Nowadays, the determination of mechanical and rheological properties, from the sample interaction with an oscillating probe, is a common trend. The characterization of polymers from its deformation via force and amplitude spectroscopy, nano-indentation, and non-contact AFM imaging has been shown as a possibility (Magonov and Reneker, 1997). In the past, it has been proposed to study the change in liquid properties near a substrate, by means of non-contact AFM. By fitting the oscillation amplitude with a theoretical expression, the resulting parameters relate the damping and the liquid/substrate interfacial stiffness to the probe time-average position and to the oscillation amplitude and frequency (Lantz et al., 1999). The measured probe bending signal were compared to simulation results, obtained
with a modified driven harmonic oscillator equation accounting for the interface elasticity. More recently, the density layering of water in confined geometries and the change of viscosity, when an immersed AFM probe vibrates laterally at subnanometric separation distances from the supporting substrate, have been observed (Li et al., 2007). Some rheological properties of liquids have been deduced from this approach, for instance the viscoelastic modulus and the relaxation time of nanoconfined water and silicon oil (Li and Riedo, 2008). The film thickness and elastic modulus of metals and solid polymers were deduced from the shift of the vibration resonance modes of an AFM probe, in contact with the film surface (Crozier et al., 2000). Elastic properties of thin films, which are stressed and deformed due to the drainage of the viscous Newtonian fluid between the probe and the film (Leroy and Charlaix, 2011). Detection of a change in composition across polymeric nanocomposites and molecularly thick lubricated surfaces is viable by means of phase analysis during NC-mode AFM imaging (Bhushan and J., 2003; Scott and Bhushan, 2003).
However, the probe/sample dynamic interaction does not only concern the determination of phys- ical properties, but also the analysis of the probe dynamic response. The exclusive situation in AFM experiments, in which the probe/sample interaction can be considered as quasi-static, is force/distance spectroscopy. General Intermittent Contact (IC) and Non-Contact (NC) mode AFM scanning conditions always entail a dynamic interaction, which requires a temporal analysis and modeling for understanding. Even force spectroscopy has been analyzed as a dynamic phenomenon, when the probe vertical displacement is taken into account. For instance, in the literature, one can find a theoretical model of a non-oscillating probe interacting with a soft solid immersed in a liquid (Bowen and Cheneler, 2012). A classic mass-spring dynamic model including probe weight and inertia, had been adapted with the vdW attractive force, the fluid squeezing between the probe and the solid and a soft compliant substrate (elastohydrodynamic lubrication). Within the simulations of the probe approach, at constant vertical speed of the probe holder, the so-called snap-in phenomenon has been captured at a certain probe/sample distance. This fact consists of the sudden probe deflection that brings probe and sample into contact. Because the model had not taken into account for the sample deformation, the snap-in distance is underestimated, as the authors have also pointed out.
Concerning the probe oscillation near rigid samples, a long distance has been traveled (García and Pérez, 2002). Simulations of IC-mode AFM have been performed including vdW attractive, repulsive forces within the driven harmonic oscillator equations (García and San Paulo, 1999). Later, accounting for the capillary forces, the intermittent formation/rupture of a liquid bridge between probe and sample was observed from simulations and validated with experiments, in which the effects of relative humidity over hydrophobic and hydrophilic probes were also evaluated (Zitzler et al., 2002).
From a theoretical viewpoint, NC-AFM was the first to be analyzed in detail (Boisgard et al., 1998; Aimé et al., 1999a). The equation of a driven damped harmonic oscillator with a probe/sample interac- tion potential, approximated with sphere/plane formula, was proposed. It is given by:
¨x +ωc,0 Qc ˙x + (ωc,0) 2(ξ − x) +HpsR(ωc,0)2 6kcx2 = (ωc,0) 2W Dcos (ωDt) ,
where x is the probe position in time t, ξ is time-average gap between the probe and the sample, ωc,0
is the probe resonance frequency, kc is the cantilever stiffness, Qc is the system quality factor, R is the
probe radius, Hps is the probe/sample Hamaker constant and WD and ωD are the driving amplitude
and the driving frequency. This equation has been handled by considering a trial function with the form of an harmonic stationary motion:
x(t) = ξ + Wccos (ωDt+ φc) ,
where Wc and φc are the probe response amplitude and phase at the driving frequency ωD. Using
a variational method, the response amplitude and phase are obtained as functions of ωD and ξ. In
addition, the nonlinear behavior of the system revealed the simultaneous existence of three states, two stable and one unstable, when the probe oscillates with ωD slightly below the free resonance frequency
and the time-average gap is around ξ ≈ 10 nm. Making some simplifications, these different behaviors are not observed neither when the ωDis selected above the resonance frequency, nor when the following
dimensionless parameter takes large values:
C∗= HpsR kc(Wc,f ree)3
5.1. OVERVIEW 67 in which, Wc,f reecorresponds to the free oscillation amplitude, when ξ → ∞, at the resonance frequency
ωc,0.
The precise stability condition has been published, as well as the set of equations, representing a closed loop NC-AFM virtual machine that keeps a constant amplitude (Couturier et al., 2003). IC-AFM has been studied by including a repulsion force with a linear elastic behavior (Nony et al., 1999; Aimé et al., 2001). Nevertheless, it has been observed that contact is achieved even without the employment of a repulsive interaction term. The use of driving frequencies below the free resonance frequency provoke an increase of the oscillation amplitude and the intermittent contact (Nony et al., 2001).
The probe/sample interaction phenomenon has been delved into with the addition of a Kelvin-Voigt viscoelastic material, as the sample in the NC-AFM model (Aimé et al., 1999b; Boisgard et al., 2002). Neglecting inertial effects, this sample deformation model was included as follows:
ksη0+ νs˙η0=
HpsR
6 (x − η0)2,
where η0 is the sample displacement, ks = ΘintEs is the sample local stiffness, Θint and Es being
the effective interaction diameter of the sample surface area and its elastic modulus, νs = Θintµs is
the local friction coefficient, µs being the sample viscosity, and the relaxation time is defined as τs =
νs/ks. Explanations of the probe energy loss during oscillations were searched, suggesting that the
local deformation of the sample should be the main source. Experiments with Polystyrene of low and high molecular weights were performed, validating the preceding assumption and showing a change of probe behavior depending on the sample mechanical properties, while keeping the tip/sample interaction constant. The probe displacement was decomposed using Fourier series, and the variational method led to find analytical approximations. For long relaxation times, only the first term in the Fourier series is considered, which approximates the probe motion to a rectangular periodic function, resulting in a displacement that varies as 1/νs for highly viscous samples. In this case, the system instability arises
when the gap between the probe and the sample is:
ξ− (WD+ η0) =3 2 HpsR 6ks 1/3