We are taught from the first lectures in electrostatics that like charges repel each other. That expected behavior breaks down, however, in some colloidal suspensions. There, large colloidal particles (called macro-ions), typically all with the same charge, are surrounded by smaller, oppositely charged counterions, not to mention a polar fluid, charged surfaces on the walls of the container, and possible additional ions. In such complex fluids, many varied experiments over the past 15 years have produced evidence of attractive interactions between the macro-ions.

In many regards, latex colloidal suspensions, with their spherical particles of nearly identical diameter, are useful models for studying behavior in more complex systems (see, for instance, the article by Alice Gast and William Russel in Physics Today, December 1998, page 24). Attractions in such systems at short length scales—on the order of a few nanometers—are well established and attributable to correlations among multivalent counterions; such attraction in biological systems is discussed in the article by William Gelbart and coauthors in Physics Today, September 2000, page 38. But in some systems, attraction has been observed at length scales of several microns, and an understanding of that behavior has been elusive. Recently, Todd Squires of Harvard University and Michael Brenner of MIT have proposed an explanation for some of the experimental results, incorporating hitherto overlooked effects of hydrodynamics. 1 Their model produces quantitative agreement with measurements, by Amy Larsen and David Grier at the University of Chicago, of the behavior of two colloidal particles near a single wall. 2  

A typical charge-stabilized latex colloidal suspension consists of spheres with diameters on the order of 1 µm or smaller, dispersed in water. Sulfate or other groups on the spheres dissociate in solution, producing a large surface charge density on the spheres. The spheres are surrounded by counterions that are many times more numerous and much smaller than the spheres. As a result, the dynamics of the counterions have a significantly faster time scale than that of the spheres. Averaging over the counterion dynamics yields a screened, exponentially decreasing electrostatic interaction between the spheres—the Derjaguin-Landau-Verwey-Overbeek (DLVO) model. Adding salt introduces more ions in solution, which decreases the screening length.

The mean-field DLVO theory has been shown to produce purely repulsive interactions between the spheres. It consequently fails to explain the various experimental hints that there is some sort of effective attraction. Many of these hints have come from observations made in bulk colloidal systems. For example, although one would expect repulsive interactions to spread the particles out evenly, Norio Ise and coworkers at Kyoto University have seen empty regions in deionized suspensions (characterized by large screening lengths). And analyzing the equilibrium distribution of particles in suspension, Seth Fraden at Brandeis University and Mauricio Carbajal-Tinoco (now at Mexico’s National Polytechnic Institute) and colleagues independently derived attractive pair potentials for the particles.

Instead of looking at bulk systems, Grier’s group at Chicago has looked at pairs of particles, positioning them with optical tweezers and then releasing them and observing their motion. Those experiments showed that whereas the behavior of two isolated spheres is well described by standard DLVO theory, when confined between two walls—or even just near one wall—the spheres behave as though they experience an effective attractive interaction.

The evidence for long-range attraction when screening lengths are long and when two spheres are confined, but not when isolated, has led many researchers to look toward many-body effects, nonlinear interactions, and other factors for possible statistical mechanical explanations. Recognizing that some of the experimental systems were not in equilibrium, Squires and Brenner have introduced a new effect for consideration: hydrodynamics. They have examined the case of two charged spheres near a single wall as investigated in Grier’s experiments. Figure 1 illustrates the effect: As one sphere moves away from a similarly charged glass wall due to electrostatic repulsion, it entrains the nearby fluid to follow along. Fluid moving in behind the sphere can drag a second nearby sphere toward the first. Thus the observed motion of the spheres toward each other may be attributable to hydrodynamic flows that arise from the proximity of the two spheres to the wall, rather than arising from an attractive interaction between the spheres themselves.

Figure 1. Hydrodynamic flows can draw colloidal particles toward each other, even when they have like charges. Here, the left sphere is moving away from the wall at the bottom, dragging nearby fluid with it as indicated by the black lines. As the surrounding fluid flows in behind the sphere, it pulls the nearby right sphere along, toward the other. Similar motion toward each other occurs when both spheres are moving away from the wall.

Figure 1. Hydrodynamic flows can draw colloidal particles toward each other, even when they have like charges. Here, the left sphere is moving away from the wall at the bottom, dragging nearby fluid with it as indicated by the black lines. As the surrounding fluid flows in behind the sphere, it pulls the nearby right sphere along, toward the other. Similar motion toward each other occurs when both spheres are moving away from the wall.

Close modal

Squires and Brenner have been able to quantify the effects of their hydrodynamic considerations by using standard hydrodynamic theory coupled to the DLVO screened Coulombic interaction between the spheres and between each sphere and the wall. They came up with an analytic expression for the hydrodynamic effects, which they then combined with diffusive Brownian motion in simulations.

The theory has only one free parameter, the surface charge density on the wall. With a judicious choice for its value, Squires and Brenner can quantitatively account for the experimental observations of the relative motion of the spheres. 2 Figure 2 shows the results of the model calculations along with the experimental observations, when both are cast as being due to forces that are the gradient of an “effective potential.” (The actual interparticle forces are strictly repulsive in this model, however.) The model correctly accounts for the depth of the effective potential as well as its long range: The observed maximum effect occurs for an interparticle spacing that is several times larger than the particle diameter or the screening length, but on the order of the distance to the wall. “This is in fact one of the major clues that tipped us off to the hydrodynamic effect,” notes Brenner.

Figure 2. With hydrodynamic effects incorporated, simulations of the motion of two charged spheres 0.65 µm in diameter near a similarly charged wall quantitatively account for the observed motion. Here, the simulated 1 (red) and the observed 2 (gray) motions have been analyzed as if they were due to forces given by the gradients of an effective potential U that depends on the interparticle separation r, for two values of the distance h from the wall (offset for clarity).

Figure 2. With hydrodynamic effects incorporated, simulations of the motion of two charged spheres 0.65 µm in diameter near a similarly charged wall quantitatively account for the observed motion. Here, the simulated 1 (red) and the observed 2 (gray) motions have been analyzed as if they were due to forces given by the gradients of an effective potential U that depends on the interparticle separation r, for two values of the distance h from the wall (offset for clarity).

Close modal

The wall’s charge density in the earlier one-wall measurements isn’t known, and so the charge density used in the model hydrodynamic calculations can’t be verified without further experiments. Still, says Grier, the value is reasonable: “It’s right in the sweet spot.”

The hydrodynamic interactions between the spheres arise only in nonequilibrium situations in which the spheres are moving. “If you could pin the two spheres and measure the force between them, you’d find pure repulsion,” says Squires.

The case of two spheres near one wall is the only case Squires and Brenner have examined quantitatively, although they do make a qualitative extension to the case of two spheres between two parallel confining walls. For two spheres that are precisely in the midplane between the walls, the hydrodynamic effects from each wall will cancel by symmetry. But Squires and Brenner argue that if the spheres are slightly off center, which is almost inevitable experimentally, the hydrodynamic effects won’t exactly cancel and could produce an apparent attraction of the magnitude observed in earlier measurements by Grier with John Crocker.

Because the hydrodynamic theory is applicable only to nonequilibrium measurements, there is no direct implication for the bulk equilibrium measurements that have also shown evidence for attractions between colloidal particles. To the extent that Squires and Brenner’s model does indeed account for the observed two-sphere behavior near a single wall, it indirectly sets limits on the influence of electrostatic or other effects that could also be playing a role, both in that system and in the bulk. Meanwhile, a consensus is emerging that many-body effects originating in counterion correlations are at the heart of the observed attraction in the bulk.

The jury is still out on the value of wall charge density, which will determine the extent to which hydrodynamic effects account for the observed behavior of two charged spheres near a wall. Grier is working closely with Squires and Brenner to reevaluate his old experiments and perform new ones aimed at settling this issue, for both one-wall and two-wall systems.

1.
T. M.
Squires
,
M. P.
Brenner
,
Phys. Rev. Lett.
85
,
4976
(
2000
) .
2.
A. E.
Larsen
,
D. G.
Grier
,
Nature
385
,
230
(
1997
) .