Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Richard O'Rorke^{a},
Andreas Winkler^{b},
David Collins^{c} and
Ye Ai*^{a}
^{a}Pillar of Engineering Product Development, Singapore University of Technology and Design, Singapore. E-mail: aiye@sutd.edu.sg
^{b}Leibniz Institute for Solid State and Materials Research (IFW), Dresden, Germany
^{c}Department of Biomedical Engineering, University of Melbourne, Parkville, Victoria, Australia

Received
12th December 2019
, Accepted 12th March 2020

First published on 20th March 2020

Surface acoustic waves can induce force gradients on the length scales of micro- and nanoparticles, allowing precise manipulation for particle capture, alignment and sorting activities. These waves typically occupy a spatial region much larger than a single particle, resulting in batch manipulation. Circular arc transducers can focus a SAW into a narrow beam on the order of the particle diameter for highly localised, single-particle manipulation by exciting wavelets which propagate to a common focal point. The anisotropic nature of SAW substrates, however, elongates and shifts the focal region. Acousto-microfluidic applications are highly dependent on the morphology of the underlying substrate displacement and, thus, become dependent on the microchannel position relative to the circular arc transducer. This requires either direct measurement or computational modelling of the SAW displacement field. We show that the directly measured elongation and shift in the focal region are recapitulated by an analytical model of beam steering, derived from a simulated slowness curve for 128° Y-cut lithium niobate. We show how the negative effects of beam steering can be negated by adjusting the curvature of arced transducers according to the slowness curve of the substrate, for which we present a simple function for convenient implementation in computational design software. Slowness-curve adjusted transducers do not require direct measurement of the SAW displacement field for microchannel placement and can capture smaller particles within the streaming vortices than can circular arc IDTs.

(1) |

(2) |

To capture smaller particles, one must generate a narrower acoustic beam. SAWs are typically generated by interdigitated transducers (IDTs), which resonate at a frequency f = v/λ, where v is the acoustic velocity in the substrate and λ is the acoustic wavelength, which corresponds to the pitch of the IDT. Focused SAWs are produced by IDTs with a curvature that directs wavelets to a common focal point. Circular arc IDTs have been used for particle focusing activities, but SAW substrates are anisotropic, meaning that the propagation directions of the excited wavelets are altered by beam steering.^{17} Ultimately, the steered wavelets converge to an elongated focal region that is displaced from the geometric centre of the IDT arc. The effects of beam steering can be negated by adjusting the IDT curvature according to the slowness curve of the substrate. Such slowness-curve-adjusted IDTs have been explored to produce a focused spot,^{18} nebulization,^{19} fluidic jetting^{20} and acoustic tweezing,^{21} but their use in microfluidic particle capture has not been explored. They offer a distinct advantage over circular arc IDTs by correcting the travel time for wavelets propagating from the electrode to the focal point so that wavelets interfere constructively at the geometric centre of the IDT. Circular IDTs, in contrast, require specialised tools to measure the substrate displacement and determine the focal point.^{12,22} The location of the focal point is critical in particle capture studies, since the acoustic beam width is narrowest at the focal point and this influences the balance of the drag force and ARF. The design of slowness-curve-adjusted IDTs requires knowledge of the slowness curve for the substrate material, which can be obtained laboriously from the stiffness matrix by solving the surface Green's function,^{18,23} finding Eigenvalues to the substrate Christoffel matrix^{24,25} or by numerical simulation.^{26}

This work explores the use of slowness-curve-adjusted IDTs for particle capture in acoustic streaming vortices. IDTs are designed using a simulated slowness curve, to which we fit a convenient function to simplify IDT design in CAD software. The effects of IDT design parameters (number of finger pairs, focal length and angle subtended at the focal point) are discussed in relation to a numerical model of SAW propagation which has been validated using laser Doppler vibrometry measurements of real devices. A subset of IDT designs was selected for particle focusing studies to compare particle capture in streaming vortices generated by circular arc and slowness-curve-adjusted IDTs.

Slowness-curve adjusted IDTs are also defined by the curvature of the innermost finger however, the curvature is an integer number of acoustic wavelengths in all propagation directions, i.e. L_{0} = nλ(ϕ), where illustrated in Fig. 1a. The curvature thus follows the slowness curve of the substrate. The slowness-curve adjusted IDT geometry was defined in SolidWorks (2015 x64) with L_{0} = 10λ(ϕ); λ(ϕ) was calculated from the simulated slowness curve for λ_{0} = 20 μm.

Fig. 1 (a) Schematic of circular arc (black) and slowness-curve-adjusted (grey) IDT structures. (b) Illustration of FEM simulation domain. |

Surface-normal displacement profiles were exported to code in MATLAB for analysis. The focal length (defined as the distance in X between the IDT and the point of minimum beam width), minimum beam width and peak displacement amplitude were evaluated for 0 < θ < 60° (where L_{0} = 10λ_{0}), 2λ_{0} < L_{0} < 14λ_{0}, (where θ = 60°) and λ_{0} = 20 μm.

(3) |

Fig. 2 (a) Simulated phase velocity as a function of propagation direction, alongside eqn (3) and literature values.^{1} (b) Plot of beam steering angle (Eqn (4)), Γ, as a function of propagation direction. (c) and (d) illustration of power flow from (a) a circular arc IDT and (b) a slowness-curve-adjusted IDT (θ = 60°) showing the effect of beam steering in the former; circular markers indicate the individual focal points for each power flow vector. |

Eqn (3) is plotted alongside simulated values in Fig. 2a, with the 95% confidence interval shaded; fitted parameters are given in ESI Table S1,† which yielded goodness of fit r^{2} = 0.999. The simulated phase velocities are in excellent agreement with published values,^{1} which are also plotted for comparison.

Beam steering causes the power flow angle to deviate from the propagation direction by an angle Γ, defined as the angle between the propagation direction and the group velocity vector, which is the normal vector to (the tangent of) the slowness curve at the intersection with the propagation vector (see ESI Fig. S2†). This angle is zero for the propagation directions ϕ = 0°, ϕ = 52.8° and ϕ = 90°, and non-zero for intermediate angles. The beam steering angle, Γ, is a function of the wavevector, and is given by:^{29}

(4) |

Eqn (4) was evaluated for f = 185 MHz using wavevectors obtained from eqn (3) and is plotted in Fig. 2b. These beam steering angles were used to plot power flow vectors emanating from circular arc and slowness-curve-adjusted IDTs (with θ = 60°, L_{0} = 10λ_{0} and λ_{0} = 20 μm) in Fig. 2c and d, respectively. The intersections of the power flow vectors with the crystallographic X axis are indicated by circular markers and the geometric focal point is located at x = 0. For circular arc IDTs, the focal length and focal region are both extended by beam steering. In contrast, the curvature of slowness-curve-adjusted IDTs negates beam steering and all power flow vectors converge at the geometric focal point. The simulated power flow vectors correspond precisely with the near-field regions in the measured displacement fields (normalised to a maximum value of one), which are given in Fig. 3a (the power flow vectors are plotted as dotted white lines). The focal length is clearly extended for circular arc IDTs, in contrast to slowness-curve-adjusted IDTs (the x-location of the geometric focal point is indicated by the vertical dashed white lines). Fig. 3b shows the simulated displacement fields (normalised to a maximum value of one) for circular arc and slowness-curve-adjusted IDTs, which accurately predict the extended near-field region of circular arc IDTs.

Previous work on slowness-curve-adjusted IDTs has approximated the curvature, R, of the innermost IDT finger to a function of the substrate anisotropy parameter, γ, and propagation direction, ϕ:^{29}

(5) |

(6) |

It should be noted that eqn (5) assumes a constant γ however, the anisotropy parameter for 128° Y-cut lithium niobate varies significantly, as shown in Fig. 4a. Nonetheless, eqn (5) can provide a sufficient approximation to the slowness-curve-adjusted curvatures. By extracting the curvatures, R(ϕ), used here from the simulated slowness curve according to:

(7) |

Fig. 4 Plots of (a) anisotropy parameter, γ, and (b) the curvature, R, of the innermost IDT finger as a function of propagation direction, ϕ; the solid line in b is for a fitted value of γ = −0.31. |

Slowness-curve-adjusted IDTs exhibit consistently larger displacement amplitudes and smaller beam widths than circular arc IDTs for θ > 20°, which indicates superior focusing. The morphologies of the measured displacements fields are consistent with the simulated fields however, we note that the measured peak displacement amplitudes were actually larger for circular arc IDTs than slowness-curve-adjusted IDTs (134 pm and 82.4 pm for circular arc and slowness-curve-adjusted IDTs, respectively). This small difference in measured displacement amplitudes is likely a result of a difference in the electrical impedance of the devices (S11 minima were −5.48 dB and −3.20 dB for circular arc and slowness-curve-adjusted IDTs). The relationship between the number of finger pairs, aperture and electrical impedance has been well characterised for straight IDT devices.^{30–32} We may extend this analysis to curved IDTs by dividing the IDT into segments across which the curvature is negligible; then, the overall IDT impedance is the parallel sum of individual segment impedances. Given that the crystallographic X direction exhibits the strongest electromechanical coupling coefficient, we expect the impedance to be lowest for the segments oriented in this direction. Although the variation in impedance across the arc of the IDT will depend on the anisotropy of the electromechanical coupling coefficient, the lowest impedance will dominate the overall IDT impedance. As such, we expect to see little impact on the overall impedance by altering either the arc (θ) or the curvature of the IDT. The morphologies of the simulated displacement fields agree exceptionally well with the measured displacement fields, which validates the use of simulated displacement fields to investigate the design of focused IDTs.

Not surprisingly, the focal length of circular arc IDTs increases with aperture, regardless of whether the aperture was altered by varying θ or L_{0}. In contrast, the focal length converges at the geometric focal length for slowness-curve-adjusted IDTs. Whilst this is consistent with our expectations of beam steering effects, slowness-curve-adjusted IDTs also exhibit a linear relationship between focal length and aperture for a < 100 μm (corresponding to θ < 40°). We speculate that as the aperture approaches several acoustic wavelengths, diffraction will dominate the anticipated focusing behaviour.

An alternative, more straightforward approach is to use slowness-curve-adjusted IDT designs, as we now demonstrate. Fluorescent micrographs of 2.1, 1 and 0.3 μm diameter particles aligned in the focused acoustic field at circular arc and slowness-curve-adjusted IDTs with θ values of 30, 45 and 60° are given in Fig. 7. Particle focusing can be seen in real-time in ESI Videos 1 and 2† for 5 μm particles at circular arc and slowness-curve-adjusted IDTs, respectively for θ = 60°. Considering first the circular arc IDTs, more particles are concentrated into streaming vortices at higher θ values. This is consistent with our simulated and measured displacement fields, which indicate that steeper displacement gradients, thus larger acoustic forces, are produced at the beam edges for higher θ values. Accordingly, we observe similar improvement in focusing by replacing the circular arc IDT design with a slowness-curve-adjusted one, as shown in Fig. 7b. Whilst both IDT designs capture 2.1 μm particles in tightly circulating vortices, slowness-curve-adjusted IDTs collect a larger portion of suspended 1 μm particles than their circular arc counterparts, and only the slowness-curve-adjusted IDTs show a certain degree of focusing of 300 nm particles. Furthermore, we see that slowness-curve-adjusted IDTs demonstrate marginally better particle focusing to circular arc IDTs where the microchannel was aligned with the actual focal region, exemplifying the practical benefits of slowness-curve-adjusted IDTs.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9ra10452f |

This journal is © The Royal Society of Chemistry 2020 |