Uniform spherical beads were used to explore the behavior of a granular system near its critical angle of repose on a conical bead pile. We found two tuning parameters that could take the system to a critical point where a simple power-law described the avalanche size distribution as predicted by self-organized criticality, which proposed that complex dynamical systems self-organize to a critical point without need for tuning. Our distributions were well described by a simple power-law with the power τ = 1.5 when dropping beads slowly onto the apex of a bead pile from a small height. However, we could also move the system from the critical point using either of two tuning parameters: the height from which the beads fell onto the top of the pile or the region over which the beads struck the pile. As the drop height increased, the system did not reach the critical point yet the resulting distributions were independent of the bead mass, coefficient of friction, or coefficient of restitution. All our apex-dropping distributions for any type of bead (glass, stainless steel, zirconium) showed universality by scaling onto a common curve with τ = 1.5 and σ = 1.0, where 1/σ is the power of the tuning parameter. From independent calculations using the moments of the distribution, we find values for τ = 1.6 ± 1.0 and σ = 0.91 ± 0.15. When beads were dropped across the surface of the pile instead of solely on the apex, then the system also moved from the critical point and again the avalanche size distributions fell on a common curve when scaled similarly using the same values of τ and σ. We also observed that an hcp structure on the base of the pile caused an emergent structure in the pile that had six faces with some fcc or hcp structure.
The oblique parameters S, T and U and their higher-order extensions (V, W and X) are observables that combine electroweak precision data to quantify deviation from the Standard Model. These parameters were calculated at one loop in the basis-independent CP-violating Two-Higgs Doublet Model (2HDM). The scalar parameter space of the 2HDM was randomly sampled within limits imposed by unitarity and found to produce values of the oblique parameters within experimental bounds, with the exception of T. The experimental limits on T were used to predict information about the mass of the charged Higgs boson and the difference in mass between the charged Higgs boson and the heaviest neutral Higgs boson. In particular, it was found that the 2HDM predicts -600 GeV < mH± - m3 < 100 GeV, with values of mH± > 250 GeV being preferred. The mass scale of the new physics (MNP) produced by random sampling was consistently fairly high, with the average of the scalar masses falling between 400 and 800 GeV for Y2 = mW2, although the model can be tuned to produce a light neutral Higgs mass (~120 GeV). Hence, the values produced for V, W and X fell well within 0.01 of zero, confirming the robustness of the linear expansion approximation. Taking the CP-conserving limit of the model was found to not significantly affect the values generated for the oblique parameters.
The index of refraction for D2O at common wavelengths was measured for several temperatures at atmospheric pressure. While heavy water's refractive index was precisely measured decades ago using the transition lines of elements, those wavelengths are seldom used now that inexpensive lasers provide a range of available wavelengths. We review those measurements, note some inconsistencies between research groups, and fit the best of the literature data to a simple equation that allows an easy calculation for the refractive index of D2O with an accuracy of ±0.0002 at any visible wavelength and between (278 and 359) K. To verify the equation, we then compare the calculated refractive index to our measured values for three He-Ne laser wavelengths (543.5, 594.1, and 632.8) nm over a temperature range from (288 to 338) K and find good agreement.
The reverse Pluronic, triblock copolymer 17R4 is formed from poly(propylene oxide) (PPO) and poly(ethylene oxide) (PEO): PPO14-PEO24-PPO14, where the subscripts denote the number of monomers in each block. In water, 17R4 shows both a transition to aggregated micellar species at lower temperatures and a separation into copolymer-rich and copolymer-poor liquid phases at higher temperatures. For 17R4 in H2O and in D2O, we have determined (1) the phase boundaries corresponding to the micellization line, (2) the cloud point curves marking the onset of phase separation at various compositions, and (3) the coexistence curves for the phase separation (the compositions of coexisting phases). In both H2O and in D2O, 17R4 exhibits coexistence curves with lower consolute temperatures and compositions that differ from the minima in the cloud point curves; we take this as an indication of the polydispersity of the micellar species. The coexistence curves for compositions near the critical composition are described well by an Ising model. For 17R4 in both H2O and D2O, the critical composition is 0.22 ± 0.01 in volume fraction. The critical temperatures differ: 44.8 degrees C in H2O and 43.6 degrees C in D2O. The cloud point curve for the 17R4/D2O is as much as 9 degrees C lower than in H2O.
One-way or unidirectional coupling is a striking example of how topological considerations — the parity of an array of multistable elements combined with periodic boundary conditions — can qualitatively influence dynamics. Here we introduce a simple electronic model of one-way coupling in one and two dimensions and experimentally compare it to an improved mechanical model and an ideal mathematical model. In two dimensions, computation and experiment reveal richer one-way coupling phenomenology: in media where two-way coupling would dissipate all excitations, one-way coupling enables soliton-like waves to propagate in different directions with different speeds.
To elucidate induced smectic A and smectic B phases in binary nematic liquid crystal mixtures, a generalized thermodynamic model has been developed in the framework of a combined Flory-Huggins free energy for isotropic mixing, Maier-Saupe free energy for orientational ordering, McMillan free energy for smectic ordering, Chandrasekhar-Clark free energy for hexagonal ordering, and phase field free energy for crystal solidification. Although nematic constituents have no smectic phase, the complexation between these constituent liquid crystal molecules in their mixture resulted in a more stable ordered phase such as smectic A or B phases. Various phase transitions of crystal-smectic, smectic-nematic, and nematic-isotropic phases have been determined by minimizing the above combined free energies with respect to each order parameter of these mesophases. By changing the strengths of anisotropic interaction and hexagonal interaction parameters, the present model captures the induced smectic A or smectic B phases of the binary nematic mixtures. Of particular importance is the fact that the calculated phase diagrams show remarkable agreement with the experimental phase diagrams of binary nematic liquid crystal mixtures involving induced smectic A orinduced smectic B phase.
We describe the theory, design, and construction of simple electromechanical devices that automatically and continually track celestial objects. As Earth rotates and revolves, a star tracker always points at a star or other object fixed to the celestial sphere, such as the center of the Milky Way galaxy. A planet tracker can fixate on any celestial object, including planets, the Sun, or the Moon. A sidereal clock mechanism drives the star tracker, and software which encoding astronomical algorithms controls an inexpensive robot that drives the planet tracker. The star tracker acts like a gyroscope, rigidly oriented in space, despite Earth's motion. Both trackers indicate the passing of time just like clocks and calendars. The resulting lecture, hallway, or museum displays promote awareness of and excitement about our place in the universe.
We study the classical dynamics of two bodies, a massive line segment or slash (/) and a massive point or dot (.), interacting gravitationally. For this slashdot (/.) body problem, we derive algebraic expressions for the force and torque on the slash, which greatly facilitate analysis. The diverse dynamics include a stable synchronous orbit, generic chaotic orbits, sequences of unstable periodic orbits, spin stabilized orbits, and spin-orbit coupling that can unbind the slash and dot. The extension of the slash provides an extra degree of freedom that enables the interplay between rotation and revolution. In this way, the slashdot body problem exhibits some of the richness of the three body problem with only two bodies and serves as a valuable prototype for more realistic systems. Applications include the dynamics of asteroid-moonlet pairs and asteroid rotation and escape rates.
When β-lactoglobulin in low pH aqueous solutions is exposed to high temperature for extended time, spherulites composed of amyloid fibrils of the β-lactoglobulin protein form. Many of these spherulites have fibrils that radiate out from a centre and, under crossed polarisers, exhibit a symmetric Maltese Cross structure. However, a significant fraction (50 of 101 observed spherulites) of β-lactoglobulin spherulites formed under these conditions demonstrate various forms of irregularity in apparent structure. The irregularities of spherulites structures were qualitatively investigated by comparing optical microscopy images observed under crossed polarisers to computationally produced images of various internal structures. In this way, inner spherulite structures are inferred from microscopy images. Modeled structures that were found to produce computed images similar to some of the experimentally viewed images include fibrils curving as they radiate from a single nucleation point; multiple spherulites nucleating in close proximity to one another; and fibrils curving in opposite directions above and below a single nucleation point.
We have experimentally realized unidirectional or one-way coupling in a mechanical array by powering the coupling with flowing water. In cyclic arrays with an even number of elements, soliton-like waves spontaneously form but eventually annihilate in pairs, leaving a spatially alternating static attractor. In cyclic arrays with an odd number of elements, this alternating attractor is topologically impossible, and a single soliton always remains to propagate indefinitely. Our experiments with 14 and 15-element arrays highlight the dynamical importance of both noise and disorder and are further elucidated by our computer simulations.
A surprising number of physics problems are well suited to "embarrassingly parallel" computations that do not require complicated software algorithms or specialized hardware. As faculty and students at small institutions, we are readily incorporating parallel computing in diverse levels of our curricula, and we are embracing the opportunity to utilize high performance computing to attack contemporary research problems in summer research, senior theses, and course work. This article describes how we do this in three significant examples: spatiotemporal patterns of one-way coupled oscillators, ray-tracing in curved spacetime, and solar escape as a three-body problem.
We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.
Measurements of the coexistence curve and turbidity were made on different molecular mass samples of the branched polymer-solvent system 8-arm star polystyrene in methylcyclohexane near its critical point. We confirmed that these systems belong in the Ising universality class. The location of the critical temperature and composition as well as the correlation length, susceptibility, and coexistence curve amplitudes were found to depend on molecular mass and the degree of branching. The coexistence curve diameter had an asymmetry that followed a "complete scaling" approach. All the coexistence curve data could be scaled onto a common curve with one adjustable parameter. We found the coexistence curve amplitude to be about 12% larger for branched than linear polystyrenes of the same molecular mass in either solvent cyclohexane or methylcyclohexane. The twoscale- factor universality ratio R was found to be independent of molecular mass or degree of branching.
The heat capacity of the liquid-liquid mixture nitrobenzene-dodecane has been measured for the first time near its upper critical consolute point using an adiabatic calorimeter. The theoretical expression for the heat capacity near the critical point was applied to our combined data runs. The critical exponent α was determined to be 0.124±0.006, which was consistent with theoretical predictions. When α was fixed at its theoretical value of 0.11, our value for the amplitude ratio A+/A- = 0.58±0.02 was consistent with experimental determinations and theoretical predictions. However, the two-scale-factor universality ratio X, now consistent among experiments and theories with a value between 0.019-0.020, was violated in this system when using a previously published value for the correlation length.
We investigate generalized seeding of the attracting states of Abelian sandpile automata and find there exists a class of global perturbations of such automata that are completely removed by the natural local dynamics. We derive a general form for such self-erasing perturbations and demonstrate that they can be highly nontrivial. This phenomenon provides a new conceptual framework for studying such automata and suggests possible applications for data protection and encryption.
In this paper, a phase diagram is developed for the molar mixtures of nematic liquid crystals of 5CB and MBBA. In order to understand the interaction of the two systems, dielectric permittivities ε|| and ε⊥ were measured for mixtures of various concentrations. The usual assumption is that in the absence of chemical reactions the bulk physical properties add up as a weighted sum of the individual properties. Our dielectric permittivity data clearly show a correlation to the phase diagram and the existence of the induced phase. In order to understand the interactions from a fundamental level, we modeled the 5CB and MBBA molecules using a Silicon Graphics O2 workstation running the software Spartan 5.1. Different electrical surfaces were calculated for a geometrically optimized molecule. Our investigations support the idea of strong charge interactions between the nematic systems.
The turbidity of the liquid-liquid mixture methanol-cyclohexane has been measured very near its critical point and used to test competing theoretical predictions and to determine the critical correlation-correction exponent η. By measuring the ratio of the transmitted to incident light intensities over five decades in reduced temperature, we are able to determine that Ferrell's theoretical prediction for the turbidity explains the data with the correlation length amplitude ξ0=0.330±0.003 nm and critical exponents η=0.041±0.005 and ν=0.632±0.002. These values are consistent with the values measured before for ξ0 in this system and with the exponents predicted by theory. The data allow five different theoretical expressions to be tested and to select two as begin equivalent when very close to the critical point.
We study a cellular automaton derived from the phenomenon of magnetic flux creep in two-dimensional granular superconductors. We model the superconductor as an array of Josephson junctions evolving according to a set of coupled ordinary differential equations. In the limit of slowly increasing magnetic field, we reduce these equations to a simple cellular automaton. The resulting discrete dynamics, a stylized version of the continuous dynamics of the differential equations, is equivalent to the dynamics of a gradient sand pile automaton. We study the dynamics as we vary the symmetry of the underlying lattice and the shape of its boundary. We find that the "simplest" realization of the automaton, on a square lattice with commensurate boundaries, results in especially simple dynamics, while "generic" realizations exhibit more complicated dynamics characterized by statistics with broad distributions, even in the absence of noise or disorder.
We study the electric field-induced first-order transition from a homeotropic smectic A structure into a polydomain texture that occurs through nucleation of toric focal conic domains (TFCDs). The process involves two steps: first nucleation of TFCDs of a size larger than a critical radius a*, and then a steady growth of TFCD to a secondary critical radius a** when surface anchoring effects become dominant and cause a transition from a circular TFCD to an elongated stripe domain (SD). Studies are performed for pure smectic A materials and for smectic A doped with kunipia nanoparticles. Non-destructive 3D imaging with the fluorescence confocal polarizing microscopy (FCPM) shows that the field-induced TFCDs can nucleate in the smectic A bulk. Clay particles reduce the energy barrier for nucleation as they distort the smectic A layers and thus increase the ground state energy. Simple elastic models of TFCD and SD allow us to describe the qualitative features of the observed phenomena.
We analyze solar escape as a special case of the restricted three-body problem. We systematically vary the parameters of our model solar system to show how optimal launch angle and minimum escape speed depend on the mass and size of Earth. In some cases, it is best to launch near the direction of Earth's motion, but slightly outward; in other cases, it is best to launch near the perpendicular to Earth's motion, but inward, toward Sun (so as to obtain a solar gravity assist). Between direct escapes for high launch speeds and trapped trajectories for low launch speeds is an irregular band of chaotic orbits that reveals something of the true complexity of solar escape and the three-body problem.
Self-organized criticality has been proposed to explain complex dynamical systems near their critical points. This experiment examined a monodisperse conical bead pile and how the distribution of avalanches is affected by the pattern of beads glued on a base, by the size or shape of the base, and by the height at which each bead was dropped onto the pile. By measuring the number of avalanches of a given size that occurred during the experiment, the resulting distribution could be compared to a power law description. When the beads were dropped from a small height, all of the data were consistent with a simple power-law of exponent 1.5, which is the mean-field model value. The data showed that neither the bead pattern on the base nor the base size or shape significantly affected the power-law behavior. This is the first time that the mean-field exponent has been observed in a granular pile. However, when the bead is dropped from different heights then the power-law description breaks down and a power-law times an exponential is more appropriate. We found a scaling relationship in the distribution of avalanches for different heights and relate our data to an energy dissipation model. We both confirm self-organized criticality and observe deviations from it.
Our objective was to study mixtures of nematic liquid crystals with dissimilar dielectric anisotropies but similar phase properties. Using light scattering and microscopy, we have established the phase boundaries and transition widths of mixtures of 4'-n-pentyl-4-cyanobiphenyl and 4'-methoxybenzylidene-4-butylaniline. In addition to the isotropic-nematic transition, there is a second induced phase for certain concentrations, which we conclude is an induced smectic B phase. Recent theoretical works provide a model for nematic to induced smectic A transition by combining Flory-Huggins and Maier-Saupe-McMillan theories. From our phase transition data and the application of the above theoretical framework, we conclude that there is a possibility of strong interaction between the two mesogens that produces the smectic B phase.
We present a simple nonlinear system that exhibits multiple distinct stochastic resonances. By adjusting the noise and coupling of an array of underdamped, monostable oscillators, we modify the array's natural frequencies so that the spectral response of a typical oscillator in an array of N oscillators exhibits N - 1 different stochastic resonances. Such families of resonances may elucidate and facilitate a variety of noise-mediated cooperative phenomena, such as Noise Enhanced Propagation, in a broad class of similar nonlinear systems.
Both the heat capacity and the turbidity of the liquid-liquid mixture succinonitrile-water near its upper critical consolute point were measured and two amplitude relations were tested. Using an adiabatic calorimeter to measure the heat capacity and the transmitted light intensity to determine the turbidity, precise and reproducible data determined the critical exponents α, η, and γ, consistent with theoretical predictions. The correlation length ξo = 1.68±0.004 nm was determined from the turbidity experiment while the heat capacity amplitudes were A+ = 0.0543±0.0004 J/(cm3K) in the one-phase region and A– = 0.1013±0.0004 J/(cm3K) in the two-phase region. The amplitude ratio A+/A– = 0.536±0.005 was consistent with other experimental determinations in liquid-liquid mixtures or liquid-vapor systems, and with recent theoretical predictions. The two-scale-factor universality ratio Χ, now consistent among experiments and theories with a value between 0.017 and 0.020, was determined to be 0.0187±0.0013.
The heat capacity of the liquid-liquid mixture perfluoroheptane and 2,2,4-trimethylpentane (also known as iso-octane) has been measured for the first time near its upper critical consolute point using an adiabatic calorimeter. The theoretical expression for the heat capacity near the critical point was applied to our combined data runs. The critical exponent α was determined to be 0.106±0.026, which agreed with theoretical predictions. When α was fixed at its theoretical value of 0.11, our value for the amplutde ratio A+/A– = 0.59±0.05 was consistent with experimental determinations and theoretical predictions. However, the two-scale-factor universality ratio χ, now consistent among experiments and theories with a value between 0.019 and 0.020, was violated in this sytem when using the published value for the correlation length.
A ground based (1-g) experiment is in progress that measures the turbidity of the density-matched, binary fluid mixture methanol-cyclohexane extremely close to its liquid-liquid critical point. By covering the range of reduced temperatures t = (T-Tc) / Tc from 10-8 to 10-2, the turbidity measurements should allow the Green-Fisher critical exponent eta to be determined. This paper reports measurements showing ±0.1 percent precision of the transmitted and reference intensities, and ±4 μK temperature control near the critical temperature of 320 K. Preliminary turbidity data show a non-zero eta consistent with theoretical predictions. No experiment has precisely determined a value of the critical exponent η, yet its value is significant to theorists in critical phenomena. Relatively simple critical phenomena, as in the liquid-liquid system studied here, serve as model systems for more complex behavior near a critical point.
A homeotropically aligned nematic liquid crystal with positive dielectric and diamagnetic anisotropies is subjected to a destabilizing AC electric field E in the bend geometry in the presence of a stabilizing magnetic field B. When the applied voltage V is gradually increased at a given frequency, the distortion that results above a threshold Vth is spatially periodic with the wavevector depending on the electric frequency f. Sudden application of a voltage step, Vs, higher than Vth causes a temporal evolution of the director field, which finally attains the homogeneously distorted HD state; the nature of temporal evolution depends on Vs. If Vs is slightly higher than Vth, the transient deformation is periodic and the wavevector of periodicity depends on f. When Vs is high enough, the transition to HD occurs via a turbulent state.
We designed and constructed an array of ten forced damped nonlinear pendulums. We drove the pivot of the pendulums in a circle and torsionally coupled them with springs. We analyzed the motion using digitized videotape. The behavior of the real array closely mirrored the behavior of its computer simulation. For a homogeneous array of identical pendulums, the spatiotemporal dynamics was chaotic; for a heterogeneous array of nonidentical pendulums, the spatiotemporal dynamics was periodic. Such temporally fixed but spatially varying chaos control has been called "disorder taming chaos".
A homeotropically aligned nematic is subjected to the action of an ac electric field applied in the sample plane. With progressively increasing electric voltage, walls move away from the electrodes, approach each other and merge. A subsequent decrease of voltage to zero causes the reverse process to occur except for hysteresis. The hysteresis width is employed to estimate the adhesion surface energy density of the walls; the surface energy density is of the same order as the anisotropy in surface tension of nematics. The wall thickness diminishes with increasing voltage. This shows that the observed walls are similar to those produced by magnetic fields. The walls exhibit curvature in the sample plane, the undulation in a wall being regular at sufficiently elevated frequencies. The walls are decorated along their length by a zigzag defect pattern which is being reported in the bend Freedericksz geometry for the first time. Some of the observations are explained qualitatively.
We use noise to extend signal propagation in one and two-dimensional arrays of two-way coupled bistable oscillators. In a numerical model, we sinusoidally force one end of a chain of noisy oscillators. We record a signal-to-noise ratio at each oscillator. We demonstrate that moderate noise significantly extends the propagation of the sinusoidal input. Both the optimal noise and the maximum propagation length scale like the square root of the coupling. We obtain similar results with two-dimensional arrays. The simplicity of the model suggests the generality of the phenomenon.
The heat capacity of the liquid-liquid mixture aniline-cyclohexane has been measured for the first time near its upper critical consolute point using an adiabatic calorimeter. Two data runs provide heat capacity data that are fitted by equations with background terms and a critical term. The critical exponent alpha was determined to be 0.104±0.011, consistent with theoretical predictions. When alpha was fixed at its theoretical value of 0.11 to determine the critical amplitudes A+ and A-, our value for the amplitude ratio A+/A- = 0.50±0.03 was consistent with most experimental determinations in liquid-liquid mixtures, but was slightly larger than either theoretical predictions or recent experimental values in liquid-vapor systems. The two-scale-factor universality ratio χ, now consistent among experiments and theories with a value between 0.019 and 0.020, is consistent in this system using one published value for the correlation length, but not with another.
Investigations are reported on the electric field induced orientational transitions in the bend Freedericksz geometry under the action of a stabilizing magnetic field. When the magnetic field is strong enough, the deformation above electric threshold is periodic with the periodicity disappearing at a higher voltage. The alignment does not remain homeotropic below threshold and the sample exhibits pretransitional biaxiality. Every transition is discontinuous and accompanied by hysteresis. A form of scaling appears to hold for all the observed thresholds. The thresholds and the direction of the wavevector are frequency dependent showing that the instability mechanism involves electrical conductivity.
We study a coupled array of torqued damped nonlinear pendulums. Disordering this system can convert chaotic spatiotemporal evolution into periodic motion. Here, in numerical experiments, we elucidate and quantify this phenomenon. For each of several types of disorder, we find an optimal magnitude of disorder which minimizes the system's largest Lyapunov exponent.
We report the effect of electric frequency on deformation threshold in the bend Freedericksz geometry in the presence of a stabilizing magnetic field applied normal to the plates with the destabilizing electric field impressed parallel to the sample. In general, the observed threshold and deformation above it are strongly dependent on frequency and magnetic strength associated with a pretransitional field-induced biaxiality. The periodic deformation observed under a strong magnetic field has wavevector along the electric field at low frequencies. Above a cut-off frequency, the direction of the wavevector becomes normal to the electric field. Hysterisis is present between increase and decrease of voltage. At DC or low frequency excitation, there is clear evidence of hydrodynamic flow which can become turbulent for some values of parameters.
The heat capacity of the binary liquid mixture triethylamine-water has been measured near its lower critical consolute point using a scanning, adiabatic calorimeter. Two data runs are analyzed to provide heat capacity and enthalpy data that are fitted by equations with background terms and a critical term that includes correction to scaling. The critical exponent a was determined to be 0.107±0.006, consistent with theoretical predictions. When a was fixed at 0.11 to determine various amplitudes consistently, our values of A+ and A- agreed with a previous heat capacity measurement, but the value of A+ was inconsistent with values determined by density or refractive index measurements. While our value for the amplitude ratio A+/A- = 0.56±0.02 was consistent with other recent experimental determinations in binary liquid mixtures, it was slightly larger than either theoretical predictions or recent experimental values in liquid-vapor systems. The correction to scaling amplitude ratio D+/D- = 0.5±0.1 was half of that predicted. As a result of several more precise theoretical calculations and experimental determinations, the two-scale-factor universality ratio χ, which we found to be 0.019±0.003, now is consistent among experiments and theories. A new "universal" amplitude ratio involving the amplitudes for the specific heat was tested. Our determination of = -0.5±0.1 and = -1.1±0.1 is smaller in magnitude than predicted and is the first such determination in a binary fluid mixture.