Touching Spheres Research Articles

The motion of a self-propelling two-sphere swimmer in a weakly viscoelastic fluid

We study analytically the propulsion of a force- and torque-free swimmer composed of two counterrotating spheres of differing radii through a viscoelastic fluid described by the Giesekus constitutive model. Our analysis includes both swimmers composed of directly touching spheres and those composed of spheres separated by some finite distance. The propulsion speed of the swimmer is calculated by first expanding the equations of motion and the Giesekus constitutive model as a power series in the Weissenberg number, and then using the Lorentz reciprocal theorem to determine the first-order propulsion speed using the known flow fields for rotating and translating two-sphere geometries at zeroth order. We calculate the relative rotation speeds of the two spheres necessary to maintain the torque-free condition, including an approximate correction for fluid elasticity. The impact of the separation distance between the two spheres, the ratio of their radii, and the value of the Giesekus mobility parameter α on the propulsion speed are all examined; we find that the propulsion speed of the swimmer is maximized for two touching spheres with a radius ratio of approximately 0.7, with a Giesekus mobility parameter of α=0, corresponding to an Oldroyd-B fluid. We also quantify how the increased shear-thinning in the fluid, represented by increasing values of α, results in a significant decrease in the swimmer speed. Finally, through calculations of the fluid stresses around the two-sphere swimmer, we demonstrate the development of enhanced hoop stresses around the smaller sphere, which drive the expulsion of stretched fluid behind the smaller sphere and induce motion of the swimmer in the direction of the larger sphere.

On smoothness conditions and selection of the edge of a scattering surface in one class of 3D time-optimal problems

A class of time-optimal problems in three-dimensional space with a spherical velocity vectogram is considered. The target set is a parametrically defined smooth curve. Numerical and analytical approaches to constructing the bisector of the target set — the scattering surface — in the time-optimal problem are proposed. The algorithms are based on formulas for the edge points of the scattering surface, written in terms of curve invariants. It is shown that these points form the edge of the bisector and lie at the centers of the touching spheres to the curve. A theorem on sufficient conditions for the smoothness of a scattering surface is proven. The equations of the tangent plane to the bisector are found for those points from which exactly two optimal trajectories emerge. An example of solving a time-optimal problem in the form of a set of level surfaces of the optimal result function is given, highlighting the surface of their non-smoothness.

Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry- Breaking of a Polarizable Dimer in Non-Uniform Fields.

A theoretical framework is presented for calculating the polarization, electro-rotation, travelling-wave dielectrophoresis, electro-hydrodynamics and induced-charge electroosmotic flow fields around a freely suspended conducting dimer (two touching spheres) exposed to non-uniform direct current (DC) or alternating current (AC) electric fields. The analysis is based on employing the classical (linearized) Poisson–Nernst–Planck (PNP) formulation under the standard linearized ‘weak-field’ assumption and using the tangent-sphere coordinate system. Explicit expressions are first derived for the axisymmetric AC electric potential governed by the Robin (mixed) boundary condition applied on the dimer surface depending on the resistance–capacitance circuit (RC) forcing frequency. Dimer electro-rotation due to two orthogonal (out-of-phase) uniform AC fields and the corresponding mobility problem of a polarizable dimer exposed to a travelling-wave electric excitation are also analyzed. We present an explicit solution for the non-linear induced-charge electroosmotic (ICEO) flow problem of a free polarized dimer in terms of the corresponding Stokes stream function determined by the Helmholtz–Smoluchowski velocity slip. Next, we demonstrate how the same framework can be used to obtain an exact solution for the electro-hydrodynamic (EHD) problem of a polarizable sphere lying next to a conducting planar electrode. Finally, we present a new solution for the induced-charge mobility of a Janus dimer composed of two fused spherical colloids, one perfectly conducting and one dielectrically coated. So far, most of the available electrokinetic theoretical studies involving polarizable nano/micro shapes dealt with convex configurations (e.g., spheres, spheroids, ellipsoids) and as such the newly obtained electrostatic AC solution for a dimer provides a useful extension for similar concave colloids and engineered particles.

Frustrated Bearings.

In a bearing state, touching spheres (disks in two dimensions) roll on each other without slip. Here we frustrate a system of touching spheres by imposing two different bearing states on opposite sides and search for the configurations of lowest energy dissipation. If the dissipation between contacts of spheres is viscous (with random damping constants), the angular momentum continuously changes from one bearing state to the other. For Coulomb friction (with random friction coefficients) in two dimensions, a sharp line separates the two bearing states and we show that this line corresponds to the minimum cut. Astonishingly, however, in three dimensions intermediate bearing domains that are not synchronized with either side are energetically more favorable than the minimum-cut surface. Instead of a sharp cut, the steady state displays a fragmented structure. This novel type of state of minimum dissipation is characterized by a spanning network of slipless contacts that reaches every sphere. Such a situation becomes possible because in three dimensions bearing states have four degrees of freedom.

Effect of spherical pores coalescence on the overall conductivity of a material.

The problem about steady-state temperature distribution in a homogeneous isotropic medium containing a pore or an insulating inhomogeneity formed by two coalesced spheres of the same radius, under arbitrarily oriented uniform heat flux, is solved analytically. The limiting case of two touching spheres is analyzed separately. The solution is obtained in the form of converged integrals that can be calculated using Gauss-Laguerre quadrature rule. The temperature on the inhomogeneity's surface is used to determine components of the resistivity contribution tensor for the insulating inhomogeneity of the mentioned shape. An interesting observation is that the extreme values of these components are achieved when the spheres are already slightly coalesced.

Densest helical structures of hard spheres in narrow confinement: An analytic derivation

The emergence of helicity from the densest possible packings of equal-sized hard spheres in narrow cylindrical confinement can be understood in terms of a density maximization of repeating microconfigurations. At any cylinder-to-sphere diameter ratio D∈(1+3/2,2), a sphere can only be in contact with its nearest and second nearest neighbors along the vertical z-axis, and the densest possible helical structures are results of a minimized vertical separation between the first sphere and the third sphere for every consecutive triplet of spheres. By considering a density maximization of all microscopic triplets of mutually touching spheres, we show, by both analytical and numerical means, that the single helix at D∈(1+3/2,1+43/7) corresponds to a repetition of the same triplet configuration and that the double helix at D∈(1+43/7,2) corresponds to an alternation between two triplet configurations. The resulting analytic expressions for the positions of spheres in these helical structures could serve as a theoretical basis for developing novel chiral materials.

On the minimum of the mean-squared error in 2-means clustering

We study the minimum mean-squared error for 2-means clustering when the outcomes of the vector-valued random variable to be clustered are on two touching spheres of unit radius in $n$-dimensional Euclidean space and the underlying probability distribution is the normalized surface measure. For simplicity, we only consider the asymptotics of large sample sizes and replace empirical samples by the probability measure. The concrete question addressed here is whether a minimizer for the mean-squared error identifies the two individual spheres as clusters. Indeed, in dimensions $n \ge 3$, the minimum of the mean-squared error is achieved by a partition that separates the two spheres and has unit distance between the points in each cluster and the respective mean. In dimension $n=2$, however, the minimizer fails to identify the individual spheres; an optimal partition is obtained by a separating hyperplane that does not contain the point at which the spheres touch.

Aggregate shape determination via light scattering by aligned and randomly oriented polydisperse aggregates

We present calculations of light scattered by simulated aligned and randomly oriented fractal aggregates of touching spheres. The alignment by an applied electric field is in the direction yielding the minimum energy associated with the polarizability of the aggregate. This direction is also within a few degrees of the direction associated with the smallest inertial eigenvalue. Aggregates were generated to have a fractal dimension of 1.78 (characteristic of diffusion–limited cluster-cluster aggregation) and to simulate the range of aggregate sizes (30 to 2000) and size distributions for post-flame generated soot by a range of hydrocarbon fuels. More nearly monodisperse aggregates were also generated to simulate the size distributions obtained via mobility or mass classification. We show that a ratio of slopes computed from small angle light scattering intensities (Guinier plots) is related to the shape parameter A31, which is the ratio of the largest to the smallest principle radii of gyration of the inertia tensor. A geometric interpretation of the overall shape is presented based on the semi-axes of an ellipsoid. Results on the correlation between the maximum structure factor ratio {S(q)}a/ {S(q)}r for the monodisperse and polydisperse clusters and the average value of A31 indicate that light scattering measurements in the q range of fractal behavior would also be feasible for shape characterization. Our calculations indicate that light scattering measurements have a good potential for characterizing aggregate shape with an advantage of being more sensitive to shape than mobility measurements.

Two-spheres method for evaluating the metrological structural resolution in dimensional computed tomography

In recent years, x-ray computed tomography has been successfully applied as an innovative coordinate measurement technology for dimensional metrology. An important characteristic to be evaluated when testing the metrological performances of computed tomography systems is the metrological structural resolution for dimensional measurements, which describes the size of the smallest structure that can still be measured within error limits to be specified. The ‘two-spheres’ concept allows for the investigation of the metrological structural resolution by using a simple reference standard consisting of two touching spheres with the same nominal diameter. This work is aimed at defining and validating an enhanced method based on the ‘two-spheres’ concept and on a new measurement strategy. Advantages in using this method are discussed and a selection of the factors influencing the results are evaluated through experimental and simulation analyses.

A general method for determining molecular interfaces and layers

A general and direct computational scheme to locate the surface separating arbitrarily shaped domains made up of molecules (or any other particles) has been developed and is described and illustrated for several, both artificial and physical examples. The proposed scheme consists of two modules: (i) triangulation and (ii) assignment of simplices to domains. Three different triangulation methods are employed, viz., the Delaunay triangulation, regular triangulation, and quasi-triangulation. In the triangulated system, the assignment step is carried out in two different ways, one based on the characteristic metric of a particular triangulation procedure and the other on the concept of a touching sphere. Some of the combinations of the triangulation and assignment steps lead to methods already used by others to find interfacial or surface molecules, namely the alpha-shape-based method of Usabiaga nad Duque [Phys. Rev. E 79 (2009) 046709] and GITIM of Sega et al. [J. Chem. Phys. 138 (2013) 044110]. The resulting surface is defined not only as a discrete set of particles, but it is build up of facets of the triangulation forming a broken line in two dimensions or a polyhedral surface in three dimensions. Individual molecular layers are identified in a very straightforward manner, starting with the interfacial layer itself and proceeding into the interior of the phase. The proposed scheme is illustrated first by identifying border molecules of pre-sampled domains of several shapes in a plane and then applied to five physically meaningful examples: thin films, near critical water, liquid water slab in an electric field, liquid water at a solid wall, and water at condition of electric-field-induced jetting. Performance of the considered methods is critically assessed. Treatment of domains forming percolating clusters through periodic boundary conditions is also described along with the determination of their periodicity and dimensionality.

Phase diagram of heteronuclear Janus dumbbells.

Using aggregation-volume-bias Monte Carlo simulations along with successive umbrella sampling and histogram re-weighting, we study the phase diagram of a system of dumbbells formed by two touching spheres having variable sizes, as well as different interaction properties. The first sphere (h) interacts with all other spheres belonging to different dumbbells with a hard-sphere potential. The second sphere (s) interacts via a square-well interaction with other s spheres belonging to different dumbbells and with a hard-sphere potential with all remaining h spheres. We focus on the region where the s sphere is larger than the h sphere, as measured by a parameter 1 ≤ α ≤ 2 controlling the relative size of the two spheres. As α → 2 a simple fluid of square-well spheres is recovered, whereas α → 1 corresponds to the Janus dumbbell limit, where the h and s spheres have equal sizes. Many phase diagrams falling into three classes are observed, depending on the value of α. The 1.8 ≤ α ≤ 2 is dominated by a gas-liquid phase separation very similar to that of a pure square-well fluid with varied critical temperature and density. When 1.3 ≤ α ≤ 1.8 we find a progressive destabilization of the gas-liquid phase diagram by the onset of self-assembled structures, that eventually lead to a metastability of the gas-liquid transition below α = 1.2.

Lattice Boltzmann methods in porous media simulations: From laminar to turbulent flow

The lattice Boltzmann method has become a popular tool for determining the correlations for drag force and permeability in porous media for a wide range of Reynolds numbers and solid volume fractions. In order to achieve accurate and relevant results, it is important not only to implement very efficient code, but also to choose the most appropriate simulation setup. Moreover, it is essential to accurately evaluate the boundary conditions and collision models that are effective from the Stokes regime to the inertial and turbulent flow regimes. In this paper, we compare various no-slip boundary schemes and collision operators to assess their efficiency and accuracy. Instead of assuming a constant volume force driving the flow, a periodic pressure drop boundary condition is employed to mimic the pressure-driven flow through the simple sphere pack in a periodic domain. We first consider the convergence rates of various boundary conditions with different collision operators in the Stokes flow regime. Additionally, we choose different boundary conditions that are representatives of first- to third-order schemes at curved boundaries in order to evaluate their convergence rates numerically for both inertial and turbulent flow. We find that the multi-reflection boundary condition yields second order for inertial flow while it converges with third order in the Stokes regime. Taking into account the both computational cost and accuracy requirements, we choose the central linear interpolation bounce-back scheme in combination with the two-relaxation-time collision model. This combination is characterized by providing viscosity independent results and second-order spatial convergence. This method is applied to perform simulations of touching spheres arranged in a simple cubic array. Full- and reduced-stencil lattice models, i.e., the D3Q27 and D3Q19, respectively, are compared and the drag force and friction factor results are presented for Reynolds numbers in the range of 0.001 to 2,477. The drag forces computed using these two different lattice models have a relative difference below 3% for the highest Reynolds number considered in this study. Using the evaluation results, we demonstrate the flexibility of the models and software in two large scale computations, first a flow through an unstructured packing of spherical particles, and second for the turbulent flow over a permeable bed.

Prediction and Control of Slip-Free Rotation States in Sphere Assemblies.

We study fixed assemblies of touching spheres that can individually rotate. From any initial state, sliding friction drives an assembly toward a slip-free rotation state. For bipartite assemblies, which have only even loops, this state has at least four degrees of freedom. For exactly four degrees of freedom, we analytically predict the final state, which we prove to be independent of the strength of sliding friction, from an arbitrary initial one. With a tabletop experiment, we show how to impose any slip-free rotation state by only controlling two spheres, regardless of the total number.

Determination of minimum impact parameter by modified touching spheres schemes for intermediate energy Coulomb excitation experiments

The energy-independent touching spheres schemes commonly used for the determination of the safe minimum value of the impact parameter for Coulomb excitation experiments are modified through the inclusion of an energy-dependent term. The touching spheres+3fm scheme after modification emerges out to be the best one while touching spheres+4fm scheme is found to be better in its unmodified form.

Modified interaction radius and touching spheres schemes for the determination of impact parameter in Coulomb excitation experiments

The parametrization schemes based on interaction radius and touching spheres (ts) commonly used for the determination of minimum value of impact parameter [Formula: see text] in the above barrier Coulomb excitation experiments are modified by incorporating an energy-dependent term. The modification leads to about 13% improvement in the value of [Formula: see text] for a variety of projectile target combinations at above Coulomb barrier incident energies.

Competitive effects between stationary chemical reaction centres: a theory based on off-center monopoles.

The subject of this paper is competitive effects between multiple reaction sinks. A theory based on off-center monopoles is developed for the steady-state diffusion equation and for the convection-diffusion equation with a constant flow field. The dipolar approximation for the diffusion equation with two equal reaction centres is compared with the exact solution. The former turns out to be remarkably accurate, even for two touching spheres. Numerical evidence is presented to show that the same holds for larger clusters (with more than two spheres). The theory is extended to the convection-diffusion equation with a constant flow field. As one increases the convective velocity, the competitive effects between the reactive centres gradually become less significant. This is demonstrated for a number of cluster configurations. At high flow velocities, the current methodology breaks down. Fixing this problem will be the subject of future research. The current method is useful as an easy-to-use tool for the calibration of other more complicated models in mass and/or heat transfer.

Numerical analysis of the external slow flows of a viscous fluid using the R-function method

In this paper the problem of calculating the external slow flow of viscous incompressible fluid past bodies of revolution is considered. A numerical method, based on the joint use of the R-function structural method of V. L. Rvachev for building the structure of the boundary-value problem solution and the Galerkin projection method for approximating the indeterminate components of the structure, is proposed for solving the problem. The problem of slow viscous incompressible fluid flow past an ellipsoid of revolution is numerically solved using the proposed approach to test the developed method. The solution obtained is compared with the known exact solution of the problem in question. In addition, the problems of flow past two touching spheres and past two joined ellipsoids are solved.

Coarse- and fine-grid numerical behavior of MRT/TRT lattice-Boltzmann schemes in regular and random sphere packings

We analyze the intrinsic impact of free-tunable combinations of the relaxation rates controlling viscosity-independent accuracy of the multiple-relaxation-times (MRT) lattice-Boltzmann models. Preserving all MRT degrees of freedom, we formulate the parametrization conditions which enable the MRT schemes to provide viscosity-independent truncation errors for steady state solutions, and support them with the second- and third-order accurate (“linear” and “parabolic”, respectively) boundary schemes. The parabolic schemes demonstrate the advanced accuracy with weak dependency on the relaxation rates, as confirmed by the simulations with the D3Q15 model in three regular arrays (SC, BCC, FCC) of touching spheres. Yet, the low-order, bounce-back boundary rule remains appealing for pore-scale simulations where the precise distance to the boundaries is undetermined. However, the effective accuracy of the bounce-back crucially depends on the free-tunable combinations of the relaxation rates. We find that the combinations of the kinematic viscosity rate with the available “ghost” antisymmetric collision mode rates mainly impact the accuracy of the bounce-back scheme. As the first step, we reduce them to the one combination (presented by so-called “magic” parameter Λ in the frame of the two-relaxation-times (TRT) model), and study its impact on the accuracy of the drag force/permeability computations with the D3Q19 velocity set in two different, dense, random packings of 8000 spheres each. We also run the simulations in the regular (BCC and FCC) packings of the same porosity for the broad range of the discretization resolutions, ranging from 5 to 750 lattice nodes per sphere diameter. A special attention is given to the discretization procedure resulting in significantly reduced scatter of the data obtained at low resolutions. The results reveal the identical Λ-dependency versus the discretization resolution in all four packings, regular and random. While very small Λ values overestimate the drag measurements several-fold on the coarse grids, Λ>1 may overestimate the permeability at the same extent. In low resolution region we provide practical guidelines, extending previously known solutions for the straight/diagonal Poiseuille flow. Analysis of the high-resolution region reveals the collapse of the solutions obtained with all the considered Λ values with the average rate of −1.3, followed by their common, smooth, first-order convergence with the rate of −1.0 as the best, towards the reference solutions provided by the “parabolic” schemes. High-quality power-law fits estimate that the bounce-back would reach their accuracy (obtained at about 200 nodes per sphere) for two-order magnitude higher grid resolution.

Rotne–Prager–Yamakawa approximation for different-sized particles in application to macromolecular bead models

AbstractThe Rotne–Prager–Yamakawa (RPY) approximation is a commonly used approach to model the hydrodynamic interactions between small spherical particles suspended in a viscous fluid at a low Reynolds number. However, when the particles overlap, the RPY tensors lose their positive definiteness, which leads to numerical problems in the Brownian dynamics simulations as well as errors in calculations of the hydrodynamic properties of rigid macromolecules using bead modelling. These problems can be avoided by using regularizing corrections to the RPY tensors; so far, however, these corrections have only been derived for equal-sized particles. Here we show how to generalize the RPY approach to the case of overlapping spherical particles of different radii and present the complete set of mobility matrices for such a system. In contrast to previous ad hoc approaches, our method relies on the direct integration of force densities over the sphere surfaces and thus automatically provides the correct limiting behaviour of the mobilities for the touching spheres and for a complete overlap, with one sphere immersed in the other one. This approach can then be used to calculate hydrodynamic properties of complex macromolecules using bead models with overlapping, different-sized beads, which we illustrate with an example.

The Effect of Orientation on the Mobility and Dynamic Shape Factor of Charged Axially Symmetric Particles in an Electric Field

The mobility of a nonspherical particle is a function of both particle shape and orientation. Thus, unlike spherical particles, the mobility, through its orientation, depends on the magnitude of the electric field. In this work, we develop a general theory, based on an extension of the work of Happel and Brenner (1965), for the orientation-averaged mobility applicable to any axially symmetric particle for which the friction tensor and the polarization energy are known. By using a Boltzmann probability distribution for the orientation, we employ a tensor formulation for computing the orientation-averaged mobility rather than a scalar analysis previously employed by Kim et al. (2007) for nanowires. The resulting equation for the average electrical mobility is much simpler than the expression based on the scalar approach, and can be applied to any axially symmetric structures such as rods, ellipsoids, and touching spheres. The theory is applied to the specific case of nanowires and the experimental results o...