# 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 measured variation in the fixed sample is, of course, noise.

- 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.
- After optimizing all settings, track the beads for about 3 minutes and save the centroids with a sampling rate of $ T $ samples per second.
- Use the Matlab function
`track`

to separate the centroids into individual trajectories, $ \vec r_n(t) $, where $ t = nT $ - Compute the sum and the difference of the trajectories for two particles, $ \vec r_+(t) = \vec r_1(t) + \vec r_2(t) $ and $ \vec r_-(t) = \vec r_1(t) - \vec r_2(t) $.
- 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 \rangle $ for intervals $ \tau=nT $ up to about ten percent of the total track length.

Why does the MSD of the sum trajectory increase while the difference trajectory stays about the same? Can you take advantage of this property to decrease the error in measurements of unknown samples?

### Estimating the diffusion coefficient by tracking suspended microspheres

According to theory,^{[1]}^{[2]}^{[3]}^{[4]} 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 particles of a known size suspended in a solvent of known viscosity.
- Using the formula above, estimate the diffusion coefficient. How closely does your estimate agree with the predicted value?
- Track unknown particles and estimate the diffusion coefficients.

- 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.

## References

- ↑ A. Einstein, On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat, Annalen der Physik (1905).
- ↑ E. Frey and K. Kroy, Brownian motion: a paradigm of soft matter and biological physics, Ann. Phys. (2005). Published on the 100th anniversary of Einstein’s paper, this reference chronicles the history of Brownian motion from 1905 to the present.
- ↑ R. Newburgh, Einstein, Perrin, and the reality of atoms: 1905 revisited, Am. J. Phys. (2006). A modern replication of Perrin's experiment. Has a good, concise appendix with both the Einstein and Langevin derivations.
- ↑ M. Haw, Colloidal suspensions, Brownian motion, molecular reality: a short history, J. Phys. Condens. Matter (2002).