The stochastic dynamics of micron and nanoscale elastic cantilevers in fluid: fluctuations from dissipation

TL;DR

This paper develops a thermodynamic and analytical framework to quantify the stochastic Brownian motion of micron and nanoscale cantilevers in fluid, relevant for improving atomic force microscopy and multi-cantilever sensing.

Contribution

It introduces a general thermodynamic approach and deterministic finite element methods to analyze stochastic fluctuations of arbitrary cantilever geometries and arrays, accounting for fluid-induced dissipation.

Findings

Analytical spectral density of noise force for long slender cantilevers.

Quantification of autocorrelation and noise spectra for single and coupled cantilevers.

Demonstration of noise reduction via correlated measurements of nanoscale cantilever arrays.

Abstract

The stochastic dynamics of micron and nanoscale cantilevers immersed in a viscous fluid are quantified. Analytical results are presented for long slender cantilevers driven by Brownian noise. The spectral density of the noise force is not assumed to be white and the frequency dependence is determined from the fluctuation-dissipation theorem. The analytical results are shown to be useful for the micron scale cantilevers that are commonly used in atomic force microscopy. A general thermodynamic approach is developed that is valid for cantilevers of arbitrary geometry as well as for arrays of multiple cantilevers whose stochastic motion is coupled through the fluid. It is shown that the fluctuation-dissipation theorem permits the calculation of stochastic quantities via straightforward deterministic methods. The thermodynamic approach is used with deterministic finite element numerical simulations to quantify the autocorrelation and noise spectrum of cantilever fluctuations for a single micron scale cantilever and the cross-correlations and noise spectra of fluctuations for an array of two experimentally motivated nanoscale cantilevers as a function of cantilever separation. The results are used to quantify the noise reduction possible using correlated measurements with two closely spaced nanoscale cantilevers.