Disclaimer: these examples are mostly taken from my pre-Eikomagos career
Rheology is the study of mechanical properties of fluids and soft solids, i.e. viscoelasticity. These parameters are routinely measured by mechanically disturbing the sample and measuring the forces involved. A macroscopic approach like this can however not capture local properties of a heterogeneous sample. In microrheology a sample is doped with small particles which are agitated naturally by the thermal motion of the molecules (Brownian motion). Physical parameters can be extraced by statistical analysis of the paths travelled by the particles
For a Newtonian fluid, the statistics are described by:
〈 Δ2(t) 〉 = 2 𝔻 D t
The expectation value of the displacement squared is linear in time with the diffusivity D and dimensionality 𝔻 as scaling factors. The displacement can be measured imaging and tracking the particles over time.
Figure 1 shows 〈 Δ2(t) 〉 at 0° and 20°C. Also shown is the predicted displacement using the physical parameters obtained using a bulk experiment for reference. The agreement is excellent.
The result for 20° seems like it might be an underestimate at longer times, although it is based on a single experiment. Thinking about this, we realised the way our tracking deals with collisions between particles might introduce a bias.
A simulation was run to investigate. The video first shows the ground truth (the parameters used to run the experiment). Then the displacement squared using an increasing amount of particles (shown in the legend) to gather statistics. Since it is a simulation, we know the identity of each particle. The result is in perfect agreement with the ground truth. If we include “collisions” and the way our tracking deals with those, we can see that it indeed introduces a bias.
The bias can be avoided by changing the tracking algorithm[*], but the finite field of view will induce a similar effect. For Newtonian fluids, we can simply use short time scales only. However, for general viscoelastic fluids, we need longer time scales to be able to capture the physical parameters - a trade-off will need to be made.
[*] one can also forego tracking and use correlation functions