Equal-sized Spheres Research Articles

On the motion of spheres in a fluid at low Reynolds numbers. (4th Report. Two equal-sized spheres moving in parallel with each other).

The velocities, angular velocities and hydrodynamic forces experienced by equalsized spheres freely-falling in parallel with each other in glycerol are measured using the`falling sphere'technique at low Reynolds numbers. The width of the test tank with a square cross section was 100 times as large as the diameter of the sphere. The width of the test tank with a square cross section was 100 times as large as the diameter of the sphere. The measurements, with an estimated uncertainty of ±1.4%, show that two equal-sized spheres moving in parallel with each other interact hydrodynamically at a distance within six times the diameter of the sphere at Reynolds numbers below 1.55, and that the drag force experienced by one of the two spheres is well approximated by the formula proposed here at low Reynolds numbers.

Note on the direct calculation of mobility functions for two equal-sized spheres in Stokes flow

A simple, direct method is presented for the calculation of mobility functions for the translational and rotational velocities and stresslets of two equal-sized spheres in unbounded low-Reynolds-number flow when the ambient velocity field is a superposition of a uniform stream, a vorticity and a rate-of-strain field. Our numerical procedure furnishes accurate values for touching spheres and coefficients for the near-field asymptotic expansions. The singular behaviour of the mobility functions is clarified. These results have been used to determine accurately the coefficient of theO(c2) term in the expression for the bulk stress in a suspension of spheres (6.95c2instead of 7.6c2).

A model for soil surface roughness influence on the spectral response of bare soils in the visible and near-infrared range

A mathematical model dealing with the influence of soil surface roughness on soil reflectance in the visible and near-infrared range is discussed. The model is based on the assumption that the reflectance from anisotropic rough soil surfaces is strongly correlated with the area of shadowed soil fragments, and therefore depends on the degree of soil roughness as well as on the illumination and the viewing geometry. A rough soil surface is simulated by equal-sized spheres arranged such that their centers form a square grid on a freely sloping plane. This model converts the bare soil reflectance data obtained for smooth samples into data relating to any natural rough surface states under conditions of unlimited illumination, which are defined by the solar altitude, the angle of a slope, and the sloping of the soil surface relating to the sunbeams direction. A simple parameter of the state of soil roughness expressing a proportion of the aggregates' and clods' area in top view in a given soil surface area is used here. The correctness of this model was evaluated on spectral data from 12 air-dried undisturbed soil samples obtained by means of a field spectrophotometer in outdoor conditions at solar altitudes from 31° to 59°. The measurements were taken for soil surfaces sloping at angles of 0°, 10°, 20°, and 30° forward and backward to the direction of sunbeams. Linear regression analysis indicates that for the studied spectral range the correlation coefficient of the relationship between all the 368 data point measured and calculated using this model reaches values of approximately 0.99. It has been found that relation between measured and predicted reflectance coefficient is similar for all the tested soils.

Rate of heat conduction from a heated sphere to a matrix containing passive spheres of a different conductivity

Using a multipole expansion scheme, we calculate the rate of heat conduction Q from a heated sphere with constant surface temperature to a matrix of conductivity k, containing a dilute dispersion of equal-sized passive spheres of conductivity αk, and whose temperature far away from the heated sphere vanishes. Values are presented for Q for various choices of α and λ, the ratio of the radius of a passive sphere to that of the heated sphere, and asymptotic expressions are derived for the cases λ≪1 and λ≫1, respectively. When the passive spheres are very small compared to the heated sphere, the composite can be viewed as an effective continuum on the length scale of the latter with an effective thermal conductivity relative to the matrix given by Maxwell’s expression, k*=1+3βc, where β=(α−1)/(α+2) and c is the inclusion volume fraction assumed to be small. We find, however, that this effective continuum approach can still be applied with a surprising degree of accuracy over the whole range of α even when λ is as large as O(10), provided that the effective thermal conductivity is allowed to vary with r, the distance from the center of the heated sphere, according to the expression k*≡1+3βρ(r), where ρ(r) denotes the probability that a point in the composite lies within a passive sphere. It is shown, furthermore, that this effective continuum approach becomes exact to O(α−1) for all λ.

Effective properties of two-phase disordered composite media: II. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres.

We evaluate third-order bounds on the effective conductivity ${\ensuremath{\sigma}}_{e}$ and effective bulk modulus ${K}_{e}$ of a random dispersion of equal-sized impenetrable spheres in a matrix up to sphere volume fractions near the random close-packing value. The third-order bounds, which incorporate an integral ${\ensuremath{\zeta}}_{2}$ that depends upon the three-point probability function of the two-phase medium, are shown to significantly improve upon second-order Hashin-Shtrikman bounds, which do not utilize this information, for a wide range of phase property values and volume fractions. The physical significance of the microstructural parameter ${\ensuremath{\zeta}}_{2}$ for general microstructures is briefly discussed. The third-order bounds on ${\ensuremath{\sigma}}_{e}$ and ${K}_{e}$ are found to be sharp enough to yield good estimates of the bulk properties for a wide range of sphere volume fractions, even when the phase property values differ by as much as two orders of magnitude. Moreover, when the spheres are highly conducting or highly rigid relative to the matrix, the third-order lower bound on the respective effective property provides a useful estimate of it for a wide range of sphere volume fractions.

Bulk properties of two-phase disordered media. I. Cluster expansion for the effective dielectric constant of dispersions of penetrable spheres

We derive a cluster expansion for the effective dielectric constant ε* of a dispersion of equal-sized spheres distributed with arbitrary degree of impenetrability. The degree of impenetrability is characterized by some parameter λ whose value varies between zero (in the case of randomly centered spheres, i.e., fully penetrable spheres) and unity (in the instance of totally impenetrable spheres). This generalizes the results of Felderhof, Ford, and Cohen who obtain a cluster expansion for ε* for the specific case of a dispersion of totally impenetrable spheres, i.e., the instance λ=1. We describe the physical significance of the contributions to the average polarization of the two-phase system which arise from inclusion-overlap effects. Using these results, we obtain a density expansion for ε*, which is exact through second order in the number density ρ, and give the physical interpretations of all of the cluster integrals that arise here. The use of a certain family of equilibrium sphere distributions is suggested in order to systematically study the effects of details of the microstructure on ε* through second order in ρ. We show, furthermore, that the second-order term can be written as a sum of the contribution from a reference system of totally impenetrable spheres and an excess contribution, which only involves effects due to overlap of pairs of inclusions. We also obtain an expansion for ε* which is exact through second order in φ2, where φ2 is the sphere volume fraction. We evaluate, for concreteness, some of the integrals that arise in this study, for arbitrary λ, in the permeable-sphere model and in the penetrable concentric-shell model introduced in this study.

Percus-Yevick results for a binary mixture of hard spheres with non-additive diameters

We have carried out extensive calculations, within the Percus-Yevick approximation, for a binary mixture of hard spheres with negative departures from additivity in their hard sphere diameters. The results presented in this work are restricted to the case of equal size spheres at equimolar composition. We have calculated the virial equation of state and show that the more negative the non-additive parameter, the better the agreement between our results and the Monte Carlo results of Adams and McDonald. Moreover, we find no evidence of a phase transition for this system. Our calculations of the radial distribution functions gij(r) show clearly the important changes brought about by non-additivity, in particular, to g 12(r). We have also calculated the number-concentration partial structure factors, which highlights the ordering effects taking place in the system as a results of non-additivity. Our results indicate that, if size difference between the hard spheres is included, non-additive hard spheres with...

Toward an alignment of the actin molecule within the actin filament

An understanding of the specific interactions of actin with regulatory proteins has been limited by the lack of information about the structure of the actin filament. Molecular actin has been studied in actin-DNase I complexes by single crystal X-ray analysis, to a resolution of about 0.6nm, and in the electron microscope where two dimensional actin sheets have been reconstructed to a maximum resolution of 1.5nm. While these studies have shown something of the structure of individual actin molecules, essential information about the orientation of actin in the filament is still unavailable.The work of Egelman & DeRosier has, however, suggested a method which could be used to provide an initial quantitative estimate of the orientation of actin within the filament. This method involves the quantitative comparison of computed diffraction data from single actin filaments with diffraction data derived from synthetic filaments constructed using the molecular model of actin as a building block. Their preliminary work was conducted using a model consisting of two juxtaposed spheres of equal size.

Translational diffusion of axisymmetrical particles in shear flow

The motion of axisymmetric particles undergoing translational and rotary Brownian motion and subjected to a simple shear flow is studied theoretically and experimentally. In general, translational diffusion is affected by rotary Brownian motion because the displacement of particles depends on their orientation. From solutions of the translational convective diffusion equation, calculations have been made of the mean-square displacement dyadic for particles in an unbounded simple shear flow. The components of this dyadic can be expressed in terms of integrals containing orientational averages. Analytical results for high rotary Péclet number are worked out. It is found that at large times the diffusion in the equatorial plane of the shear field can be described by an effective diffusion coefficient D eff. Because the effective diffusion constant depends only weakly on the orbit constant, it is to be expected that for arbitrary Péclet numbers, D eff does not deviate much from the value at high Péclet numbers. Similarly, diffusion in the direction perpendicular to the equatorial plane can be, at large times, described by an effective diffusion constant as well. Experiments performed with a newly designed traveling microtube apparatus confirm certain aspects of the theory. Measurements have been made of the mean-square displacement 〈 x 3 ∗2 〉 in the direction of flow of doublets consisting of touching equal-sized polystyrene latex spheres, subjected to Poiseuille flow, with a rotary Péclet number of order unity. It was found that at short times 〈 x 3 ∗2 〉 varied linearly with time, in agreement with a generalization of the early results of Einstein; at large times, the displacement was found to increase cubically with time and the diffusion could be described by an effective diffusion coefficient, which value was in good agreement with theory.

The effective conductivity of a periodic array of spheres

We consider the problem of determining the effective conductivity k * of a composite material consisting of equal-sized spheres of conductivity α arranged in a cubic array within a homogeneous matrix of unit conduc­tivity. We modify Zuzovski & Brenner’s (1977) method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by McKenzie et al . (1978) using a method originally devised by Rayleigh (1892). On solving these equations we derive expressions for k * to O ( c 9 ) ─ c being the volume fraction of the spheres ─ for simple, body-centred and face-centred cubic arrays, and also obtain numerical values for k * over the whole range of α and c . We show that these results for cubic arrays can be used to estimate k * for random arrays of identical spheres. For arrays of highly conducting and nearly touching spheres, Batchelor & O’Brien (1977) showed that k * ∼ {─ K 1 In (1 ─ χ ) ─ K 2 ( α = ∞, χ = ( c / c max ) ⅓ → 1), 2 K 1 In α ─ K ʹ 2 ( χ = 1, α → ∞), where c max corresponds to the volume fraction when the spheres are actually touching each other, and determined K 1 for the three cubic arrays. Our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants K 2 and Kʹ 2 thereby obtained do not quite satisfy the relation Kʹ 2 = K 1 (3.9 ─ In 2) + K 2 given by Batchelor & O’Brien. We have been unable to find the reason for this slight discrepancy.

Morphology of bovine fibrinogen monomers and fibrin oligomers

Molecules of bovine fibrinogen and small oligomers of thrombin-induced fibrin have been examined by electron microscopy after contrast enhancement by rotary shadowing and by negative staining. Rotary-shadowed molecules, in addition to exhibiting the well-known trinodular structure, showed a spine of fairly uniform width connecting the central nodule with the terminal protrusions. In specimens negatively stained with uranyl formate or acetate most of the molecules had a gently sigmoidal configuration, all of like handedness. The spines were visibly prominent and frequently had a slight central bulge. The terminal protrusions appeared with a distinctly oblong shape and were oriented parallel to each other, with the long dimension tilted about 45 ° to the axis of the contiguous part of the spine. After exposure of the fibrinogen molecules to millimolar Ca 2+ the central bulge became more noticeable and each terminal protrusion took on the form of two contiguous, equal-sized spheres. Significant dimensions are: overall length of molecule, 45 nm; length and width of spine, 31 nm × 3 nm; length and width of terminal protrusions, 9 nm × 4 nm; diameter of Ca 2+-induced spheres, ~ 4 nm. Specimens from fibrinogen solutions to which minute amounts of thrombin had been added (and the macroscopic clots removed) contained oligomeric assemblies that were small enough to be resolvable into their fibrin subunits. They consisted of two chains that were parallel and were similar to each other in that they never differed by more than one in their subunit number. The elements in one chain of an oligomer appeared to be bound to those of the other chain with a half-molecule staggered overlap. They retained the lengths they had as fibrinogen molecules.

A three dimensionally periodic, eleven coordinated, dense packing of symmetry equivalent spheres

Symmetry equivalent equal size spheres can be packed in an array filling 71.87% of space. Each sphere in this packing has eleven neighbors at unit distance. The density of this tetragonal close packing is second only to closest packing with twelve coordination. Anions in numerous compounds, including some which are solid electrolytes, arrange themselves in this pattern. It occurs among others in rutiles, β-BeO, Li 4GeO 4 and CaSO 4.

Binary mixture of hard spheres with attractive square-well interaction between unlike spheres

We report calculations of partial structure factors, concentration and particle density fluctuations for a binary mixture of equal-sized hard spheres at equimolar composition, where spheres of different species interact via an attractive square well. This model exhibits the qualitative features attributed to liquid binary alloys with compound-forming tendencies.

Ｃｏ‐ＷおよびＮｉ‐Ｐ合金めっきの構造について 熱力学的消極作用説に対する反論

A structural study of Co75W25 (atomic %) and Ni100-xPx (x=16-25) alloys both prepared from aqueous solutions has been made to see whether they have nonequilibrium structures. Their structural models were constructed and examined by the correlation method. The model for the Co-W alloy consists of close packed tetrahedra, each composed of four hard spheres of equal size. The tetrahedron contains three Co atoms and one W atom of 2.55A in diameter. In modeling the Ni-P alloys, however, Ni and P atoms were treated as non-hard spheres of different mean diameters; 2.61Å and 1.88Å, respectively. Furthermore, Ni-P alloys, irrespective of their alloy compositions, were treated as the mixture of structural units Ni2P and isolated Ni atoms. These units and atoms were randomly deposited on a plane surface of (100) in the bcc structure. Interference functions calculated from the above models agree quite well with x-ray diffraction data of these alloys. The atomic arrangement in these models forms distorted icosahedra which forbid crystallinity in the alloys, indicating that they indeed have nonequilibrium structures. It should be noticed that the present result violates the thermodynamical depolarization theory, which does not allow the deposition from aqueous solutions of nonequilibrium tungsten or phosphorus alloys.

Chains of spheres in shear and electric fields

The behaviour of electrically formed linear chains of rigid conducting spheres of equal size suspended in a Newtonian, dielectric fluid was studied in simple shear at rateG and in uniform electric fields of strengthEo directed across the planes of shear. At lowG (Eo = 0) the chains rotated as single rigid axisymmetric particles; at highG they broke up into smaller rigid fragments. WhenE0 ≠ 0, (G = 0), they became oriented with their axes parallel to the electric field direction. In combined shear and electric field they behaved either as rigid or as flexible particles depending on the interparticle gaps.

The kinetics of flowing dispersions: VIII. Doublets of rigid spheres (theoretical)

A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutral, equal-sized spheres in a viscous fluid undergoing uniform shear flows is presented. The method is based on the matrix formulation of the hydrodynamic resistances by Brenner and O'Neill, and it is illustrated for simple shear and planar and axisymmetric extensional flows, for which the trajectories of pairs of sphere centers are calculated. It is shown that, in simple shear, trajectories are either open or closed; in extensional flows, all trajectories must be open. This has interesting implications in the dispersion of aggregates by shear. Although the translational and rotational behavior of interacting spheres is characterized by the type of flow, the behavior of the spheres in each flow is calculable from flow-independent quantities. Special emphasis is placed on the collision behavior of doublets. Initially separate spheres can never touch, but they can come into very close proximity, especially during equatorial encounters. In simple shear, translating spheres rotate about their own axes, which, in a doublet, causes one sphere to roll relative to the other except when they touch or are rigidly connected. The rotations of free and connected doublets and dumbbells are related to those of prolate spheroids.

The kinetics of flowing dispersions: IX. Doublets of rigid spheres (experimental)

Experiments to test hydrodynamic interaction theories of a pairs of spheres of equal size in a viscous liquid were made in simple shear. Excellent agreement with theory was found for the trajectories of the spheres which could be characterized by two measurable parameters for permanent and separating doublets colliding both equatorially and nonequatorially. Agreement with theory was also found for the angular velocities of each sphere and their relative rolling velocities in equatorial collisions. Also determined were the periods of rotations of permanent nontouching and touching doublets for which average equivalent spheroidal axis ratios were calculated and found to be in reasonable agreement with theory. Touching doublets (which could not be formed without the aid of electric fields) separated because of surface roughness, which, although microscopically small, forced spheres initially moving on closed trajectories into open ones, thus changing their paths of recession from those of approach. During such irreversible encounters the spheres locked onto one another temporarily to form a rigid doublet until separation occurred when their line of centers was normal to the direction of the shear flow. Irreversibility was reduced by coating the spheres with very viscous liquids miscible with the medium. Doublets formed in electric fields were held together by an adhesive bond due to polymerized oil in the interparticle gap formed during dielectric breakdown caused by electric field intensification between the approaching particles. The bond could be broken above a critical shear rate, from which estimates of the adhesive bond strength could be made.

Streamlines around single spheres and trajectories of pairs of spheres in two-dimensional creeping flows

A general two-dimensional linear flow having pure shear (or two-dimensional extensional) flow as one limit and pure rotation as the other, with simple shear flow as an intermediate case is discussed. Calculations of the streamlines around a rigid, neutrally buoyant sphere in creeping flow indicate that there are both closed and open streamlines in the flow regimes between pure and simple shear, whereas only closed streamlines exist between simple shear and pure rotation. The only flow in which closed streamlines do not exist is pure shear. The trajectories of two neutral spheres of equal size in this family of flows have also been calculated. Closed trajectories do not exist in a large regime of flows near pure shear. The significance of these findings with particular reference to hydrodynamic dispersion is discussed.

The microrheology of colloidal dispersions: IV. Pairs of interacting spheres in shear flow

The effect of interaction forces on the trajectories of pairs of equal-sized spheres in shear flow is calculated. Depending on the functional form of the interaction potential there are three modes of behavior: (i) the spheres can collide and form a touching doublet, (ii) they can be captured in an energy minimum, orbiting around each other after capture, or (iii) the interactions are transitory and the spheres separate. For interactions determined jointly by van der Waals attraction and electrostatic repulsion, we have calculated the conditions under which (i) touching doublets from (i.e., in the so-called primary energy minimum) and the system coagulates, (ii) nontouching doublets form (in a secondary minimum) and the system flocculates, or (iii) no permanent doublets form and the system is stable. The results indicate that when Brownian motion can be neglected, dilute spherical sols can be unstable at low and high shear rates, but stable in between.

Theoretical calculation of dense random packings of equal and non-equal sized hard spheres applications to amorphous metallic alloys

A model describing the structure of amorphous metallic alloys is proposed using a packing of non-equal-sized hard spheres. Models containing up to 1000 spheres were generated by computation. Hypotheses guiding the calculation are chosen in order to obtain a model as compact as possible and to take into account affinity between metal and metalloids: therefore two small spheres are not allowed to be in hard contact. This method of calculation is first tested by building up heaps of equal-sized spheres. Both the radial distribution function and the interference function are calculated for each model and the variations of these curves with the number of spheres are studied. In the case of two sizes of spheres heaps calculations are devoted to the study of the so-called ‘shoulder interference function’ which characterizes many amorphous alloys such as NiP, PdSi, etc. It is shown that the existence of such a shoulder depends on two parameters: the ratio of sphere diameters and the relative concentration of small and large spheres. These results are in good agreement with some recent experimental observations. This shoulder is well-marked when the diameter ratio reaches 10% and when the concentration in small spheres lies in the range 10–15%. In such a structure (e.g. amorphous NiP and PdSi alloys) it is observed that small spheres are always surrounded by nine large spheres such as P or Si atoms in crystalline Ni 3P or Pd 3Si. Large spheres, i.e. metallic atoms, form distorted icosahedra, the distortion of which varies from one to the other. The short-range order has a five-fold symmetry which is not compatible with any long-range order.