Many-Body Electrostatic Forces Between Colloidal Particles at Vanishing Ionic Strength

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Jason W. Merrill, Sunil K. Sainis, and Eric R. Dufresne

Physical Review Letters 103 (2009) 138301

wiki entry by Emily Russell, Fall 2010

The article can be found here.


This paper reports a striking demonstration that effective pair potentials do not tell the full story in colloidal systems, and that furthermore, a constant surface potential is a better assumption than a constant surface charge, at least in some cases. The authors are able to observe many-body effects on the forces in systems of small numbers of colloidal particles, and use a simple Poisson-Boltzmann model to predict these effects from the pair potentials.


The experiments were carried out using 600nm-radius PMMA colloids in nonpolar hexadecane as the solvent. NaAOT, a surfactant, was added, which forms reverse micelles, increasing the particle charge, and decreasing the screening length.

The particles were positioned using optical tweezers. Three configurations were studied: pairs of particles; an equilateral triangle of particles (each pair of which had been measured previously); and a hexagonal arrangement of seven particles. The particles were released and tracked to determine their drift velocities; individual drift velocities were converted to the velocity of the breathing mode of the system, and the force on this breathing mode determined by <math> f = k_B T v_d / D </math>. (With a moment's thought, it is not obvious that this equation should apply for many-body modes; this is addressed in an earlier paper by the group, Statistics of Particle Trajectories at Short Time-Intervals Reveal fN-Scale Colloidal Forces.) The force was calculated for a range of particle separations.

The forces observed in isolated particle pairs were fit to obtain the surface potential and screening length. These parameters were then input to the linearized Poisson-Boltzmann equation, which was numerically solved with constant-potential boundary conditions to predict the forces in the triangular and hexagonal situations, taking into account the many-body effects. Forces were also predicted assuming only pairwise interactions, and these two predictions were compared to the measured forces.


The results are very nicely summed up in the main figure of the paper, Fig. 1.

Fig. 1. Direct measurement of nonpairwise electrostatic interactions.—Forces on beads in pair (first column), equilateral (second column), and hexagonal (third column) configurations, at AOT concentrations of 10 (first row) and 0.5 mM (second row). The arrows on the particle configurations in the first row indicate the form of the breathing modes used for analysis. Breathing modes are normalized so that the sum of the squares of the particle displacements is unity. For the pair measurements, different colored points represent different pairs in the same sample. The dotted, solid, and dashed lines are fits to constant charge density, constant potential, and a simple approximation of constant potential based on Eq. (3), respectively. For the equilateral and hexagonal configurations, black points are measured forces on the breathing mode, and red points are a direct pairwise sum of the measured pair forces. The red lines are pairwise sums of the constant potential pair fits. Constant potential predictions for the force on the breathing mode based on fits to the pair data are shown as black lines. The solid line is based on the full numerical solution, while the dashed line is based on Eq. (3).

At short screening length (<math>\kappa a = 0.58 </math>, high surfactant concentrations), the pairwise and many-body predictions are similar; for the hexagonal configuration, only a small deviation is observed from the pairwise prediction. The dramatic results are obvious in the measurements at long screening length (<math>\kappa a = 0.14 </math>, low surfactant concentration), where the measurements are fit very well by the calculations taking into account many-body effects, the forces being substantially smaller in the triangular and hexagonal configurations than a naive pairwise calculation would predict. The authors emphasize that the potential and screening length obtained from the fits to the pair data are the only parameters input to the calculations for the triangular and hexagonal configurations.

It turns out that it is not the non-linearity of the Poisson-Boltzmann equation itself which makes many-body effects important in this system; indeed, the authors find that the linearized PB equation is sufficient to predict the forces. Instead, the surface charges of the particles change depending on the presence of other particles in order to maintain a constant surface charge. The pairwise prediction would be appropriate only if the surface charges themselves remained constant. A simple model of the particles as a central point charge, whose magnitude changes to maintain a constant potential at the particle surface, gives results close to the full numerical solutions.

The authors reference the work of Verwey and Overbeek, who first suggested that constant surface potential may be a more appropriate assumption for colloidal particles than constant surface charge, based on an equilibrium between the surface and bulk concentrations of ions. They also point out that the majority of models in colloids have used constant surface charge boundary conditions, so that some models may need to be revisited.


If there is a weakness in this paper, it is that the experiments were only performed once - repetition with different particles to fill out a few more data points would have made the data even more convincing (but let's not tell Jason I said that). I also found it odd that they did not discuss the screening lengths found for the two surfactant concentrations; experiments exploring a range of screening lengths and indicating more precisely at what point many-body interactions become important would have been informative. Otherwise, the paper is written clearly, reporting experiments which are easy to understand and using straightforward but informative models, and arguing well that particle interactions are modulated by the proximity of other charged particles.

Of course, these results might not be so damaging to the world of colloid research using pairwise models as it at first seems; the results are only significant when the screening length is long compared to the particle separation. In aqueous solutions, dominated by short screening lengths, the many-body effects are much less significant.