A rapid and meshless analytical model of acoustofluidic pressure fields for waveguide design

Abstract

Acoustofluidics has a strong pedigree in microscale manipulation, with particle and cell separation and patterning arising from acoustic pressure gradients. Acoustic waveguides are a promising candidate for localizing force fields in microfluidic devices, for which computational modelling is an important design tool. Meshed finite element analysis is a popular approach for this, yet its computation time increases rapidly when complex geometries are used, limiting its usefulness. Here, we present an analytical model of the acoustic pressure field in a microchannel arising from a surface acoustic wave (SAW) boundary condition that computes in milliseconds and provide the simulation code in the supplementary material. Unlike finite element analysis, the computation time of our model is independent of microchannel or waveguide shape, making it ideal for designing and optimising microscale waveguide structures. We provide experimental validation of our model with cases including near-field acoustic patterning of microparticles from a travelling SAW and two-dimensional patterning from a standing SAW and explore the design of waveguides for localised particle or cell capture.

FIG. 1.

Conceptual illustration of (a) spherical waves generated at a point source (*) on a SAW device propagating towards an image plane. (b) An example waveguide (white regions are acoustically active). (c) An illustration of the pressure field, ⟨P⟩, in a PDMS channel arising from a standing SAW coupled through the waveguide in (b) and an enlarged image of the pressure field in (c). Both (b) and (d) measure 15 λ_SAW × 15 λ_SAW.

FIG. 2.

Outline of the analytical model: a waveguide geometry defines the acoustic source area for the boundary condition, which is convolved with a SWF to give the pressure field, ⟨P⟩, in the region defined by the channel geometry. In this example, the boundary condition is a standing SAW in X.

FIG. 3.

(a) Plots of computation time (squares) and convergence parameter, C, (circles) vs resolution for a 2 × 2λ simulation area; insets show the pressure field for 4, 10, 30, and 50 pixels per λ as examples. (b) Plots of computation time as a function of the simulation area at a converged resolution of 30 px per λ; insets show example normalised pressure fields, ⟨P⟩, for square simulation domains measuring 2 × 2, 5 × 5, 10 × 10, and 20 × 20λ, corresponding to areas of 4, 25, 100 and 400 λ², respectively.

FIG. 4.

(a) Experimental image of near-field particle alignment adjacent to a PDMS wall under the action of a travelling SAW in X (Multimedia view). Simulated normalised pressure fields, ⟨P⟩, in the XY and XZ planes of an equivalent liquid channel (measuring 8 λ_SAW in X, 7 λ_SAW in Y, and 20 μm in Z; scale bars are one acoustic wavelength) are obtained using (b) our spherical wave model (SWM) and (c) FEM. Simulation results are shown in the XY plane at a height of 5 μm above the substrate, corresponding to the height of pressure nodes in the channel. The spacings between adjacent nodes are plotted in (d). Multimedia view: https://doi.org/10.1063/1.5021117.1

FIG. 5.

(a) Experimental image of microbead alignment in a confined microchannel (channel width = 2 λ_SAW) under the action of a standing SAW in X. Simulated normalised pressure fields, ⟨P⟩, are shown in the XY and XZ planes (for the left half of the channel in the XY plane) of an equivalent liquid channel (measuring 20λ_SAW in X, 2λ_SAW in Y, and 20 μm in Z; scale bars are one acoustic wavelength) obtained using (b) the spherical wave model and (c) finite element analysis. Simulation results are shown in the XY plane at a height of 15 μm above the substrate, corresponding to the height of pressure nodes in the channel. The spacings between adjacent nodes in X and Y are plotted in (d) and (e).

FIG. 6.

Simulated normalised pressure fields, ⟨P⟩, in the XY, XZ, and YZ planes in a micro-reactor channel layout for a standing SAW in X, with acoustic coupling: (a) across the entire channel, (b) through a rectangular waveguide, (c) through a circular waveguide, and (d) through a torus waveguide. Solid white lines indicate the waveguide geometry, dotted white lines indicate the locations of the XY, XZ, and ZX slices, and the scale bars are one acoustic wavelength, λ = 60 μm.

FIG. 7.

Simulated normalised pressure fields, ⟨P⟩, in the XY, XZ, and YZ planes for a 1D (horizontal) standing SAW coupled through cylindrical waveguides with diameters of (a) 2λ_SAW, (b) λ_SAW, and (c) 0.5λ_SAW. The liquid channel measures 3λ in X and Y and 20 μm in Z, and scale bars are one acoustic wavelength, λ = 60 μm. A torus waveguide was used in (d-f) with an inner diameter equal to half the outer diameter. The waveguide geometry is highlighted by a solid white line, and the locations of the planes evaluated are indicated with dotted white lines.