Pair Of Spheres Research Articles

Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions

The dynamics of macroscopically homogeneous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics. We show that the complex dynamics of sheared suspensions can be characterized as a chaotic motion in phase space and determine the dependence of the largest Lyapunov exponent on the volume fraction ϕ. We also offer evidence that the chaotic motion is responsible for the loss of memory in the evolution of the system and demonstrate this loss of correlation in phase space. The loss of memory at the microscopic level of individual particles is also shown in terms of the autocorrelation functions for the two transverse velocity components. Moreover, a negative correlation in the transverse particle velocities is seen to exist at the lower concentrations, an effect which we explain on the basis of the dynamics of two isolated spheres undergoing simple shear. In addition, we calculate the probability distribution function of the transverse velocity fluctuations and observe, with increasing ϕ, a transition from exponential to Gaussian distributions.The simulations include a non-hydrodynamic repulsive interaction between the spheres which qualitatively models the effects of surface roughness and other irreversible effects, such as residual Brownian displacements, that become particularly important whenever pairs of spheres are nearly touching. We investigate, for very dilute suspensions, the effects of such a non-hydrodynamic interparticle force on the scaling of the particle tracer diffusion coefficients Dy and Dz, respectively, along and normal to the plane of shear, and show that, when this force is very short-ranged, both are proportional to ϕ2 as ϕ → 0. In contrast, when the range of the non-hydrodynamic interaction is increased, we observe a crossover in the dependence of Dy on ϕ, from ϕ2 to ϕ as ϕ → 0. We also estimate that a similar crossover exists for Dz but at a value of ϕ one order of magnitude lower than that which we were able to reach in our simulations.

The visuomotor transformation for arm movement accounts for 3-D eye orientation and retinal geometry.

To point or reach to a visual target, you need to know its direction relative to your shoulder. That direction can be computed from the retinal image, if the brain also knows the orientation of the eyeball, head, and clavicle. To be geometrically exact, the neural computation would have to involve rotary operations.1 When the eye was turned 30° up, for instance, the brain would take the locations of all objects relative to the eye and rotate them 30° down to find the locations relative to the head. Some theories2,3,5–7 suggest that the brain might approximate that rotation by a simpler vector addition, merely shifting all retinal locations by the same vector. But that strategy would lead to marked errors in some situations. FIGURE 1A shows five earthfixed, horizontal arcs wrapped around a cylinder centered on an eyeball. In front of the eye are two spheres and 90° to the right are two cubes, one sphere-cube pair at eye level and the other 30° up. When the eye fixates the eye-level sphere, the retinal image of the eye-level cube falls on the eye’s horizontal meridian (FIG. 1B). But when the eye fixates the sphere at 30° up, the retinal image of the corresponding cube falls well below the horizontal meridian (FIG. 1C). That is, both pairs of spheres and cubes are horizontally separated, but in one case their retinal images are not, owing to the rotation of the eye. If the brain tried to compute these objects’ locations relative to the head or the shoulder simply by shifting all their retinal locations by a fixed vector, it would misestimate the elevations of at least some objects, and would misaim its eye and arm movements. Here we study whether the visuomotor transformation for arm movements uses a vector-additive strategy or correctly accounts for the rotary geometry of retinal projection.

Interaction between large spheres immersed in small spheres: remarkable effects due to a trace amount of medium-sized spheres

The interaction induced between a pair of large hard spheres immersed in a binary hard-sphere mixture is analyzed using integral equation theories. The mixture consists of dense small spheres containing a trace amount of medium-sized spheres that are smaller than the large ones but much larger than the small ones. With a sufficiently high (but still very low) concentration of the medium-sized spheres, a strongly attractive interaction reaching about two diameters of the medium-sized spheres arises, which makes the interaction range much longer than in a single-component system of the small spheres.

Interparticle force in polydisperse electrorheological fluids: Beyond the dipole approximation

We have developed a multiple image method to compute the interparticle force for a polydisperse electrorheological (ER) fluid. We apply the formalism to a pair of dielectric spheres of different dielectric constants and calculate the force as a function of the separation. The results show that the point-dipole (PD) approximation errs considerably because many-body and multipolar interactions are ignored. The PD approximation becomes even worse when the dielectric contrast between the particles and the host medium is large. From the results, we show that the dipole-induced-dipole (DID) model yields very good agreements with the multiple image results for a wide range of dielectric contrasts and polydispersity. The DID model accounts for multipolar interaction partially and is simple to use in computer simulation of polydisperse ER fluids.

Nonlinear ER effects in an ac applied field

The electric field used in most electrorheological (ER) experiments is usually quite high, and nonlinear ER effects have been theoretically predicted and experimentally measured recently. A direct method of measuring the nonlinear ER effects is to examine the frequency dependence of the same effects. For a sinusoidal applied field, we calculate the ac response which generally includes higher harmonics. In is work, we develop a multiple image formula, and calculate the total dipole moments of a pair of dielectric spheres, embedded in a nonlinear host. The higher harmonics due to the nonlinearity are calculated systematically.

The effects of electrical fields on growth of titania particles formed in a CH 4–O 2 diffusion flame

The effects of a DC electric field on the growth processes of titania particles formed in a premixed CH 4–O 2 flame were investigated for three basic types of electrode geometry, namely pairs of needle tips, spheres and plates. The conclusions are based on measurements of the primary particle diameter, agglomerate mobility distribution and fractal dimension, agglomerate charge distribution, as well as flame temperature profile and electrical current, wherever possible with spatial resolution. Changes in primary particle diameter and rutile content were found to correlate well in magnitude and sign with shifts in flame temperature profile induced by the electric field. When measured just above the flame, needle electrodes were found to increase agglomerate sizes (in terms of average number of primary particles per agglomerate) by up to 3-fold, explainable by a redistribution of aerosol mass into more but smaller primary particles (as a result of lower flame temperatures). Electrostatic enhancement did not play a significant role as particle charge distributions remained bipolar and little affected by the field. Plate electrodes lead to a slowing of agglomerate growth which compensated the increased number of smaller primary particles, resulting in a little net change in number of primary particles as compared to no field. This is caused by separation of the aerosol into a positively and a negatively charged plume as suggested by Vemury et al., however without addition of charge carriers. Aside from the slowing of coagulation in the plate configuration, which can be considered a primary electrostatic process, the effects of electrical fields resulted mostly from secondary mechanisms, especially shifts in local flame temperature, which reduced or enhance the sintering rate in primary particles. Their magnitude depended on applied potential difference alone, not on electrode polarity.

Diffusion into a pair of reactive spheres with first-order reaction

Exact series solutions for the reaction rates of two identical spherical sinks immersed in an infinite medium are presented for both reactant-impenetrable and -penetrable particles based on the method of twin spherical expansion. The reaction rates are explicitly expressed in terms of the interparticle distance from reaction- to diffusion-limited conditions. The effect of the ratio of the diffusivities inside and outside the sink is investigated. The existence of the second particle influences the reaction rate most substantially for the external diffusion-limited condition. The exact solutions are used to evaluate the accuracy of simple, analytical approximations obtained by including the first few multipole contributions only. The result by keeping terms up to quadrupole level gives values in excellent agreement with the exact solution.

Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer.

We have measured the interparticle potential between pairs of micron-sized silica spheres induced by adsorbed polyethylene oxide polymer using a line-scanned optical tweezer. We found this long-range steric repulsion to be exponential over the range of energies (0.1k(B)T-5k(B)T) and polymer molecular weights (452,000-1,580,000) studied, and that the potential scaled with the polymer's radius of gyration R(G). The potential's exponential decay length was about 0.6R(G) and its range was about 4R(G), although both parameters varied significantly from one pair of spheres to another. The potential's exponential prefactor was greater than mean-field predictions.

Some steady axisymmetric vortex flows past a sphere

Steady, inviscid, axisymmetric vortex flows past a sphere are obtained numerically as solutions of a partial differential equation for the stream function. The solutions found include vortex rings, bounded vortices attached to the sphere and infinite vortex tubes. Four families of attached vortices are described: vortex wakes behind the sphere, spherically annular vortices surrounding the spherical obstacle (which can be given analytically), bands of vorticity around the sphere and symmetric pairs of vortices fore and aft of the sphere. Each attached vortex leads to a one-parameter family of vortex rings, analogous to the connection between Hill's spherical vortex and the vortex rings of Norbury.

Lattice Thermal Conduction across Disordered Interfaces

Lattice thermal conduction across the disordered interfaces in multilayered materials has been studied. The model consists of an array of slabs of scattering centers (mass defects) embedded in a host material. The layered distribution of the mass defects can be characterized by the structure factor in the scattering probability due to the slabs. We assume that the physical arrangement of the scattering centers within a slab is quenched disorder and is approximately described by the structure factor of a homogeneous liquid interacting via hard sphere pair potentials. By varying the packing fraction of the scattering centers from low density to high density, the calculated thermal conductivity exhibits the minimum which reflects the local structural organization of the scattering centers. The relation to thermoelectric energy conversion has also been discussed.

Development and Application of the Particle Simulation Method for Fiber and Platelike Particle Dispersed Systems.

A particle simulation method is proposed for computing the dynamics of fiber and platelike particle dispersed systems. In this method, a particle is modeled by using arrays of spheres. Considering the deformation of the particle, each pair of adjacent spheres is connected with three types of springs; stretch, bend, and twist. The motion of the particles in flow is followed by solving the translational and rotational equations of motion for each constituent sphere. The mobility matrix for each particle is calculated to obtain the hydrodynamic force and torque exerted on each sphere. For the hydrodynamic interaction among particles, the near-field lubrication force is considered when the separation between spheres belonging different particles is close, but the far-field part is neglected. The method was applied to predicting the rheological properties of particle dispersed systems and the motion of fillers during an injection molding process of thermoplastics.

Conformal curvatures of curves in

We define a complete set of conformal invariants for pairs of spheres in ▪ and obtain from these the expressions of the conformal curvatures of curves in ( n + 1)-space in terms of the Euclidean invariants.

Gravity-driven instability in a spherical Hele-Shaw cell.

A pair of concentric spheres separated by a small gap form a spherical Hele-Shaw cell. In this cell an interfacial instability arises when two immiscible fluids flow. We derive the equation of motion for the interface perturbation amplitudes, including both pressure and gravity drivings, using a mode coupling approach. Linear stability analysis shows that mode growth rates depend upon interface perimeter and gravitational force. Mode coupling analysis reveals the formation of fingering structures presenting a tendency toward finger tip-sharpening.

Determining a Pair of Concentric Spheres for Assessing Sphericity Errors Using Discrete Voronoi Diagrams

In this paper, we propose a new procedure for computing a pair of concentric spheres with a minimum radial separation for assessing sphericity errors by constructing discrete farthest and nearest Voronoi diagrams. The properties of 3D Voronoi diagrams are exploited to determine a pair of concentric spheres with minimum radial separation such that all the data points are bounded by these two spheres. A voxel map is used to construct a discrete Voronoi diagram. The convergence of the proposed procedure to the true global minimum sphericity error is guaranteed. This procedure works for various forms of spherical measurement data, including scattered measurement points. The proposed procedure is simple in implementation and effective in computation.

Interparticle force in polydisperse electrorheological fluids

We have developed a multiple image method and an integral equation approach to compute the interparticle force for a polydisperse electrorheological (ER) fluid in which the suspending particles can have various sizes. As an illustration, we apply the formalism to a pair of dielectric spheres of different radii and calculate the force as a function of the separation. The results show that the point-dipole approximation errs considerably because many-body and multipolar interactions are ignored. The approximation becomes even worse when the particle sizes differ too much in polydisperse ER fluids. From the results, a dipole-induced-dipole (DID) model is proposed for computer simulation. The DID model accounts for many-body and multipolar interactions partially, but is simpler than the coupled-dipole model because a self-consistent solution is not needed.

Dynamic Mobility of Particles with Thick Double Layers in a Nondilute Suspension

The dynamic mobility of a nondilute suspension of spherical particles is investigated in the case where the thickness of the electrical double layer around each particle is comparable to the particle radius. A formula is obtained for the O(φ) correction in a random suspension of particles with volume fraction φ, involving an integral over the dynamic mobility of a pair of spheres. This formula is then evaluated using both analytical approximations and numerical results previously obtained for the pair mobilities and valid for low surface potentials. The effect of double-layer thickness on the O(φ) coefficient is most pronounced at low frequencies, and lessens once the hydrodynamic penetration depth is smaller than the particle radius. Various approximations are considered that use the O(φ) result to predict the dynamic mobility in concentrated suspensions, and at high frequencies these approximations are shown to give results qualitatively different from those of recent cell models.

When like charges attract: interactions and dynamics in charge-stabilized colloidal suspensions

Measured pair interactions among highly charged colloidal spheres dispersed in simple electrolytes yield several surprises. Isolated pairs of like-charged spheres are found to repel each other, much as predicted by conventional theory. The same spheres, however, can develop a strong and long-ranged attraction for each other when confined either by charged glass walls, or by neighbouring spheres. Such like-charge attractions are inconsistent with the mean-field theory for macroionic interactions. These and other experimental observations further constrain recently proposed extensions to the mean-field theory.

Comment on "Monte carlo study of structural ordering in charged colloids using a long-range attractive interaction"

A recent theoretical analysis [B. V. R. Tata and N. Ise, Phys. Rev. E 58, 2237 (1998)] of interactions and phase transitions in charge-stabilized colloidal suspensions made reference to our previously published measurements [J. C. Crocker and D. G. Grier, Phys. Rev. Lett. 73, 352 (1994); 77, 1897 (1996); A. E. Larson and D. G. Grier, Nature (London) 385, 230 (1997)] of colloidal interactions. Tata and Ise claim that our measurements cannot distinguish between predictions of the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory and those of the competing theory due to Sogami and Ise (SI). We demonstrate that the DLVO theory accurately describes the measured interactions between isolated pairs of like-charged spheres, while the SI theory fails both quantitatively and qualitatively to describe our data.

624 任意位置にある2粒子の鉛直管内沈降(GS-5 二相流)

The present work is aimed at the investigation of settling of two particles at arbitrary positions in a stagnant viscous fluid inside a vertical cylinder. The ratio of the particle diameter to the cylinder diameter is in the range of 0.0086-0.0174, and the particle Reynolds number is in the range of 0.025-0.20. The cylinder Reynolds number is assumed finite. Various motions of a pair of spheres are shown experimentally and theoretically. They moved horizontally due to the inertial effect. The theoretical results are found to be in good agreement with the experimental results.

Model for dynamics in supercooled water.

We propose a phenomenological model for the intermediate scattering function (ISF) associated with density fluctuation in low temperature water. The motivation is twofold: to extract various physical parameters associated with the ISF computed from extended simple-point-charge model water at supercooled temperatures, and to apply this model to analyze high resolution inelastic x-ray scattering data of water in the future. The ISF of the center of mass of low temperature water computed from 10 M-step molecular dynamics (MD) data shows clearly time-separated two-step relaxation with a well-defined plateau in-between. We interpret this result as due to the formation of a stable hydrogen-bonded, tetrahedrally coordinated cage around a typical molecule in low temperature water. We thus model the long-time cage relaxation by the well-known Kohlrausch form exp[-(t/tau)(beta)] with an amplitude factor which is a k-dependent Debye-Waller factor A(k), and treat the short-time relaxation as due to molecular collisional motions within the cage. The latter motions can be described by the generalized Enskog equation, taking into account the confinement effect of the cage. We shall show that the effect of the confinement changes the collisional dynamics by modifying certain input parameters in the kinetic theory by a factor [1-A(k)](1/2). We solve the generalized Enskog equation approximately but analytically by a Q-dependent triple relaxation time kinetic model. This kinetic model was previously shown to account for the large k behavior of Rayleigh-Brillouin scattering from moderately dense, simple fluids. We find that our model fits well with the MD generated collective as well as single-particle ISFs. For the short-time collisional dynamics, we fix values of the hard sphere diameter sigma and pair correlation function at contact g(sigma), without introducing any adjustable parameters. The calculated ISFs reproduce the correct Brillouin peak frequencies at low k values. From the long-time dynamics, we deduce values of the Debye-Waller factor A(k), the Kohlrausch exponent beta(k), and the cage relaxation time tau(k).