Abstract

Near vertical optically trapped dimers, composed of pairs of microspheres, and constructed in situ, were imaged in bright-field in flow and at rest, and with displacement $\Delta z$ from the transverse $xy$ imaging plane of an inverted microscope. Image first central moments ${\mu _{01}}$ were measured, and their dependence on the imposed flow velocity of the surrounding fluid was calculated. This dependence was related to the at-rest restricted diffusion statistics. It was assumed that, for small perturbations, the torque $T$ on the dimer was proportional to the velocity of flow $v$ and resulting angular deflection $\Delta \theta$ so that $T \propto v \propto \Delta \theta$. Displacements $\Delta z$ at which $v \propto \Delta {\mu _{01}} \propto \Delta \theta$, which are typically off focus, were examined in more detail; in this range, $\Delta \theta = h\Delta {\mu _{01}}$. The hydrodynamics of the dimer were modeled as that of a prolate ellipsoid, and the constant of proportionality $h$ was determined by comparing the short-time mean-squared variation measured during diffusion to that predicted by the model calculation: ${h^2}\langle \Delta\mu_{01}^2(t)\rangle = \langle \Delta {\theta ^2}(t)\rangle$. With $h$ determined, the optical trap stiffness ${k_\theta}$ was determined from the long-time restricted diffusion of the dimer. The measured ${k_\theta}$ and $\Delta \theta$ can then be used compute torque: $T = {k_\theta}\Delta \theta$, potentially enabling the near vertical optically trapped dimer to be used as a torque probe.

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