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2022 | OriginalPaper | Chapter

3. Motion of Chiral and Achiral Structures at Low Re

Author : Johannes Sachs

Published in: Motion, Symmetry & Spectroscopy of Chiral Nanostructures

Publisher: Springer International Publishing

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Abstract

This chapter is based on and contains excerpts and figures from the articles “Role of symmetry in driven propulsion at low Reynolds number” [1] and “Characterization of active matter in dense suspensions with heterodyne laser Doppler velocimetry” [2]. Contributions of coauthors are indicated.

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previous chapter Fundamentals of Chiral Nanostructures in Fluids

next chapter Chiroptical Spectroscopy of Single Chiral and Achiral Nanoparticles

Many textbooks, e.g. see [20], define the principal axes of either rotation, translation or coupling tensors based on eigenvectors that diagonalize the corresponding hydrodynamic resistance tensors, whereas here its is defined in terms of the hydrodynamic mobility, which is the inverse of the resistance tensors (see (2.10)). For arbitrary shapes the two definitions yield different frames.

The entries of \(\boldsymbol{{\mathfrak C}}\) are determined by \({\mathfrak C}_{i} \equiv \mathscr {G}_{i i} /\left( \mathscr {F}_{i} \ell \right) \) on the diagonal and \({\mathfrak C}_{i j}=\frac{1}{2 \ell }\left( \mathscr {G}_{i j} / \mathscr {F}_{j}+\mathscr {G}_{j i} / \mathscr {F}_{i}\right) \) with \(i\ne j\) and no summation over repeating indices for off-diagonal elements.

For the 99.5% pure glycerol that has been used, a temperature increase from 25\(^{\circ }\)C to 29\(^{\circ }\)C corresponds to a decrease in dynamic viscosity \(\eta \) by a factor 1.4.

The index i in \(\widehat{R}_i\) indicates a rotation about \(\pi \) around the i-th principal axis of rotation \(\boldsymbol{e}_i\).

Here z indicates the optical axis of the microscope.

The resolution limit for visible light and an NA \(=\) 1 is approximately \(r_{\mathrm {res}} = \frac{\lambda }{2\mathrm {NA}} \approx 250\,\) nm.

In practice the artificial reference beam was realized by shifting the frequency of the illumination beam with an acousto-optical modulator (APE GmbH) by a desired Bragg frequency \(\omega _{\mathrm {Bragg}}\).

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Title: Motion of Chiral and Achiral Structures at Low Re
Author: Johannes Sachs
Publisher: Springer International Publishing
Book: Motion, Symmetry & Spectroscopy of Chiral Nanostructures
Print ISBN: 978-3-030-88688-2

Electronic ISBN: 978-3-030-88689-9

Copyright Year: 2022
DOI: https://doi.org/10.1007/978-3-030-88689-9_3