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Complex Fluids in Biological Systems pp 113–157Cite as

Theoretical Microrheology

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Part of the book series: Biological and Medical Physics, Biomedical Engineering ((BIOMEDICAL))

Abstract

The importance of microrheology in the study of biological systems has a long and rich history, tracing its roots to the work of the botanist Robert Brown in the early nineteenth century. Indeed, passive microrheology and Brownian motion are one and the same. Brown’s observation of microscopic pollen grains dancing about in water was initially thought to reveal some sort of “fundamental life force.” However, upon further investigation, it turned out the motion depended only on the microscopically small size of the particles. The mysterious phenomenon went unexplained until the turn of the next century when Einstein and Perrin utilized Brownian motion to prove the atomic nature of matter. In addition to this profound result, the foundation of modern-day passive microrheology had been laid. Einstein combined the theory of diffusion with the Stokes’ solutions for creeping flow to yield the Stokes–Einstein relation connecting observable particle motion—diffusion—to material properties: the viscosity. Perrin’s experiments confirmed the theory. But Einstein’s arguments and the Stokes–Einstein relation rely on the existence of equilibrium and other narrow criteria. New approaches have extended the idea of tracking the motion of a Brownian particle to understand material properties far beyond this limited model. These advancements are critical to the study of many biological systems which conduct much of their function in a nonequilibrium condition. In this chapter we will see how one can study biological systems from a rheological perspective, showing the unique role played by microrheology in understanding such systems. In a sense, Brown’s initial hypothesis was not too far off the mark: rather than being driven by life, Brownian motion plays the role of the invisible hand that drives many of the processes required for life to proceed and indeed may have played a role in the very origin of the life process.

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Notes

  1. 1.

    Equation (3.5) is appropriate for consideration of fluid relaxation in the sense of a steady flow. It should not be confused with the unsteady Stokes equations which, used in conjunction with a Langevin equation, are appropriate for the study of vorticity diffusion in molecular liquids and the time-dependent Stokes drag. Further discussion of the topic may be found in e.g. Russel (1981) [73].

  2. 2.

    Colloid motion and the associated Stokes drag reach steady state over a short but finite time. However, over times relevant to study of diffusion coefficients, the unsteadiness in particle motion is superfluous. Further discussion of the topic may be found in, e.g., Russel (1981) [73].

  3. 3.

    Batchelor combined statistical mechanics theory and the hydrodynamics of Stokes flow to derive expressions and some values for the coefficients K ij , C, and S ij by modeling three systems: particle velocity during sedimentation, gradient diffusion, and tracer diffusion in a polydisperse suspension. For each an average of the appropriate hydrodynamic functions for particle motion is weighted by the distribution of positions of particle pairs [6, 69].

  4. 4.

    The idea of a fixed force or fixed velocity is an approximation, the accuracy of which is dictated by the uniformity of the applied field or the stiffness of the optical trap. Such approximations can be made quite accurate [37, 38].

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  1. Cornell University, Ithaca, NY, USA

    Roseanna N. Zia

  2. California Institute of Technology, Pasadena, CA, USA

    John F. Brady

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    Saverio E. Spagnolie

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Zia, R.N., Brady, J.F. (2015). Theoretical Microrheology. In: Spagnolie, S. (eds) Complex Fluids in Biological Systems. Biological and Medical Physics, Biomedical Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2065-5_3

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