Introduction

The significant heat accumulation occurring during the operation of electronic devices necessitates efficient cooling systems1. Consequently, two-phase cooling based on microchannels has become a popular research direction in the field of electronic device cooling due to its excellent heat dissipation capability and compact structural design2,3,4. The cooling effect can be further improved by active and passive methods, including applying acoustic5,6, electric7, and magnetic fields8, as well as the customization of micro- and nanostructured surfaces9,10,11,12. Compared to passive methods, active methods offer superior control, flexible adjustment, and less surface dependence13,14. However, among active methods, the application scenarios of magnetic fields requiring magnetic fluids and electric fields necessitating high voltage conditions are constrained. Therefore, acoustic-enhanced heat transfer technology has obtained increasing attention due to its advantages of flexible manipulation, easy implementation, and high safety15,16.

Acoustic waves exhibit excellent directionality, penetrability, and focusing capabilities within the medium17. The unique acoustic cavitation effect and the acoustic radiation force facilitate the activation of numerous nucleation sites, enhance the frequency of bubble detachment, and reduce the diameter of detached bubbles under noninvasive conditions, thus realizing the boiling heat transfer performance enhancement18,19,20,21. However, relying heavily on high-power ultrasonic transducers to achieve high heat transfer coefficients, traditional acoustic field enhancement techniques for boiling heat transfer encounter several limitations. Firstly, integrating bulky transducers within compact heat exchangers poses a significant challenge. Secondly, operating at heightened power levels significantly increases the system’s power consumption and generates substantial heat, jeopardizing the transducer’s capability to maintain stable performance over a prolonged period. Furthermore, conventional acoustic-enhanced boiling heat transfer techniques exhibit limited effectiveness in augmenting the critical heat flux, ultimately resulting in their inability to mitigate the onset of thermal crises. Therefore, it is of great significance to investigate the utilization of low-power acoustic excitation to enhance critical heat flux within compact spaces.

Herein, we presented a design of a low-power acoustic-enhanced compact heat exchanger for contactless control of bubble behavior through the utilization of acoustic radiation force. The strong acoustic radiation force generated by a thin piezoelectric ceramic during the boiling process facilitated swift bubble detachment and migration on the silicon wafer surface. This not only reduced the size of the bubbles but also reduced the possibility of bubble agglomeration. Consequently, the improved bubble behavior enhanced heat transfer performance, evidenced by the increase in CHF and reduction in wafer surface temperature. The acoustic radiation force generated by the thin piezoelectric ceramic exhibited variations in direction and intensity with frequency, thereby enabling the possibility of tuning the frequency to achieve optimal cooling performance. Additionally, the intensity of the acoustic excitation power is directly correlated with the magnitude of critical heat flux. Based on a thorough frequency-power analysis, we achieved long-term stable operation of acoustic-enhanced boiling heat transfer, offering a viable technical solution for the application of acoustic field in two-phase cooling of electronic devices.

Results

Fabrication and development of ALCHE

The flow boiling heat transfer performance was intricately intertwined with bubble behavior22,23,24,25. To improve bubble behavior during the flow boiling process, we fabricated ALCHE using thin piezoelectric ceramic (Fig. 1a, Supplementary Fig. 1). The entire heat transfer mechanism was illustrated in Fig. 1b, where the coolant flowed in from the inlet, exchanged heat with the silicon wafers within the microchannel, and flowed out from the outlet. Compared to the bulky ultrasonic transducers, piezoelectric ceramic was thin and compact, facilitating their easy integration into heat exchangers (Fig. 1c). The silicon wafer was heated by a copper electrode (Supplementary Fig. 2). During the flow boiling process, a significant number of bubbles formed on the silicon wafer surface, and the bubble behavior was modulated by the acoustic field generated by the piezoelectric ceramic (Fig. 1d). Driven by low-power acoustic field, the acoustic radiation force acting on bubbles at the nucleation site facilitated their rapid detachment and migration (Fig. 1e, f). With acoustics ON, a significant reduction of bubble size at the nucleation site was observed. Small bubbles were ejected from the nucleation site, while large bubbles originally adhering to the heated surface underwent swift movement (Fig. 1h and Supplementary Fig. 3, Supplementary Movie 1).

Fig. 1: Fabrication and development of ALCHE.
figure 1

a Schematic diagram of ALCHE, which consists of a cover plate, microchannel, silicon chip, and piezoelectric ceramic. b Schematic diagram of heat exchange process inside ALCHE. The piezoelectric ceramic is bonded to the inlet side of the heat exchanger. c Physical diagram of the ultrasonic transducer and the piezoelectric ceramic. d Schematic diagram of the effect of acoustic waves on the flow boiling heat transfer process. The cold fluid (blue) absorbs heat from the silicon wafer, transitioning into a hot fluid (red) within the microchannel. During this process, heat absorption above the silicon wafer induces bubble formation. The acoustic waves generated by the piezoelectric ceramic change the behavior of the bubbles. e Schematic diagram of the acceleration effect of sound waves on air bubbles. The blue circles in the figure represent the air bubbles, and the shading of the circles represents the accelerated movement of the bubble. f Comparison of force analyses on bubbles with Acoustics ON and OFF. The blue circles in the figure represent the air bubbles, Far represents the acoustic radiation force on the bubbles. g Comparison of visualization images of bubble migration process with Acoustics ON and OFF. The acoustic frequency (fa) is 1.5 kHz, and the acoustic power (Pa) is 7.5 W. The pink dashed line is used to mark the position of the bubble at the beginning and end of the migration. h Comparison of bubble detachment visualization images at nucleation sites with Acoustics ON and OFF (fa = 1.5 kHz, Pa = 7.5 W). i Statistical results of bubble detachment size and time. j Statistical results of bubble migration velocity. The positions at the two ends of the rectangular box correspond to 25% and 75% migration velocity, and the upper and lower edges are the maximum and minimum migration velocity, respectively.

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Statistical analysis of bubble detachment at the nucleation site (Supplementary Fig. 4) revealed a drastic reduction in both bubble diameter and detachment size with acoustics ON. Specifically, the average bubble detachment size and time were reduced by 85.2% and 79.4%, respectively (Fig. 1i). Meanwhile, the time between bubble detachment from the nucleation site and its disappearance in the frame was drastically shortened (Fig. 1g, Supplementary Fig. 5), and the migration velocity of the bubbles on the silicon wafer was calculated based on the distance and time. Compared to that with acoustics OFF, the migration velocity with acoustics ON increased by an average of 225% (Fig. 1j). During the boiling heat transfer process, the reduction in bubble detachment size and enhancement in migration velocity contribute significantly to delaying the formation of vapor films, thus facilitating the achievement of higher heat flux. Additionally, the accelerated detachment frequency of the bubbles accelerates the rewetting rate of the nucleation site, facilitating the increase of the heat transfer coefficient. Consequently, the low-power acoustic field generated by piezoelectric ceramic can effectively optimize bubble behavior during the boiling heat transfer process, which is conducive to the boiling heat transfer performance.

Flow boiling heat transfer performance of ALCHE

To validate the excellent heat transfer performance of ALCHE, the heat flux curves, and heat transfer coefficient curves at different flow rates were shown in Fig. 2a, b and Supplementary Fig. 6. The low-power acoustic field indeed effectively enhanced CHF and maximum HTC at different flow rates. Especially at a low flow rate (0.1 m s−1), the acoustic field (1.5 kHz, 7.5 W) elevated the CHF and the maximum HTC by 42.1% and 38.1%, respectively. Meanwhile, the introduction of the acoustic field didn’t have any impact on the pressure drop (Fig. 2c). The acoustic enhancement analysis in Fig. 1 was mainly based on the changes in bubble behavior occurring at individual nucleation sites, to ensure the low-power acoustic field influence on the bubbles was covering the entire silicon surface, images and videos of bubbles across the silicon wafer at the same heat flux and flow rate were given as shown in Fig. 2d and Supplementary Movie 2, where the acoustic field accelerated the detachment and migration of bubbles from the entire surface of the silicon wafer.

Fig. 2: Flow boiling heat transfer performance of ALCHE.
figure 2

a1 Heat flux curves with acoustics ON and OFF at 0.1 m s−1 flow rate. a2 Heat flux curves with acoustics ON and OFF at 0.3 m s−1 flow rate. b1 Heat transfer coefficient with acoustics ON and OFF at 0.1 m s−1 flow rate. b2 Heat transfer coefficient with acoustics ON and OFF at 0.3 m s−1 flow rate. c Pressure drop with acoustics ON and OFF under different flow rates. The error of the pressure drop is attributed to minor instabilities in the pump’s performance during liquid conveyance. d Bubble visualization images with acoustics ON and OFF (fa = 1.5 kHz, Pa = 7.5 W, V = 0.1 m s−1, q = 20 W cm−2). e Bubble size distribution with acoustics ON and OFF (fa = 1.5 kHz, Pa = 7.5 W, V = 0.1 m s−1, q = 20 W cm−2), the horizontal axis represents the logarithmic scaling of bubble area pixel number distribution (lg). The area of the bubbles is directly proportional to the number of pixels. f Bubble numbers with acoustics ON and OFF (fa = 1.5 kHz, Pa = 7.5 W, V = 0.1 m s−1, q = 20 W cm−2). g Schematic diagram of bubble images with acoustics ON and OFF at high heat flux (fa = 1.5 kHz, Pa = 7.5 W, V = 0.1 m s−1, q = 50 W cm−2). The blue circles in the figure represent the air bubbles, Far represents the acoustic radiation force on the bubbles. h Temperature decreases with acoustics ON at various heating powers (fa = 1.5 kHz, Pa = 7.5 W). i Cyclic operation of the piezoelectric ceramic under different heat fluxes (fa = 1.5 kHz, Pa = 7.5 W, V = 0.1 m s−1). A total of 20 cyclic operations were performed, with each cycle consisting of a two-minute period: 1 min with acoustics OFF and 1 min with acoustics ON.

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To quantify the enhancement effect, image processing on the visualization images was performed to count the size distribution and number of bubbles on the whole wafer surface (Supplementary Fig. 7). The statistics summarized results of bubble size at different moments showed that the acoustic field reduced the bubble size and increased the number of bubbles on the surface, which meant the bubble detachment promotion (Fig. 2e, f). In addition, compared with the bubble information recorded in single-frame images, the trajectories recorded in multi-frame images could effectively illustrate the migration velocity and direction of bubbles on the entire silicon surface, based on the bubble tracking algorithm, bubble migration trajectories over multiple time periods indicated that the 1.5 kHz acoustic field accelerated bubble migration across the silicon surface (Fig. 3h). At high heat flux, the acoustic radiation force further reduced bubble sizes and expedited their migration, thus minimizing bubble coalescence and delaying the formation of vapor film, ultimately facilitating higher heat flux (Fig. 2g and the Supplementary Movie 3). Additionally, this acoustic radiation force hastened the rewetting rate of nucleation sites and enhanced the quenching heat flux resulting from the rewetting process, leading to the surface temperature decrease across various heat flow ranges (Fig. 2h). To verify the performance repeatability of the acoustic field, 20 cyclic operations at different heat fluxes were recorded (Fig. 2i). ALCHE was found to maintain repeatable heat transfer performance after 20 cyclic operations.

Fig. 3: Cooling mechanism regulated by acoustic frequency.
figure 3

a Temperature decrease with acoustics ON at various acoustic frequencies (V = 0.1 m s−1, q = 40 W cm−2). b Bubble size distribution with various acoustic frequencies (V = 0.1 m s−1, q = 20 W cm−2), the horizontal axis represents the logarithmic scaling of bubble area pixel number distribution (lg). c Bubble numbers with various acoustic frequencies (V = 0.1 m s−1, q = 20 W cm−2). d Simulation of the displacement of ALCHE with acoustics of 1.5 kHz. x and y represent the width and length direction of the microchannel. The blue circles in the figure represent the air bubbles. The black arrows represent the bubble migration direction. e Simulation of the displacement of ALCHE with acoustics of 7.5 kHz. The blue circles in the figure represent the air bubbles. The black arrows represent the bubble migration direction. f Visualization image of the bubble migration direction (V = 0.1 m s−1, q = 20 W cm−2). The pink and blue arrows represent the bubble migration direction. g Schematic diagram of bubble tracking algorithm. t and t + Δt represent the image of a particular frame and the image of the next frame. The blue circles in the figure represent the air bubbles. Black arrows represent bubble migration distances (d). Tm (pink dashed line) represents the maximum displacement of the bubble, Tn (yellow dashed line) represents the neighborhood threshold of a given bubble, Tq (orange dashed line) represents the quasi-rigidity threshold. h Bubble trajectory length statistical results with acoustics off and on (V = 0.1 m s−1, q = 20 W cm−2). The statistical distribution was obtained from Fig. 3i. i Bubble trajectories with acoustics off and on (V = 0.1 m s−1, q = 20 W cm−2). Each image records four sets of bubble trajectories. The time interval for each trajectory is 11.25 ms. The black, pink and blue arrows represent the bubble migration direction.

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Cooling mechanism regulated by the acoustic frequency

To investigate the mechanism of frequency effect on the heat transfer performance, the analysis was conducted on the performance of various acoustic frequencies at the same acoustic power (Fig. 2a, b). Among the test frequencies, the 1.5 kHz acoustic field exhibited the most significant improvement in CHF and maximum HTC, resulting in the most pronounced temperature reduction (Fig. 3a), followed by the 4.5 kHz acoustic field, whereas the remaining frequencies displayed subpar performance. The statistical results of bubble size distribution at different times (Supplementary Fig. 8) were summarized and polynomial fitted (Supplementary Table 1) to obtain the fitted curves (Supplementary Fig. 9). Subsequently, a compilation of the fitted results across different frequencies was presented in Fig. 3b. It was obvious that the bubbles of the 1.5 kHz exhibited a smaller size and higher count (Fig. 3c and Supplementary Fig. 10), indicating that bubble detachment occurred at a smaller size and with a faster frequency.

To further explore the effect of frequency on heat transfer performance from a force-based perspective, the time-averaged acoustic radiation force acting on the bubble is formulated in the following equation26:

$${\overrightarrow{{F}_{{ar}}}=-\left\langle V(t)\vec{\nabla }{p}_{s}(\vec{x},t)\right\rangle }_{T}$$
(1)

where V(t) represents the instantaneous volume of the bubble at time t, while ps(\(\vec{x}\),t) is the instantaneous acoustic pressure at location \(\vec{x}\) and time t; the symbol T means the time-average value over one circle of an acoustic period.

Based on Eq. 1, it was observed that the acoustic radiation force’s influence on the heat transfer performance of a bubble of a specific volume encompassed two key aspects: the magnitude of the acoustic radiation force and the direction of the acoustic pressure field. These factors varied depending on the acoustic frequency. Taking 1.5 kHz and 7.5 kHz as an example, the bubbles detached from the wafer surface showed a smaller size distribution at 1.5 kHz, implying that bubbles are subjected to a larger radiation force to detach, whereas bubbles of the same size were not able to detach due to a weaker acoustic radiation force at 7.5 kHz.

The wafer surface displacement was obtained (Fig. 3d, e) by simulating the piezoelectric effect of ALCHE, providing insights into the acoustic pressure field distribution. The results revealed that the acoustic pressure field direction was consistent across positions at 1.5 kHz, promoting rapid bubble movement from upstream to downstream on the silicon wafer. Conversely, the acoustic pressure field directed upstream bubbles towards the middle of the downstream region at 7.5 kHz, which could prematurely form a large vapor film on the silicon wafer at higher heat flux, impeding further heat transfer enhancement. Experimental images verified the accuracy of the simulation results, the acoustic pressure field directions at both frequencies were consistent (Fig. 3f, i and Supplementary Movie 4). From the dynamic trajectories recorded at multiple time intervals, the direction of bubble migration at 1.5 kHz was consistent, with longer migration distances at the same time interval, while the direction of bubble migration at 7.5 kHz tended to be clustered in the middle, with shorter migration distance (Fig. 3h). The combination of a strong acoustic radiation force and favorable directionality contributed to the superior heat transfer performance exhibited at the 1.5 kHz frequency.

Heat transfer performance control by acoustic power

To investigate the effect of acoustic power on the heat transfer performance of ALCHE, the heat transfer performance of ALCHE with different acoustic powers was examined at 1.5 kHz (Fig. 4a, b and Supplementary Fig. 11). As acoustic power increased, the heat transfer performance of ALCHE exhibited increasing improvements in both CHF and HTC. At an acoustic power of 10.5 W and an inlet flow rate of 0.1 m s−1, CHF and the maximum HTC of ALCHE were enhanced by 47.1% and 47.3%, respectively. The pressure drop of ALCHE was essentially stable at different heat fluxes and acoustic powers (Fig. 4c). The enhanced acoustic power resulted in a stronger acoustic radiation force, causing bubbles to detach with smaller size, faster frequency, and faster migration (Fig. 4d). This phenomenon delayed bubble aggregation, ultimately enabling a higher CHF. Under various flow velocities, the heat dissipated by the system was at least five times greater than the total power of the pump and the acoustics (Supplementary Fig. 12). More importantly, the piezoelectric ceramic with low-power operation generated less heat. Therefore, the piezoelectric ceramic provided stable control of temperature fluctuations for nearly 3 h under an inlet flow rate of 0.1 m s−1 and a high heat flux of 55 W cm−2 (Fig. 4e). Furthermore, a comparison of this work with past literature was shown in Supplementary Fig. 13. Compared to the reported literature values, this work achieved a significant increase in CHF at extremely low acoustic power levels.

Fig. 4: Boiling heat transfer performance of ALCHE at different acoustic powers.
figure 4

a1 Heat flux curves at different acoustic powers at 0.1 m s−1 flow rate. a2 Heat flux curves at different acoustic powers at 0.3 m s−1 flow rate. b1 Heat transfer coefficient at different acoustic powers at 0.1 m s−1 flow rate. b2 Heat transfer coefficient at different acoustic powers at 0.3 m s−1 flow rate. c Pressure drop at different acoustic powers under different flow rates. d Effect of sound power intensity on bubble behavior (V = 0.1 m s−1, q = 20 W cm−2). Schematic diagrams and visualization images of the bubbles corresponding to 4.5 W and 10.5 W acoustic power are included in the figure. The blue circles in the figure represent the air bubbles. The blue dashed line represents the initial position of bubble migration. e Long-term stable operation of piezoelectric ceramic (fa = 1.5 kHz, Pa = 7.5 W, V = 0.1 m s−1, q = 55 W cm−2). The heat flux has exceeded the CHF of acoustic OFF.

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Conclusions

In summary, we designed ALCHE utilizing piezoelectric ceramic. The strong acoustic radiation force generated by the piezoelectric ceramic propelled bubbles from nucleation sites, achieving smaller bubble detachment sizes, higher detachment frequencies, and faster migration speed. Consequently, this approach led to a significant improvement in the heat exchanger’s heat transfer performance. Specifically, when the acoustic frequency and power were 1.5 kHz and 7.5 W, CHF and maximum HTC were enhanced by 42.1% and 38.1%, respectively. Furthermore, the mechanisms of acoustic frequency and acoustic power on the heat transfer performance of ALCHE were explored. The magnitude of the acoustic radiation force and the direction of the acoustic pressure field, which were closely related to heat transfer properties, varied with frequency. Through the combination of experimental observation of bubble images and numerical simulation of displacement, the 1.5 kHz acoustic field was found to possess a particularly strong acoustic radiation force and favorable acoustic pressure field direction, contributing to superior heat transfer performance. Furthermore, an increase in acoustic power resulted in a faster bubble detachment and migration due to the stronger acoustic radiation force, thereby facilitating higher CHF. Based on the analysis of acoustic frequency and power mechanism, piezoelectric ceramic achieved long-time stable operation of heat exchanger under high heat flux. The design of this ALCHE not only offers an approach for advancing acoustic field-enhanced boiling heat transfer technology but also provides robust technical support for exploring performance optimization in the heat dissipation domain.

Methods

Cooling test platform

The cooling test platform could be divided into three parts: fluid circulation section, test section, and data acquisition section (Supplementary Fig. 14). In the fluid circulation section, the coolant (ethanol) was heated and cooled from the storage tank and flowed to the test section. The inlet temperature was controlled at 20 °C. The coolant absorbed heat from the wafer surface in the test section and flowed back to the circulation section. The volumetric flow rate was measured by a gear flow meter with a filter (50 μm pore size) installed upstream of the flow meter to prevent contaminants from flowing into the test section.

In the test section (Fig. 1b), the coolant exchanged heat with the wafers in the microchannel, and the contact heat transfer area of the silicon wafer (P-doped N-type) was 1 × 1 cm2. the wafer was heated by a DC power supply (HCP1023H, Henghui, China). The heat flux and HTC processes and details are shown in Supplementary Note 1. The reliability of the experimental system was evaluated by comparing the Nusselt number of the single-phase heat transfer experiment with the Sider-Tate formula and the heat loss of the test section (Supplementary Fig. 15), and the maximum error between the Nusselt number of the experimental and the Sider-Tate formula was 9.8%, while the experimental heat loss was controlled within 7.1%, which verified the reliability of the experimental system. The acoustic field of the piezoelectric ceramic was generated by a signal generator (DG5100, PINZHI, China) and a high-voltage amplifier (HA-820A, PINZHI, China).

In the data acquisition section, the temperature and pressure drop between the inlet and outlet as well as the temperature of the silicon wafer surface were collected by the data acquisition system (NATIONAL INSTRUMENTS, cDAQ-9174). Furthermore, a high-speed camera (Nac Memrecam HX-6E) was used to observe and record the bubble behavior.

Fabrication of acoustic heat exchanger (ALCHE)

ALCHE primarily comprised a transparent cover plate, microchannel substrate, silicon wafer, and piezoelectric ceramic wafer (Fig. 1a). The transparent cover plate was fabricated from polymethylmethacrylate (PMMA, 30 mm × 70 mm × 1 mm), while the microchannel substrate was made of epoxy glass fiber (30 mm × 100 mm × 2 mm). The silicon wafer (P-doped N-type) had dimensions of 20 mm × 10 mm × 0.5 mm, and the piezoelectric ceramic wafer was made of PZT (Lead zirconate titanate) (30 mm × 40 mm × 1 mm).

The transparent cover, microchannel substrate, and piezoelectric ceramic wafer were tightly adhered to by adhesive to ensure the hermeticity of the flow process and the propagation of acoustic waves. The silicon wafer was heated through copper electrodes on both sides, which were fabricated in a three-step process (Supplementary Fig. 2). Initially, the wafer surface was cleaned using a plasma gun, and then a titanium layer was sputtered on both sides of the wafer to serve as a bonding layer. Finally, copper electrodes were sputtered onto both sides (magnetron sputtering machine, MSP-300B, CHUANGSHIWEINA, China). The bottom of the silicon wafer was sealed with an adiabatic adhesive to minimize heat loss.

Bubble image processing

To explain the mechanism of acoustic field-enhanced boiling heat transfer performance, it was necessary to extract the number of bubbles and bubble size distribution from the images. The accuracy of these measurements hinged closely on the quality of the bubble images. Consequently, the following image processing techniques were applied to improve the image quality: acutance enhancement, brightness, and contrast adjustment, binarization, and hole filling (Supplementary Fig. 7). The purpose of each step in this process was outlined below: Firstly, acutance enhancement was used to clarify the boundary between the vapor and liquid phases, ensuring a distinct separation between the two regions. Subsequently, brightness and contrast adjustments were performed to amplify the pixel value difference between the vapor and liquid phase regions. Next, binarization transformed the vapor-phase and liquid-phase regions into distinct white and black areas. Finally, a hole-filling process was implemented to mitigate reflections on the bubble surface during filming. The main objective of this entire process was to produce a more dependable image of the bubbles, enabling a more reliable counting of bubble size and number. The image processing and recognition code were all conducted in Matlab software.

When counting the bubble size distribution and number, to avoid the possible extreme results of bubble distribution and number at a certain moment, a visualization image was taken each 100 ms in 400 ms time intervals, and the aggregated data of bubble size distribution at five moments was used as the result of bubble size distribution, and the average of bubble number at five moments was used as the result of bubble number. Additionally, the bubble size was expressed by the number of pixel points per closed white area.

Vibration simulation of ALCHE

The vibration of ALCHE is simulated by COMSOL to obtain the displacement distribution on the silicon surface and the electrical response and mechanical vibration are governed by the following equations27:

$${T}_{{ij}}={C}_{{ijkl}}^{E} \, {{\cdot }} \, {S}_{{kl}}-{e}_{{ijk}}^{T} \, {{\cdot }} \, {E}_{k}$$
(2)
$${D}_{i}={e}_{{ikl}} \, {{\cdot }}{S}_{{kl}}+{\varepsilon }_{{ij}}^{S} \, {{\cdot }} \, {E}_{k}$$
(3)

where the stress vector Tij, the strain vector Skl, the electrical displacement Di, and the electric field Ek are coupled; \({C}_{{ijkl}}^{E}\), \({e}_{{ikl}}\), and \({\varepsilon }_{{ij}}^{S}\) represent the elasticity matrix, the piezoelectric matrix, and the permittivity matrix, respectively.

The ‘Solid Mechanics’ module and the ‘Electrostatics’ module were coupled to simulate the piezoelectric effect and the vibration of ALCHE. The governing equations of ‘Solid Mechanics’ are as follows:

$$-\rho {\omega }^{2}{{\boldsymbol{u}}}=\nabla \, {{\cdot }} \, {{\boldsymbol{S}}}-{{{\boldsymbol{F}}}}_{{{\boldsymbol{v}}}}{e}^{i\varphi }$$
(4)

where ρ is the density, ω is the angular frequency, u is the displacement vector, S is the stress tensor, FV is the volume force vector, and φ is the phase. The governing equations of ‘Electrostatics’ are as follows:

$$E=-\nabla V$$
(5)
$$\nabla \, {{\cdot }} \, \left({\varepsilon }_{0}{\varepsilon }_{r}E\right)={\rho }_{v}$$
(6)

where E is the electric field, V is the electric potential, ε0 is the permittivity of vacuum, εr is the relative permittivity of the material, and ρv is the space charge density. The“Piezoelectric Effect” Multiphysics condition was applied to the boundary between the PZT field and the microchannel substrate field to couple the mechanical vibration and the electrical behavior in the microchannel substrate. The simulation was solved in the frequency domain.

Bubble tracking algorithm

To investigate the effect of acoustic field on the migration speed and direction of bubbles on the whole silicon wafer, we used Matlab to implement a multi-bubble tracking algorithm based on two-frame images, the algorithm was proposed by Baek and Lee28, and Zhou29 has already applied the algorithm to bubble tracking procedure. The acquisition of the trajectory needed to satisfy two features proposed by the algorithm:

  1. (1)

    Maximum displacement. The time interval between the two frames of the image was Δt, the maximum velocity of the bubble was Um during the time interval of Δt, and the maximum displacement of the bubble Tm = UmΔt. Therefore, the bubble displacement could not be larger than the threshold Tm between two frames, as shown in Fig. 4g–i.

  2. (2)

    Quasi-rigidity condition. A small region of bubbles showed a similar pattern of motion (Fig. 4g-ii, iii), with displacement vectors of similar magnitude and direction. This phenomenon was almost rigid in a small region.

The bubble tracking algorithm could be implemented based on the above two features, the process was as follows (Supplementary Fig. 16): The central coordinates of each bubble were taken out from the processed two frame images, and then the candidate bubble central points in the second frame were selected according to the bubble central points in the first frame. The candidate bubble central points should satisfy the maximum displacement feature:

$${d}_{{ij}}=\left|{x}_{i}-{y}_{j}\right| < {T}_{{{\rm{m}}}}$$
(7)

where xi denoted the bubble central point in the first frame, yj denoted the bubble central point in the second frame, dij denoted the displacement from the bubble central point xi to yj, and Tm denoted the maximum displacement.

$${T}_{{{\rm{m}}}}={U}_{{{\rm{m}}}}\Delta t$$
(8)

where Um was the maximum bubble velocity of the bubble, and Δt was the time interval between two continuous frames.

The trajectory was determined by iteratively evaluating the match probability Pij and the no-match probability Pi* from the bubble in the first frame image to the candidate bubble in the second frame image. The match probability was a match degree from point yj to point xi, and the no-match probability was a no-match degree that xi had no matching point in the second frame. For point xi, the following equation should be satisfied at each iteration:

$$\sum_{i}{P}_{{ij}}+{P}_{i}^{* }=1$$
(9)

For simplicity, the initial values of match probability and no-match probability at the beginning of the iteration were:

$${P}_{{ij}}^{(0)}={P}_{i}^{* (0)}=\frac{1}{N+1}$$
(10)

Where N was the number of candidate bubble centroids corresponding to bubble centroid xi.

The quasi-rigidity condition was used for a small region of bubbles in the vicinity of the bubble central point xi, and the point xk should be in the neighborhood of xi:

$$\left|{x}_{i}-{y}_{k}\right| \, < \, {T}_{{{\rm{n}}}}$$
(11)

where Tn denoted the neighborhood threshold. Tn was set to the deciles of all results corresponding to xi.

The iterative update probability of the quasi-rigidity condition needed to satisfy the similar displacement magnitude and direction:

$$\left|{d}_{{ij}}-{d}_{{kl}}\right| < {T}_{{{\rm{q}}}}$$
(12)

where the quasi-rigidity threshold Tq = Tm sin φ, and the magnitude of φ was chosen according to the actual situation. Taking Fig. 4g-iv as an example, |dij–d22 | < Tq, while |dij–d21 | > Tq, then d22 satisfied the quasi-rigidity principle.

When Eqs. 7 and 11 were satisfied, iterative updating was performed using Eq. 12, and for the bubble central point xi, the match probability after iterative updating was:

$${\widetilde{P}}_{{ij}}^{(n)}={A \, {{\cdot }} \, P}_{{ij}}^{(n-1)}+B \, \, {{\cdot }} \, \sum_{k}\sum_{l}{P}_{{kl}}^{(n-1)}$$
(13)

where A (<1) and B (>1) were accelerated convergence factors. In this paper, A and B were set to 0.3 and 3, respectively. \({\widetilde{p}}_{{ij}}^{(n)}\) denoted the non-normalized match probability and n was the iteration step number.

To still satisfy Eq. 9 during the iteration procedure, the probabilities need to be normalized, and the formulas for the normalized match probability and no-match probability were as follows:

$${P}_{{ij}}^{(n)}=\frac{{\widetilde{P}}_{{ij}}^{(n)}}{{\sum }_{j}{\widetilde{P}}_{{ij}}^{(n)}+{P}_{i}^{* (n-1)}},\,{P}_{i}^{* (n)}=\frac{{P}_{i}^{* (n-1)}}{{\sum }_{j}{\widetilde{P}}_{{ij}}^{(n)}+{P}_{i}^{* (n-1)}}$$
(14)

After 20 iterations, the correct matching probability of bubble trajectories tended to be 1, and the incorrect matching probability of bubble trajectories tended to be 0. Therefore, the result with the largest probability could be selected as the final trajectory and the trajectory results could be plotted by Matlab.