Procedure: Particle tracking

Materials

• Microscope
• Computer with Matlab and latest version of particle tracker
• Sample with fixed particles (preferably with characteristics similar to the ones you plan to measure)
• Sample with known size particles suspended in a solvent of known viscosity (1μm spheres in water, for example)

Characterizing instrument stability

The accuracy of optical particle tracking may be limited by mechanical and optical phenomena. Vibration and drift are a source of additive noise. Shot noise and CCD readout noise in the image of a particle bring about uncertainty in the estimate of its centroid. Excessive vibration can frequently be corrected by improving the mechanical support structure of the instrument. Most stages can be locked to reduce drift. Shot noise is fundamental; however, its effect can be minimized by ensuring that the optical system is functioning at peak efficiency.

Before attempting to make measurements with particle tracking, it is essential to determine the performance characteristics of the instrument to be used. This can be accomplished by measuring a specimen with known characteristics. Perhaps the most foolproof choice is a sample with fixed particles. Any variation measured in this sample is noise.

File:Stability Plot.tif

1. Bring a slide with fixed beads into focus. Choose a slide with beads that are as similar to those you plan to measure as possible. Find a field of view that contains at least two beads.
2. After optimizing all settings, track the beads for about 3 minutes and save the centroids with a sampling rate of $ T $ samples per second.
3. Use the Matlab function track to separate the centroids into individual trajectories, $ \vec r_n(t) $, where $ t = nT $
4. Compute the sum and the difference of the trajectories for two particles, $ \vec r_sum(t) = \vec r_1(t) + \vec r_2(t) $ and $ \vec r_difference(t) = \vec r_1(t) - \vec r_2(t) $.
5. Compute and plot the mean squared displacement of $ r_+ $ and $ r_- $ as a function of time interval, $ \left \langle {\left | \vec r(t+\tau)-\vec r(t) \right \vert}^2 \right $ for intervals $ \tau=nT $ up to about ten percent of the total track length.

Estimating the diffusion coefficient by tracking suspended microspheres

According to theory, the mean squared displacement of a suspended particle is proportional to the time interval as: $ \left \langle {\left | \vec r(t+\tau)-\vec r(t) \right \vert}^2 \right \rangle=2Dd\tau $, where r(t) = position, d = number of dimensions, D = diffusion coefficient, and $ \tau $= time interval.

• Track some 3μm (Samples A & B) and 5μm (Samples C & D) microspheres.
• Estimate the diffusion coefficient of these microspheres suspended in solutions of varying viscosities. For reference, Sample A contains 3μm spheres suspended in water.
• Consider how many particles you should track and for how long. What is the uncertainty in your estimate?

See: this page for more discussion of Brownian motion and a Matlab simulation.