Embedded Microbubbles for Acoustic Manipulation of Single Cells and Microfluidic Applications

Acoustically excited microstructures have demonstrated significant potential for small-scale biomedical applications by overcoming major microfluidic limitations. Recently, the application of oscillating microbubbles has demonstrated their superiority over acoustically excited solid structures due to their enhanced acoustic streaming at low input power. However, their limited temporal stability hinders their direct applicability for industrial or clinical purposes. Here, we introduce the embedded microbubble, a novel acoustofluidic design based on the combination of solid structures (poly(dimethylsiloxane)) and microbubbles (air-filled cavity) to combine the benefits of both approaches while minimizing their drawbacks. We investigate the influence of various design parameters and geometrical features through numerical simulations and experimentally evaluate their manipulation capabilities. Finally, we demonstrate the capabilities of our design for microfluidic applications by investigating its mixing performance as well as through the controlled rotational manipulation of individual HeLa cells.


Contents
To achieve the same mode shape of the wall as for a wall thickness of T = 5 µm (see Figure 2), one has to excite the wall with T = 10 µm at 123.6 kHz. Slightly lower and higher frequencies lead to a different mode shape.

Supporting Texts
A. Text ST-1.

Supporting Information ST-1: Numerical model
We used the Solid Mechanics interface to model the solid components of the model (glass, PDMS). To account for damping, we chose to model the glass as a linear elastic material with complex Lamé parameters. The PDMS was modelled as a linear elastic material with Poisson's ratio and complex Young's modulus to account for damping in our system. Please refer to table S-1 for all material parameters.

: this value has been utilised to distinguish between individual resonance peaks. It does not represent the actual Q-factor of the material
The mechanical vibrations were implemented in the model by means of a prescribed displacement boundary condition within the solid mechanics interface. The displacement was applied over the whole length of the bottom of the glass plate and its amplitude was set to 10 nm. The Thermoviscous Acoustics interface was applied to both channels within the PDMS. The materials inside both channels were modelled with the standard material parameters provided by the software. The Thermoviscous Acoustic-Structure Boundary interface was implemented to couple the Solid Mechanics interface and the Thermoviscous Acoustics interface at the Water-PDMS and the Air-PDMS interfaces. To incorporate acoustic streaming in our simulations, we followed the procedure described in (3,4). We assigned the Creeping Flow interface to the water domain. Volume Forces were set on the whole domain. Finally, a Pressure Point Constraint was set to an arbitrary point of the domain boundary (p0 = 0[P a]).

B. Text ST-2.
Supporting Information ST-2: ARDE lag reduction Aspect ratio dependent etching (ARDE) lag has been reduced through iterative parameter optimisations. Figure S-3 a) shows ARDE lag as typically observed in deep reactive ion etching, where wider trenches are etched deeper than narrow structures. This observation is further represented by the red squares in Figure S-3 b) denoting the etch depth for trenches with varying widths T in relation to the depth D achieved in unconfined areas of the processed wafer. Following an approach introduced by Lai et al., (5) we adjusted the durations of the two process steps, i.e., of tD for the deposition or passivation and tE for the etching, to allow for equal etch rates between trenches with varying widths and, subsequently, the reduction of ARDE lag (see blue triangles in Figure S-3 b) ). It is important to highlight that the derived durations are closely related to the aspect ratio dependency of the individual steps, with the polymer deposition and silicon etching being mostly chemically induced while the initial polymer etching at the bottom of the trench relies on physical processes.

Supporting Information ST-3: Minnaert-Strasberg frequency in PDMS
To account for the PDMS environment, the derivation of the free spherical microbubble presented in the manuscript can be adjusted as follows: (6) , [1] where γ is the polytropic coefficient, P0 is the bubble's gas pressure, V0 is the bubble's volume, C0 is the capacitance, and ρ PDMS is the density of the surrounding PDMS. By assuming equal conditions and volumes for the spherical bubbles in water and PDMS, the natural frequency of a spherical bubble in PDMS can be calculated as f PDMS,0 = f0 ρ Water ρ PDMS ≈ 38.6 kHz, [2] with the previously derived natural frequency of the free bubble in water f0 ≈ 38 kHz, the density of water ρ Water = 1000 kg m −3 , and the density of PDMS ρ PDMS = 970 kg m −3 . As the change in the surrounding material properties does not influence the electrostatic capacitance of our microbubbles, the natural resonance frequency of a prolate spheroid in PDMS with a volume equivalent to an embedded microbubble with air chamber length L = 500 µm can be derived as (7) f PDMS,P = CP C0 f PDMS,0 ≈ 43.2 kHz, [3] with a capacitance ratio CP /C0 = 1.25 as presented in the manuscript. While the change in surrounding media only let to a minor adjustment of the of the bubbles resonance frequency, it is important to highlight the possibly limited significance of this derivation, as the initial formulations used for the calculation of the bubble's natural frequency are intended for liquid environments and, hence, the results should be evaluated with the necessary caution.

D. Text ST-4.
Supporting Information ST-4: Frequency shift due to a nearby solid wall As the influence of shape-anisotropy as well as the introduction of PDMS environment have been detected as insufficient to describe the major resonance observed in numerical and experimental characterisations of embedded microbubbles, the analytical derivation is extended through the correction of the resonance frequency for nearby rigid walls, such as the glass slide enclosing our design. The presence of the rigid boundary, i.e., the glass slide, is simulated by the addition of a second spherical microbubble with equal radius R0. Its centre is located at a distance 2 d from the first one, where d is equal to the bubble radius such that their surfaces are in contact. As analytically derived by Strasberg in 1953 and numerically confirmed by Spratt in 2017, the capacity for each of these spherical bubbles can be calculated as (8,9) with the bubble radius R0 = 85 µm and the distance 2 d = 2 R0 between the bubble centres. The signs introduced by ± are required to denote the location of each bubble centre such that normal velocities midway between the bubbles, i.e., at the location of the rigid wall or glass slide, vanish. The derivation of the capacitance now allows for the correction of the wall effect for a spherical bubble's natural resonance frequency in PDMS as follows: f PDMS,W = 0.833 · f PDMS,0 ≈ 32.2 kHz, [5] where 0.833 has been obtained from M. Strasberg While the derived natural frequency decreases as expected if a rigid boundary, such as the glass slide, is introduced, it is important to highlight that this derivation relies on significant assumptions including but not limited to the previous the calculation of the natural frequency in a PDMS environment. Furthermore, it is worth noting that the resonance frequency f PDMS,W does not account for the shape-anisotropy of the air chamber in the embedded microbubbles and, hence, an increase in frequency in the range of 10 % would be expected (based on Equation (11) and (12) provided in the manuscript). However, despite such an increase, the analytical model of a microbubble's natural frequency proves to be insufficient to account for the complex relation between the various components of the embedded microbubbles and, as such, is unable to describe the resonance frequencies and resulting manipulation capabilities observed in numerical as well as experimental investigations.