Pair Of Spheres Research Articles

RESEARCH NOTE Radial distribution functions of hard sphere mixtures

A simple method is proposed to determine radial distribution functions of pure hard sphere fluids and mixtures of hard spheres of different diameters. The range of centre-centre distances of a pair of hard spheres is divided into two parts: that from the closest approach up to the distance x m of the first minimum, and that for distances x > x m. Radial distribution functions in the former interval are determined via a method proposed earlier to calculate the background correlation function; the approach is based on the evaluation of the residual chemical potential of a homo- or heteroatomic dumb-bell and chemical potentials of the individual hard spheres in the system studied. Values of the radial distribution functions (RDFs) in the latter interval are evaluated from an analytical continuation of the RDF by employing a formula for damped oscillations. A comparison with Monte Carlo data of radial distribution functions of pure fluids and equimolar binary mixtures (with diameter ratio 0·9) reveals fair accuracy for the proposed method. Because of its simplicity, the method is especially useful for determining radial distribution functions of individual pairs of multicomponent mixtures.

Pair Interactions between Heterogeneous Spheres

Surface heterogeneities on colloidal particles may cause nonuniform charge distributions, which result in electrostatic potential profiles differing from those seen when charge is distributed uniformly. This, in turn, can give rise to attractive interactions between pairs of such surfaces which may be of greater magnitude and range than van der Waals attractions. We extend previous work on interactions between pairs of heterogeneous plates to pairs of spheres, with nonuniform constant potential or constant charge boundary conditions. As in the planar case, large attractive free energies can be obtained which are not present in the equivalent uniform surfaces. Additionally, large restraining torques can be found, affecting the rotational motion. These torques are not present in uniform charge models and may go some way toward explaining restraining torques found in some experimental systems.

A further test of the Boublik et al. equations for binary hard sphere mixtures

The Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state is examined for binary hard sphere mixtures. It is argued that this equation of state is reliable for a large diameter ratio. Monte Carlo (MC) simulations are performed for many states, some of high density and large diameter ratio. At moderate densities away from the solid-liquid transition, the MC results show reasonably good agreement with the BMCSL's prediction of the compressibility factor. This confirms the earlier findings of Jackson et al. for the validity of BMCSL in the liquid phase. The failure of the BMCSL prediction in some states is due to proximity to the solid-liquid transition and not due to the large diameter ratio. The individual pair distribution functions predicted by the BMCSL in all cases, however, can be poor, especially, for a pair of large spheres at very low concentration. Comparison of the MC distribution functions and those predicted by several theories are made.

A further test of the Boublik et al. equations for binary hard sphere mixtures

The Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state is examined for binary hard sphere mixtures. It is argued that this equation of state is reliable for a large diameter ratio. Monte Carlo (MC) simulations are performed for many states, some of high density and large diameter ratio. At moderate densities away from the solid-liquid transition, the MC results show reasonably good agreement with the BMCSL's prediction of the compressibility factor. This confirms the earlier findings of Jackson et al. for the validity of BMCSL in the liquid phase. The failure of the BMCSL prediction in some states is due to proximity to the solid-liquid transition and not due to the large diameter ratio. The individual pair distribution functions predicted by the BMCSL in all cases, however, can be poor, especially, for a pair of large spheres at very low concentration. Comparison of the MC distribution functions and those predicted by several theories are made.

The nature of particle contacts in sedimentation

An experimental study of the contact of a falling sphere with a neutrally buoyant sphere in a viscous fluid is described. The experiments used a mixture of polyalklene glycol and tetrabromoethane as the Newtonian fluid, together with a pair of equal-sized Teflon■ and nylon spheres of 6.355 mm diameter. The spheres have microscopic surface roughness, which allows them to make physical contact when in close approach. This contact breaks the symmetry of the relative trajectory of the two spheres and affects the rate at which the heavy sphere moves past the neutrally buoyant one. The experimental observations verify the roll/slip model of Davis [Phys. Fluids A 4, 2607 (1992)] for the interaction of the two spheres in contact. This model assumes that contact prevents the nominal surfaces from approaching closer than a minimum separation equal to the effective roughness height, and that the tangential component of the contact force is described by solid friction theory. A friction coefficient of 0.28±0.02 provides the best agreement between theory and experiment, with some variation observed when the experiments were repeated for the same and different pairs of spheres.

Interactions and Phase Behavior of Nanosized Particles

As the concentration of silicotungstate particles (SiW12O404-) is increased at fixed background electrolyte concentration, the suspension separates into a dense, crystalline colloidal phase in equilibrium with a dilute colloidal liquid. The solubility of the particles ranges from 2.67 to 0.2 g/mL as the background 1:1 electrolyte concentration increases from 0 to 5 M. The second virial coefficient of suspensions of silicotungstate particles is a decreasing function of electrolyte concentration and, if interpreted in terms of square well or adhesive hard sphere pair potentials, suggests the particles' feel attractions are on the order of 0−2kT at the phase boundary. Despite attractions of sufficient strength, colloidal gas/colloidal liquid phase transitions are not observed. These results are interpreted as indicating particles feel an attraction with an extent which is much smaller than the core particle diameter.

Static structure factor and collective diffusion of globular proteins in concentrated aqueous solution

We report our measurement of the time average and the temporal autocorrelation function of the intensity of light scattered by the highly monomeric globular protein, bovine γII-crystallin, in aqueous solution as a function of wave number q, protein volume fraction φ, and temperature T. The time average intensity data is used to obtain the q→0 limit of the static structure factor S(φ,T), as a function of φ and T. We show that S(φ,T) may be well characterized by modeling the proteins as interacting through the Baxter adhesive hard sphere pair interaction potential. The temporal autocorrelation function data is used to determine the collective diffusion coefficient D̃(φ,T) of the proteins as a function of φ and T. We then obtain the experimental hydrodynamic factor H̃(φ,T)≡S(φ,T)[D̃(φ,T)/D0(T)], where D0(T) is the diffusion coefficient of the individual proteins in the φ→0 limit. We find that H̃ exhibits a different φ-dependence at low (φ≤0.016) and high (φ≳0.02) protein volume fractions. In the low φ domain our data for H̃ are consistent with the theoretical result for the collective diffusion in the q→0, t→0 limit. However, for φ≳0.02 we find a deviation from single exponential decay in the autocorrelation functions, and an unexpected, large change in the slope of the H̃ vs φ relation. This crossover at such low φ suggests the existence of a heretofore unappreciated length scale in the dynamics of colloid solutions. Clearly, further theoretical insights are required to understand the origin of this crossover behavior.

Free energy model for the inhomogeneous hard-body fluid: application of the Gauss-Bonnet theorem

By capturing the correct geometrical features, the fundamental-measure free energy density functional (Rosenfeld, Y., 1989, Phys. Rev. Lett., 63, 980; 1993, J. chem. Phys., 98, 8126; 1994, Phys. Rev. Lett., 72, 3831) leads to an accurate description of the general inhomogeneous simple (atomic) fluid. The key to its derivation is the convolution decomposition of the excluded volume for a pair of spheres in terms of characteristic functions for the geometry of the two individual spheres. The extension of this functional to molecular (complex) fluids is now made possible by uncovering the relation between the convolution decomposition for spheres and the Gauss-Bonnet theorem for the geometry of convex bodies (Rosenfeld, Y., 1994, Phys. Rev. E, 50, R3318). This provides (i) a free energy functional for hard particles, for which the accurate fundamental-measure functional for hard spheres is just a special case, and (ii) a simple geometrical test for its expected accuracy; also (iii) it constitutes a powerful ...

Dielectrophoretic forces and potentials induced on pairs of cells in an electric field

A combined numerical/experimental study is reported of the membrane potentials and dielectrophoretically induced forces between cells, membrane pressures, and velocity of attraction of cells under the influence of an electric field. This study was designed to explore electrical and mechanical effects produced by a field on cells in close proximity or undergoing electrically induced fusion. Laplace's equation for pairs of membrane-covered spheres in close proximity was solved numerically by the boundary element method, and the electrically induced forces on the cells and between cells were obtained by evaluating the Maxwell stress tensor. The velocity of approach of erythrocyte ghosts or fused ghosts in a 60-Hz field of 6 V/mm was measured experimentally, and the data were interpreted by using Batchelor's theory for hydrodynamic interaction of hard spheres. The numerical results show clearly the origin of the dielectrophoretic pressures and forces in fused and unfused cells and the effects of a nearby cell on the induced membrane potentials. The experimental results agree well with predictions based on the simple electrical model of the cell. The analysis shows the strong effect of hydrodynamic interactions between the cells in determining their velocity of approach.

String support and method

A stringed musical instrument of the guitar type has a plurality of strings which extend from tuning devices on a head portion, along a neck portion, to a body portion of the instrument. A plurality of string support assemblies are mounted in the material of the musical instrument adjacent to a connection between the head and neck portions of the instrument. Each of the string support assemblies includes a pair of spheres which are held in engagement with each other. One of the strings presses against a pair of spheres to position the string relative to the head and neck portions of the instrument. In one embodiment of the invention, the spheres have different diameters. In this embodiment of the invention, the string bends around the larger one of the two spheres. The spheres, whether of the same diameter or of different diameters, are disposed in a recess which extends part way through the string support. An end of the recess is blocked by material which is received in the recess. To block the end of the recess, a punch removes material from a sheet and presses it into an open end of the recess.

Acoustic scattering by a pair of spheres

If acoustic scattering by a single sphere is the most basic problem of scalar scattering, then sound scattering by a pair of spheres is next in the hierarchy of complexity. The problem has been formulated by several approaches in the past, but no actual detailed studies have been openly published so far. Two spheres insonified by plane waves at arbitrary angles of incidence are considered. The solution of this simplest of multiple-scattering problems is generated by exactly accounting for the interaction between the two spheres, which can be strong or weak depending on their separation, compositions, frequency, and directions of observation. The tools to attack this type of problem are the (forward/backward) addition theorems for the spherical wave functions, which permit the field expansions—all referred to the center of one of the spheres—by means of Wigner (3-j) symbols. The fields scattered by each sphere are obtained as pairs of (double) sums in the spherical wave functions, with coefficients that are coupled through an infinite set of two linear, complex, algebraic equations. These are then solved (by truncation) and used to obtain (i) the scattered fields and (ii) the scattering cross section of the pair of spheres. These exact results are illustrated with many plots of the form functions at various relevant incidence angles, separations, frequencies, etc. Finally, some asymptotic approximations for this problem that are analytically simple are obtained. They are displayed and compared to the exact solutions found above, with quite satisfactory results, even for the simple approximations used here. Thus the phenomenon is described, explained, graphically displayed, physically interpreted, and reduced to a simple accurate approximation in some important cases.

Depletion of membrane skeleton in red blood cell vesicles

A possible physical interpretation of the partial detachment of the membrane skeleton in the budding region of the cell membrane and consequent depletion of the membrane skeleton in red blood cell vesicles is given. The red blood cell membrane is considered to consist of the bilayer part and the membrane skeleton. The skeleton is, under normal conditions, bound to the bilayer over its whole area. It is shown that, when in such conditions it is in the expanded state, some cell shape changes can induce its partial detachment. The partial detachment of the skeleton from the bilayer is energetically favorable if the consequent decrease of the skeleton expansion energy is larger than the corresponding increase of the bilayer-skeleton binding energy. The effect of shape on the skeleton detachment is analyzed theoretically for a series of the pear class shapes, having decreasing neck diameter and ending with a parent-daughter pair of spheres. The partial detachment of the skeleton is promoted by narrowing of the cell neck, by increasing the lateral tension in the skeleton and its area expansivity modulus, and by diminishing the attraction forces between the skeleton and the bilayer. If the radius of the daughter vesicle is sufficiently small relative to the radius of the parent cell, the daughter vesicle can exist either completely underlaid with the skeleton or completely depleted of the skeleton.

Sound Wave Mobilities in a Nondilute Suspension of Spheres

The motion of particles due to the passage of a sound wave through a nondilute, monodisperse suspension of spheres is calculated correct to the first order in particle volume fraction using the macroscopic boundary integral technique (O'Brien, R. W., J. Fluid Mech. 91, 17 (1979)). This calculation uses functions describing the hydrodynamic interactions of pairs of spheres in a sound wave of arbitrary frequency. These functions are calculated using the multipole collocation technique (Weinbaum, S., Ganatos, P., and Yan, Z., Annu. Rev. Fluid Mech. 22, 275 (1990)). The results of the sound wave mobility calculation are compared with the high frequency limit which has previously been calculated by Sangani et al. (Sangani, A. S., Zhang, D. Z., and Prosperetti, A., Phys. Fluids A 3(12), 2955 (1991)). The agreement in the limit of infinitely high frequencies is found to be excellent for particle densities more than 0.52 times that of the suspending fluid. Discrepancies exist between the theories when the corrections to the high frequency limit given in Sangani et al. are used to calculate mobilities at finite frequencies. The results of this paper are designed to be the first step in developing a rigorous approach toward using ultrasonic techniques to determine the properties (size or density) of particles in a colloidal suspension.

Dynamic simulation of flow‐induced fiber fracture

AbstractA dynamic simulation method has been developed of the fracture process of a fiber in a flow field using the particle simulation method proposed i a previous paper. The fiber is modeled with bonded spheres as a fiber model. The flexibility of the fiber model is altered by changing three parameters of the stretching, bending, and twisting constants between adjacent spheres. The stress induced in each bond of the fiber model as a result of deformation is formulated using displacement of the bodn distanc, bond angle, and torsion angle fr each pair of spheres. After deformation, the fiber model fractures at the bond at which the stress surpasses the strength of the fiber. The motion of the fiber model in a flow field is determined by solving the translational and rotational motion equations for individual spheres under the hydrodynamic force and torque exerted on them. The correctness of the method and formulation was verified by comparing the simulated deflection curve of a cantilever beam (with a concentrated load at the end) with the theoretical curve. Good agreement was found in both the deflection and slope of the beam. The fracture process of a fiber after bending deformation in a two‐dimensional siimple shear flow was simulated under assumptions of an infinitely dilute system, no hydrodynamic interaction, and a low Reynolds number of a particle. The calculated critical conditions of the flow field for fiber fracture were compared with Forgacs and Mason's theoretical ones. Simulated values of the fracture condition of the fluid shear stress related to the Young's modulus of a fiber agree with theoretical ones over an aspect ratio of 15.

Optical properties of a pair of spheres: comparison of different theories

We compare the optical absorption of two touching and non-touching spheres, as calculated by different approaches. In the quasistatic limit, an exact treatment based on the conformal transformation of coordinate frames has recently been developed. On the other hand, a multipolar expansion of the rigorous solution of Maxwells equations can in principle be used for arbitrary aggregates and beyond the quasistatic limit. Our aim is to study the convergence and ascertain the accuracy of the low multipolar orders of the latter theory. Calculations were carried out for gold particles in air. A comparison of the exact theory and the dipolar approximation shows a very good agreement for spheres that are one half to one radius apart. In the case of touching spheres the convergence of the multipolar expansion is very slow and there are considerable quantitative discrepancies between the theories.

The electric field instrument on the polar satellite

The Polar satellite carries a system of four wire booms in the spacecraft spin plane and two rigid booms along the spin axis. Each of the booms has a spherical sensor at its tip along with nearby guard and stub surfaces whose potentials relative to that of their sphere are controlled by associated electronics. The potential differences between opposite sphere pairs are measured to yield the three components of the DC to >1 MHz electric field. Spheres can also be operated in a mode in which their collected current is measured to give information on the plasma density and its fluctuations. The scientific studies to be performed by this experiment as well as the mechanical and electrical properties of the detector system are described.

Fluid–fluid phase separations in nonadditive hard sphere mixtures

We investigated the phase stability of a system of nonadditive hard sphere (NAHS) mixtures with equal diameters, d, between like species and an unequal collision diameter, d(1+α), between unlike species. It is based on an analytic equation of state (EOS) which refines an earlier expression [J. Chem. Phys. 100, 9064 (1994)] within the mixed fluid phase range. The new EOS gives a reliable representation of Monte Carlo EOS data over a wide range of density, composition, and nonadditivity parameters (α). Comparisons with available computer simulations show that the new EOS predicts satisfactory phase boundaries and the critical density line. It is superior to results derived from integral equations (the Percus–Yevick, the Martynov–Sarkisov, and the modified Martynov–Sarkisov) and analytic theories (the MIX1 model, the van der Waals one-fluid model, and the scaled particle theory). The present study shows that, unless α exceeds 0.026, the fluid phase will remain fully miscible up to the freezing point of pure hard spheres. We have also investigated structural aspects of the phase stability by Monte Carlo computations. The radial distribution functions, the local mole fraction, and coordination numbers for like and unlike pairs of hard spheres exhibit significant number dependencies close to the fluid phase boundary. They provide precursory signals to an impending phase change. Finite systems used in the Monte Carlo sampling limit fluctuations in sizes and shapes of heterogeneous clusters. The observed number dependence simply reflects this fact.

Three-dimensional co-location of RNA polymerase I and DNA during interphase and mitosis by confocal microscopy.

The relative three-dimensional co-location of RNA polymerase I (RPI) and DNA was studied using confocal laser scanning microscopy during interphase and all the steps of mitosis in human cancerous cells. For each step of the cell cycle, immunolabeled RPI molecules and DNA specifically stained with chromomycin A3 were simultaneously imaged at high resolution through numerous optical sections. Then, all the data obtained were used to generate transverse sections, anaglyphs and volumic representations, which are all prerequisite approaches to a representative study of the three-dimensional organization of the nucleolus and the mitotic chromosomes. Our results indicated that in the interphasic nuclei, in which DNA is organized as a regular 3-D network, RPI was present within numerous irregular spheres arranged as several twisted necklaces. During metaphase, RPI labeling was segregated into pairs of spheres and typical crescent-shaped structures; both were centrally located within the set of chromosomes. During anaphase and telophase, a typical central and symmetric arrangement of labeled structures was systematically seen among the decondensing chromosomes, arranged as a regular cylinder and as a hollow half-sphere, respectively. This typical 3-D organization of structures containing RPI relative to DNA is another strong example of the non-random organization of the genome during interphase and mitosis.

Scattering and absorption cross sections of compounded spheres I Theory for external aggregation

The cross section for total scattering by clusters of spheres is derived from an integration, over a closed spherical surface, of the scattered Poynting flux associated with the different pairs of spheres in the ensemble. With the use of the addition theorem for vector spherical harmonics the integral can be evaluated analytically. The pairwise cross sections can be rearranged into an expression for the scattering cross section of sphere aggregates that is analogous to that obtained from Lorenz–Mie theory for a single sphere. The latter formulation, however, is more difficult to treat numerically than is the summation over pairwise cross sections.

Calculation of total cross sections of multiple-sphere clusters

A method for calculating the extinction, absorption, and scattering cross sections of clusters of neighboring spheres for both fixed and random orientations is developed. The analysis employs the superposition formulation for radiative interactions among spheres, in which the total field from the cluster is expressed as a superposition of vector spherical harmonic expansions about each of the spheres in the cluster. Through the use of addition theorems a matrix equation for the expansion coefficients is obtained. Further application of addition theorems on the inverse of the coefficient matrix is shown to yield analytical expressions for the orientation-averaged total cross sections of the sphere cluster. Calculations of the cross sections of pairs of spheres and fractal aggregates of several spheres are presented. It is found that a dipole representation of the field in each sphere does not adequately predict the absorption cross section of clusters of small-size-parameter spheres when the spheres are highly conducting. For this situation several multipole orders are required for an accurate calculation of the absorption cross section. In addition, the predicted absorption of sphere clusters can be significantly greater than that estimated from the sum of the isolated-sphere cross sections.