s A Discussion Regarding the Application of the Hertz Contact Theory on Biological Samples in AFM Nanoindentation Experiments

Background: Atomic Force Microscopy (AFM) Nanoindentation procedure regarding biological samples poses significant challenges with respect to the accuracy of the provided results. These challenges are related to the inhomogeneity of biological samples, various uncertainties in experimental methods and certain approximations regarding the theoretical analysis. The most commonly used theoretical model for data processing at the linear elastic regime regarding biological samples is the Hertz model. Objective: This paper focuses on the investigation of the resulting errors of the basic equation of the Hertz theory that depend on the ratio, indentation depth/indenter’s radius regarding the Young’s modulus calculation. Methods: An extended new equation is derived which takes into account the influence of the indentation depth/indenter’s radius ratio on the calculation of the Young’s modulus and can be easily used for calculations. The derived equation is further combined with equations which take into account the shape of the sample. Results: Several examples in the literature that do not take into account the value of the ratio indentation depth/indenter’s radius are reported and the related errors are calculated and discussed. Moreover, a rational explanation, regarding the extended differences of the Young’s modulus calculations using the same experimental results when these are processed using the Hertz model and the Oliver & Pharr analysis (which is the general model that applies for any axisymmetric indenter) is provided. Conclusion: A complete and reliable theoretical tool was developed (that takes into account the indentation depth/indenter’s radius ratio and the shape of the sample) which can be generally applied in order to reduce the errors produced by the current methodology (Hertz model).