Simulation of Surface Acoustic Waves

It is widely accepted that acoustic problems have a vast range of uses in different physical applications and the study of life science. In the context of microfluidic, the applications encompass disease diagnostics, biochemical analysis, food indus- try, drug delivery, medical science, military, and aid to build the lab-on-chip (LOC) system [270, 89]. Results from numerical experiments can provide insight to the underlying physics of acoustic problems. Here we focus our attention on the devel- opment of a computational tool to aid the design of acoustic diagnostic devices (e.g. lab-on-chip). Through the in-depth study of acoustic fields inside a micro droplet, transducer design and related SAW parameters can be identified and improved, while also reducing the uncertainty from the system. Minor physical differences due to parameterization generally might not be observable in physical experiments but can be revealed by micro-scale numerical modelling [72]. Furthermore, numer- ical modelling is a low cost and economical efficient option. Once the connection between numerical and physical experiments is established, we can perform the

numerical test before setting up experiments with expensive devices [274].

In the pioneering works of Eckart and Lord Rayleigh [139, 268], the motion of fluid flow and the acoustic streaming caused by high intensity sound attenuation was introduced, and has since been further developed in various fields. That acoustic streaming phenomena inside micro liquid actuated by SAW can be used as a driving force to pump liquids was presented in [242, 243], in which the height of the channel and droplet, and amplitude of the waves was considered. In articles [299] and [330], it was also shown that the SAW can be exploited to drive a liquid droplet in a positive direction on a flat surface such as 128 degree Y cut lithium niobate (LiNbO3), where droplets with different viscosities were taken into account.

In recent years, Shiokawa, Matsui and Ueda [291] led experiments regarding the manipulation of liquid droplets by applying SAW. The reaction of the droplets was discovered to be intimately dependent on the material and chemical condition of the solid surface. It was also found that the material from the droplets was ejected due to the strong SAW streaming force. Nonetheless, in their numerical assumptions, the viscosity damping and the internal reflections from solid-fluid interface and fluid-air interface are neglected [292].

Alghane et al. [17] applied the finite volume method in OpenFOAM [221] to examine the motion of acoustic waves in a liquid droplet sitting on a LiNbO3 sur-

face. The excitation of the fluid droplet was observed by solving the laminar incom- pressible Navier-Stokes equations. Furthermore, the article presumed that the SAW propagating inside the substrate and fluid obey certain displacement equations that limited the variation of motion of SAW before it incidents on the fluid object. In [283], two coupled-field analysis are provided. The work discusses the Nyborg’s streaming theory and the fluid-solid interaction finite element formulation, it also introduces the idea of the complex SAW number to represent the exponential decay of leaky Rayleigh waves.

Figure 1.2: Acoustic streaming in liquid droplet caused by SAW [17]

Figure 1.2 illustrates the construction for such a system. A SAW that is gen- erated from the alternating electric field through the interdigital transducer (IDT) will propagate alongside the solid-air interface until it reaches the solid-liquid in- terface. It will then change its form to that of a leaky SAW due to the longitudinal component of SAW being emitted into the droplet at a Rayleigh angle of ΘR. The

leaky SAW will produce longitudinal waves that will propagate into the fluid, and will further generate a body force acting on the fluid. The attenuation of magnitude of the longitudinal waves depends on the materials of the solid surface as well as the density of the liquid.

In articles [17, 291, 283, 222, 339], the nonlinear body force derived from the Reynolds’ stress was mentioned as the main driving force behind the motion of the fluid (e.g acoustic streaming). The phenomenon is a result of the attenuation of the high intensity acoustic energy of the wave propagating in the medium. However, the Reynolds’ stress did not result from any computational modelling processes but the assumption of the nonlinear acoustic streaming force. In 2012, Alghane,

et al. [18] conducted experimental and numerical simulations on confined droplets

actuated by SAW. In this work, the relationship between the height of the droplet, attenuation length of the SAW and the streaming flow were investigated, in which the attenuation of SAW was represented in closed form. The deformation of the droplet subject to high power SAW has been investigated by Schindler and his co-authors in [284], but the SAW radiation was neglected inside the droplet.

In 2013, Quintero and Simonetti [260] demonstrated the numerical simulation on how acoustic pressure fields, on the surface of a substrate, propagate into a droplet, with a frequency of 3.5MHz. In the assumption of Rayleigh wave transit to Stoneley wave instead of leaky surface acoustic wave (LSAW) inside the fluid, their model adopt the closed formula extracted from [323]. The coherent structure is not observed from simulation against [17]. The reflected waves from the internal surface of a droplet are not considered in their setup as well as the amount of energy dissipated due to solid-fluid interface.

In most recent times, Collins et al. [106] have demonstrated the effectiveness of size-selective particle concentration process from acoustic streaming generated by high-frequency SAW. The numerical solution simulated based on analytical forms have shown good agreement with the experimental results.

In [87], the acoustic pressure field has been numerically observed inside flat droplets actuated by SAW. The multiple reflections from the internal of a droplet shape chaotic cavity is considered. The asymmetry of droplet deformation is con- firmed, but the mechanism related to the acoustic pressure and streaming have not yet been fully unveiled [274], the authors consider the viscosity as an important factor to support the existence of acoustic streaming inside a droplet, also a crite- rion to differentiate various types of acoustic streaming pattern. They presume the cavity of a droplet is an acoustic chaotic field in their setup.

The standard finite element numerical formulation for acoustic problems can be computationally expensive and unable to fully resolve the small wavelengths. Finite element modelling of such a problem also has been limited in its ability to handle the broad range of timescales. In particular, direct time integration techniques are computationally expensive because of the need to resolve the smallest timescale. To avoid the need of solving the 4 dimensional wave equation, the problem is expanded to polychromatic waves via Fourier transformation in both the time and frequency domain with respect to the input signal, solving the 3D Helmholtz equation in- stead. As shown in Figure 1.3b , it is presumed that the computational domain of a droplet (shown in purple) is a hemispherical shape with some contact angles. A SAW passes along a substrate then interacts with the droplet. To determine the boundary conditions (BCs) on the droplet, the incoming Rayleigh waves are expressed as a closed form analytical equation based on developments from [270] and [320] (1.3a). From a physical perspective, the incident Rayleigh wave carries energies while propagating through the substrate, and starts leaking energies into

the micro fluid. In this presumption, the Rayleigh wave generated from the input signal will be treated as a typical type of standing waves [245, 251]. The Rayleigh SAW travels with speed cs, the wave speed is proportional to the stiffness of the

material of the substrate. However, the wave which leaks to the droplet is a pro- gressive wave, since the speed of the wave in the substrate and fluid are different. As a result, the wave observed in the fluid is a polychromatic wave.

A Fourier transform is then applied to the analytical equation of Rayleigh wave to establish the BCs. This allows us to solve for the propagation of the acoustic waves in the fluid droplet in the frequency domain, consequently it yields the pressure and velocity fields in the fluid which serves to calculate the acoustic Reynolds’ stress and radiation force [187]. Theoretically, these nonlinear time averaged stress terms can be applied to solve the problem in the slow time scale using a direct time integration of Navier-Stokes equation for fluid flow to investigate the evolution of the droplet, specifically taking care of the surface tension. This final part of the problem is strongly nonlinear as a result of the evolving droplet geometry. Thus, the calculations of the acoustic wave in the fluid droplet are repeated for each time step at the slow time scale. The physical phenomena of fluid vortices and capillary wave inside the droplet incurred by acoustic actuation and streaming yet remain a challenging area, and have not yet been fully understood [89].

Figure 1.3: Surface acoustic waves application. (a) SAW actuation of a liquid drop on a LiNbO3 piezoelectric surface, showing the leaky Rayleigh waves in the drop.

(b) Illustration of the different timescales. grey = fast time scale, blue = medium time scale, Cyan = slow time scale