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Functionalized AFM probes for force spectroscopy: eigenmode shapes and stiffness calibration through thermal noise measurements

Justine Laurent, Audrey Steinberger and Ludovic Bellon,  Nanotechnology 24, 225504 (2013)

doi: 10.1088/0957-4484/24/22/225504

The functionalization of an atomic force microscope (AFM) cantilever with a colloidal bead is a widely used technique when the geometry between the probe and the sample must be controlled, particularly in force spectroscopy. But some questions remain: how does a bead glued at the end of a cantilever influence its mechanical response? And more importantly for quantitative measurements, can we still determine the stiffness of the AFM probe with traditional techniques? In this paper, the influence of the colloidal mass loading on the eigenmode shape and resonant frequency is investigated by measuring the thermal noise on rectangular AFM microcantilevers with and without beads attached at their extremities. The experiments are performed with a home-made ultra-sensitive AFM, based on differential interferometry. The focused beam from the interferometer probes the cantilever at different positions and the spatial shapes of the modes are determined up to the fifth resonance, without external excitation. The results clearly demonstrate that the first eigenmode is almost unchanged by mass loading. However the oscillation behavior of higher resonances presents a marked difference: with a particle glued at its extremity, the nodes of the modes are displaced towards the free end of the cantilever. These results are compared to an analytical model taking into account the mass and inertial moment of the load in an Euler–Bernoulli framework, where the normalization of the eigenmodes is explicitly worked out in order to allow a quantitative prediction of the thermal noise amplitude of each mode. A good agreement between the experimental results and the analytical model is demonstrated, allowing a clean calibration of the probe stiffness.

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Dissipation of micro-cantilevers as a function of air pressure and metallic coating

Tianjun Li, Ludovic Bellon,  EPL 98, 14004 (2012)

doi: 10.1209/0295-5075/98/14004

In this letter, we characterize the internal dissipation of coated micro-cantilevers through their mechanical thermal noise. Using a home-made interferometric setup, we achieve a resolution down to 10-14 m/rtHz in the measurement of their deflection. With the use of the fluctuation dissipation theorem and of the Kramers-Kronig relations, we rebuilt the full mechanical response function from the measured noise spectrum, and investigate frequency dependent dissipation as a function of the air pressure and of the nature of the metallic coatings. Using different thicknesses of gold coatings, we demonstrate that the internal viscoelastic damping is solely due to the dissipation in the bulk of the coating.

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Carbon nanotubes adhesion and nanomechanical behavior from peeling force spectroscopy

Julien Buchoux, Ludovic Bellon, Sophie Marsaudon, Jean-Pierre Aimé, European Journal of Physics B 84, 69–77 (2011)

doi: 10.1140/epjb/e2011-20204-1

Applications based on Single Walled Carbon Nanotube (SWNT) are good example of the great need to continuously develop metrology methods in the field of nanotechnology. Contact and interface properties are key parameters that determine the efficiency of SWNT functionalized nanomaterials and nanodevices. In this work we have taken advantage of a good control of the SWNT growth processes at an atomic force microscope (AFM) tip apex and the use of a low noise (10-13 m/rtHz) AFM to investigate the mechanical behavior of a SWNT touching a surface. By simultaneously recording static and dynamic properties of SWNT, we show that the contact corresponds to a peeling geometry, and extract quantities such as adhesion energy per unit length, curvature and bending rigidity of the nanotube. A complete picture of the local shape of the SWNT and its mechanical behavior is provided.

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Exploring nano-mechanics through thermal fluctuations

L. Bellon, Habilitation à Diriger des recherches de L’École Normale Supérieure de Lyon, 2010

oai: tel.archives-ouvertes.fr/tel-00541336

Abstract

This mémoire presents my current research interests in micro and nano-mechanics in a comprehensive manuscript. Our experimental device is first presented: this atomic force microscope, designed and realized in the Laboratoire de Physique de l’ENS Lyon, is based on a quadrature phase differential interferometer. It features a very high resolution (down to 10 fm/rtHz) in the measurement of deflexion, down to low frequencies and on a huge input range. The dual output of the interferometer implies a specific handling to interface common scanning probe microscope controllers. We developed analog circuitries to tackle static (contact mode) and dynamic (tapping mode) operations, and we demonstrate their performance by imaging a simple calibration sample.

As a first application, we used the high sensitivity of our interferometer to study the mechanical behavior of micro-cantilevers from their fluctuations. The keystone of the analysis is the Fluctuation-Dissipation Theorem (FDT), relating the thermal noise spectrum to the dissipative part of the response. We apply this strategy to confront Sader’s model for viscous dissipation with measurements on raw silicon cantilevers in air, demonstrating an excellent agreement. When a gold coating is added, the thermal noise is strongly modified, presenting a 1/f like trend at low frequencies: we show that this behavior is due to a viscoelastic damping, and we provide a quantitative phenomenological model. We also characterize the mechanical properties of cantilevers (stiffness and Elastic Moduli) from a mapping of the thermal noise on their surface. This analysis validates the description of the system in term of its normal modes of oscillations in an Euler-Bernoulli framework for flexion and in Saint-Venant approach for torsion, but points toward a refined model for the dispersion relation of torsional modes.

Finally, we present peeling experiments on a single wall carbon nanotube attached to the cantilever tip. It is pushed against a flat substrate, and we measure the quasi-static force as well as the dynamic stiffness using an analysis of the thermal noise during this process. The most striking feature of these two observables is a plateau curve for a large range of compression, the values of which are substrate dependent. We use the Elastica to describe the shape of the nanotube, and a simple energy of adhesion per unit length Ea to describe the interaction with the substrate. We analytically derive a complete description of the expected behavior in the limit of long nanotubes. The analysis of the experimental data within this simple framework naturally leads to every quantity of interest in the problem: the force plateau is a direct measurement of the energy of adhesion Ea for each substrate, and we easily determine the mechanical properties of the nanotube itself.

Keywords

thermal noise, Atomic Force Microscopy (AFM), cantilever, Sader model, interferometry, dissipation, carbon nanotube, high precision, metrology, FDT, Kramers-Kronig, Elastica, stiffness, viscoleasticity, coating

Jury:

  1. -Arezki Boudaoud – Président du jury 
  2. -Joël Chevrier – Rapporteur
  3. -Matteo Ciccotti – Rapporteur
  4. -Philippe Poncharal – Rapporteur
  5. -Mark Rutland – Examinateur

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Steady state fluctuation relations for systems driven by an external random force

J.R. Gomez-Solano, L. Bellon, A. Petrosyan and S. Ciliberto, Europhysics Letters 89, 60003 (2010)

doi: 10.1209/0295-5075/89/60003

We experimentally study the fluctuations of the work done by an external Gaussian random force on two different stochastic systems coupled to a thermal bath: a colloidal particle in an optical trap and an atomic force microscopy cantilever. We determine the corresponding probability density functions for different random forcing amplitudes ranging from a small fraction to several times the amplitude of the thermal noise. In both systems for sufficiently weak forcing amplitudes the work fluctuations satisfy the usual steady state fluctuation theorem. As the forcing amplitude drives the system far from equilibrium, deviations of the fluctuation theorem increase monotonically. The deviations can be recasted to a single master curve which only depends on the kind of stochastic external force.

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Direct measurement of spatial modes of a micro-cantilever from thermal noise

P. Paolino, B. Tiribilli and L. Bellon, Journal of Applied Physics 106, 094313 (2009)

doi: 10.1063/1.3245394

Measurements of the deflection induced by thermal noise have been performed on a rectangular atomic force microscope cantilever in air. The detection method, based on polarization interferometry, can achieve a resolution of 10-14 m/Hz1/2 in the frequency range 1 kHz – 800 kHz. The focused beam from the interferometer probes the cantilever at different positions along its length and the spatial modes’ shapes are determined up to the fourth resonance, without external excitation. Results are in good agreement with theoretically expected behavior. From this analysis accurate determination of the elastic constant of the cantilever is also achieved.

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Frequency dependence of viscous and viscoelastic dissipation in coated micro-cantilevers from noise measurement

P. Paolino and L. Bellon, Nanotechnology 20, 405705 (2009)

doi: 10.1088/0957-4484/20/40/405705

We measure the mechanical thermal noise of soft silicon atomic force microscopy cantilevers. Using an interferometric setup, we obtain a resolution down to 10-14 m/Hz1/2 on a wide spectral range (3 Hz to 105 Hz). The low frequency behavior depends dramatically on the presence of a reflective coating: almost flat spectrums for uncoated cantilevers versus 1/f like trend for coated ones. The addition of a viscoelastic term in models of the mechanical system can account for this observation. Use of Kramers-Kronig relations validate this approach with a complete determination of the response of the cantilever: a power law with a small coefficient is found for the frequency dependence of viscoelasticity due to the coating, whereas the viscous damping due to the surrounding atmosphere is accurately described by the Sader model. 

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Bruit thermique et dissipation visqueuse d’un microlevier

P. Paolino and L. Bellon, poster presented in the 11ème forum des microscopies à sonde locale, Hardelot, France 16-19 mars 2009.

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Nous introduisons un cadre théorique simple pour prédire le bruit thermique d’un microlevier dans un fluide visqueux : nous utilisons l’approche de Sader pour décrire l’effet du fluide environnant (masse ajoutée et traînée visqueuse) et le Théorème Fluctuation-Dissipation appliqué aux modes propres de vibrations du système pour dériver une expression générique du spectre des fluctuations thermiques. Cette prédiction est comparée à une mesure expérimentale sur un levier commercial de microscopie à force atomique dans une gamme de fréquence couvrant les  premières résonances. Un excellent accord est obtenu sur l’ensemble du spectre, avec pour seul paramètre ajustable l’épaisseur du microlevier. En utilisant les relations de Kramers-Kronig, nous reconstruisons également la fonction réponse mécanique complète du levier et démontrons la pertinence du modèle de Sader en dehors des résonances.

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Bruit thermique et dissipation d’un microlevier

Pierdomenico Paolino, PhD Thesis, École Normale Supérieure de Lyon (2008)

hal: tel-00423692

 En microscopie à force atomique (AFM), l’étude des échantillons est réalisée à l’aide d’une pointe montée sur un microlevier. Le coeur de la technique est la mesure de la force d’interaction pointe-surface, directement proportionnelle à la déflexion du levier. Plus généralement, la compréhension profonde des propriétés mécaniques des microstructures joue un rôle significatif dans le développement des microsystèmes électromécaniques (MEMS), ou encore de capteurs chimiques ou biologiques miniatures.

Au delà du dispositif traditionnel de mesure de déflexion angulaire, nous avons conçu et réalisé un AFM avec une détection interférométrique différentielle (entre la base encastrée et l’extrémité libre du levier). La résolution ultime est de 10-14 m/Hz1/2, la mesure est de plus intrinsèquement calibrée, indifférente aux dérives thermiques lentes et sans limitation de la plage d’amplitude de la déflexion. 

Grâce à notre dispositif, nous mesurons le bruit thermique le long du levier. Une reconstruction de la forme spatiale des quatre premiers modes propres en flexion révèle un excellent accord avec le modèle de poutre de Euler-Bernoulli. Un ajustement simultané sur les quatre résonances thermiquement excitées est réalisé à l’aide d’un seul paramètre libre : la raideur du levier, qui est ainsi mesurée avec une grande précision et robustesse. 

Les fluctuations thermiques de déflexion à basse fréquence démontrent qu’un modèle d’oscillateur harmonique avec dissipation visqueuse n’est plus pertinent hors résonance. De plus, on observe des différences substantielles entre les leviers avec et sans revêtement métallique. Pour ces derniers, l’approche hydrodynamique de Sader rend compte fidèlement du comportement des fluctuations en dessous de la résonance dans l’air. La présence du revêtement introduit une deuxième source de dissipation : la viscoélasticité. Elle se manifeste comme un bruit en 1/f à basse fréquence. L’utilisation du Théorème Fluctuation-Dissipation (TFD) et des relations de Kramers-Kronig permettent une caractérisation complète de la réponse du levier à l’aide des spectres de fluctuations. Une estimation quantitative de la viscoélasticité et de sa dépendance en fréquence est notamment obtenue.

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Thermal noise of microcantilevers in viscous fluids

L. Bellon, Journal of Applied Physics 104, 104906 (2008)

doi: 10.1063/1.3021102

We present a simple theoretical framework to describe the thermal noise of a microscopic mechanical beam in a viscous fluid: we use the Sader approach to describe the effect of thesurrounding fluid (added mass and viscous drag) and the fluctuation dissipation theorem for each flexural modes of the system to derive a general expression for the power spectrum density offluctuations. This prediction is compared with an experimental measurement on a commercial atomic force microscopy cantilever in a frequency range covering the two first resonances. A very good agreement is found on the whole spectrum, with no adjustable parameters but the thickness of the cantilever.

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