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Analytical Developments for the Measurement of Viscoelastic Properties with the Atomic Force Microscope Open Access

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The atomic force microscope (AFM) is a relatively new technology (invented in the 1980’s) and since then there has been a substantial technological sophistication which has allowed to achieve exciting results such as high resolution imaging of matter at the nanoscale and sometimes even with atomic resolution. Besides the remarkable imaging capabilities of the atomic force microscope, it has also been used as a tool for mechanical characterization.This level of technological sophistication however has not been accompanied accordingly by a strong theoretical framework describing the relationship between the observables that the AFM provides and unambiguous mechanical information.Generally, the theoretical framework used in AFM is the Hertzian mechanics which describes the indentation of a spherical elastic body (the AFM tip) and an elastic surface (the sample). However, several applications demand going beyond the Hertzian approximation, for example the characterization of the mechanical response of polymer components in fuel-cells and organic solar cells, analysis of the viscoelastic response of living cells for early cancer detection and studies of the dynamics of polymeric components in small-scale drug delivery approaches.In these cases, a theoretical framework able to consider the history-dependent nature of some materials is needed. Although the theory of linear viscoelasticity and the classical contact mechanics theory is mature, the implementation of that knowledge in the context of AFM is very rudimentary. There is a sense of urgency in filling the gap between the rapid technological advances and the underlying theory for mechanical characterization of history-dependent samples. The models that are usually employed in AFM theory to describe the deformation of viscoelastic samples by spherical indenters -as is normally the case of an AFM tip- are often oversimplifications that obey mathematical convenience but compromise physical accuracy. As a result, it is customary to find in the AFM literature analytical equations that claim to offer a simple relationship between the AFM observables and mechanical properties. However as aforementioned, those equations are derived with unphysical assumptions therefore their applicability is questionable.Following that sense of urgency, this project has focused on the ambitious goal of applying rigorously classical linear viscoelastic theory and contact mechanics to the AFM context, to relate measurable quantities to unambiguous history-dependent mechanical properties. Those developments are going to be limited to AFM contact-based methods (force spectroscopy and contact resonance) and the most popular dynamic method (tapping mode AFM).

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