Kelvin probe force microscopy (KPFM) is a powerful tool for studying contact electrification (CE) at the nanoscale, but converting KPFM voltage maps to charge density maps is nontrivial due to long-range forces and complex system geometry. Here we present a strategy using finite-element method (FEM) simulations to determine the Green's function of the KPFM probe/insulator/ground system, which allows us to quantitatively extract surface charge. Testing our approach with synthetic data, we find that accounting for the atomic force microscope (AFM) tip, cone, and cantilever is necessary to recover a known input and that existing methods lead to gross miscalculation or even the incorrect sign of the underlying charge. Applying it to experimental data, we demonstrate its capacity to extract realistic surface charge densities and fine details from contact-charged surfaces. Our method gives a straightforward recipe to convert qualitative KPFM voltage data into quantitative charge data over a range of experimental conditions, enabling quantitative CE at the nanoscale.

Kelvin probe force microscopy (KPFM) is a powerful tool for studying contact electrification at the nanoscale, but converting KPFM voltage maps to charge density maps is non-trivial due to longrange forces and complex system geometry. Here we present a strategy using finite element method (FEM) simulations to determine the Green's function of the KPFM probe/insulator/ground system, which allows us to quantitatively extract surface charge. Testing our approach with synthetic data, we find that accounting for the AFM tip, cone and cantilever are necessary to recover a known input, and that commonly applied heuristics and approximations lead to gross miscalculation. Applying it to experimental data, we demonstrate its capacity to extract realistic surface charge densities and fine details from contact charged surfaces. Our method gives a straightforward recipe to convert qualitative KPFM voltage data into quantitative charge data over a range of experimental conditions, enabling quantitative contact electrification experiments at the nanoscale.