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

Abstract

The stochastic dynamics of micron and nanoscale cantilevers immersed in a viscous fluidare quantified. Analytical results are presented for long slender cantilevers driven byBrownian noise. The spectral density of the noise force is not assumed to be white and thefrequency dependence of the noise force is determined from the fluctuation-dissipationtheorem. The analytical results are shown to be useful for the micron scale cantilevers thatare commonly used in atomic force microscopy. A general thermodynamic approach isdeveloped that is valid for cantilevers of arbitrary geometry as well as for arrays ofmultiple cantilevers whose stochastic motion is coupled through the fluid. It is shownthat the fluctuation-dissipation theorem permits the calculation of stochasticquantities via straightforward deterministic methods. The thermodynamic approachis used with deterministic finite element numerical simulations to quantify theauto-correlation and noise spectrum of cantilever fluctuations for a single micronscale cantilever and the cross-correlations and noise spectra of fluctuations foran array of two experimentally motivated nanoscale cantilevers as a function ofcantilever separation. The results are used to quantify the noise reduction possibleusing correlated measurements with two closely spaced nanoscale cantilevers.