Tensorial generalized Stokes–Einstein relation for anisotropic probe microrheology

Abstract

In thermal “passive” microrheology, the random Brownian motion of anisotropically shaped probe particles embedded within an isotropic viscoelastic material can be used to extract the material’s frequency-dependent linear viscoelastic modulus. We unite the existing theoretical frameworks for separately treating translational and rotational probe motion in a viscoelastic material by extending the generalized Stokes–Einstein relation (GSER) into a tensorial form that reflects simultaneous equilibrium translational and rotational fluctuations of one or more anisotropic probe particles experiencing viscoelastic drag. The tensorial GSER provides a formal basis for interpreting the complex Brownian motion of anisotropic probes in a viscoelastic material. Based on known hydrodynamic calculations of the Stokes mobility of highly symmetric shapes in a simple viscous liquid, we show simple examples of the tensorial GSER for spheroids and half-stick, half-slip Janus spheres.