All hard, convex shapes are conjectured by Ulam to pack more densely than spheres, which have a maximum packing fraction of phi = pi/ radical18 approximately 0.7405. Simple lattice packings of many shapes easily surpass this packing fraction. For regular tetrahedra, this conjecture was shown to be true only very recently; an ordered arrangement was obtained via geometric construction with phi = 0.7786 (ref. 4), which was subsequently compressed numerically to phi = 0.7820 (ref. 5), while compressing with different initial conditions led to phi = 0.8230 (ref. 6). Here we show that tetrahedra pack even more densely, and in a completely unexpected way. Following a conceptually different approach, using thermodynamic computer simulations that allow the system to evolve naturally towards high-density states, we observe that a fluid of hard tetrahedra undergoes a first-order phase transition to a dodecagonal quasicrystal, which can be compressed to a packing fraction of phi = 0.8324. By compressing a crystalline approximant of the quasicrystal, the highest packing fraction we obtain is phi = 0.8503. If quasicrystal formation is suppressed, the system remains disordered, jams and compresses to phi = 0.7858. Jamming and crystallization are both preceded by an entropy-driven transition from a simple fluid of independent tetrahedra to a complex fluid characterized by tetrahedra arranged in densely packed local motifs of pentagonal dipyramids that form a percolating network at the transition. The quasicrystal that we report represents the first example of a quasicrystal formed from hard or non-spherical particles. Our results demonstrate that particle shape and entropy can produce highly complex, ordered structures.

The centroid of a fluorophore can be determined within Ϸ1.5-nm accuracy from its focused image through fluorescence imaging with one-nanometer accuracy (FIONA). If, instead, the sample is moved away from the focus, the point-spread-function depends on both the position and 3D orientation of the fluorophore, which can be calculated by defocused orientation and position imaging (DOPI). DOPI does not always yield position accurately, but it is possible to switch back and forth between focused and defocused imaging, thereby getting the centroid and the orientation with precision. We have measured the 3D orientation and stepping behavior of single bifunctional rhodamine probes attached to one of the calmodulins of the light-chain domain (LCD) of myosin V as myosin V moves along actin. Concomitant with large and small steps, the LCD rotates and then dwells in the leading and trailing position, respectively. The probe angle relative to the barbed end of the actin (␤) averaged 128°while the LCD was in the leading state and 57°in the trailing state. The angular difference of 71°r epresents rotation of LCD around the bound motor domain and is consistent with a 37-nm forward step size of myosin V. When ␤ changes, the probe rotates ؎27°azimuthally around actin and then rotates back again on the next step. Our results remove degeneracy in angles and the appearance of nontilting lever arms that were reported.3D orientation ͉ lever arm ͉ single molecule ͉ fluorescence imaging with one-nanometer accuracy C omplementary conformational changes can be measured on single motor proteins by use of fluorescence imaging with one-nanometer accuracy (FIONA) (1-3) and by single-molecule fluorescence polarization microscopy (SMFP) (4). FIONA is a method in which the emission distribution of a single fluorophore is detected by using a charge-coupled device and fitted to a 2D Gaussian function to determine the position of the probe. The positional accuracy of the measurement, typically Ϸ1.5 nm, is generally limited by the number of collected photons (1-3). In contrast, SMFP is sensitive to the 3D orientation of a single dye's transition dipole moments (4). In SMFP, the dye is excited by multiple polarized beams, incident from different directions. The resulting emission is split with respect to its polarization and detected with avalanche photodiodes (APDs). However, by slightly defocusing the microscope objective and by using appropriate fitting routines, the defocused image of the probe can be used to determine both its position and its orientation (5-7). We call this technique defocused orientation and position imaging (DOPI). When the sample is deliberately moved Ϸ500 nm away from the best focus position, combinations of lobes and fringes appear on the charge-coupled device. These images are compared with calculated model images to obtain the best estimates of both 3D orientation and position of the probe. Because the image is spread out over a greater number of pixels in DOPI versus FIONA, DOPI inherently has poorer signal-tonoise ratio...

No abstract

Ferrofluids are familiar as colloidal suspensions of ferromagnetic nanoparticles in aqueous or organic solvents. The dispersed particles are randomly oriented but their moments become aligned if a magnetic field is applied, producing a variety of exotic and useful magnetomechanical effects. A longstanding interest and challenge has been to make such suspensions macroscopically ferromagnetic, that is having uniform magnetic alignment in the absence of a field. Here we report a fluid suspension of magnetic nanoplates that spontaneously aligns into an equilibrium nematic liquid crystal phase that is also macroscopically ferromagnetic. Its zero-field magnetization produces distinctive magnetic self-interaction effects, including liquid crystal textures of fluid block domains arranged in closed flux loops, and makes this phase highly sensitive, with it dramatically changing shape even in the Earth's magnetic field.

The stylus of an atomic force microscope is used to scribe preferred directions for liquid-crystal alignment on a polyimide-coated substrate. The opposing substrate that comprises the liquid-crystal cell is rubbed unidirectionally, resulting in a twisted nematic structure associated with each micrometer-sized pixel. The polarization of light entering from the uniformly rubbed substrate rotates with the nematic director by a different amount in each pixel, and each of the two emerging polarization eigenmodes interferes separately. Two examples are discussed: a square grating that allows only odd-order diffraction peaks and a grating that combines rotation with optical retardation to simulate a blazed grating for circularly polarized light. The gratings can be electrically switched if used with semitransparent electrodes.