# Introduction to microrheology

Written by Eric R. Weeks (back in 1999)

 Brief explanation: Complex fluids like mayonaise, complex materials like plastics, and biomaterials like cells are all somewhat squishy. To some degree, these materials can act both like fluids and like solids, depending on how you put forces on them. Conventional rheology is a way to measure the squishiness of materials: how viscous is a liquid, how elastic is a solid. For a viscoelastic system, rheology can describe the combined viscous and elastic properties of a material. However, conventional rheology requires several milliliters of the substance to be measured. Back in the mid-1990's in the David Weitz lab they developed the technique of microrheology which looks at the thermal motion of small particles embedded in a material in order to extract the bulk rheological properties. Only small amounts of material are needed for this technique. Additionally, this technique increases our microscopic understanding of these complicated materials.

Viscosity and diffusion: A simple experiment is to put small tracer particles in a viscous fluid, and study their motion. The drawing below-left is an example. The mean square displacement <dx^2> of the batch of tracer particles grows linearly in time dt, with a growth rate given by the diffusion constant D (this is the first equation at the right). The diffusion constant is related to the radius of the tracer particles a, the temperature of the fluid, and the viscosity of the fluid eta; this is given by the second equation at right. In that equation, k is the Boltzmann constant and pi=3.14159... Thus, by measuring <dx^2>, if the particle size and temperature of the fluid are known, the fluid viscosity can be determined.
 = 2D dt D = kT / (6 pi eta a) LEFT:Trajectory of a two micron diameter particle for 24 s; each dot indicates it's position at 1/30 s intervals. It started at the red end and ended up at the blue end. The scale bar is half a micron long. This is from experimental data.
 RIGHT:Click here to see an animated GIF movie showing half micron diameter particles diffusing in water. The movie shows 4 s of data. Elasticity and thermal motion: Imagine if we repeat the same experiment in a purely elastic medium. In this case, the tracer particles move due to thermal motion, but never actually diffuse through the sample. In fact, the amount of motion they can achieve is directly linked to the elasticity of the medium. Over time, they have (1/2)kT of energy per degree of freedom, and this can be converted into stored elastic energy. The amplitude of the motion is <dx^2> as dt gets very large; in a purely elastic medium this approaches a constant value for large dt. Roughly speaking, this can be considered as similar to the stretching of a spring, so thus using the formula for stored energy in a spring, we get something like the formula at right. Thus, by measuring <x^2> for large dt, an effective spring constant K can be measured, and more rigorously this would be related to the elastic moduli of the material.
 spring energy = thermal energy (1/2) K (large dt) = (1/2) kT

Putting it all together: So, by measuring the motion of small beads we could either determine the viscosity of a pure fluid, or the elasticity of a purely elastic material. In practice, the motion of small beads in a viscoelastic material can be directly related to the viscoelastic moduli characterizing the material. The same quantity is measured, the mean square displacement <dx^2>, and analyzed to determine the viscoelastic properties. In general the mean square displacement has nontrivial dependence on the lag time dt, and this reflects the fact that the viscoelastic moduli are frequency dependent. Click here to see an animated GIF movie of particles doing Brownian motion. In both cases the particles are 2 microns in diameter. The left picture shows particles moving in pure water; the right picture shows particles moving in a concentrated solution of DNA, a viscoelastic solution in other words.

### An improvement: 2-Point Microrheology

The microrheology technique, as sketched above, has a few potential flaws. First, many complex materials are inhomogeneous on small scales. For example, mayonnaise is made from small droplets of oil. If your microscopic particles are trapped in the oil droplets, they will probe the viscoelastic properties of the oil, but not that of the mayonnaise as a bulk material. This inhomogeneity problem occurs whenever your particles are smaller than the inhomogeneity size. Second, in some cases the particles themselves may cause inhomogeneities, if they disturb the complex fluid in some way.

However, these problems can be solved by examining the correlated motion of two particles located in different places. Consider the case of a simple fluid. Each particle moves randomly, due to thermal energy (Brownian motion), but if the two particles are close to each other the motion will be somewhat correlated. In other words, if one particle moves to the left, a nearby particle is more likely to move to the left also. In fact, the amount of correlation should be reduced proportional to the inverse of the distance separating the two particles, if they are far away from each other. Likewise, the same relationship holds for two particles in a purely elastic material. The magic is that the amount of correlated motion not only depends on the separation distance, but also on the viscoelastic moduli of the material.

Thus, by examining many particles, this correlated motion can be measured. Each particle will be moving with some random, uncorrelated motion, and some motion that is correlated with nearby particles. By averaging all this, the random uncorrelated motion averages to zero, leaving only the correlated motion to be measured. From this the viscoelastic moduli can be determined.

The nice feature of this is that the correlation must decay as 1/r for a homogeneous viscoelastic medium. By measuring the correlation as a function of the separation r, this 1/r relationship can be looked for. For small separations, it may not hold: this gives the length scale of the inhomogeneity. For large separations, hopefully it does hold: this verifies that the method is working successfully. This solves the problems of studying inhomogeneous media. Another nice feature is that, in practice, the particle sizes don't need to be known, and in fact you can use particles with a variety of sizes at the same time.

 This page is currently maintained by Eric Weeks. If you have questions or comments send me email: erweeks / emory.edu I wrote this webpage orginally when I was a postdoc at Harvard with the Weitz group. (A version of this page still exists on their website, but without the images for some reason. That's why I made this copy.) Back when I was at Harvard, several of my friends were involved in microrheology. Here I've preserved my original list from back in ~1999: Eric "Colloidal Glasses" Weeks Megan "Fibroblasts" Valentine Bivash "PEO" Dasgupta John "Two-Point" Crocker Dave "Pays Our Salaries" Weitz A picture of a conventional rheometer -- pretty big and bulky, isn't it? Expensive, too. Microrheology can be done by video microscopy, or light scattering.