##### Description

__Tewfik Souier__

^{1}Yusra Al-Habsi

^{1 2}1, Sultan Qaboos University, Muscat, , Oman

2, Middle East College, Muscat, , Oman

Electrostatic force microscopy EFM, that employs a biased conducting AFM tip, has been used to quantify dielectric constant, surface potential and surface charges of various nanomaterials including Graphene and Carbon nanotubes. It has also been employed for sub-surface imaging of carbon nanotubes embedded and dispersed inside a polymeric thin film.

There are several experimental approaches used in EFM. The most widely used one requires using a “small” oscillation amplitude so that the tip-sample electrostatic force can be linearized. In this approach, the gradient of the electrical force is measured directly from the phase-shift. This approaches, while simple, require a numerical and an experimental validation. Moreover, the use of the small amplitude relatively far from the sample surface (> 20 nm) deteriorates the resolution of the technique.

In this paper, we introduce an algorithm for the electrostatic force reconstruction in amplitude modulation “tapping mode” spectroscopy. The method is based on the recoding amplitude and phase-shift versus tip-sample distance while applying a bias. These two observables (amplitude and phase) are used to calculate the virial of the tip-sample interaction which is used to reconstruct the electric force. The proposed algorithm is verified by numerical simulation and works for an arbitrary amplitude of oscillation. The model is used to extract the long-range electrostatic force versus distance curves on carbon nanotubes. The model is compared to the force-gradient model (linearization) and thus for various “practical” amplitude of oscillation. Finally, the experimental force curves are used to calculate the surface potential as well as a dielectric response of the carbon nanotubes.