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

[+] Author and Article Information
David C. Lin

Laboratory of Integrative and Medical Biophysics, National Institutes of Health, 9 Memorial Drive, Building 9, Room 1E118, Bethesda, MD 20892lindavid@mail.nih.gov

Emilios K. Dimitriadis

National Institute of Biomedical Imaging and Bioengineering, National Institutes of Health, 13 South Drive, Building 13, Room 3N17, Bethesda, MD 20892dimitria@helix.nih.gov

Ferenc Horkay

Laboratory of Integrative and Medical Biophysics, National Institutes of Health, 13 South Drive, Building 13, Room 3W16, Bethesda, MD 20892horkay@helix.nih.gov

J Biomech Eng 129(6), 904-912 (Apr 06, 2007) (9 pages) doi:10.1115/1.2800826 History: Received October 19, 2006; Revised April 06, 2007

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.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

Schematic of force-displacement behavior as predicted by the Hertzian, DMT, JKR, and MD theories. Force-displacement plots are shown for the approach portion of the cycle. The sign convention is as follows: piezo displacement z toward the sample surface is positive, convex bending of the cantilever (deflection d in direction of piezo motion) is negative, and convex deformation of the sample surface δ (initial contact or pull-off in the JKR and MD theories) is also negative.

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Figure 2

Comparison of force-displacement behavior in the three adhesive contact models (DMT, JKR, and MD). Indentation is always zero at the point of contact in the DMT model (indicated by ○), but can be negative in the JKR and MD models (indicated by ◻) to allow for deflection of the sample surface toward the tip. Equally important in the JKR and MD models is the point of zero indentation (indicated by ◇), which is the reference point used to calculate indentation depth from the force-displacement data. In all three theories, the point of zero applied force (i.e., zero cantilever deflection, indicated by ▵) occurs at positive indentation depth.

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Figure 3

(a) A typical retraction force curve fitted with a power function of 3∕2 power (solid curve) and a line (dashed curve) joining the first point of the noncontact region and the first point of the power function. The negative slope of the line is indicative of significant adhesive interactions. For clarity, every fifth data point is shown. (b) Portion of a sample dataset (magnified 2×) showing a force law based on the LJ potential (solid curve) fitted to the noncontact segment of the data. The simple linear fit (dashed line) is also shown to demonstrate its comparatively poor fit when adhesive interactions are present. For clarity, every fourth point is shown. (c) Example of adhesive interactions during tip retraction that do not follow the LJ force law. The actual pull-off point (w′,d′) is clearly not the minimum, but rather the initial release point (not clear in the figure due to the small number of points plotted). The minimum in this case is the final release point. Note, however, that the force law still provides a superior fit of the data in the noncontact region than a linear function, and hence is still capable of identifying the presence of adhesive interactions. For clarity, every fourth data point is shown.

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Figure 4

Flowchart representing the comprehensive algorithm for processing AFM indentation data. Details on Hertzian analysis can be found in the previous paper (1).

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Figure 5

Results of macroscopic compression and AFM indentation tests on PVA gels. Due to the large difference in sample size between AFM and macroscopic measurements, error bars show standard deviation rather than standard error. Extend and retract curves were analyzed for each AFM dataset and all datasets were restricted to ∼25% strain.

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Figure 6

Four representative retraction curves (every fifth to eighth point is plotted and indicated by ◇) showing disparate adhesive interactions. For the fitting, all datasets were restricted to ∼25% strain. Fitted curves are represented by the solid lines. Also shown are the contact points (indicated by ○) and points of zero indentation (indicated by ●).



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