# Statistics of Particle Trajectories at Short Time-Intervals Reveal fN-Scale Colloidal Forces

Sunil K. Sainis, Vincent Germain, and Eric R. Dufresne

Physical Review Letters **99**, 018303 (2007)

wiki entry by Emily Russell, Fall 2010

The article can be found here.

# Overview and Comments

This is an elegant article. Using reasonably straightforward measurements and a minimum of assumptions, the authors demonstrate a technique of extracting forces and potentials between pairs of colloidal particles, with a finer precision than was previously possible. The exposition is clear and well presented, and the results quite compelling.

# Background

The article opens with a brief review of previous methods for obtaining interparticle forces. Atomic force microscopy can be used to measure forces mechanically down to piconewton resolution. Real-space imaging of dilute systems of particles can provide the interparticle potential based upon the probability distribution of separations, or from the radials distribution function (g(r)); however, this works only for weak, pairwise additive potentials, and the calculations can be confounded by surface effects. A more precise and robust method is clearly desirable, particularly for the purpose of understanding the weak DLVO forces between colloidal particles.

# Methods

The authors use a superdilute suspension of carboxylated polystyrene colloidal particles 1.2 <math>\mu</math>m in diameter in hexadecane with a surfactant. Bright-field imaging with a high-speed camera provides particle positions with a frequency of 500 Hz, and the by now standard Crocker and Grier algorithm of particle location is implemented in Matlab to identify the positions of the particles to a resolution of around 10 nm.

Two particles are placed at a fixed distance apart using optical tweezers; the tweezers are then blinked, alternately releasing the particles and re-positioning them. The particle trajectories are tracked while the tweezers are off.

By repeating these measurements many times for each initial separation, the authors compute the average <math>\Delta</math>r (change in separation) as a function of timestep <math>\Delta</math>t; this turns out to be linear, giving a velocity *v*. They also compute the variance of <math>\Delta</math>r as a function of timestep, which is also linear, giving a diffusion constant *D* for the particles' motion.

They give a rigorous demonstration that, so long as the time intervals studied are short enough, the conservative force on the particles can be expressed very simply as <math>F = k_B T \frac{v}{D}</math>. This is essentially equivalent to Einstein's relation for the drag coefficient, <math>D = \frac{k_B T}{k_{drag}}</math>, with the drag coefficient <math>k_{drag}</math> given by <math>\frac{F}{v} = k_{drag}</math>. The elegance of this relation is that it is independent of the details of the hydrodynamics, the solvent, or the properties of the particles, providing a very general method for determining the forces.

The authors are thus able to map out the screened Coulomb force between the two charged particles, matching quite well to a theoretical fit.