Dynamic measurement of the acoustic streaming time constant utilizing an optical tweezer

The combination of a bulk acoustic wave device and an optical trap allows for studying the buildup time of the respective acoustic forces. In particular, we are interested in the time it takes to build up the acoustic radiation force and acoustic streaming. For that, we measure the trajectory of a spherical particle in an acoustic field over time. The shape of the trajectory is determined by the acoustic radiation force and by acoustic streaming, both acting on different time scales. For that, we utilize the high temporal resolution ($\mathrm{\Delta }t=0.8\mu \mathrm{s}$) of an optical trapping setup. With our experimental parameters the acoustic radiation force on the particle and the acoustic streaming field theoretically have characteristic buildup times of $1.4\mu \mathrm{s}$ and $1.44\mathrm{ms}$, respectively. By choosing a resonance mode and a measurement position where the acoustic radiation force and acoustic streaming induced viscous drag force act in orthogonal directions, we can measure the evolution of these effects separately. Our results show that the particle is accelerated nearly instantaneously by the acoustic radiation force to a constant velocity, whereas the acceleration phase to a constant velocity by the acoustic streaming field takes significantly longer. We find that the acceleration to a constant velocity induced by streaming takes in average about 17 500 excitation periods ($\approx 4.4\mathrm{ms}$) longer to develop than the one induced by the acoustic radiation force. This duration is about four times larger than the so-called momentum diffusion time which is used to estimate the streaming buildup. In addition, this rather large difference in time can explain why a pulsed acoustic excitation can indeed prevent acoustic streaming as it has been shown in some previous experiments.

Figure 1

Sketch of device. The light-gray area is the silicon channel walls, the light-blue area is the fluid cavity, and the dark-gray block is the piezoelectric element. The total length of the device is 76 mm. All dimensions are as listed in Table 3.

Figure 2

Results for streaming simulations of a cavity-only model (a) and a model with surrounding structure (b). The color map shows the total acoustic pressure and the white arrows the streaming flow. For both simulations $f_{\text{max}}$ is the frequency of maximal acoustic energy density $E_{\text{ac}}$ for a pressure mode with 16 nodal lines. The pressure mode for both simulations is the same besides the phase shift of $\pi$. (a) Streaming simulation for cavity-only model at $f_{\text{max}}=3.987\mathrm{MHz}$. (b) Streaming simulation for whole-device model at $f_{\text{max}}=3.745\mathrm{MHz}$. The gray area is the structure around the cavity.

Figure 3

Shutter location in laser path and schematic of laser path through microscope setup; full details on the setup in [37, 41].

Figure 4

Schematic of controller timings for the shutter, the laser, and the US. During the $P_{\text{measure}}$ state the particle is not trapped by the OT. In the time interval $(0\mathrm{ms},30\mathrm{ms})$ (1) the shutter is fully opened, (2) the US is switched on, and (3) the particle is free to move. During this interval the measurement is performed.

Figure 5

Measured steady-state acoustic forces for a $4.39\mu \mathrm{m}$ particle with $f_{\text{ex}}=4.015\mathrm{MHz}$ and $V_{\text{pp}}=10.7\mathrm{V}$. The top row depicts the forces along ${\mathit{e}}_y$ and the bottom along ${\mathit{e}}_z$. The two columns correspond to two different measurement $yz$ planes at $x=-0.1\mathrm{mm}$ and $x=0.1\mathrm{mm}$, respectively.

Figure 6

Measured steady-state acoustic forces for a $2.06\mu \mathrm{m}$ particle with $f_{\text{ex}}=4.015\mathrm{MHz}$ and $V_{\text{pp}}=10.7\mathrm{V}$. The top row depicts the forces along ${\mathit{e}}_y$ and the bottom along ${\mathit{e}}_z$. The two columns correspond to two different measurement $yz$ planes at $x=0.1\mathrm{mm}$ and $x=0.1\mathrm{mm}$, respectively.

Figure 7

Measured steady-state acoustic forces when averaged over the cavity height. All values are in pN. For each plot the left $y$ axis is the measured force on the $4.39\mu \mathrm{m}$($D_4$) particle and the right one for the $2.06\mu \mathrm{m}$($D_2$) particle, respectively. The gray shaded area corresponds to the positions where the time evolution is measured. (a) $F_y$ (pN). (b) $F_z$ (pN).

Figure 8

Maximal $\mathrm{\Delta }{V}_y$ and $\mathrm{\Delta }{V}_z$ averaged over all repetitions in the time span between 35 ms and 55 ms for the three different measurement heights $\mathrm{\Delta }z=-10,0,10\mu \mathrm{m}$. The gray shaded area represents the $\mathrm{\Delta }{y}_i$ of best signal strength for $max\left[\mathrm{\Delta }{V}_z\left(t\right)\right]$. The data points of best strength are taken for the time evolution results. The wavelength marker represents the same length as in Fig. 7. (a) Data for $y$ component ($m=y$). (b) Data for $z$ component ($m=z$).

Figure 9

Time evolution of the normalized $\mathrm{\Delta }{V}_y$ (left column) and $\mathrm{\Delta }{V}_z$ (right column) for the three measurement heights $\mathrm{\Delta }z=-10,0,10\mu \mathrm{m}$ and the positions for $\mathrm{\Delta }y=-0.06,-0.05,-0.04,-0.03\mathrm{mm}$. The gray shaded area of each plot marks the time when the US is off; $t_0=\frac{1}{f_{\text{ex}}}$.