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 planarVOLUME 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 struc￾tures 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 ob￾served, 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 rotat￾ing 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 ori￾gin may be traced to a cooperative dipole rotation, which can be predicted theoretically as well as visualized experi￾mentally. 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 excel￾FIG. 3 (color). Calculated total energies of three different pla￾nar 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 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 energies following the 23± transition is due to the coopera￾tive 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 di￾rection, 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 interest￾ing 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 projec￾tion 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 ra￾tio of 4:1, we found that in most cases, a small sphere is surrounded by five large spheres to form a five-sided lo￾cal 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 ex￾amples 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