Like-charged Spheres Research Articles

Collision efficiency of like-charged spheres settling in a quiescent environment

We study the gravity-induced collisions of charged spheres of dielectric materials dispersed in a gaseous medium. When the gap thickness between the surfaces of two spheres is shorter than the mean free path of the surrounding fluid medium, continuum assumptions for the hydrodynamics interactions are no longer valid, and the non-continuum lubrication interactions result in surface-to-surface contact in finite time. Two like-charged dielectric spheres attract each other at close separations for a wide range of size and charge ratio values. We use trajectory analysis to calculate the collision rate and, thus, explore the role of electrostatic interactions in the collision dynamics of a pair of like-charged dielectric spheres. We present the modifications of pair trajectories due to electrostatic forces and show how collision efficiencies vary with the non-dimensional parameter capturing the relative strength of the electrostatic force to gravity as well as the charge ratio and size ratio.

Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres

We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force, we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres increases when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximized through an optimal lowering of the voltage on the larger sphere at an optimal sphere separation.

Numerical and analytical verifications of the electrostatic attraction between two like-charged conducting spheres

We calculate the capacitance coefficients for the problem of two conducting spheres using a numerical (boundary element) method as well as two analytical approaches. The consistency of the results serves as an independent verification of the counter-intuitive analytical result that like-charged spheres almost always attract at close approach. We further support this claim by reproducing the analytical surface charge density with the numerical method. Practical implementations are given in the Supporting Information.

Mechanism of attraction between like-charged particles in aqueous solution

Although it has been long known that like-charged particles attract one another in aqueous media, the mechanism underlying this counter-intuitive phenomenon has remained controversial. We tested the hypothesis put forth long ago by Langmuir and again by Feynman and by Ise, that the attraction between like-charged entities lies in an intermediate of unlike charges. Tests were facilitated by the observation that the attractive forces could be confirmed between widely separated particles of macroscopic size. Two approaches showed comparable results. In the first, pH-sensitive dyes showed intermediate zones of opposite charge: an accumulation of protons was found between negatively charged spheres, whereas between positively charged spheres the intermediate zone contained OH−groups. In the second and complementary approach, microelectrode measurements showed that in between negatively charged spheres, the electrical potential was relatively positive, whereas between positively charged spheres it was relatively negative. Hence, both approaches confirm theoretical expectations. The large number of unlike charges lying in between the like-charged spheres may come from the build-up of the recently reported “exclusion zone” surrounding each particle.

Attraction of like-charged macroions in the strong-coupling limit.

Like-charged macroions attract each other as a result of strong electrostatic correlations in the presence of multivalent counterions or at low temperatures. We investigate the effective electrostatic interaction between i) two like-charged rods and ii) two like-charged spheres using the recently introduced strong-coupling theory, which becomes asymptotically exact in the limit of large coupling parameter (i.e. for large counterion valency, low temperature, or high surface charge density on macroions). In contrast to previous applications of the strong-coupling theory, we deal with curved surfaces and an additional parameter, referred to as Manning parameter, is introduced, which measures the ratio between the radius of curvature of macroions to the Gouy-Chapman length. This parameter, together with the size of the confining box enclosing the two macroions and their neutralizing counterions, controls the counterion-condensation process that directly affects the effective interactions. For sufficiently large Manning parameters (weakly-curved surfaces), we find a strong long-ranged attraction between two macroions that form a closely-packed bound state with small surface-to-surface separation of the order of the counterion diameter in agreement with recent simulations results. For small Manning parameters (highly-curved surfaces), on the other hand, the equilibrium separation increases and the macroions unbind from each other as the confinement volume increases to infinity. This occurs via a continuous universal unbinding transition for two charged rods at a threshold Manning parameter of Epsilon c = 2/3, while the transition is strongly discontinuous for spheres because of a pronounced potential barrier at intermediate distances. Unlike the cylindrical case, the attractive forces between spheres disappear slowly for increasing confinement volume due to the complete de-condensation of counterions. Scaling arguments suggest that for moderate values of coupling parameter, strong-coupling predictions remain valid for sufficiently small surface-to-surface separations.

Dynamic self-assembly of rings of charged metallic spheres.

This Letter describes dynamic self-assembly in a system of stainless steel spheres ( approximately 1 mm in diameter) rolling on a flat dielectric surface under the influence of an external magnetic field that rotates parallel to the plane of the surface. As the spheres move, they charge triboelectrically. Self-assembly is mediated by two types of electrostatic interactions among these charges: (i) attraction between negatively charged regions of the surface and positively charged spheres and (ii) repulsion between the like-charged spheres. The spheres organize into highly ordered rings as a result of these electrostatic interactions.

On interaction of two closely spaced charged conducting spheres

Electrostatic interaction between two charged conducting spheres is analyzed in the case of a small spacing between them, when the polarization effects are significant. It is shown that short-range polarization forces result in the attraction of the like-charged spheres. At a sufficiently small spacing, this attraction replaces repulsive forces acting on like charges.

Strong electrostatic interactions in spherical colloidal systems.

We investigate spherical macroions in the strong Coulomb coupling regime within the primitive model in salt-free environment. We first show that the ground state of an isolated colloid is naturally overcharged by simple electrostatic arguments illustrated by the Gillespie rule. We furthermore demonstrate that in the strong Coulomb coupling this mechanism leads to ionized states and thus to long range attractions between like-charged spheres. We use molecular dynamics simulations to study in detail the counterion distribution for one and two highly charged colloids for the ground state as well as for finite temperatures. We compare our results in terms of a simple version of a Wigner crystal theory and find excellent qualitative and quantitative agreement.

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.

Interactions between charged spheres in divalent counterion solution

We simulate model systems of charged spherical particles in their counterion solution and measure the thermodynamic pressure and the pair distribution function from which we derive effective potentials of mean force. For a system with only electrostatic and hard core interactions, we investigate the effective potential between two like-charged spheres in divalent counterion solution as a function of concentration. We find a strong attractive interaction for high concentration and a global repulsive effective interaction for dilute systems. The results indicate a first order phase transition in sphere–counterion density as a function of global concentration and the effective sphere–sphere potentials in the dilute (solvated) regime suggest significant density fluctuations due to short range local minima in the effective energy surface. Our results arise from a minimal approach model of several recent experiments on polystyrene latex particles in monovalent counterion solution.

Long-range electrostatic attraction between like-charge spheres in a charged pore

The existence of long-range attractive electrostatic forces between particles of like charge is one of the great current controversies ofcolloid science. The established theory (Derjaguin–Landau–Vervey–Overbeek; DLVO) of colloidal interactions predicts that an isolated pair of like-charged colloidal spheres in an electrolyte should experience a purely repulsive screened electrostatic (coulombic) interaction1,2. Direct measurements of such interactions have shown quantitative agreement with DLVO theory3,4,5. Recent experiments, however, provide evidence that the effective interparticle potential can have a long-range attractive component in more concentrated suspensions6,7 and for particles confined by charged glass walls3,5,8,9,10. It is apparent that the long-range attraction in concentrated systems is due to multi-body interactions and may have a similar explanation to the attraction observed for otherwise confined colloids. Theoretical explanations have been proposed11,12,13 but remain the subject of controversy14,15. Here we present a quantitative theoretical explanation of these attractive forces between confined colloidal particles, based on direct solutions of the nonlinear Poisson–Boltzmann equation for two like-charged spheres confined in a cylindrical charged pore. The calculations show that the attraction may be explained by the redistribution of the electric double layers of ions and counterions in solution around the spheres, owing to the presence of the wall; there is thus no need to revise the established concepts underlying theories of colloidal interactions.