μFlow: A Computational Platform for Microfluidic Hall-Effect Magnetic Bead Detection with Parametric Design Optimization.

Microfluidic Hall-effect biosensors detect superparamagnetic bead labels as they flow past a thin-film Hall element in a microchannel. Designing one couples bead magnetization, stray-field distribution, Hall transport, and channel flow across 14 parameters that finite-element solvers explore only at minutes to hours per configuration. We present a coupled analytical-numerical framework for this signal chain: Clausius-Mossotti bead magnetization with a volume fraction correction, a point-dipole stray field, a volume-averaged Hall voltage, Poiseuille transport, and a Johnson-Nyquist and Hooge 1/f noise model, evaluated across 12 sensor presets compiled from the literature, spanning graphene, III-V semiconductors, Si CMOS, bismuth, and topological insulators; any other platform can be defined from user-supplied transport parameters. Benchmarked against a companion COMSOL Multiphysics 6.0 study, the framework reproduces the Hall voltage to within 4.8% at a favorable bead-to-sensor area ratio and deviates by 22% and 15% at off-optimum geometries, consistent with the point-dipole near-field limit at h/rb=1. Three design rules follow: a signal-to-noise ridge at sensor widths comparable to the bead diameter (w*≈db; area ratios 0.4-1.0 at constant voltage, 0.5-2.6 at constant current), matching reported single-bead geometries; a material choice that must be made under an explicit electrical drive constraint; and a sampling-limited flow-velocity window. Predicted signals agree at the order-of-magnitude level with published InAs and Si CMOS experiments. We release the model as a freely accessible, no-install browser implementation with a built-in 2D axisymmetric magnetostatic finite-element (FEM) solver that maps where the dipole approximation degrades.