Microrheology

Passive microrheology

Passive microrheology uses the thermal energy (kT) to move the tracers, although recent evidence suggest that active random forces inside cells may instead move the tracers in a diffusive-like manner. 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.

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 ( ω ) = G ′ ( ω ) + i G ″ ( ω ) with G′(ω) the elastic (conservative) part and G″(ω) the viscous (dissipative) part and ω=2πf the pulsation. The GSER is as follows:

G ~ ( s ) = k B T π a s ⟨ Δ r ~ 2 ( s ) ⟩

with

G ~ ( s ) : Laplace transform of G k_B: Boltzmann constant T: temperature in kelvins s: the Laplace frequency a: the radius of the tracer ⟨ Δ 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 ω 2 > can be found, and then related to the real and imaginary parts of the response function, α ( ω ) . The response function leads directly to a calculation of the complex shear modulus, G ( ω ) via:

G ( ω ) = 1 6 π a α ( ω )

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. More recently, it has been developed into Force spectrum microscopy to measure contributions of random active motor proteins to diffusive motion in the cytoskeleton.