Electrode-Geometry Control of Normal Electric-Field Distributions and Electrostatic Loading on Sessile-Droplet Interfaces: A Finite-Difference Study with a Finite-Element Cross-Check and Surrogate-Assisted Design Exploration

Electrohydrodynamic emission from a sessile droplet depends on a coupled balance among electric traction, capillarity, gravity, charge transport and liquid motion. The present work addresses only the electrostatic-loading part of that problem. Verification-backed axisymmetric and three-dimensional Laplace solvers are used to map how parallel-plate, on-axis-pin, off-axis-pin and bipolar double-pin electrodes redistribute the normal electric field over a prescribed conducting water-droplet interface. The primary response is the dimensionless electric capillary number, CaE = ε0En2Rv/γ. To compare geometries on a common voltage scale, V1 is defined as the applied voltage at which the peak prescribed-interface loading reaches CaE = 1. V1 is a normalisation voltage and not a jetting or stability threshold. The axisymmetric solver reproduces the exact conducting-hemisphere solution to within 0.07% at the finest grid. The three-dimensional finite-difference results are mesh-assessed, and their normalised surface-field topology is cross-checked against an independently implemented Galerkin finite-element model. At 4 kV, the finite parallel-plate cell produces an apex enhancement of 3.24 relative to V/H. Replacing the plate with an on-axis 1 mm pin reduces the apex field by 39.5%, which corresponds to a 63% reduction in CaE, and increases V1 from approximately 5.0 to 8.2 kV. Lateral pin displacement moves the surface-field maximum away from the apex and produces a broad nominal plateau near d = 5–7 mm, although the sub-grid steering distance remains sensitive to mesh and extraction settings. The bipolar double-pin configuration produces two symmetric surface-field maxima together with a near-null at the apex. This topology, but not its absolute magnitude, is reproduced by the finite-element cross-check. A Gaussian-process model interpolates the one-dimensional offset family accurately under leave-one-offset-out validation (R2 = 0.999). Four Bayesian-optimisation trials locate the broad steering plateau but show no visible evaluation-count advantage over random sampling in this one-dimensional test. A three-mesh study gives a reported field-magnitude mesh-sensitivity estimate of approximately 6.2% at the finest grid (rising to about 9.5% at the h = 0.20 mm production mesh) for the representative three-dimensional case, and an indicative combined-uncertainty band of approximately 10% is shown for V1 in the exploratory trade-off plot. Illustrative Young–Laplace profiles at contact angles of 70° to 110° preserve the comparative pin-versus-plate field reduction, whereas V1 varies by up to approximately 50%. A simplified Peek-law screening estimate places corona inception (the pin being cathodic) in the approximate range of 4.8–10 kV, which is comparable to the on-axis-pin V1, so gas discharge may intervene before large electrocapillary loading is reached in ambient air. The results establish electrode geometry as a controllable electrostatic-loading parameter while explicitly deferring coupled stability analysis and experimental validation. By resolving this loading on a single exact-solution-verified basis, the study quantifies electrode geometry as a control parameter that idealised enhancement factors and the nominal gap field cannot capture and provides a verified fixed-interface reference state for subsequent coupled electrohydrodynamic modelling.