Real-Time Estimation of Drag Coefficients of a Magnetically Driven Microprobe in Aqueous Solutions Near 3D Objects

We present a method for the real-time estimation of the spatially correlated drag coefficient of a magnetically driven scanning microprobe in aqueous environments near three-dimensional (3D) objects. As the microprobe approaches a surface, the drag coefficient, which characterizes fluidic resistance, varies with the probe's distance from the surface. This spatial dependence is influenced by the 3D geometry and mechanical properties of the object, which may be unknown a priori. We detail the system design, theoretical performance analysis, and experimental validation of both a calibration procedure and the proposed real-time estimation technique. In the calibration experiment, a spherical magnetic probe (radius: $2.25 \mu ~\mathrm{m}$) was positioned at nine discrete heights above a flat culture dish surface. Brownian motion at each height was recorded, and the corresponding variance was used to compute the position-dependent drag coefficient. In the second experiment, real-time estimation was validated as the probe was controlled to approach three different 3D surfaces: a flat culture dish, a spherical glass bead, and a live cell. The results demonstrate the feasibility and effectiveness of the proposed method for the real-time estimation of the spatially varying fluidic drag in complex microenvironments.