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^{1}, Milad Radiom

^{1}, Brian A. Robbins

^{1}, John Y. Walz

^{1}, Mark R. Paul

^{1}and William A. Ducker

^{1,a)}

### Abstract

We validate a theoretical approach for analyzing correlations in the fluctuations of two cantilevers in terms of a deterministic model, using the fluctuation-dissipation theorem [M. R. Paul and M. C. Cross, Phys. Rev. Lett. **92**, 235501 (2004)]. The validation has been made possible through measurement of the correlations between the thermally stimulated vibrations of two closely spaced micrometer-scale cantilevers in fluid. Validation of the theory enables development of a method for characterizing fluids, which we call correlation force spectrometry.

The work described in this paper was funded by the National Science Foundation via Award Number CBET-0959228 and by Virginia Tech.

### Key Topics

- Viscosity
- 5.0
- Thermal noise
- 4.0
- Acoustic noise spectra
- 3.0
- Boltzmann equations
- 3.0
- Electric measurements
- 3.0

## Figures

(Color online) (a) Light microscope image of closely spaced AFM cantilevers seen from above and the side. The cantilevers are 200 *μ*m long and 40 *μ*m wide. (b) Schematic of the cantilevers separated by a distance *s*, and the shape of the free end of the simulated cantilever. The real cantilever has a flat 10 *μ*m end and then slopes back at about 59°; this slope is simulated by two equal steps for computational simplicity. (c) Schematic of the detection system.

(Color online) (a) Light microscope image of closely spaced AFM cantilevers seen from above and the side. The cantilevers are 200 *μ*m long and 40 *μ*m wide. (b) Schematic of the cantilevers separated by a distance *s*, and the shape of the free end of the simulated cantilever. The real cantilever has a flat 10 *μ*m end and then slopes back at about 59°; this slope is simulated by two equal steps for computational simplicity. (c) Schematic of the detection system.

(Color online) (a) Auto-correlation. (b) Cross-correlation. (c) Noise spectrum, G_{12}. These results are for a pair of commercial AFM cantilevers (ORC8 B: length = 200 *μ*m, width = 40 *μ*m, and Nm^{−1}) separated by 8 *μ*m in liquid water at 23 °C. In (a) and (b), the experimental measurements are shown using data symbols and the theoretical prediction is given by the solid line. The left axis shows the correlation function normalized by *k* _{b} *T*/*k* where *k _{b} * is Boltzmann’s constant,

*T*is the temperature, and the right axis shows the same data in units of nm

^{2}using the measured value of the spring constant. In (b), the single error bar shows the range of data in three repeat experiments at this time lag. Each repeat experiment has a different pair of cantilevers. In (c), the noisy line is the experimental measurement and the smooth line is the theoretical prediction.

(Color online) (a) Auto-correlation. (b) Cross-correlation. (c) Noise spectrum, G_{12}. These results are for a pair of commercial AFM cantilevers (ORC8 B: length = 200 *μ*m, width = 40 *μ*m, and Nm^{−1}) separated by 8 *μ*m in liquid water at 23 °C. In (a) and (b), the experimental measurements are shown using data symbols and the theoretical prediction is given by the solid line. The left axis shows the correlation function normalized by *k* _{b} *T*/*k* where *k _{b} * is Boltzmann’s constant,

*T*is the temperature, and the right axis shows the same data in units of nm

^{2}using the measured value of the spring constant. In (b), the single error bar shows the range of data in three repeat experiments at this time lag. Each repeat experiment has a different pair of cantilevers. In (c), the noisy line is the experimental measurement and the smooth line is the theoretical prediction.

(Color online) Autocorrelation of equilibrium fluctuations in cantilever displacement for an AFM cantilever in a series of Newtonian fluids. The left axis is normalized by *k* _{b} *T*/*k* and the right axis is in units of nm^{2}.

(Color online) Autocorrelation of equilibrium fluctuations in cantilever displacement for an AFM cantilever in a series of Newtonian fluids. The left axis is normalized by *k* _{b} *T*/*k* and the right axis is in units of nm^{2}.

(Color online) Cross-correlation of equilibrium fluctuations in cantilever displacement for a pair of AFM cantilevers in a series of Newtonian fluids. Experimental measurements are shown as data symbols and theoretical predictions are shown by the solid lines. The left axis is normalized by *k* _{b} *T*/*k* and the right axis is in units of nm^{2}.

(Color online) Cross-correlation of equilibrium fluctuations in cantilever displacement for a pair of AFM cantilevers in a series of Newtonian fluids. Experimental measurements are shown as data symbols and theoretical predictions are shown by the solid lines. The left axis is normalized by *k* _{b} *T*/*k* and the right axis is in units of nm^{2}.

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