Ⅴ OLUME85. NUMBER25 PHYSICAL REVIEW LETTERS i 8 DECEMBER 2000 Planar Magnetic Colloidal Crystals Weijia Wen, Lingyun Zhang, and Ping Sheng Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China ( Received 14 June 2000) We report a novel form of planar magnetic colloidal crystals formed by coated magnetic microspheres floating on a liquid meniscus. Under an external magnetic field, the magnetic interaction and the"" interaction, due to the weight of rticles projected along the surface tangent, yields not only the triangular lattice with a variable lattice constant, but also all the other planar crystal symmetries such as the oblique, centered-rectangular, rectangular, and square lattices. By using two different sized magnetic particles, local formations of 2D quasicrystallites with fivefold symmetry are also observed. PACS numbers: 82.70. Kj, 64.70 Kb, 75.50. Mm, 83.80.Gv Since its discovery more than two decades ago, colloidal spheres [19]. The scanning electron microscope images crystals have blossomed into a fertile area of research indicate the Ni coating to be uniformly deposited, shown tion of three-dimensional mesocrystals, i.e., crystals with formed by dispersing 52-um-sized coated mi crystals were crons [1-11]. More recently, two-dimensional, or planar, diameter of I cm. The bottle ed in a glass bottle with a lattice constants ranging from submicrons to tens of mi- the surface of glycerin, contai placed in a rotatable colloidal crystals have been observed through a number aluminum stage located in the center region of a pair of of self-assembly techniques such as magnetic hole formed Holmholz coils, where the magnetic strength of coils was with nonmagnetic particles in a ferrofluid [12], field- adjusted by a computer-controlled current source. In order induced assembly of floating magnetic particles [13]. to change direction of magnetic field relative to the liquid electric-field-induced planar crystal [14], and surfactant- surface, the Helmholz coil can rotate freely along its di- mediated colloid 15]. In particular, two- ameter. Lattice formation and transitions were monitored dimensional magnetic colloidal crystals have afforded and recorded by a video system fundamental studies on 2D melting and crystallization, In the absence of a magnetic field, the coated micro- mediated with the hexatic phase [16] spheres aggregate in the center region of the slightly curved In this Letter, we report the unexpected discovery that liquid surface, visible in the upper inset in Fig. 1. Whe in a certain parameter range of monodispersed magnetic a perpendicular magnetic field was applied, the spheres particles, two-dimensional(2D)crystals can be formed move radially outward and form a stable hexagonal lattice on a fluid surface with not just the(hexagonal) triangular The entire process of hexagonal lattice formation is shown lattice, but also with all the other planar crystal symme- in the lower insets in Fig. 1, where the lattice constant is tries such as the oblique, centered-rectangular, rectangular, noted to increase monotonically as a function of the field and square lattices [17]. These lattice structures, some of strength. This behavior clearly indicates a competition be- which are metastable, can be reversibly tuned by adjusting tween the repulsive magnetic interaction and the"attrac the polar and azimuthal angles of the magnetic field rela- tive"interaction due to the weight of the particles projected ive to the surface normal and the symmetry direction of along the surface tangent. Such competition is possible the 2D lattices. Furthermore, by using two different sized because the attractive and repulsive interactions are on the magnetic particles, local formations of 2D quasicrystal- same order due to the fact that the magnetic interaction lites with fivefold symmetry were observed. Theoretical which depends on the coating thickness, and the weight of predictions based on energy considerations are shown to the sphere, which depends on the sphere di in good agreement with the experiments separately controlled in our system. Quantitative predic The spherical magnetic particles are fabricated by coat- tions based on this simple picture, given below, are shown ing 52(+2)-um and 26(+2)-um-sized glass spheres with to give excellent agreement with the experiment. 2-um and 1.5-um-thick nickel layers, respectively. In B the boundary condition and the Laplace order to obtain magnetic microspheres with controllable formula, the sha surface can be deduced as moments, we selected uniform glass microspheres with zoLlo(Ar)-1/o(Aro)-ll, where z denotes the sur- two different sizes as the initial cores and coated a thin face height, with z=0 at the center of the surface r =0, layer of nickel using the electroless plating technique [18]. Io(x)is the zeroth order modified Bessel function of the The nickel-coated microspheres were heated in a vacuum first kind, A=peg/o, where pe=1.26 x 10 kg/m3 chamber at 400C for 2 h and then annealed at 550C denotes the mass density of glycerin, g denotes the for 3 h. The annealed microspheres possess a small mag- gravitational acceleration, and o=63. 4 mJ/m2 denotes netic moment, on the order of 10 emu for the larger the surface tension. Here the maximum depth of liquid 0031-9007/00/85(25)/5464(4)$15.00 O 2000 The American Physical Society
VOLUME 85, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 DECEMBER 2000 Planar Magnetic Colloidal Crystals Weijia Wen, Lingyun Zhang, and Ping Sheng Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (Received 14 June 2000) We report a novel form of planar magnetic colloidal crystals formed by coated magnetic microspheres floating on a liquid meniscus. Under an external magnetic field, the balance between the repulsive magnetic interaction and the “attractive” interaction, due to the weight of the particles projected along the surface tangent, yields not only the triangular lattice with a variable lattice constant, but also all the other planar crystal symmetries such as the oblique, centered-rectangular, rectangular, and square lattices. By using two different sized magnetic particles, local formations of 2D quasicrystallites with fivefold symmetry are also observed. PACS numbers: 82.70.Kj, 64.70.Kb, 75.50.Mm, 83.80.Gv Since its discovery more than two decades ago, colloidal crystals have blossomed into a fertile area of research encompassing diverse approaches for controlled fabrication of three-dimensional mesocrystals, i.e., crystals with lattice constants ranging from submicrons to tens of microns [1–11]. More recently, two-dimensional, or planar, colloidal crystals have been observed through a number of self-assembly techniques such as magnetic hole formed with nonmagnetic particles in a ferrofluid [12], fieldinduced assembly of floating magnetic particles [13], electric-field-induced planar crystal [14], and surfactantmediated colloid crystals [15]. In particular, twodimensional magnetic colloidal crystals have afforded fundamental studies on 2D melting and crystallization, mediated with the hexatic phase [16]. In this Letter, we report the unexpected discovery that in a certain parameter range of monodispersed magnetic particles, two-dimensional (2D) crystals can be formed on a fluid surface with not just the (hexagonal) triangular lattice, but also with all the other planar crystal symmetries such as the oblique, centered-rectangular, rectangular, and square lattices [17]. These lattice structures, some of which are metastable, can be reversibly tuned by adjusting the polar and azimuthal angles of the magnetic field relative to the surface normal and the symmetry direction of the 2D lattices. Furthermore, by using two different sized magnetic particles, local formations of 2D quasicrystallites with fivefold symmetry were observed. Theoretical predictions based on energy considerations are shown to be in good agreement with the experiments. The spherical magnetic particles are fabricated by coating 5262-mm and 2662-mm-sized glass spheres with 2-mm and 1.5-mm-thick nickel layers, respectively. In order to obtain magnetic microspheres with controllable moments, we selected uniform glass microspheres with two different sizes as the initial cores and coated a thin layer of nickel using the electroless plating technique [18]. The nickel-coated microspheres were heated in a vacuum chamber at 400 ±C for 2 h and then annealed at 550 ±C for 3 h. The annealed microspheres possess a small magnetic moment, on the order of 1026 emu for the larger spheres [19]. The scanning electron microscope images indicate the Ni coating to be uniformly deposited, shown in the lower inset in Fig. 1. Planar colloidal crystals were formed by dispersing 52-mm-sized coated microspheres on the surface of glycerin, contained in a glass bottle with a diameter of 1 cm. The bottle was placed in a rotatable aluminum stage located in the center region of a pair of Holmholz coils, where the magnetic strength of coils was adjusted by a computer-controlled current source. In order to change direction of magnetic field relative to the liquid surface, the Helmholz coil can rotate freely along its diameter. Lattice formation and transitions were monitored and recorded by a video system. In the absence of a magnetic field, the coated microspheres aggregate in the center region of the slightly curved liquid surface, visible in the upper inset in Fig. 1. When a perpendicular magnetic field was applied, the spheres move radially outward and form a stable hexagonal lattice. The entire process of hexagonal lattice formation is shown in the lower insets in Fig. 1, where the lattice constant is noted to increase monotonically as a function of the field strength. This behavior clearly indicates a competition between the repulsive magnetic interaction and the “attractive” interaction due to the weight of the particles projected along the surface tangent. Such competition is possible because the attractive and repulsive interactions are on the same order due to the fact that the magnetic interaction, which depends on the coating thickness, and the weight of the sphere, which depends on the sphere diameter, can be separately controlled in our system. Quantitative predictions based on this simple picture, given below, are shown to give excellent agreement with the experiment. By using the boundary condition and the Laplace formula, the shape of surface can be deduced as z z0I0lr 2 1I0lr0 2 1, where z denotes the surface height, with z 0 at the center of the surface r 0, I0x is the zeroth order modified Bessel function of the first kind, l p rgs, where r 1.26 3 103 kgm3 denotes the mass density of glycerin, g denotes the gravitational acceleration, and s 63.4 mJm2 denotes the surface tension. Here the maximum depth of liquid 5464 0031-90070085(25)5464(4)$15.00 © 2000 The American Physical Society
VOLUME 85. NUmber 25 PHYSICAL REVIEW LETTERS i 8 DECEMBER 2000 Experimental Theoretical 150 iquid 130 110 》 SiO2 6080100120140160180 B(Gauss) FIG. 1(color). Variation of the hexagonal lattice constant as a function of the external field. Solid line is theory: solid circle denote measured values. Pictures of the hexagonal lattice under(a)17 G,(b)28 G, and(c)80 G. A cross-sectional picture of the coated sphere is shown in the lower-right inset. The curved liquid meniscus is shown in the upper-left inset surface, zo, and the radius of the container ro =0.5 cm be the most favorable for normal applied field. The pre are given experimentally. The maximum height zo is dicted lattice constant variation as a function of the ap- noted to depend on the weight of the coated spheres. plied field is in good agreement with the experiment,as Its value can be measured by photographical means. A shown in Fig. 1. Here the only parameters involved are inset in Fig. 1 perimental and material parameters, e.g., nickel's magne- e magnetostatic energy density Um consists of the tization of 55 emu/g of magnetic dipole-dipole interaction and the ex- Structural transitions were realized by tilting the mag field netic field at an angle 0 away from the z direction and l·i,(;·;)(·;) rotating an angle relative to the x axis as defined in the upper inset in Fig. 2. A pictorial summary of all the different structures observed. and their occurrence in the h·近 8, coordinates, is shown in Fig. 2. With increasing 8 () and fixed =0, the lattice constant along the x axis decreases while that along the y direction increases to where ui i xh is the total magnetic dipole moment form the centered rectangular structure, as shown in at site i, Imi is the permanent dipole moment of sphere i, Fig. 2(c). Starting at 0=27, lattice instability sets in x is the magnetic susceptibility, and h is the external field. Some of the spheres are attracted by their neighbors to Here ri denotes the vector pointing from the position of form short chains as indicated by the arrows in Fig. 2(d) the ith dipole to that of the jth dipole. To find the equilib- Further increase in 0 leads to perfectly equally space rium orientation state of the dipoles, we start from different straight chains aligned along the field direction, as shown initial random configurations and use dissipative spin dy- in Fig. 2(f). Besides the equal spacing, it is noteworthy namics to evolve towards the configuration of minimum that the neighboring chains are also shifted by half a energy. When the magnetic energy is considered together diameter in respect to each other. with the gravitational potential of the microspheres on the To obtain other planar crystal structures, we start from fluid surface, the lattice structure and the lattice constant the case shown in Fig. 2(b) and then rotate the center stage are determined by the condition of minimum total energy. holder along the p direction. If the rotation is very slow, This is so because the thermal effect is totally negligible e.g., less than a few degrees per minute, then the whole in our system due to the size of the microspheres. The pattern would just rotate in step with the magnetic field esults of our calculations show the triangular lattice to and there is no change in the lattice structure. However, 465
VOLUME 85, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 DECEMBER 2000 FIG. 1 (color). Variation of the hexagonal lattice constant as a function of the external field. Solid line is theory: solid circles denote measured values. Pictures of the hexagonal lattice under (a) 17 G, (b) 28 G, and (c) 80 G. A cross-sectional picture of the coated sphere is shown in the lower-right inset. The curved liquid meniscus is shown in the upper-left inset. surface, z0, and the radius of the container r0 0.5 cm are given experimentally. The maximum height z0 is noted to depend on the weight of the coated spheres. Its value can be measured by photographical means. A picture of the curved fluid surface is shown in the upper inset in Fig. 1. The magnetostatic energy density Um consists of the energy of magnetic dipole-dipole interaction and the external field, Um 1 V µX i,j ui ? u j r 3 ij 2 3 ui ? rij u j ? rij r 5 ij 2 X i h ? ui ∂ , (1) where ui m i 1 xh is the total magnetic dipole moment at site i, m i is the permanent dipole moment of sphere i, x is the magnetic susceptibility, and h is the external field. Here rij denotes the vector pointing from the position of the ith dipole to that of the jth dipole. To find the equilibrium orientation state of the dipoles, we start from different initial random configurations and use dissipative spin dynamics to evolve towards the configuration of minimum energy. When the magnetic energy is considered together with the gravitational potential of the microspheres on the fluid surface, the lattice structure and the lattice constant are determined by the condition of minimum total energy. This is so because the thermal effect is totally negligible in our system due to the size of the microspheres. The results of our calculations show the triangular lattice to be the most favorable for normal applied field. The predicted lattice constant variation as a function of the applied field is in good agreement with the experiment, as shown in Fig. 1. Here the only parameters involved are jm j 1026 emu and x 0.008, both estimated from experimental and material parameters, e.g., nickel’s magnetization of 55 emug. Structural transitions were realized by tilting the magnetic field at an angle u away from the z direction and rotating an angle f relative to the x axis as defined in the upper inset in Fig. 2. A pictorial summary of all the different structures observed, and their occurrence in the u, f coordinates, is shown in Fig. 2. With increasing u and fixed f 0, the lattice constant along the x axis decreases while that along the y direction increases to form the centered rectangular structure, as shown in Fig. 2(c). Starting at u 27±, lattice instability sets in. Some of the spheres are attracted by their neighbors to form short chains as indicated by the arrows in Fig. 2(d). Further increase in u leads to perfectly equally spaced, straight chains aligned along the field direction, as shown in Fig. 2(f). Besides the equal spacing, it is noteworthy that the neighboring chains are also shifted by half a diameter in respect to each other. To obtain other planar crystal structures, we start from the case shown in Fig. 2(b) and then rotate the center stage holder along the w direction. If the rotation is very slow, e.g., less than a few degrees per minute, then the whole pattern would just rotate in step with the magnetic field, and there is no change in the lattice structure. However, 5465
VOLUME 85. NUmber 25 PHYSICAL REVIEW LETTERS i 8 DECEMBER 2000 60 ·2· 600 3 ::::: entered-rectangular 0(degree) FIG. 3(color). Calculated total energies of three different pla nar structures. The crossing points, denoted a and b, are the e values at which the structural transitions occur. They are in excellent agreement with the experiment. The insets give a di rect visualization of the cooperative dipole rotation associated with the increase in total energies. lent agreement with the experimentally measured values of 23 and 57, respectively. The large increase in the FIG.2(color). The formation of different planar lattice struc- energies following the 23 transition is due to the coopera tures along the polar angle 8, shown downward. and tive rotation of the magnetic dipoles from a predominantly the azimuthal angle of the magnetic field. Here the z axis is vertical orientation to a predominantly flat orientation. By the surface normal at the center of the liquid meniscus, and the defining an order parameter, S=(M)=2imiz/N,for x axis is defined in the uppermost picture the spin magnetization component along the vertical di- rection, this cooperative rotation of the dipole orientation a rotation rate faster than 1/ s, for example, would result can be easily quantified to vary from $=0.9 at 0=27 in structure change. These facts indicate a long relaxation to S=0.07 at 6= 36. However, perhaps more interest time for the observed structures. By rotating at a rate of ing is the direct visualization of this dipole rotation. This about 2/ s to = 20, a planar oblique structure is ob- was realized by dispersing nanosized nickel particles on served, as shown in Fig. 2(g). The square and rectangular each sphere. These nanoparticles would tend to aggregate planar structures are obtained by starting either from (c) at the magnetic poles. Under a vertical magnetic field, the and(d) and then rotating along the o direction to(h) and nanoparticles project upward, shown in the lower-left inset (i), respectively, or by starting from(g) and further rotat- in Fig. 3. When the dipoles are rotated, the nanoparticles ing along the 0 direction. These structures are metastable, form an elongated steak aligned along the in-plane projec- in the sense that if strongly perturbed, they would go back tion of the local magnetic field, shown in the two upper -left to the =0 states. Nevertheless, all planar structures insets in Fig 3 can be obtained uniquely and repeatably as a function of 8 By mixing the 52-and 26-um-sized spheres in a ra- and d tio of 4: 1, we found that in most cases, a small sphere is o The lattice instability starting at 0=27 is instrumental surrounded by five large spheres to form a five-sided lo- or all the lattice structures formed subsequently. Its ori- cal formation under a vertical magnetic field. The overall gin may be traced to a cooperative dipole rotation, which structure is amorphous. However, we have unexpectedly can be predicted theoretically as well as visualized experi- found that in some areas there can be beautiful textbook ex- mentally. In Fig. 3, the respective energies of three lattice amples of quasicrystal formations with fivefold rotational structures-hexagonal, centered-rectangular, and chain- symmetry, as shown in Fig 4. To our knowledge, this is are shown. At points a and b, indicated by arrows, the the first time that such a 2D pattern has been reproduced nergy curves cross, implying structural transitions should as a force-balanced, natural-occurring metastable state occur at 22(from hexagonal to centered-rectangular) and We have demonstrated that by using coated spheres with 55(to the chain structure). These values are in excel- weak magnetic moments, many different forms of planar
VOLUME 85, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 DECEMBER 2000 FIG. 2 (color). The formation of different planar lattice structures along the polar angle u, shown increasing downward, and the azimuthal angle f of the magnetic field. Here the z axis is the surface normal at the center of the liquid meniscus, and the x axis is defined in the uppermost picture. a rotation rate faster than 1±s, for example, would result in structure change. These facts indicate a long relaxation time for the observed structures. By rotating at a rate of about 2±s to w 20±, a planar oblique structure is observed, as shown in Fig. 2(g). The square and rectangular planar structures are obtained by starting either from (c) and (d) and then rotating along the f direction to (h) and (i), respectively, or by starting from (g) and further rotating along the u direction. These structures are metastable, in the sense that if strongly perturbed, they would go back to the f 0 states. Nevertheless, all planar structures can be obtained uniquely and repeatably as a function of u and f. The lattice instability starting at u 27± is instrumental for all the lattice structures formed subsequently. Its origin may be traced to a cooperative dipole rotation, which can be predicted theoretically as well as visualized experimentally. In Fig. 3, the respective energies of three lattice structures—hexagonal, centered-rectangular, and chain— are shown. At points a and b, indicated by arrows, the energy curves cross, implying structural transitions should occur at 22± (from hexagonal to centered-rectangular) and 55± (to the chain structure). These values are in excelFIG. 3 (color). Calculated total energies of three different planar structures. The crossing points, denoted a and b, are the u values at which the structural transitions occur. They are in excellent agreement with the experiment. The insets give a direct visualization of the cooperative dipole rotation associated with the increase in total energies. lent agreement with the experimentally measured values of 23± and 57±, respectively. The large increase in the energies following the 23± transition is due to the cooperative rotation of the magnetic dipoles from a predominantly vertical orientation to a predominantly flat orientation. By defining an order parameter, S M P i mizN, for the spin magnetization component along the vertical direction, this cooperative rotation of the dipole orientation can be easily quantified to vary from S 0.9 at u 27± to S 0.07 at u 36±. However, perhaps more interesting is the direct visualization of this dipole rotation. This was realized by dispersing nanosized nickel particles on each sphere. These nanoparticles would tend to aggregate at the magnetic poles. Under a vertical magnetic field, the nanoparticles project upward, shown in the lower-left inset in Fig. 3. When the dipoles are rotated, the nanoparticles form an elongated steak aligned along the in-plane projection of the local magnetic field, shown in the two upper-left insets in Fig. 3. By mixing the 52- and 26-mm-sized spheres in a ratio of 4:1, we found that in most cases, a small sphere is surrounded by five large spheres to form a five-sided local formation under a vertical magnetic field. The overall structure is amorphous. However, we have unexpectedly found that in some areas there can be beautiful textbook examples of quasicrystal formations with fivefold rotational symmetry, as shown in Fig. 4. To our knowledge, this is the first time that such a 2D pattern has been reproduced as a force-balanced, natural-occurring metastable state. We have demonstrated that by using coated spheres with weak magnetic moments, many different forms of planar 5466
Ⅴ OLUME85. NUMBER25 PHYSICAL REVIEW LETTERS i 8 DECEMBER 2000 (4 P. Yang, T. Deng, D. Zhao, P Feng, D. Pine, B F. Chmelka G M. Whitesides, and G.D. Stucky, Science 281, 2244 65A.V. Blaaderen, R. Ruel, and P. wiltzius, Nature(London) 385,321(199 6]J. Zhu, M. Li, R. Rogers, w. Meyer, R H Ottewill, W. B Russel, and P. M. Chaikin, Nature (London) 387, 883 [7 Z. Cheng, W.B. Russel, and p. M. Chaikin, Nature (Lon- don)401,893(1999) 8]AE. Larsen and D G. Grier, Nature(London) 385, 230 FIG 4(color). Left panel: a picture of the locally fivefold symmetric structure observed at one area of the overall [9] T.J. Chen, R.N. Zitter, and R. Tao, Phys. Rev. Lett. 68 orphous planar lattice with two different sized coated 255(1992) There is an erfection in the structure, at [10] L. Zhou, W. Wen, and P. Sheng, Phys. Rev. Lett. 81, 1509 about the 7 o'clock position, where a large sphere is located at (1998) he position for a small sphere. Rig uasicrystalline structure, such as that shown in[20]. [11]w. Wen, N. Wang, H Ma, Z Lin, W.Y. Tam, C.T. a drawing of a a distinct similarity with the picture in the left panel is noted and P. Sheng, Phys. Rev. Lett. 82, 4248(1999) [12] A.T. Skjeltorp, Phys. Rev. Lett. 51, 2306(1983);J Phys.55,2587(1984) crystals may be obtained with ease. These planar magnetic [13] M. Golosovsky, Y. Saado, and D. Davidov, AppL. Phys lattices with tunable lattice constants may present some po- Lett.75,4l68(1999) tential applications, such as a modulation of surface waves [14] S. Yeh, M. Seul, and B I. Shralman, Nature(London)386, and as a template for growing 2D photonic crystals 57(1997 We thank N. Wang, Z Q. Zhang, and X.X. Zhang for [5]L. Ramos, T.C. Lubensky, N. Dan, P. Nelson, and D A useful comments and helpful discussions. This work is Weitz, Science 286, 2325(1999) partially supported by DAG Grant No. DAG99/00SC32. [16]K Zahn, R. Lenke, and G. Maret, Phys. Rev.Lett.82, 2721 (1999) [17] C. Hammond, The Basics of Crystallography and Diffrac- tion(Oxford University Press, New York, 1997) [18] W.Y. Tam, G.H. Yi, W. Wen, H. Ma, M.M.T. Loy, and [1 P.N. Pusey and w. van Megen, Nature(London)320, 340 P. Sheng, Phys. Rev. Lett. 78, 2987(1997) (1986 [19] We can use smaller microspheres than those used here 2]A D. Dinsmore, J C. Crocker, and A G. Yodth, Curr Opin microspheres small in diameter are Colloid Interface Sci. 3, 5(1998) more difficult to coat and are also hard to control due to [3] Special issue on"From dynamics to devices: directed self- increased Brownian motion assembly of colloidal materials, "edited by D G. Grier [20] C. Janot, Quasicrystals, A Primer(Clarendon Press, Ox [MRS Bull. 23, No 10, 21(1998)1
VOLUME 85, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 DECEMBER 2000 FIG. 4 (color). Left panel: a picture of the locally fivefold symmetric structure observed at one area of the overall amorphous planar lattice with two different sized coated microspheres. There is an imperfection in the structure, at about the 7 o’clock position, where a large sphere is located at the position for a small sphere. Right panel: a drawing of a fivefold quasicrystalline structure, such as that shown in [20]. A distinct similarity with the picture in the left panel is noted. crystals may be obtained with ease. These planar magnetic lattices with tunable lattice constants may present some potential applications, such as a modulation of surface waves and as a template for growing 2D photonic crystals. We thank N. Wang, Z. Q. Zhang, and X. X. Zhang for useful comments and helpful discussions. This work is partially supported by DAG Grant No. DAG99/00.SC32. [1] P. N. Pusey and W. van Megen, Nature (London) 320, 340 (1986). [2] A. D. Dinsmore, J. C. Crocker, and A. G. Yodth, Curr. Opin. Colloid Interface Sci. 3, 5 (1998). [3] Special issue on “From dynamics to devices: directed selfassembly of colloidal materials,” edited by D. G. Grier [MRS Bull. 23, No. 10, 21 (1998)]. [4] P. Yang, T. Deng, D. Zhao, P. Feng, D. Pine, B. F. Chmelka, G. M. Whitesides, and G. D. Stucky, Science 281, 2244 (1998). [5] A. V. Blaaderen, R. Ruel, and P. Wiltzius, Nature (London) 385, 321 (1997). [6] J. Zhu, M. Li, R. Rogers, W. Meyer, R. H. Ottewill, W. B. Russel, and P. M. Chaikin, Nature (London) 387, 883 (1997). [7] Z. Cheng, W. B. Russel, and P. M. Chaikin, Nature (London) 401, 893 (1999). [8] A. E. Larsen and D. G. Grier, Nature (London) 385, 230 (1997). [9] T. J. Chen, R. N. Zitter, and R. Tao, Phys. Rev. Lett. 68, 255 (1992). [10] L. Zhou, W. Wen, and P. Sheng, Phys. Rev. Lett. 81, 1509 (1998). [11] W. Wen, N. Wang, H. Ma, Z. Lin, W. Y. Tam, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 82, 4248 (1999). [12] A. T. Skjeltorp, Phys. Rev. Lett. 51, 2306 (1983); J. Appl. Phys. 55, 2587 (1984). [13] M. Golosovsky, Y. Saado, and D. Davidov, Appl. Phys. Lett. 75, 4168 (1999). [14] S. Yeh, M. Seul, and B. I. Shralman, Nature (London) 386, 57 (1997). [15] L. Ramos, T. C. Lubensky, N. Dan, P. Nelson, and D. A. Weitz, Science 286, 2325 (1999). [16] K. Zahn, R. Lenke, and G. Maret, Phys. Rev. Lett. 82, 2721 (1999). [17] C. Hammond, The Basics of Crystallography and Diffraction (Oxford University Press, New York, 1997). [18] W. Y. Tam, G. H. Yi, W. Wen, H. Ma, M. M. T. Loy, and P. Sheng, Phys. Rev. Lett. 78, 2987 (1997). [19] We can use smaller microspheres than those used here. However, microspheres smaller than 1 mm in diameter are more difficult to coat and are also hard to control due to increased Brownian motion. [20] C. Janot, Quasicrystals, A Primer (Clarendon Press, Oxford, 1992). 5467
