Simulating the electrostatic patch force in experimental geometries

Abstract

Potential patches are responsible for a force between closely-spaced objects that forms a parasitic contribution to sensitive force measurements. Existing analytical models cannot account for the patch force in the 3D geometries of real experiments. Here, we present a finite-element method model to evaluate the impact of patches in geometries with roughness, edges, and curvature. First, we test our model against the plate-plate and sphere-plate geometries, for which the exact solutions are known. Then, we apply it to more complicated geometries for which analytical solution are challenging, and finally we extend it to handle AFM-measured rough surfaces. Patch textures are generated as a Voronoi diagram representing crystalline grains, or may be imported from potentials measured in Kelvin Probe Force Microscopy experiments. This work provides a reliable estimation of the parasitic contribution from random potential patches in realistic experimental geometries, which may be of relevance to Casimir force measurements or gravitational wave interferometers.

Figures (8)

• Figure 1: Schematic of the patch simulation method. Two independent Voronoi patterns representing patch textures are generated. They are projected onto the top and bottom surfaces of a square volume bounded by two parametric surfaces representing the experimental geometry. Each patch is assigned a random, normally distributed potential value. The finite element mesh of the surface interpolates the potentials from both of the patch textures, and a swept mesh is used for the volume. The electrostatic problem is solved and the total energy is extracted for a given distance $d$. The model then iterates over all desired distances to extract the distance-dependence.
• Figure 2: A: The pressure exerted by potential patches between two plates (blue), and a sphere-plate (green) as evaluated by our finite element method (FEM) and the numerical evaluation of the analytical expressions. B: Force-distance scaling exponent $F \propto d^n$ for the patch force between two parallel plates (blue) and a sphere and plate (green), evaluated in our FEM model and by the numerical evaluation of the analytical expressions. The errorbars denote the standard deviation of the scaling over $15$ random patch textures. C: Schematic of electric field lines between potential patches in a parallel-plate system (top plate not shown). In the limit of large patches, $\ell \gg d$, the interaction between patches on the same plate is small compared to the interaction of the patches with the other plate. By contrast, in the limit of small patches, $\ell \ll d$, the interaction between patches on the same plate is much stronger than the interaction with the other plate.
• Figure 3: Patch pressure between two flat plates for various patch sizes (colors) and plate separations, evaluated by our FEM method. In the limit of large patches, $d \ll \ell$, all curves follow the (patchless) capacitor case (dashed black line). In the limit of small patches, $d \gg \ell$, the pressure is proportional to $d^{-4}$.
• Figure 4: The pressure exerted by potential patches between surfaces of various geometries. All curves are averages of $15$ random patch textures. All spherical surfaces follow a radius of 250nm, the edge has a step height $2\ell$, and the tip has a Lorentzian shape with height $\ell$ and a half-width at half-maximum of $\ell$. The wavy surface is the identical to the one shown in Fig. \ref{['FigSimulationschematic']}.
• Figure 5: A: Measurement of an Al surface, with height from a regular AFM measurement (left panel) and potential from a KPFM measurement (right panel). B: COMSOL model of the combined AFM and KPFM measurements, to be used as the bottom or top surface in our model for evaluation of the patch force. C: Pressure exerted by potential patches based on AFM and KPFM measurements. The shaded area indicates the spread between various $2\times 2$ µm measurement areas close by on the same sample. The patch pressure in geometries with roughness is much less than the pressure between two perfectly plates (analytical model, dashed line).