# 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.

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 $\mu$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 $\Delta$r (change in separation) as a function of timestep $\Delta$t; this turns out to be linear, giving a velocity v. They also compute the variance of $\Delta$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 $F = k_B T \frac{v}{D}$. This is essentially equivalent to Einstein's relation for the drag coefficient, $D = \frac{k_B T}{k_{drag}}$, with the drag coefficient $k_{drag}$ given by $\frac{F}{v} = k_{drag}$. 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.