Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(iii) / 17 / ii / Solution

For a no-slip spherical tracer of radius $a$ in an isotropic incompressible elastic continuum, a small translation produces restoring force $6\pi aG\mathbf{x}$. ThusCombining this with the equilibrium three-dimensional positional variance givesEquivalently, if $P_{\infty }$ is the stationary three-dimensional displacement-MSD plateau, $G=k_{B}T/(\pi a{P}_{\infty })$. For a single measured coordinate, $⟨x_i^2⟩=k_{B}T/k$ and $G=k_{B}T/(6\pi a⟨x_i^2⟩)$.

This is passive microrheology. Calibrate bead radius, position sensitivity, temperature and localization noise; subtract drift and test that a plateau is resolved. The continuum must be meaningful on the bead scale, the bead must couple without slip, and thermal equilibrium must hold. Boundaries, compressibility, active motion, probe attachment and unresolved relaxation can change the inferred response. An imposed optical trap adds stiffness and must be removed or independently calibrated rather than mistaken for material elasticity.