Studies of Contact Mechanics with the QCM

Abstract

The quartz crystal microbalance can serve as high-frequency probe of the microcontacts formed between the crystal surface and a solid object touching it. On a simplistic level, the load can be approximated by an assembly of point masses, springs, and dashpots. The Sauerbrey model, leading to a decrease in frequency, is recovered if small particles are rigidly attached to the crystal. In another limiting case, the particles are so heavy that inertia holds them in place in the laboratory frame. The spheres exert a restoring force onto the crystal, thereby increasing the stiffness of the composite resonator. The resonance frequency increases in proportion to the lateral spring constant of the sphere-plate contacts. A third limiting case is represented by particles attached to the crystal via a dashpot. Within this model (extensively used in nanotribology) the dashpot increases the bandwidth. The momentum relaxation time τ _S (“slip time”) is calculated from the ratio of the increase in bandwidth and the decrease in frequency, ΔГ/(− Δf).

The force-displacement relations in contact mechanics are often nonlinear. A prominent example is the transition from stick to slip. Even for nonlinear interactions, there is a strictly quantitative relationship between the shifts of frequency and bandwidth, Δf and ΔГ, on the one hand, and the force acting on the crystal, F(t), on the other. Δf and ΔГ are proportional to the in-phase and the out-of-phase component of F(t), respectively. Evidently, F(t) cannot be explicitly derived from Δf and ΔГ. Still, any contact-mechanical model (like the Mindlin model of partial slip) can be tested by comparing the predicted and the measured values of Δf and ΔГ . Further experimental constraints stem from the measurement of the amplitude dependence of the resonance parameters.

Contacts mechanics in the MHz range is much different from its low-frequency counterpart. For instance, static friction coefficients probed with MHz excitation are often much above 1. Contact mechanics at short time scales should be of substantial practical relevance.