Diffusion Again

Let's consider again in greater detail the irregular jiggling displacements of an uncharged particle floating in hemolymph or within a microdevice, as all those irregularly bouncing water molecules strike the particle. We can calculate the particle's average or root mean square displacements at any intervals we choose. Then we can compare our observations with theoretical calculations.

First some history. In 1905, Einstein showed how small water molecules could jiggle particles like pollen grains observable under a microscope. In his classic paper: 'On the motion of small particles suspended in a stationary liquid according to the molecular kinetic theory of heat,' Einstein used statistics to show that many molecular "beatings" combined to bounce larger particles around. For particles smaller than about twenty micrometers across, the impacts falling equally on all sides failed to average out, thus giving the particle a kick in some direction. As we know, no particle anticipates where it may be kicked to next.

Particles experience viscous drag. Drag depends on mass, how fast a particle moves and a coefficient describing the viscosity of the hemolymph. If our particle's a sphere, we can use Stokes Law. Our particle, therefore, also feels this rapidly fluctuating force that averages to zero over enough time. If we multiply our particle's equation of motion by its displaced distances averaged over time, and we apply the equipartition law, we get Einstein's equation for diffusion. This equation says explicitly what the root mean square displacement is in parameters we can measure. However, we must watch our particle for a long time compared to the shorter intervals between hits. Einstein and Smoluchowski asked: in a time interval how far does our particle move from where it starts?

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