# Two-particle interfacial microrheology

Vikram Prasad, Skanda Vivek, Ken Desmond, Thibaut Divoux, Marquise Hopson, Stephan Koehler, Paul Martin, Justin Pye, James Sebel, Gopal Subedi, & Eric Weeks

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### Introduction to microrheology (one and two-particle)

Microrheology is a technique that involves putting tracer particles in a solution whose viscoelasticity you want to measure. Passive microrheology involves looking at the thermal diffusion of these particles, and from the Stokes-Einstein relation shown below, the viscoelasticity of the material can be determined. Here, dr is the displacement of a tracer particle, D is the diffusion coefficient, and eta is the viscosity of the material in question. This equation is true for viscous materials, but can be generalized to viscoelastic materials easily.
 = 6D dt D = kT / (6 pi eta a)

Microrheology resources:

This technique has been very powerful in estimating the viscoelasticity of materials like actin solutions, biopolymers, hydrogels etc. However, the technique has certain limitations. For example, if the material in question has the consistency of swiss cheese, with many pores in it, then the tracer particles could get trapped in these pores. We would then be measuring the properties of the holes in the cheese, rather than the cheese itself. To overcome these difficulties, a new technique called two-particle microrheology was developed. This technique looks at the correlated thermal motion of particles in the material. How is this different from looking at single particle motions? For starters, the motion of one bead creates a strain field(or flow field, if you want to be pedantic) in the medium surrounding it. This strain field will clearly affect the motion of another particle in the medium. The correlated motion simply tells us how this strain field is propogated in the medium, if we do the correlations at different separations. The larger the separation between the particles, we would expect the correlation to be less pronounced. In fact, in 3-dimensions, the correlations die out as 1/R (R being the separation between the particles), independent of the nature of the material.
 = 6 ~ 1/R

Here D_rr represents the longitudinal component of the correlated motions (the component parallel to the line joining the centers of the spheres). A similar expression can be written for the transverse component that is perpendicular to the line joining the centers of the spheres. The seminal paper to read about two-particle microrheology is :

### Two-particle microrheology at a fluid-fluid interface

What I have been researching for the past year is microrheology at an interface. The problem is thus: similar to the measurements in 3D, we should be able to extend two-particle microrheology to measure the viscoelasticity of an interface. However, complications arise due to the fact that an interface is always in contact with a reservoir of bulk fluid. This causes the Stokes-Einstein equation to be modified in quasi 2D systems, to the following:
 = 4D dt D = kT *(complicated logarithmic correction)/ (4 pi eta_s )
where eta_s is the viscosity of the interface. It has units different from bulk viscosity (the ratio of the surface to bulk viscosity has units of length)

The coupling between the interface and the bulk fluid reservoirs also causes the two-particle correlations to be modified from the 1/R behavior described in the previous section. While this has been described by theory, no experiments to date have performed 2-particle microrheology at an interface. We look at the correlated motion of polystyrene beads at an air-water interface inhabited by Human Serum Albumin (HSA) molecules. We find that the correlated motion depends sensitively on the surface viscosity eta_s of the interface. Details about this behavior can be found in our paper:

### Microrheology of soap films

A soap film is truly a quasi 2-d viscous system, since it consists of an extremely thin layer of water, and two surfactant layers that buffer the water from the air above and below it. The thickness of the soap film can range from a couple of nanometers to many microns thick, but if it reflects light in the optical spectrum, then the size is probably beween 100nm-5 microns. Shown below is a picture that is taken of an 18-hr old soap film (I made it with a solution of 60% glycerol, 40% water and 2% solution of the kitchen detergent DAWN)

Since the soap film looks lightish-blue in color, it is of the order of a few hundred nanometers. Typically, the soap film starts off as micron-sized or slightly larger, but over time it drains of water, and starts thinning. Experiments are in progress to entrain spheres (colloids) and rods (bacteria) in these soap films and to observe their Brownian motion.

Soap films are extremely thin, but our work shows that in some cases three dimensional effects due to their finite thickness are important. When the soap films are really thin (less than a micron thick), they behave as 2D liquids with a 2D viscosity. However, clearly this will not be true if the layer of liquid is extremely thick; in such a case you'd have to see 3D liquid behavior. In our work, we identify precisely where the transition from 2D to 3D behavior occurs as the thickness of the soap film is gradually increased. We do so by putting spherical particles of size d in soap films of thickness h, and observing the thermal Brownian motion of these particles with a microscope. This is done for different values of h/d ranging from 0.5-15. We use the particles' motion to measure a 2D viscosity of the soap films. Unphysical values of the 2D viscosity are found for soap films with h/d > 7, leading us to propose that soap films transition from 2D to 3D fluid-like behavior at a critical ratio h/d = 7. Details about this can be found in our papers:

For more information, please contact Eric Weeks: <weeks(at)physics.emory.edu> or Vikram Prasad: <vprasad(at)physics.emory.edu>