Microrheology

Microrheology^[1] is a technique used to measure the rheological properties of a medium, such as microviscosity, via the measurement of the trajectory of a flow tracer (a micrometre-sized particle). It is a new way of doing rheology, traditionally done using a rheometer. 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.^{[2] [3]}

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.^[4] 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.

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  Typical trajectory of a Brownian particle (simulation)

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  Two examples of MSD: one for a purely viscous fluid (free diffusion) and one for a viscolelastic fluid (trapped by elastic network)

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  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).^[5]

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 $G(\omega )=G'(\omega )+iG''(\omega )\,$ with G′(ω) the elastic (conservative) part and G″(ω) the viscous (dissipative) part and ω=2πf the pulsation. The GSER is as follows:

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

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

$\langle \Delta {\tilde {r}}^{{2}}(s)\rangle$: 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.^[6] 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, $\alpha (\omega )$.^[7] 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 ^{[8][9][10][11][12]} or optical tweezers^[13][14][15] 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.^[4]

References

1. Mason, Thomas G. & Weitz, David A. (1995). "Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids". Physical Review Letters. 74: 7. Bibcode:1995PhRvL..74.1250M. doi:10.1103/physrevlett.74.1250.
2. Crocker, John C.; Valentine, M. T.; Weeks, Eric R.; Gisler, T.; et al. (2000). "Two-Point Microrheology of Inhomogeneous Soft Materials". Physical Review Letters. 85 (4): 888–891. Bibcode:2000PhRvL..85..888C. doi:10.1103/PhysRevLett.85.888.
3. Levine, Alex J. & Lubensky, T. C. (2000). "One- and Two-Particle Microrheology". Physical Review Letters. 85 (8): 1774–1777. arXiv:cond-mat/0004103. Bibcode:2000PhRvL..85.1774L. doi:10.1103/PhysRevLett.85.1774.
4. Guo, Ming; et al. (2014). "Probing the Stochastic, Motor-Driven Properties of the Cytoplasm Using Force Spectrum Microscopy". Cell (158): 822–832. doi:10.1016/j.cell.2014.06.051.
5. Bellour, M.; Skouri, M.; Munch, J.-P.; Hébraud, P. (2002). "Brownian motion of particles embedded in a solution of giant micelles". European Physical Journal E. 8: 431–436. Bibcode:2002EPJE....8..431B. doi:10.1140/epje/i2002-10026-0.
6. Schnurr, B.; Gittes, F.; MacKintosh, F. C. & Schmidt, C. F. (1997). "Determining Microscopic Viscoelasticity in Flexible and Semiflexible Polymer Networks from Thermal Fluctuations". Macromolecules. 30 (25): 7781–7792. arXiv:cond-mat/9709231. Bibcode:1997MaMol..30.7781S. doi:10.1021/ma970555n.
7. Gittes, F.; Schnurr, B.; Olmsted, P. D.; MacKintosh, F. C.; et al. (1997). "Determining Microscopic Viscoelasticity in Flexible and Semiflexible Polymer Networks from Thermal Fluctuations". Physical Review Letters. 79 (17): 3286–3289. arXiv:cond-mat/9709228. Bibcode:1997PhRvL..79.3286G. doi:10.1103/PhysRevLett.79.3286.
8. A.R. Bausch; et al. (1999). "Measurement of local viscoelasticity and forces in living cells by magnetic tweezers". Biophysical Journal. 76 (1 Pt 1): 573–9. Bibcode:1999BpJ....76..573B. doi:10.1016/S0006-3495(99)77225-5. PMC 1302547. PMID 9876170.
9. K.S. Zaner & P.A. Valberg (1989). "Viscoelasticity of F-actin measured with magnetic microparticles". Journal of Cell Biology. 109 (5): 2233–43. doi:10.1083/jcb.109.5.2233. PMC 2115855. PMID 2808527.
10. F.Ziemann; J. Radler & E. Sackmann (1994). "Local measurements of viscoelastic moduli of entangled actin networks using an oscillating magnetic bead micro-rheometer". Biophysical Journal. 66 (6): 2210–6. Bibcode:1994BpJ....66.2210Z. doi:10.1016/S0006-3495(94)81017-3. PMC 1275947. PMID 8075354.
11. F.G. Schmidt; F. Ziemann & E. Sackmann (1996). "Shear field mapping in actin networks by using magnetic tweezers". European Biophysics Journal. 24: 348. doi:10.1007/bf00180376.
12. F. Amblard; et al. (1996). "Subdiffusion and Anomalous Local Viscoelasticity in Actin Networks". Physical Review Letters. 77 (21): 4470–4473. Bibcode:1996PhRvL..77.4470A. doi:10.1103/PhysRevLett.77.4470. PMID 10062546.
13. E. Helfer; et al. (2000). "Microrheology of Biopolymer-Membrane Complexes". Physical Review Letters. 85 (2): 457–60. Bibcode:2000PhRvL..85..457H. doi:10.1103/PhysRevLett.85.457. PMID 10991307.
14. Manlio Tassieri; et al. (2012). "Microrheology with optical tweezers: data analysis". New Journal of Physics. 14 (11): 115032. Bibcode:2012NJPh...14k5032T. doi:10.1088/1367-2630/14/11/115032.
15. David Engström; Michael C.M. Varney; Martin Persson; Rahul P. Trivedi; et al. (2012). "Unconventional structure-assisted optical manipulation of high-index nanowires in liquid crystals". Optics Express. 20 (7): 7741–7748. Bibcode:2012OExpr..20.7741E. doi:10.1364/OE.20.007741.

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
• Microrheology Overview with Animations