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Role of a finite exposure time on measuring an elastic modulus using microrheology

Chemical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

^*Author to whom correspondence should be addressed. Email address: pdoyle@mit.edu

Phys. Rev. E 71, 041106 – Published 25 April, 2005

DOI: https://doi.org/10.1103/PhysRevE.71.041106

Abstract

The role of a finite exposure time $σ$ on measuring rheological properties using microrheology techniques is theoretically investigated. We concentrate on studying fluid models displaying a plateau in the mean-squared displacement (MSD) of the embedded probe particle. A model is developed to compare the resulting experimentally measured MSD of the particle to its expected value in the fluid model. A plateau MSD is greatly modified in a measurement when $σ$ is greater than the plateau onset time. Moreover, apparent dynamics drastically differ from the true dynamics at frequencies $ω ≲ σ^{- 1}$ . These results quantify when and how a finite exposure time effects the measured MSD of a probe particle which can then alter the extracted rheological properties and physical interpretations.

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References (19)

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In Fig 1(d), one cannot see $\bar{α} (σ, σ) \to α$ as $σ \to 0$ . In that limit, both $t$ and $σ$ are tending to 0 together since we set $t = σ$ . One can show, however, that $\bar{α} (t, σ) = α (t) + O ({(σ ∕ t)}^{ν})$ with $ν > 0$ .
Note that the plateau scaled value obtained from Eq. (24) is $1 - τ_{b} ∕ τ_{m}$ .
We use $1 ∕ τ_{+} + 1 ∕ τ_{-} = 1 ∕ τ_{m}$ . Note that this approximation returns the inertialess limit for $σ = 0$ : $⟨ Δ {\bar{x}}^{2} (t, σ = 0) ⟩ ∕ ⟨ Δ x_{p}^{2} ⟩ ≃ t ∕ τ_{m} + 1$ .
At the concentration used ( $[CTAB] ≃ 0.3 M$ and $[K Br] ≃ 1 M$ ) and at $T ≃ 35 ° C$ , the solution is Maxwellian in the vicinity of a relaxation time $τ_{m} ≃ 0.1 s$ with a high-frequency elastic plateau at $G ≃ 1000 Pa$ . The Brownian time can be calculated with $τ_{b} = 2 a^{2} ρ ∕ (9 G τ_{m}) ≃ 10^{- 13} s$ , where $ρ ≃ 1000 kg ∕ m^{3}$ is the particle density.

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Role of a finite exposure time on measuring an elastic modulus using microrheology

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