Interactions of quadrupolar nematic colloids. Phys. Rev. E 77, (2008). [2] Smalyukh, I., Shiyanovskii,S. and Lavrentovich, O. D. Three-dimensional imaging of orientational order by fluorescence confocal polarizing microscopy. Chem. Phys. Lett. 336, (2001). [3] Lubensky, T. C., Pettey, D., Currier, N., Stark, H. Topological defects and interactions in nematic emulsions, Phys.Rev.E 57, 610 (1998). [4] Shiyanovski, S. V., Smalyukh,I. I. and Lavrentovich, O. D. Computer simulations and fluorescence confocal polarizing microscopy of structures in cholesteric liquid crystals, pp , in Defects in Liquid Crystals: Computer Simulations, Theory and Experiments, O.D. Lavrentovich, P. Passini, C. Zannoni, and S. Zumer (eds.), NATO Science Series, Kluwer Academic Publishers (2001). 6

7 [5] Poulin, P., Stark, H., Lubensky, T. C. and Weitz, D. A. Novel colloidal interactions in anisotropic fluids. Science 275, 1770 (1997). [6] Stark, H., Physics of colloidal dispersions in nematic liquid crystals.phys. Rep. 351, (2001). [7] Škarabot, M., Ravnik, M., Žumer, S., Tkalec, U., Poberaj, I., Babič, D., Osterman, N. and Muševič, I. Two-dimensional dipolar nematic colloidal crystals. Phys. Rev. E 76, (2007). [8] Ognysta, U., Nych, A., Nazarenko, V., Muševič, I., Škarabot, M., Ravnik, M., Žumer, S.Poberaj, I., Babič, D. 2D interactions and binary crystals of dipolar and quadrupolar nematic colloids. Phys. Rev. Lett. 100, (2008). [9] Gu, Y., Abbott, N. L. Observation of Saturn-ring defects around soild microspheres in nematic liquid crystals. Phys. Rev. Lett. 85, (2000). [10] Žumer, S. 3D assemblies of colloidal particles in homogeneous and inhomogeneous nematics. Keynote lecture at Discussion Metting on Liquid Crystal Theory and Modeling, 29th-30th October, Oxford,

8 FIGURES left left right right z=-5mm +z z=-2mm -z z=1mm z=4mm FIG. S1. Up/down orientation of a hyperbolic topological defect. The position of the point defect, close to a silica microsphere in a nematic liquid crystal, can be discriminated by scanning the focal plane of the microscope along the z-direction, which is also the topological dipole direction. The left image shows the hyperbolic point defect. For the left-particle it is on the top panel, at z = 5 µm. For the right particle, it is on the bottom panel, at z = 4 µm. 8

9 FIG. S2. Director profile in quasi 2D checkerboard lattice. Schematic drawing of the nematic director profile along the vertical cross-section of the quasi-2d lattice presented in Fig.1,d of the paper. The voids are occupied by colloidal particles. Each particle is accompanied by a hyperbolic hedgehog defect. The direction of topological dipoles, formed by the particle and its companion defect, is alternating from particle to particle. 9

10 a b z x FIG. S3. Calculated cross-section of the FCPM intensity from checkerboard structure. a, FCPM intensity is calculated directly from the director distribution shown in Fig. S2. The polarization of the excitation light and the polarization of detected light are both along the x-axis. b, The same image as in a, but overlayed with the director field from Figure S2. 10

11 a b FIG. S4. Calculated FCPM images, where the point spread function (PSF) of the confocal microscope is taken into account. a, Figure S3a after convolving the calculated FCPM intensity with approximate PSF kernel of the confocal microscope. The kernel is shown in the inset. b. The same as a, but the director field has been superposed. 11

12 t=0 t=20s t=24s This colloid is out of plane FIG. S5. Time sequence of microscope images of self-assembly of dipolar colloidal chains in a planar cell of 5CB. Each colloidal chain is assembled from 2.32 µm silica microspheres, treated with DMOAP to induce strong anchoring of LC molecules perpendicular to the surface. The thickness of the planar cell is h = 8 µm. Top panel: two dipolar chains are assembled with their topological dipoles antiparallel. When the third chain is added with its topological dipole also antiparallel to the neighboring dipole, it is attracted only at one end, as seen in the lowest panel. At this end, the colloidal particle in the middle chain is clearly pushed out of plane. The upper end of the third chain is clearly repelled from the pair. Three dipolar chains are therefore in a frustrated situation, which is a evidence of the breakdown of a simple electrostatic analog for nematic colloidal interactions. 12

13 5 μm z = -3 μm z = 0 μm z = 3 μm FIG. S6. Dipolar colloidal chains in a very thick planar cell of ZLI Each colloidal chain is assembled from 4.32 µm silica microspheres, treated with DMOAP to induce strong anchoring of LC molecules perpendicular to the surface. The thickness of the planar cell is h = 14 µm. The images are taken at different positions of the focal plane of the microscope. 13

14 a glass plates thin mica sheet (~2 m) 4 m 4 m b single cell double cell t=0 t=39s t=44s t=62s FIG. S7. Assembly of a dipolar nematic colloidal crystal in nematic LC cell with two separate planar nematic compartments. a, Schematic geometry of the cell. The cell is assembled from two glass plates coated with ITO and planar and rubbed alignment layers of polyimide. In one half, the cell was horizontally separated by a 2µm thick sheet of mica. Using 4µm microbeads to keep the surface separation, the two horizontal compartments were created on top of each other. These two compartments merged into a thicker ( 10µm) compartment filled with the same nematic LC. Mica provided good planar alignment along the crystal symmetry axis, which was determined by analyzing the birefringence of the thin mica sheet between crossed polarizers. b, Sequence of images, demonstrating laser-tweezers dragging of colloidal platelets in upper and lower compartments across the edge of the mica separating sheet. In the final position, four colloidal chains are assembled in a small 3D dipolar colloidal cluster, whereas the remaining two chains are only loosely attached to this cluster. 14

15 CAPTIONS TO MOVIES S1 TO S9 Caption to Movie S1 Assembly of dipolar colloidal chains.three silica particles in the nematic liquid crystals are assembled into a single colloidal chain that is oriented perpendicular to the plane of the screen. The small red cross is the optical trap. Each colloidal particle is a topological dipole, and all dipoles have the same direction. The images are taken under crossed polarizers and the LC appears dark because the optical axis runs along the direction, perpendicular to the screen. Caption to Movie S2 Electrostriction of a quasi-2d checkerboard colloidal crystal.a quasi-2d colloidal crystallite in a nematic liquid crystal, assembled into the checkerboard structure, shrinks, when the electric field is applied in a direction perpendicular to the crystallite. Several volts are applied across several micrometer-thick nematic LC. Caption to Movie S3 Topologically frustrated 3D colloidal trio.three colloidal chains, each of them made of three colloidal particles (like those in Movie S1) are brought close to each other by the laser tweezers, and then they start to interact. In a final position, we clearly see a frustrated trio of colloidal chains. The trio is tilted with respect to the vertical to the screen and the positions of each chain are arranged in a bow-like manner. Caption to Movie S4 Laser tweezers assisted assembly of a 3D colloidal crystal in a nematic LC. Assembly of two colloidal blocks in the nematic LC, each of them has three colloidal floors. A block of 2 x 2 x 3 particles is attracted to the block of 2 x 4 x 3 particles, finally forming a 2 x 6 x 3 colloidal crystal, bound-together by structural forces, mediated by the nematic LC. 15

16 Caption to Movie S5 Spontaneous attraction of 3D colloidal blocks in the nematic LC. A colloidal block of 2 x 4 x 3 silica microspheres is attracted to another 2 x 4 x 3 colloidal block and self-assembles into the final colloidal crystal of 6 x 4 x 3 microspheres. The crystal is bound by structural forces, mediated by the nematic LC. Caption to Movie S6 Self-assembly of two 3D colloidal blocks into a colloidal crystal in the nematic LC. A colloidal block of 2 x 6 x 3 silica microspheres is attracted to a colloidal block of 4 x 6 x 3 silica microspheres, assembling into the final colloidal crystal of 6 x 6 x 3 particles. The crystal is bound by structural forces, mediated by the nematic LC. Caption to Movie S7 3D confocal image of the tetragonal colloidal crystal in the nematic LC. 3D reconstruction of the fluorescence confocal polarizing microscopy images of a 6 x 6 x 3 colloidal crystal in the nematic LC. The tetragonal structure is clearly demonstrated. Caption to Movie S8 Dragging a 3D colloidal crystal by light of the laser tweezers. A small colloidal crystal of 3 x 3 x 3 silica microspheres is assembled in the nematic liquid crystal. The particles are so strongly bound together by the elastic forces, mediated by the nematic LC, that the crystal can be dragged through the liquid crystal by the laser tweezers. Caption to Movie S9 Rotating a 3D colloidal crystal in the nematic LC by an electric field. The electric field is applied to the colloidal crystal of 4 x 4 x 3 silica microspheres, assembled in the nematic liquid crystal with negative dielectric anisotropy. Above a certain threshold, the crystal rotates, becuse the LC molecules collectively tilt away from the field direction. The 16

17 angle of rotation-tilting is fully and reversibly controlled by the electric field. The shape of the crystal is preserved because of strong particle binding by the elastic forces of the nematic LC. 17