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Resonances arising from hydrodynamic memory in Brownian motion

Nature volume 478, pages 85–88 (2011)Cite this article

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Abstract

Observation of the Brownian motion of a small probe interacting with its environment provides one of the main strategies for characterizing soft matter1,2,3,4. Essentially, two counteracting forces govern the motion of the Brownian particle. First, the particle is driven by rapid collisions with the surrounding solvent molecules, referred to as thermal noise. Second, the friction between the particle and the viscous solvent damps its motion. Conventionally, the thermal force is assumed to be random and characterized by a Gaussian white noise spectrum. The friction is assumed to be given by the Stokes drag, suggesting that motion is overdamped at long times in particle tracking experiments, when inertia becomes negligible. However, as the particle receives momentum from the fluctuating fluid molecules, it also displaces the fluid in its immediate vicinity. The entrained fluid acts back on the particle and gives rise to long-range correlations5,6. This hydrodynamic ‘memory’ translates to thermal forces, which have a coloured, that is, non-white, noise spectrum. One hundred years after Perrin’s pioneering experiments on Brownian motion7,8,9, direct experimental observation of this colour is still elusive10. Here we measure the spectrum of thermal noise by confining the Brownian fluctuations of a microsphere in a strong optical trap. We show that hydrodynamic correlations result in a resonant peak in the power spectral density of the sphere’s positional fluctuations, in strong contrast to overdamped systems. Furthermore, we demonstrate different strategies to achieve peak amplification. By analogy with microcantilever-based sensors11,12, our results reveal that the particle–fluid–trap system can be considered a nanomechanical resonator in which the intrinsic hydrodynamic backflow enhances resonance. Therefore, instead of being treated as a disturbance, details in thermal noise could be exploited for the development of new types of sensor and particle-based assay in lab-on-a-chip applications13,14.

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Figure 1: Characteristic time scales of a Brownian particle confined by the three-dimensional (3D) harmonic potential of an optical trap.
Figure 2: The colour of thermal force.
Figure 3: Enhancing resonance and sensitivity to particle size.
Figure 4: Transition to the inertial regime.

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Acknowledgements

S.J. acknowledges the Swiss National Science Foundation (SNF; grant nos 200021-113529 and 206021-121396). M.G. is supported by NCCR Nanoscale Science and the German Academic Exchange Service (DAAD) and F.M.M. is supported by the National Competence Center in Biomedical Imaging (NCCBI). M.B. and G.F. acknowledge support from the SNF (grant no. PP0022_119006). We thank W. Öffner and R. Ko˝szali for technical help and U. Aebi, B. U. Felderhof, H. Flyvbjerg, S. Melchionna, E. Sackmann and R. G. Winkler for discussions.

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Author notes

  1. Thomas Franosch and Matthias Grimm: These authors contributed equally to this work.

Authors and Affiliations

  1. Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany

    Thomas Franosch

  2. M. E. Müller Institute for Structural Biology, Biozentrum, University of Basel, Klingelbergstrasse 70, Basel 4056, Switzerland

    Matthias Grimm & Sylvia Jeney

  3. Laboratory of Physics of Complex Matter, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

    Matthias Grimm, Flavio M. Mor, László Forró & Sylvia Jeney

  4. Institute of Theoretical Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

    Maxim Belushkin & Giuseppe Foffi

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Contributions

T.F. and M.G. contributed to the planning of the experiments, designed parts of the data analysis software, derived the theory, fitted the theory to the data and interpreted the data. M.B. performed the simulations, analysed the numerical results and contributed to the fitting of the data. F.M.M. contributed to the optimization of the experimental set-up. M.B. and G.F. devised, implemented and tested the numerical simulations. L.F. contributed to the planning of the experiments. S.J. constructed and characterized the experimental set-up; designed, planned and carried out the experiments; designed the data analysis software; and interpreted the data. All authors contributed to, discussed and commented on the manuscript.

Corresponding author

Correspondence to Sylvia Jeney.

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The authors declare no competing financial interests.

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This file contains Supplementary Text and Data, Supplementary Figures 1 – 7 with legends and additional references. (PDF 1943 kb)

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Franosch, T., Grimm, M., Belushkin, M. et al. Resonances arising from hydrodynamic memory in Brownian motion. Nature 478, 85–88 (2011). https://doi.org/10.1038/nature10498

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