Colloids: A Surprisingly Attractive Couple

David G. Grier
The James Franck Institute and Department of Physics
The University of Chicago
5640 S. Ellis Ave., Chicago, IL 60637

**Figure:** The pair interaction potential between unconfined pairs of charged colloidal spheres is purely repulsive, as can be seen in (a). Data are plotted in units of the thermal energy k_B T. This pair potential was measured between two polystyrene sulfate spheres of radius $a = 0.482~\mu$ m suspended in deionized water. Measurements were made by manipulating the spheres with optical tweezers and tracking their motions with digital video microscopy as described in reference [6]. The solid line is the DLVO theory's prediction for this system. (b) When the same pair of spheres is confined by parallel glass walls separated by $3.5~\mu$ m, an attractive minimum develops in the pair potential. The unconfined potential from (a) is overplotted as a dashed line to emphasize the confinement's influence. Data reproduced from reference [6]. Attractions of comparable range and strength have been implicated in the formation of metastable superheated colloidal crystals such as the ones shown in (c). The scale bar indicates $10~\mu$ m. In this case the confinement responsible for the many-sphere cohesion appears to be provided by the spheres themselves. Photograph from reference [9].
$\begin{figure} \centering\includegraphics[width=2.7in]{figures/both.eps}\\ [1ex]... ...ring\includegraphics[width=2.7in]{figures/crystal.eps}\vspace{1ex}\end{figure}$

A vast number of industrial and natural processes depend on controlling the interactions between micrometre-sized colloidal particles dispersed in fluid suspensions. For instance, colloidal milk fat and proteins aggregate into either cheese or yoghurt depending on how they are handled; colloidal paint pigments must remain suspended for years in a can, yet should coagulate rapidly into a tough coating when spread on a wall. For just such reasons, colloidal interactions and their ramifications have been researched intensively over two centuries.

And yet, over the past two decades, one of the fundamental tenets of colloid science come under attack: contrary to the lessons of long experience it now seems that like-charged colloidal particles sometimes attract each other. Reports of this extraordinary effect and efforts to explain it have sparked a rancorous debate. Its resolution may at last be at hand, in part because of results from the numerical study by Bowen and Sharif on page XXX of this issue [1].

Although many processes affect colloidal behaviour, the phenomena under contention depend on three energy scales - those set by van der Waals attraction, by the randomizing influence of thermal energy, and by the hierarchy of electrostatic interactions among highly-charged colloidal particles and the singly-charged simple ions around them. The balance of these three determines the properties of a large class of colloidal suspensions.

Van der Waals attraction arises from sympathetic fluctuations in particles' electron distributions. It causes colloid to aggregate, and is partly responsible for dust's tenacious grip on a television screen. But van der Waals attraction only exceeds thermal energies for colloidal particles very near contact, at separations smaller than a few nanometres - More widely separated particles are saved from its grip by thermal collisions with surrounding fluid molecules. Preventing particles from colliding therefore prevents them from aggregating. The conventional wisdom, worked out more than 50 years ago by Derjaguin, Landau, Verwey and Overbeek (DLVO), is that just a few static charges on colloidal particles' surfaces can cause repulsions strong enough to keep them stably separated [2].

The system of equations describing interactions among the spheres, solvent and simple ions is so intrinsically nonlinear that it resists analytical treatment to this day. The DLVO formulation avoids this complexity by linearizing the equations and averaging out ionic fluctuations. Despite these simplifying approximations, the DLVO theory accounts quite well for the stability and properties of charge-stabilized colloidal suspensions and serves as the standard model for colloidal electrostatic interactions.

The problem is that some highly-charged colloidal particles do not behave as DLVO says they should. Inspired by the alternative Sogami-Ise theory for colloidal electrostatic interactions, Norio Ise and his collaborators carefully investigated the structure of colloidal suspensions. Unlike the DLVO theory, the Sogami-Ise theory predicts that like-charged spheres can attract each other. In observations spanning fifteen years, they recorded a variety of phenomena apparently inconsistent with the DLVO theory, ranging from anomalously small lattice constants in colloidal crystals [3] to large stable voids in colloidal fluids [4]. They interpreted their observations as evidence for long-ranged attractions between like-charged spheres. Other researchers suggested more conventional explanations, however, and controversy ensued.

Because many-body behavior often obscures underlying pair interactions, measurements of bulk properties leave room for disagreement. Only in the past five years have techniques been developed capable of measuring the fantastically small forces between individual colloidal particles. These measurements have begun to tell a coherent story: isolated pairs of like-charged spheres are found to repel each other much as predicted by the DLVO theory [5,6,7,8,9]. But spheres confined by glass walls [6,9,10] or by a concentration of other spheres [9] develop long-ranged attractions inconsistent with DLVO. Attractions seem to be favored by highly charged spheres in very low salt concentrations - circumstances under which the DLVO approximations might be expected to fail.

The apparent breakdown of linear superposition (pairs repel but groups cohere) implicates nonlinearity as the culprit. But a slew of alternative explanations have been proposed, including the inherently linear Sogami-Ise theory and mechanisms involving fluctuations in the simple ion distributions. Differences between these theories hinge on the experimentally invisible simple ions.

Bowen and Sharif's numerical study demonstrates that nonlinearity - long-neglected - can indeed explain the observed attractions but is not the whole story. The simple ion distributions around an isolated pair of spheres mediates repulsive interaction, even in their nonlinear calculations. Confining walls, however, redistribute the simple ions so as to mediate a long-wavelength attraction. Comparable redistributions in the DLVO theory's approximations turn out to be too subtle to induce attractions. So both nonlinearity and confinement are needed. Temporal fluctuations are not considered in these calculations, and therefore are not necessary to generate attractions.

Knowing the ingredients for like-charge colloidal attractions is a big step toward understanding their origin and predicting their ramifications. But we still do not understand how nonlinearity and geometric confinement conspire to produce simple ion distributions conducive to long-ranged cohesions between neighboring spheres. Numerical simulations, like physical experiments, offer insights into how particular systems behave under particular circumstances; they offer no predictive insights into the behaviour of other systems.

What geometries support long-ranged like-charged colloidal interactions? How can we turn them on and off? Could they affect smaller macroions such as proteins and DNA? By identifying what processes are at work, Bowen and Sharif have poised us on the brink of being able to answer such questions.

Bibliography

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