Determination of Young's modulus of samples of arbitrary thickness from force distance curves : numerical investigations and simple approximate formulae

We present simple expressions for load required to indent a layer of arbitrary thickness with a conical, paraboloidal or cylindrical punch. A rigid substrate underneath the sample leads to an increase of load required for indentation. This effect has to be corrected for to prevent overestimation of Young’s modulus from indentation measurements, such as force - distance curves recorded with the Atomic Force Microscope (AFM). The problems of the frictionless contact of an axisymmetric punch and an isotropic, linear-elastic layer are reducible to Fredholm integral equations of the second kind. We solved them numerically and used the Remez algorithm to obtain piecewise polynomial approximations of the load - indentation relation for samples that are either in frictionless contact with the rigid substrate or bonded to it. Their relative error due to approximation is negligible and uniformly spread. Combining the numerical approximations with asymptotic solutions for very thin layers, we obtained equations appropriate for samples of arbitrary thickness. They were implemented in a new version of AtomicJ, our free, open source application for analysis of AFM recordings