Stochastic Aspects of Diffusion:
Colloidal Beads

Colloids: The term colloidal dispersion covers a broad class of materials which fall in the size range of approximately 1nm-10 microns. They are important in a wide variety of applications ranging from paints to drug delivery. Spherical polymeric colloids also serve as a nice model system for fundamental studies of diffusive processes. Using fluorescence microscopy we can track the trajectory of a single bead in solution and see the random trajectory in the figure to the right. Click here for a movie showing the diffusion and trajectories of several beads.

Diffusion from a point source: We can use the trajectories of the beads to calculate the diffusion of a point source of probability . Specifially, we want to determine the probability that a bead with a position ri (0) at time=0 has moved during an interval time=t to a new location ri (t) = ri (0) + dri (t). The probability of this displacement, P(dri,t), is described by a Gaussian with a zero mean. This is discussed in sections 2.9 and 4.9 of Deen.

Stokes-Einstein equation: The diffusion coefficient of the spherical colloidal spheres is well described by the Stokes Einstein equation: DAB = kT/(6 pi mu rA) where mu is the solvent viscosity and r A is the diameter of the bead (see page 19 of Deen). We recall that the mean square displacement of a single bead (without flow) scales as DAB. Physically then we expect that putting the same size beads in a more viscous solution will slow down their motion. This is qualitatively shown in the following movie where we show the trajectories in 0.5 micron diameter beads in solutions with viscosities of 1cp, 2cp and 4 cp (left to right). The scale bar in the movies is 10 microns and the numbers denote time in seconds. The Stokes-Einstein relation also shows that the velocity of the beads will increase as we make them smaller. This is qualitatively shown in the following movie where we show the trajectories in 1 micron, 0.5 micron and 0.1 micron (left to right) beads in water. The scale bar in the movies is 10 microns and the numbers denote time in seconds. The smallest colloid in the previous movies is about 2 orders of magnitude larger than most simple fluids. Thus we can now imagine how rapidly molecules in simple fluids fluctuate under ambient conditions!