Correction of Microrheological Measurements of Soft Samples with Atomic Force Microscopy for the Hydrodynamic Drag on the Cantilever

Abstract

Force measurements with atomic force microscopy (AFM) in liquid are subjected to the hydrodynamic drag force artifact (Fd) due to viscous friction of the cantilever with the liquid. This artifact may be especially relevant in microrheological studies of soft samples. Common approaches estimate Fd at a certain distance above the sample and subtract its value from the contact force measured on the sample. However, this procedure can underestimate Fd at contact. The aim of this work was to assess the effect of the hydrodynamic drag in microrheological AFM measurements of soft samples in liquid at low Reynolds numbers (Re < 1). Drag forces of water on rectangular and V-shaped cantilevers were measured in noncontact when subjecting the substrate to low-amplitude (35 nm) sinusoidal oscillations at different frequencies (1−200 Hz) and tip−substrate distances (h) (0.2−3 μm). Fd increased proportionally with the relative velocity (v). Moreover, the drag factor b(h) defined as Fd/v rose when the cantilever approached the substrate. Thus, the hydrodynamic drag exhibited locally a pure viscous behavior. Drag factor dependence on distance was well-fitted (r2 ≈ 0.95) by the scaled spherical model b(h) = 6πηaeff2/(h + heff), where η is the viscosity of the liquid, aeff is the effective radius of the cantilever, and heff is the effective height of the tip. Drag factor at contact was estimated as b(0) = 1.38 × 10-6 (rectangular) and 1.55 × 10-6 Ns/m (V-shaped) by extrapolating b(h) to h = 0. Drag factor measured at 2 μm underestimated b(0) by 30−50%. Thus, correction of the hydrodynamic artifact with drag factor measured a few micrometers above the surface could result in substantial errors in AFM microrheological measurements of soft samples. Our results suggest that drag artifact in contact microrheological measurements under low Re can be accurately estimated by b(0). Precise correction of drag artifact could lead to an improvement in scan speed in contact AFM imaging and in pulling speed in force spectroscopy studies.