Z. angew. Math. Phys. 59 (2008) 702–714 0044-2275/08/040702-13 DOI 10.1007/s00033-007-7041-7 c  2007 Birkh¨auser Verlag, Basel Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Variational treatment of electrostatic interaction force in atomic force microscopy E. Shmoylova ∗ , A. Dorfmann and S. Potapenko Abstract. In this paper we introduce the mathematical model for the electrostatic interaction force between an atomic force microscope (AFM) tip and a sample surface. We formulate the electrostatic potential problem in Sobolev spaces and find the corresponding weak solution in terms of the integral potential, which can be approximated numerically by generalized Fourier series and used to find the interaction force between an AFM tip and a sample surface. The formulation of the problem in a weak (Sobolev) space setting allows us to determine the force for AFM tips of arbitrary shape. Efficiency of the method is illustrated using numerical examples for the spherical and tetrahedral AFM tips. Mathematics Subject Classification (2000). 35J05, 42C15, 31B05, 31B10. Keywords. Atomic force microscope, Sobolev spaces, generalized Fourier series. 1. Introduction Surface potential imaging is one of the AFM techniques used to characterize electri- cal properties of material. The application areas include electrical failure analysis, detecting contact potential difference, mapping relative strength and direction of electric polarization. In addition to potential imaging and testing electrical con- tinuity, a conductive AFM tip with an applied voltage can be used to modify electrical properties of material locally, e.g. polarization of ferroelectric films. It is also used for characterization of properties of modern polymers, for example, for investigation of ionic diffusion mechanism of some polymer membranes. A conductive AFM tip interacts with the sample surface through long-range Coulomb forces. A number of mathematical models, involving different tip ge- ometries, has been developed lately to describe these interacting forces. However, simplifying assumptions with respect to the tip geometry have been made. For example, Terris et al. [1] restricted the mathematical formulation to a tip; Pan et al. [2] assumed a tip geometry in the shape of a hyperboloid; Hudlet et al. [3] limited the analytical formulation to an axisymmetric but otherwise arbitrary Corresponding author: e-mail: Elena.Shmoylova@tufts.edu