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The experimental method is based on the detection of the local electric force gradient at different values of the tip-sample distance. The value of the dielectric permittivity is then calculated by means of the Equivalent Charge Method (ECM). The idea of the ECM is to find a discrete charge distribution (Nc charge points qi at a distance zi on the axis x = 0) that will create a given potential V at the tip surface. Contrary to some analytical expressions, numerical simulations based on the ECM make it possible to carry out quantification of the dielectric permittivity whatever the thickness of the film, the radius of the tip and the tip-sample distance. In this way we can simulate different tip-sample configurations and calculate the corresponding forces and force gradients. By combining this results with the experimental measurements we can quantitatively determine the dielectric permittivity.

This approach can be easily extended, by using the so-called double pass method, in order to obtain quantitative dielectric images. Thus, during the first scan the topography is acquired whereas during the second scan the tip sample distance is kept at a constant value and the dielectric interaction between them is measured. In this way, the topography is completely decoupled from the dielectric response. This method gives not only good dielectric contrast but also excellent lateral resolution and quantitative values of the dielectric permittivity. This approach opens the door for quantitative studies of the dielectric response in polymers, nano-structured materials and soft-matter in general.

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Principle of EFM microscopy
using the double pass method

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a) Representation of the charges
modelling the tip over a metallic plate.
b) The potential created in air and in the dielectric
(h = 100nm, eps = 4) by a tip (R = 130nm, theta = 15º).

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Dielectric map at 70ºC for an heterogeneous
poly(vinyl acetate)/polystyrene (PVAc/PS) blend.

Representative papers:

"Determination of the nanoscale dielectric constant by means of a double pass method using Electrostatic Force Microscopy", C. Riedel, R. Arinero, Ph. Tordjeman, M. Ramonda, G. Lévêque, G. A. Schwartz, D. G. de Oteyza, A. Alegría, J. Colmenero. Journal of Applied Physics, 106, 024315 (2009).

"Nanodielectric mapping of a model polystyrene-poly(vinyl acetate) blend by electrostatic force microscopy", C. Riedel, R. Arinero, Ph. Tordjeman, G. Lévêque, G. A. Schwartz, A. Alegría, J. Colmenero. Physical Review E, 81, 010801 (2010).