Steady flows above a quartz crystal resonator driven at elevated amplitude

A steady flow of liquid was observed above the surface of a quartz crystal microbalance under conditions where the oscillation amplitude exceeded 10 nm. The streaming flow occurs parallel to the displacement vector and is directed towards the center of the plate. It is expected to have applications in acoustic sensing, in microfluidics, and in micromechanics in a wider sense. The flow is caused by the nonlinear term in the Navier-Stokes equation, which can produce a nonzero time-averaged force from a periodic velocity field. Central to the explanation are the flexural admixtures to the resonator's mode of vibration. Unlike pressure-driven flows, the acoustically driven steady flow attains its maximum velocity at a distance of a few hundred nanometers from the surface. It is therefore efficient in breaking bonds between adsorbed particles and the resonator surface. As a side aspect, the flow pattern amounts to a diagnostic tool, which gives access to the pattern of vibration. In particular, it leads to an estimate of the magnitude of the flexural admixtures to the thickness-shear vibration.

Figure 1

The steady flow (dashed arrows) occurs parallel to the displacement direction. It is directed towards the center of the plate. Double arrows indicate oscillatory motion. Its normal component is independent of $z$ because of incompressibility. Its tangential component decays within less than 1 $\mu \mathrm{m}$. The combination of a normal oscillatory motion with a strong gradient in oscillatory tangential motion gives rise to steady forces.

Figure 2

The $z$ dependence of the oscillatory tangential velocity [$v_{us,x}$($z$, $t$ = 0), dot-dashed, green, Eq. (7)], of the steady tangential force [$f_{s,x}$($z$), dashed, red, Eq. (10)], and of the steady velocity [($v_{s,x}$($z$), solid, black, Eq. (13)].

Figure 3

Most practical “thickness-shear resonators” are designed such that the amplitude of oscillation is large in the center and decays towards the edge. The amplitude distribution is sketched at the top of (a). This distribution is achieved by making the plate slightly thicker in the center than at the edge, for instance with electrodes applied to the center only (thick black lines). For reasons of volume conservation (more precisely, because of the nonzero Poisson ratio), an in-plane gradient in the amplitude of tangential oscillation produces a periodic bending of the plate. In air, the plate bends convexly in regions where the gradient in tangential displacement would produce a compression [dashed line in (a)]. However, when the resonator is immersed in a liquid, a similar volume-conservation argument applies to the liquid phase. The liquid pushes the plate downwards because it resists compression, as well. The plate bends downwards for that reason (b).

Figure 4

Velocity of the steady flow versus oscillation amplitude as determined from a video of the flow (see Supplemental Material [36]). The cell height was 5 mm. The data are compatible with a quadratic scaling of velocity with amplitude (dashed line).

Figure 5

Comparison of the steady velocities between overtone orders. The dashed line denotes a power law of $n^{-3/2}$.

Figure 6

In some case, particles were found to deposit in ring-shaped areas, where the distance between the rings was about twice the thickness of the crystal. The particles used to produce the above image were glass spheres with a diameter of 5 μm. The light source was a helium-neon laser, hence the red color. The amplitude of oscillation was 32 nm. The rings developed after the resonator had been driven at this amplitude for about an hour. Tentatively, they are explained as areas of lower-than-average amplitude. The width of the image corresponds to 10 mm.

Figure 7

Determination of the crystallographic $x$ axis by conoscopy. (a) Some crystals have a flat part at the edge, indicating the direction of the $x$ axis. (b),(c) For round crystals, the crystallographic $x$ axis can be found with a polarizing microscope, using the conoscopic mode. One observes colored rings. The $z$ axis corresponds to the center of these rings. (c) shows the direction of the $x$ axis. The radius of the field of view in (b) corresponds to about 30°.