Robust Strategies for Automated AFM Force Curve Analysis—II: Adhesion-Influenced Indentation of Soft, Elastic Materials

Abstract

In the first of this two-part discourse on the extraction of elastic properties from atomic force microscopy (AFM) data, a scheme for automating the analysis of force-distance curves was introduced and experimentally validated for the Hertzian (i.e., linearly elastic and noninteractive probe-sample pairs) indentation of soft, inhomogeneous materials. In the presence of probe-sample adhesive interactions, which are common especially during retraction of the rigid tip from soft materials, the Hertzian models are no longer adequate. A number of theories (e.g., Johnson-Kendall-Roberts and Derjaguin-Muller-Toporov), covering the full range of sample compliance relative to adhesive force and tip radius, are available for analysis of such data. We incorporated Pietrement and Troyon's approximation (2000, "General Equations Describing Elastic Indentation Depth and Normal Contact Stiffness Versus Load," J. Colloid Interface Sci., 226(1), pp. 166-171) of the Maugis-Dugdale model into the automated procedure. The scheme developed for the processing of Hertzian data was extended to allow for adhesive contact by applying the Pietrement-Troyon equation. Retraction force-displacement data from the indentation of polyvinyl alcohol gels were processed using the customized software. Many of the retraction curves exhibited strong adhesive interactions that were absent in extension. We compared the values of Young's modulus extracted from the retraction data to the values obtained from the extension data and from macroscopic uniaxial compression tests. Application of adhesive contact models and the automated scheme to the retraction curves yielded average values of Young's modulus close to those obtained with Hertzian models for the extension curves. The Pietrement-Troyon equation provided a good fit to the data as indicated by small values of the mean-square error. The Maugis-Dugdale theory is capable of accurately modeling adhesive contact between a rigid spherical indenter and a soft, elastic sample. Pietrement and Troyon's empirical equation greatly simplifies the theory and renders it compatible with the general automation strategies that we developed for Hertzian analysis. Our comprehensive algorithm for automated extraction of Young's moduli from AFM indentation data has been expanded to recognize the presence of either adhesive or Hertzian behavior and apply the appropriate contact model.

Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology

“…The force measured upon close approach wasfitted with the Hertzian contact model following the procedure presented in [34,35] to obtain E * (Fig. 4c) and the position at which the contact starts to become elastic.…”

The use of the PeakForceTMquantitative nanomechanical mapping AFM-based method for high-resolution Young's modulus measurement of polymers

“…The most commonly used models for the calculation of Young's modulus by AFM techniques (where the predominant contact is between a spherical tip of defined radius and a flat surface) are Hertzian, DerjaguinMuller-Toporov (DMT) and Johnson-Kendall-Roberts (JKR) [10]. In practice, the surface is rarely flat, and the tip apex may differ from an ideal sphere, leading to errors in the calculated modulus values.…”

Bayesian Change-Point Analysis for Atomic Force Microscopy and Soft Material Indentation

“…Outside of the linear-elastic materials we consider here, it is of interest to apply the methodology to viscoelastic materials (e.g., biopolymers) which return to their pre-contact state slowly over time (Lin et al, 2007b). In such cases, the amount of induced deformation depends not only on the indenter geometry, but also on the rate of indentation.…”