Calculation of standard viscoelastic responses with multiple retardation times through analysis of static force spectroscopy AFM data

Abstract

We explore the physics of an atomic force microscopy (AFM) cantilever tip interacting with a generalized viscoelastic sample containing an arbitrary number of characteristic times, when the cantilever's base is driven with constant velocity toward the sample. This mode of operation, often called static force spectroscopy (SFS), can be harnessed to thoroughly analyze time-dependent viscoelastic information frequently overlooked in experiments. We generalize the solution of previous authors who have studied the standard linear solid model, and offer a solution applicable to any linear viscoelastic model. This generalization is crucial for the prediction of the model's response over wide ranges of time-scale. As a demonstration, successful predictions of harmonic functions (e.g., loss tangent) over a wide frequency range are obtained through analysis of simulated SFS results. In addition, we show that analysis through the generalized solution and previous expressions is no longer valid when the force does not grow linearly in time, so we also deliver an alternate route for extracting the viscoelastic information, which does not rely on the force linearity assumption. Despite the large amount of theoretical content (included for theoretical rigor's sake), the practical user can also benefit from the new procedures offered and the corresponding explanations.

Imaging of viscoelastic soft matter with small indentation using higher eigenmodes in single-eigenmode amplitude-modulation atomic force microscopy

“…According to it, the tip–sample force is a functional of the sample deformation h , i.e., the force at the current time t , F ts ( t ), depends on the history of the surface deformation at all previous times ξ, from ξ = 0 to ξ = t . This definition of tip–sample force emphasizes the history-dependent behavior of the material, therefore the tip–sample force not only depends on tip position but also on tip velocity and higher displacement derivatives, in addition to force derivatives [11,16]. …”

“…When the material is probed at infinitely low loading rates, the springs in the Maxwell arms do not experience any deformation at all because the dashpots yield (recall that the force exerted by the dashpots is proportional to the deformation velocity), and the only element ruling the mechanical behavior is the rubbery modulus ( G e spring). At this extreme, no energy dissipation takes place and the material behaves in a soft-elastic manner [11,15–17]. On the other hand, when the material is probed at extremely high loading rates, the dashpots do not deform and the behavior of the mechanical model is ruled by the summation of all the individual springs in parallel.…”

“…This, in turn, results in a faster sample deformation, which in a viscoelastic material causes larger reaction forces that oppose the downward motion of the tip into the sample. These larger forces cause greater perturbation of the cantilever oscillation, reducing its ability to indent the sample [11]. …”

Material property analytical relations for the case of an AFM probe tapping a viscoelastic surface containing multiple characteristic times

“…Within AFM, quantitative characterization of viscoelastic materials is usually performed through contact-mode methods. Contact-resonance AFM, force-modulation AFM and static force spectroscopy are the most popular examples in this category [ 9 – 13 ]. The permanent-contact nature of these methods offers an important advantage in mechanical characterization.…”

Acquisition of time–frequency localized mechanical properties of biofilms and single cells with high spatial resolution

“…In this case we should analyze the AFM data in the light of an appropriate theoretical framework that considers the history-dependent nature of biofilms. This distinct nature can be approximately captured by rheological models comprised by (elastic) springs and (viscous) dashpots39,40 . The springs reproduce the elastic response of the specimen, whereas the dashpots consider the energy dissipated through the mechanical deformation.…”