Modeling AFM cantilever fluctuations
Résumé
Atomic Force Microscopy (AFM) has become a widely used technique to measure mechanical properties of biological samples. Usually, a given load is applied to the surface, and the force is registered as a function of the indentation. This procedure can be complemented with other methods that are less intrusive and mainly, more relevant to the system's dynamics. In particular, the different oscillating modes of a cantilever represent an alternative way of obtaining information about the sample and its surrounding medium. We propose a spring-mass system to model the symmetric modes of the cantilever, allowing us to determine parameters related to the elasticity and the viscosity of the sample. We focus on four different situations: a) free oscillations in air, b) free oscillations in liquid, c) cantilever tip in contact with a stiff surface, and d) in contact with a soft material. The first two, while representing a good starting point to construct the model, are critical when calibrating the spring constant of the cantilever. In this case, the deflection due to thermal fluctuations gives information about the stiffness of the beam, as well as the viscosity of the medium where it's immersed in. Next, we are concerned about knowing the properties of a given sample, when the tip establishes contact with it. For a stiff and elastic surface the cantilever passes from a clamped-free configuration to a clamped-clamped one, arising to a shift of the resonance frequency to larger values. To better mimic a biological system, i.e., an animal cell, we must consider a very soft material, leading to a change in the boundary conditions. The theoretical mechanical modeling is in a good agreement with experimental data obtained for a commercially available AFM soft cantilever (theoretical spring constant k=0.06 N/m). The same situations described above are considered. In order to have a free oscillator, the cantilever is immersed either in air or in water and it's placed far from the surface, so its fluctuations are only caused by thermal noise. The contact with a stiff surface is established using a common glass coverslip, while for the soft material we prepared different samples of Agar whose Young's modulus value is similar to the one that animal cells present.