Ruled variety

Microrheology is a technique to measure the rheological properties of a medium, such as microviscosity, via the measurement of the trajectory of a flow tracer (a micrometre-size particle). It is a new way of doing rheology, traditionally done using a rheometer. The size of the tracer is around a micrometre. There are two types of microrheology: passive microrheology and active microrheology. Passive microrheology uses inherent thermal energy to move the tracers, whereas active microrheology uses externally applied forces, such as from a magnetic field or an optical tweezer, to do so. Microrheology can be further differentiated into 1- and 2-particle methods .

Passive microrheology

Passive microrheology uses the thermal energy (kT) to move the tracers. The trajectories of the tracers are measured optically either by microscopy or by diffusing-wave spectroscopy (DWS). From the mean squared displacement with respect to time (noted MSD or <Δr²> ), one can calculate the visco-elastic moduli G′(ω) and G″(ω) using the generalized Stokes–Einstein relation (GSER). Here is a view of the trajectory of a particle of micrometer size.

• 
  Typical trajectory of a Brownian particle (simulation)
• 
  Two examples of MSD: one for a purely viscous fluid (free diffusion) and one for a viscolelastic fluid (trapped by elastic network)
• 
  Animation of a particle in a polymer-like network

Observing the MSD for a wide range of time scales gives information on the microstructure of the medium where are diffusing the tracers. If the tracers are having a free diffusion, one can deduce that the medium is purely viscous. If the tracers are having a sub-diffusive mean trajectory, it indicates that the medium presents some viscoelastic properties. For example, in a polymer network, the tracer may be trapped. The excursion δ of the tracer is related to the elastic modulus G′ with the relation G′ = k_BT/(6πaδ²).

Microrheology is another way to do linear rheology. Since the force involved is very weak (order of 10⁻¹⁵ N), microrheology is guaranteed to be in the so-called linear region of the strain/stress relationship. It is also able to measure very small volumes (biological cell).

Given the complex viscoelastic modulus $G(\omega )=G^{\prime }(\omega )+i{G}^{″}(\omega )$ with G′(ω) the elastic (conservative) part and G″(ω) the viscous (dissipative) part and ω=2πf the pulsation. The GSER is as follows:

$\tilde{G}(s)=\frac{k_{\mathrm{B}}T}{\pi as⟨\mathrm{\Delta }{\tilde{r}}^2(s)⟩}$

with

$\tilde{G}(s)$: Laplace transform of G

k_B: Boltzmann constant

T: temperature in kelvins

s: the Laplace frequency

a: the radius of the tracer

$⟨\mathrm{\Delta }{\tilde{r}}^2(s)⟩$: the Laplace transform of the mean squared displacement

A related method of passive microrheology involves the tracking positions of a particle at a high frequency, often with a quadrant photodiode. From the position, $x(t)$, the power spectrum, $<x_{\omega }^2>$ can be found, and then related to the real and imaginary parts of the response function, $\alpha (\omega )$. The response function leads directly to a calculation of the complex shear modulus, $G(\omega )$ via:

$G(\omega )=\frac{1}{6\pi a\alpha (\omega )}$

Active microrheology

Active microrheology may use a magnetic field or optical tweezers to apply a force on the tracer and then find the stress/strain relation.

References

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External links

• Harvard Weitz Lab page
• Review of microrheology in optical tweezers
• Review on microrheology
• Illustrated description of microrheology and a microrheology analysis, with movies