Figure 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).Reuse & Permissions

Figure 2

Charge reduction in the presence of a third particle.—Ratio of the charge per particle on a set of three constant potential spheres arranged in an equilateral triangle of side length $s$ to the charge per particle on a pair of similar spheres separated by length $s$ for six values of the ionic strength: $\kappa a=0$, 0.2, 0.4, 0.6, 0.8, and 1.0. The solid line represents a numerical solution of the LPBE, and the dashed line represents an analytical approximation based on a single point charge at the center of each sphere [Eqs. (4, 5)].Reuse & Permissions