Simulating the electrostatic patch force in experimental geometries

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.

💡 Research Summary

The paper presents a comprehensive finite‑element method (FEM) framework for calculating the electrostatic force generated by surface potential patches in realistic three‑dimensional experimental geometries. Potential patches—spatial variations of the work function caused by grain orientation, adsorbates, or defects—produce a parasitic force even when the mean potential difference between two bodies is nulled. This force is a known systematic error in precision measurements such as Casimir‑force experiments, gravitational‑wave interferometers, tests of the inverse‑square law, and atomic‑force microscopy. Existing analytical treatments are limited to simple configurations (parallel plates, sphere‑plate) and often rely on the proximity‑force approximation (PFA) or cumbersome bispherical coordinates, making them unsuitable for the complex shapes encountered in modern experiments.

To overcome these limitations, the authors construct a FEM model that can handle arbitrary parametric surfaces. Two independent Voronoi‑generated patch textures are projected onto the top and bottom surfaces of a simulation domain of side length L. The number of patches n determines a characteristic patch size ℓ = L/√n; each patch is assigned a random voltage drawn from a normal distribution with standard deviation σV (typically 100 mV). The potentials are interpolated onto a 256 × 256 grid, and a swept hexahedral mesh fills the volume between the surfaces. Zero‑normal‑flux boundary conditions on the side walls emulate an infinite extension of the plates. By solving the electrostatic problem for a series of separations d, the total electrostatic energy is obtained and differentiated to yield the force.

The model is first validated against known analytical results for parallel plates and sphere‑plate configurations. Fifteen independent random patch realizations are generated for each geometry; the FEM‑computed forces agree closely with numerical evaluations of the analytical expressions, especially at small separations where the “capacitor‑like” upper bound P = ½ ε0 σV²/d² is recovered. The distance scaling transitions from F ∝ d⁻¹ (sphere‑plate, d ≪ ℓ) and F ∝ d⁻² (parallel plates, d ≪ ℓ) to the dipole‑dominated regimes F ∝ d⁻³ (sphere‑plate, d ≫ ℓ) and F ∝ d⁻⁴ (parallel plates, d ≫ ℓ), in line with the theory of random dipole distributions on surfaces.

Having established accuracy, the authors explore a variety of more complex geometries: concentric spheres, a plate with a step edge, a cylindrical surface facing a plate, a plate‑tip configuration, and a plate‑plate case with curvature. All geometries are defined by parametric functions f(s₁,s₂) and share the same minimal separation d for fair comparison. The results show that the largest patch pressure occurs for geometries where every point on one surface is at the minimal distance from the opposite surface (parallel plates, concentric spheres). The smallest pressure is observed for the plate‑tip case, where only a tiny fraction of the surface is close enough to interact strongly. Intermediate geometries fall between these extremes, and the pressure‑distance exponent n varies between the parallel‑plate value (n = 2) and the sphere‑plate value (n = 1) in the large‑patch limit. At large separations the finite size of the simulation domain forces all curves toward the d⁻⁴ scaling of the parallel‑plate case, highlighting a limitation of the current implementation.

A key extension of the work is the incorporation of real surface data. The authors acquire height maps and Kelvin‑probe force microscopy (KPFM) potential maps of a 2 µm × 2 µm evaporated aluminium film (40 nm thick). These datasets are imported into COMSOL, generating a mesh that faithfully reproduces the measured roughness and voltage landscape. Simulations using this realistic input reveal a pressure‑distance relationship that is less steep than the ideal parallel‑plate prediction and resembles the plate‑tip scaling, indicating that local asperities dominate the interaction. This finding corroborates earlier suggestions that microscopic surface imperfections can significantly alter the apparent scaling of patch forces.

The paper concludes with practical recommendations: (i) use the parallel‑plate upper bound as a conservative estimate when ℓ ≫ d; (ii) apply the dipole‑based d⁻⁴ scaling for ℓ ≪ d; (iii) minimize surface roughness and employ in‑situ KPFM measurements to characterize patches; and (iv) employ the presented FEM tool—available publicly—to evaluate geometry‑specific patch contributions in any precision‑force experiment. By enabling quantitative predictions of patch‑induced forces for arbitrary 3D shapes and real surface data, this work provides a vital resource for reducing systematic errors in Casimir‑force measurements, gravitational‑wave detector calibrations, and other frontier experiments that demand sub‑percent force accuracy.