“Under the microscope one, to some extent, immediately sees a part of thermal energy in the form of mechanical energy of the moving particles.” —A. Einstein 1915

### What it Shows

Tiny latex spheres in water, viewed under a microscope, undergo a kind of random jiggling motion called *Brownian motion*—named after the botanist Robert Brown, who observed this kind of motion in 1827 when looking at tiny pollen grains. The spheres are all 1.054 micron in diameter. Each particle can be seen to exhibit a “random walk” under the influence of thermal agitation.

This type of motion had been observed in dust particles as long ago as the ancient Romans, yet a satisfying model to explain it did not emerge until the early 20th century. Albert Einstein in 1905 interpreted this phenomenon in terms of statistical mechanics, and it was this work—along with that of Marian Smoluchowski, Jean Baptiste Perrin, and others—that finally won universal consensus for the idea that matter can be modeled in terms of discrete atoms and molecules.

### How it Works

If we follow the motion of a single particle, we notice that it seems to undergo a random walk in all directions. If we divide the walk into short, equal time intervals and only consider one component—say, the x-component—of the motion, then we would observe many short random walks along that direction, and a histogram of the displacements would show a nice Gaussian distribution. Einstein and Smoluchowski showed that the square of the mean of the displacements—which is also the square of the standard deviation—is proportional to the *diffusion constant D* for that particle, and that the product of D and the *Stokes drag f* is proportional to Boltzmann’s constant k_{B} (1.38 x 10^{-23} J/K)^{[1]}.

### Setting it Up

A video camera coupled to our microscope provides a live view of the sample on the slide. The concentrated solution is readily available from VWR^{[2]} and is diluted 1:20 with distilled water at room temperature.

To avoid a situation where the slide dries up before the class has a chance to view it, the slide is usually prepared in the final minutes before class (often we will show the camera shot on the lecture hall video display as students are taking their seats). A well-slide may be used, although a flat slide with a well made of a ring of grease also works well, and can often preserve the sample for longer than the well slide. Make sure the slide is dry and that there is no net drift of the particles.

The particles often settle to the bottom of the bottle, so make sure to shake it before making the slide. Start with the microscope objective at 10X magnification to get the sample in focus, then switch to the 40X for more detail. To prolong the life of the sample, you may want to wait until the class is ready to see the demo before turning the light source on.

### Comments

Students are welcome to download video footage of our spheres in action so that they may perform their own video analysis of the motion of the particles:

- 1-micron spheres at 24 fps with a field of view of 140 microns [https://drive.google.com/file/d/1C8ELhRAsx9aVi_nkT7furkAZzcudSTEp/view?usp=sharing]
- 1-micron spheres at 3 fps with a field of view of about 80 microns [https://drive.google.com/file/d/1CAk00CC0cVZvj8QVjhxR02vcDvuUKlyZ/view?usp=sharing]

### References

"The Collected Papers of Albert Einstein Volume 2: The Swiss Years.” Edited by David C. Cassidy, Jurgen Renn, and Robert Schulmann. (Princeton University Press, 1989)

Perrin, M. Jean. “Brownian Movement and Molecular Reality.” Translated from “Annales de Chimie et de Physique, 8me Series, September 1909” by F. Soddy, M.A., F.R.S. (Taylor and Francis, London, 1910)

[1] The diffusion constant *D (m ^{2}/s)*, Stokes drag

*f (Pa*s*m)*, temperature

*T (Kelvin)*, and Boltzmann's constant

*k*are related by

_{B}(J/K)
Df = k_{B}T

The Stokes drag is f = 6πηr where *η* is the fluid viscosity and *r* is the radius of the spherical particle.

[2] The microspheres we use are VWR catalog number AA42743-4Y Poly(styrene) 2.5% (w/w) dispersion in water, latex microsphere 1 micron.