A look-up table protocol for calibrating standing SAW acoustofluidics

Abstract

The acoustic radiation force (ARF) acting on particles measures the performance of microfluidic devices driven by standing surface acoustic waves (SSAWs). However, existing ARF calibration techniques rely on image post-processing or additional equipment. This work proposes a look-up table method to determine the ARF by examining the particle acoustophoresis mode in discrete phase-modulated SSAW fields, where the phase difference between the two counter-propagating SAWs is changed at fixed time intervals. Theoretical analysis indicates that particles in a straight channel migrate laterally either in the “locked” mode or the “drift” mode, while mode switching can be observed when the interval reaches a critical value highly dependent on the ARF amplitude. A look-up table can then be established for a given SSAW device. By observing the particle acoustophoresis modes at different phase-changing intervals, the ARF amplitude can be obtained from the easily determined critical interval. The procedure is demonstrated experimentally in an SSAW acoustofluidic device and compared with the particle tracking protocol to verify the former’s effectiveness and demonstrate its operational simplicity. Inspired by the established theory, a method to improve the efficiency of particle acoustophoresis by optimizing the phase-modulating parameters is also proposed.

Appendices

Appendix A: FE Simulations

FE simulations are carried out in the y-z cross-section of the device using the commercial software COMSOL Multiphysics (v5.5, COMSOL, Stockholm, Sweden). In Sect. 4.1, the “Pressure acoustics, Transient” interface is used to study the transient acoustic field in the channel, and “Solid” and “Electrostatics” for the displacement and electric fields in the substrate. The material properties of $\hbox {LiNbO}_{3}$ are adopted from the material library of the software, and the other parameters are taken from Table 1. To reduce the computational effort, the “Impedance” boundary condition is adopted to describe the PDMS sidewalls, and “Low-reflecting” boundaries to minimize wave reflections at the side and bottom of the substrate. The governing equations and other boundary conditions are the same as those adopted in the previous study, (Ni et al. 2019) and a time step of 1/30f is chosen for a total period of $2t_{\textrm{pulse}}$. The maximum element size is 1/38 of the wavelength in the fluid, $\lambda _{\textrm{s}}/20$ at the upper surface of the substrate, and $\lambda _{\textrm{s}}/6$ in the bulk of the substrate. In calculating $\langle p_{\textrm{in}}^2 \rangle$, the transient pressure squared is averaged over a period $t_{\textrm{pulse}}$ after the SSAW field is built in the channel. A schematic of the numerical model can be found in the Supplementary Fig. S4.

In Sect. 4.3, particle trajectories are also obtained via FE analysis. The modeling method is the same as above, except that the “Pressure acoustics, Frequency domain” interface is adopted to obtain the in-channel pressure field, and “Particle tracing for fluid flow” for extracting particle trajectories. In particle tracking, the particle positions are recorded at a time step of 0.01 s for the period $t \!=\! 0$–3 s. The implementation process, constitutive equations, and boundary conditions are the same as those used in the Model S in our previous work, (Ni et al. 2019) except that the acoustic streaming effect is neglected to improve the computational speed.

Appendix B: The reflection coefficient at the channel ceiling

In SAW acoustofluidic devices like that used in this work, the acoustic pressure varies along the channel height, and this is due to the wave reflections at the channel ceiling. As two counter-propagating SAWs enter the channel area, they leak into the fluid domain at the Rayleigh angle; due to field superposition, the two beams form a standing wave pattern in the horizontal direction; if the ceiling is absent and the water domain extends upwards to infinity, in the z-direction there should exist only traveling components, and the pressure amplitude should be invariant along the channel height (within the overlapping area of the two beams).

The ceiling serves as an imperfect reflector for z-direction wave propagation. Part of the up-going waves is absorbed, and the rest is reflected, resulting in a partial standing wave along the channel height. Considering the reflection coefficient at the channel ceiling being $r_{\textrm{p}}$, an upward incident acoustic field with an amplitude of 1 induces a traveling component (amplitude: $1-r_{\textrm{p}}$) and a standing component (amplitude: $2r_{\textrm{p}}$), the average amplitude along the z-direction turns to be $r_{\textrm{p}}$. As a result, the average pressure amplitude along the z direction is 1 and the minimum is $1-r_{\textrm{p}}$. Considering the particle tracking method produces the z-averaged pressure while the look-up table method targets the minimum pressure plane, a scaling factor $\gamma =1-r_{\textrm{p}}$ should be introduced when comparing the two methods.

According to Snell’s law, the reflection coefficient of an acoustic wave incident at the Rayleigh angle $\theta _{\textrm{R}} \!=\! \sin ^{-1}\left( c_0/c_{\textrm{s}} \right)$ upon the water-PDMS interface at the channel ceiling is $r_{\textrm{p}} \!=\! \left( \frac{Z_0}{\cos \theta _{\textrm{R}}} - \frac{Z_{\textrm{PDMS}}}{\cos \theta _{\textrm{T}}} \right) / \left( \frac{Z_0}{\cos \theta _{\textrm{R}}} + \frac{Z_{\textrm{PDMS}}}{\cos \theta _{\textrm{T}}} \right)$, $Z_0 \!=\! \rho _0 c_0$ and $Z_{\textrm{PDMS}} \!=\! \rho _{\textrm{PDMS}} c_{\textrm{PDMS}}$ are the acoustic impedance of water and PDMS, respectively, and $\theta _{\textrm{T}} \!=\! \sin ^{-1}\left( c_{\textrm{PDMS}} \sin \theta _{\textrm{R}} /c_0 \right)$ is the transmission angle. Considering the device design, $r_{\textrm{p}}$ is estimated as 0.17, and the scaling factor is determined as $\gamma \!=\! 1-r_{\textrm{p}} \!=\! 0.83$.