We report on experimental demonstrations of the first sub-100 μm scale standard and inverse Chladni figures in both one- (1D) and two-dimensional (2D) fashions in liquid, and on exploiting these micro-Chladni figures for patterning microparticles on chip, via engineering multimode micromechanical resonators with rich, reconfigurable 1D and 2D mode shapes. Silica microparticles (1–8 μm in diameter) dispersed on top of resonating doubly-clamped beams (100 × 10 × 0.4 μm3) are observed to aggregate at antinodal points, forming 1D inverse Chladni figures, while microbeads atop square trampoline resonators (50 × 50 × 0.2 μm3) cluster at nodal lines/circles, creating 2D standard Chladni figures such as “,” “○,” “×,” and “\.” These observations suggest two distinct micro-Chladni figure patterning mechanisms in liquid. Combining analytical and computational modeling, we elucidate that streaming flow dominates the inverse Chladni pattern formation in the 1D beam experiments, while vibrational acceleration dictates the standard Chladni figure generation in the 2D trampoline experiments. We further demonstrate dynamical patterning, switching, and removal of 2D micro-Chladni figures in swift succession by simply controlling the excitation frequency. These results render new understandings of Chladni patterning genuinely at the microscale, as well as a non-invasive, versatile platform for manipulating micro/nanoparticles and biological objects in liquid, which may enable microdevices and functional device-liquid interfaces toward relevant sensing and biological applications.

Manipulation and precise control of micro/nanoparticles on various surfaces and in multiple phases are essential for creating functional materials, devices, and interfaces toward sensor technologies, micro/nanoelectronic, and optical devices.1–5 In interfacial and biological sciences and engineering, in particular, manipulating and patterning micro/nanoscale biological objects (e.g., cells and molecules) have been enabling biomolecule detection,6,7 drug and toxic screening,8,9 tissue engineering,10,11 and facilitating biophysical studies, such as investigating cellular interactions.12 The increasing needs for fast, non-invasive, and “programmable” techniques and tools for such manipulation and patterning of particles with micro/nanometer precision have already stimulated a number of emerging optical,13,14 electrical,10,15 and acoustic11,16 techniques, beyond the conventional template-assisted self-assembly and surface patterning7,17 that often require pre-patterned chemical functionalization, modification, and topography (e.g., wells and microfluidic channels). Optical tweezers13,14 have been powerful in manipulating dielectric spheres, cells, bacteria, and viruses with very high force and spatial resolutions, whereas they could be compromised by the potential laser-induced adverse physiological effects on the biological objects.18–20 Dielectrophoresis10,15 relies on the particle polarizability and medium conductivity to generate electrical forces, which could in turn cause undesired cell physiological effects via current-induced heating and direct electric field interaction.10,21 Surface acoustic wave (SAW) devices, or acoustic tweezers,11,16 have attracted increasing attention lately. They can position microparticles, bacteria, and cells by exploiting acoustic radiation forces. Such techniques can also pattern microparticles/cells into “dot” or “line” arrays (defined by the acoustic pressure nodes), through engineering of the interdigital transducers (IDTs) to generate interferencing SAWs.

Chladni figures,22 discovered by the German physicist and musician Ernst Chladni that sand particles sprinkled on top of a vibrating metal plate excited by a violin bow could be organized into diverse patterns in response to the resonant motions of the plate, offer new possibilities of manipulating particles using mechanical resonances. In those conventional cases, the sand particles were commonly seen accumulated at the nodes of the mode shapes (“standard Chladni figures”) due to the “bouncing” particles on the plate, while the fine shavings from the hairs of his violin bow settled at antinodes (“inverse Chladni figures”) due to the mechanical vibration-induced air streaming.23 Imagine scaling those sand particles in the classical Chladni experiments down to micro/nanoscale particles (and biological objects, e.g., cells and molecules), we can thus envision to explore and realize new microscale Chladni figures for patterning, by greatly miniaturizing the classical Chaldni plates, down to micromachined resonators with rich multimode responses. While such microscale Chladni figures would be attractive for non-invasive and chip-scale manipulation of micro/nanoscale particles and biological objects (without relying on their types, geometries, optical, electrical properties, etc.) toward relevant applications, experimental demonstration, and understanding of the mechanisms of Chladni figure patterning on genuinely microscale device platforms (i.e., sub-100 μm to sub-10 μm), especially in two-dimensional (2D) fashion, have been lacking.

In this work, we demonstrate both one- (1D) and two-dimensional (2D) micro-Chladni figures, down to 50 μm scale, via patterning microparticles (of various sizes from ∼1 to 10 μm) in an aquatic environment on two types of micromechanical resonators (doubly-clamped beams and square trampolines, as shown in Fig. 1). Quite different from the centimeter-sized surface acoustic wave (SAW) devices11,16 that often utilize acoustic radiation effect, the micro-Chladni resonators exploit the flexural vibration induced boundary streaming or acceleration to move the microparticles at micrometer scale in liquid. Our results promise a versatile micromechanical approach to non-invasively, rapidly manipulating micro/nanoscale objects in liquid, while the particle assembly can be readily reconfigured via programmed excitation frequency. This study also addresses the knowledge gap toward a better understanding of micro-Chladni figure formation in liquid adding to the previous studies,24,25 suggesting that standard and inverse Chladni figures at microscale can be dominated by different mechanisms (streaming flow or vibrational acceleration in Fig. 1), and achieved by engineering the device structures and resonance modes other than varying the particle size, which has been experimentally seen in both macroscale (i.e., classical Chladni figures) and microscale cases.

FIG. 1.

Micro-Chladni figures and patterning of microparticles in liquid via multimode micromechanical resonances. (a) A conceptual illustration (cross-sectional view) of the microparticle patterning mechanism. Microparticles are organized either at the nodes (standard Chladni figures) or antinodes (inverse Chladni figures), dominated by the mechanical vibration induced acceleration (Fa||¯) or streaming flow (Fd||¯), respectively. (b) Schematic illustrations showing manipulating microparticles into 1D inverse and 2D standard Chladni patterns by engineering resonator structures (e.g., beam and trampoline) and their diverse resonance modes.

FIG. 1.

Micro-Chladni figures and patterning of microparticles in liquid via multimode micromechanical resonances. (a) A conceptual illustration (cross-sectional view) of the microparticle patterning mechanism. Microparticles are organized either at the nodes (standard Chladni figures) or antinodes (inverse Chladni figures), dominated by the mechanical vibration induced acceleration (Fa||¯) or streaming flow (Fd||¯), respectively. (b) Schematic illustrations showing manipulating microparticles into 1D inverse and 2D standard Chladni patterns by engineering resonator structures (e.g., beam and trampoline) and their diverse resonance modes.

Close modal

We design and fabricate two types of micromechanical resonators, i.e., doubly-clamped beams and square trampolines. These devices are fabricated based on a silicon carbide-on-silicon (SiC-on-Si) platform with two steps: (i) focused ion beam (FIB) milling through the SiC layer to define the device geometries and etching windows for the subsequent Si etch; (ii) HNA etching to anisotropically etch away the Si substrate and release the device (see Sec. I in supplementary material). SiC material is selected because of its outstanding optical, mechanical properties, chemical inertness, and, more importantly, biocompatibility26 (for its potential in biological applications). Devices are miniaturized to have lateral sizes of 100 μm × 10 μm for doubly-clamped beams and 50 μm × 50 μm for square trampolines to push for realizing Chladni figures at sub-100 μm scale for manipulating and patterning microparticles. These devices exhibit robust multimode resonances from hundreds of kilohertz (kHz) up to megahertz (MHz) in aqueous environments. The devices are fully immersed in deionized (DI) water and their multiple flexural resonances are first characterized using an ultrasensitive optical interferometry system.26 This step determines the frequency-sweeping range of the piezoelectric actuator during Chladni patterning experiments (see Secs. II and III in supplementary material). Silica (SiO2) microbeads, with diameters of d = 0.96, 1.70, 3.62, 7.75 μm (coefficient of variation, CV = 2%–5%), are sonicated in DI water to be fully separated to create “microbead suspensions.” A microparticle delivery module is employed to locally transfer the microbeads onto device surface (microbeads quickly settle on the device surface within seconds) (see Fig. S3 in supplementary material). Microbead movements are recorded by a time-lapse imaging module using a high-resolution CCD camera (see Sec. II in supplementary material).

We first study 1D micro-Chladni figures and patterning in liquid using SiC doubly-clamped beams, as shown in Fig. 2(a) (here 1D refers to the fact that the nodes and antinodes in the beam devices are in 1D fashion, also in comparison with the 2D scenarios in Sec. III). When the piezoelectric excitation frequency is swept from 5 MHz back to 100 kHz, three patterns, corresponding to the 5th, 3rd, and 1st flexural modes and their mode shapes of the beam at ∼3.3 MHz, 1.0 MHz, and 0.4 MHz, are vividly visualized [Fig. 2(b)]. Microbeads are observed to quickly gather at the antinodal points within just ∼2 s [antinodes are determined by referring to the simulated mode shapes in Fig. 2(b)], forming clusters lined up in 1D arrays. These inverse Chladni figures also show good consistency across a range of microbead diameters, d = 0.96, 1.70, 3.62, 7.75 μm. By programming the excitation frequency to switch among 3.3 MHz, 1.0 MHz, and 0.4 MHz or off these modes, these microscale inverse Chladni figures of microbead patterns can be readily switched on and off and reconfigured in situ into 5, 3, and 1 clusters (see Movie I in supplementary material), demonstrating good versatility and flexibility of such a micromechanical particle patterning technique.

FIG. 2.

1D inverse micro-Chladni figures and patterning of microparticles, visualized on resonating SiC doubly-clamped beams (100 × 10 × 0.4 μm3). (a) Aerial-view SEM and top-view optical images showing doubly-clamped beam structure. (b) Microbead patterns show good agreement with the simulated mode shapes (red/blue: high/low displacement amplitude). Three 1D patterns are observed corresponding to the 1st, 3rd, and 5th flexural modes of the beam. Pattern formation shows good consistency for microbeads with various diameters (d = 0.96, 1.70, 3.62, 7.75 μm). Images taken under a 10× objective.

FIG. 2.

1D inverse micro-Chladni figures and patterning of microparticles, visualized on resonating SiC doubly-clamped beams (100 × 10 × 0.4 μm3). (a) Aerial-view SEM and top-view optical images showing doubly-clamped beam structure. (b) Microbead patterns show good agreement with the simulated mode shapes (red/blue: high/low displacement amplitude). Three 1D patterns are observed corresponding to the 1st, 3rd, and 5th flexural modes of the beam. Pattern formation shows good consistency for microbeads with various diameters (d = 0.96, 1.70, 3.62, 7.75 μm). Images taken under a 10× objective.

Close modal

We then explore 2D micro-Chladni figures and patterning in liquid using the SiC square trampoline resonators, as shown in Fig. 3(a). When a group of (about 100) 3.62 μm microbeads are dispersed to the device area and the excitation frequency is swept from 5 MHz to 100 kHz, we observe a series of 2D geometric patterns, such as “,” “○,” “×,” and “\” [shown in Fig. 3(b)], corresponding to the multiple flexural modes of the trampoline resonators [see the simulated mode shapes in Fig. 3(b)]. Interestingly, these microbeads are observed to cluster along the nodal lines/circles this time (within ∼40 s), forming standard Chladni figures. To our best knowledge, this is the first time that 2D standard Chladni figures (in shapes of “,” “○,” “×,” and “\”) and patterning of microparticles are demonstrated at 50 μm scale.

FIG. 3.

2D standard micro-Chladni figures and patterning of microparticles, visualized on resonating SiC trampoline resonators (50 × 50 × 0.2 μm3). (a) Aerial-view SEM and top-view optical images showing trampoline structure clamped at four tethers. (b) A series of 2D patterns of microparticles, such as “\,” “×,” “○,” and “,” show good agreement with the simulated mode shapes (red/blue: high/low displacement amplitude). Microbead diameter: d = 3.62 μm. Images taken under a 50× objective.

FIG. 3.

2D standard micro-Chladni figures and patterning of microparticles, visualized on resonating SiC trampoline resonators (50 × 50 × 0.2 μm3). (a) Aerial-view SEM and top-view optical images showing trampoline structure clamped at four tethers. (b) A series of 2D patterns of microparticles, such as “\,” “×,” “○,” and “,” show good agreement with the simulated mode shapes (red/blue: high/low displacement amplitude). Microbead diameter: d = 3.62 μm. Images taken under a 50× objective.

Close modal

We also show frequency-controlled, dynamic patterning, switching and erasing of these 2D micro-Chladni figures (state of erasing is defined by trampoline’s fundamental mode, see Movie II in supplementary material). This process is reversible, as microbeads could repeatedly be “grabbed onto” and “dusted off” the device surface by selectively exciting the corresponding flexural modes (the increasing number of microbeads from “” mode to “○” mode in Fig. 3 also exemplifies this phenomenon). We attribute such merits to the unique device structure (with carefully designed etching window and suspended rim) and its robust multimode resonances in liquid. We further validate that such a patterning effect holds true for microbeads with other diameters (d = 1.70 μm and 7.75 μm, see Sec. IV in supplementary material).

The 1D inverse and 2D standard micro-Chladni figures, created on the doubly-clamped beam and square trampoline resonators, respectively, suggest different Chladni patterning mechanisms, given the same microbead diameters, comparable piezoelectric excitation amplitudes, and same frequency weeping range. To understand the dynamics of microparticle movement in these two cases and their mechanisms for pattern generation, we implement careful force analysis by taking into account resonator-particle interaction forces (mostly van der Waals force and vibrational acceleration force), in addition to the boundary streaming induced Stokes drag force. Gravity effects play a negligible role at such a scale. Friction between particles and device surfaces is omitted.

To calculate streaming induced Stokes drag force, we develop a realistic finite element method (FEM) model that can best mimic the streaming flow induced by a finite, oscillating boundary in the actual experiments. In this model, the fluid velocity field (ufluid) is solved by the incompressible Navier-Stokes (N-S) equations (assuming no-slip boundary condition at fluid-device interface):

ρufluidt+ρ(ufluid)ufluid=[pI+μ(ufluid+(ufluid)T)]+F,ρufluid=0,
(1)

where ρ and μ are the density and dynamic viscosity of the fluid, I is the unit diagonal matrix, and F is the volume force (assume that no gravitational or other volume forces affect the fluid, so F = 0). Combined with a time-dependent fluid-structure interaction (FSI) module, ufluid is solved in real time. Taking into account that streaming flow is a time-averaged effect (net flow) over single vibrational cycle, its profile can be extracted by an additional time-averaging step over ufluid (see Sec. V in supplementary material). It is worth noting that our computational model has several advantages over the analytical method.24,27 First, our simulations take into account finite device geometries, leading to more accurate predictions and realistic visualization of the streaming flow velocity field. Second, our simulations apply to various resonators with complex structures. More promisingly, it could be extended to devices with 2D surfaces (3D modeling) when the streaming flow field and multiphase situations are challenging to be solved analytically.

In Fig. 4, we show an example of streaming flow velocity profile when a 100 μm-long beam is vibrating in water at its 3rd flexural mode at 1.0 MHz. We observe clear inner and outer vortices across device surface between adjacent antinodal and nodal points. A zoom-in view of inner vortices shows flow direction toward antinode on the top layer while reversed at the bottom layer.

FIG. 4.

1D inverse Chladni pattern formation. (a) Streaming flow generated by a 100 μm-long doubly-clamped beam vibrating at its 3rd flexural mode at 1.0 MHz. A cross-sectional view of the inner streaming is highlighted. (b) Schematic illustration of maximum acceleration (am) a microbead experiences on a vibrating beam. (c) Model illustration of van der Waals force (Fv) between the microbead and an infinite flat surface. (d) 3D view of maximum acceleration force (Fam) and Fv as functions of microbead diameter (d) and vibration amplitude (A). (e) 2D view of Famvs.Fv at d = 3.62 μm.

FIG. 4.

1D inverse Chladni pattern formation. (a) Streaming flow generated by a 100 μm-long doubly-clamped beam vibrating at its 3rd flexural mode at 1.0 MHz. A cross-sectional view of the inner streaming is highlighted. (b) Schematic illustration of maximum acceleration (am) a microbead experiences on a vibrating beam. (c) Model illustration of van der Waals force (Fv) between the microbead and an infinite flat surface. (d) 3D view of maximum acceleration force (Fam) and Fv as functions of microbead diameter (d) and vibration amplitude (A). (e) 2D view of Famvs.Fv at d = 3.62 μm.

Close modal

We first explain the formation of the inverse micro-Chladni figures and patterning of microbeads in the resonating 1D beam experiments. The mechanical vibration induced acceleration can be expressed as28 

a=2yt2=Aω2sin(ωt)sinnπxL,
(2)

where A is the resonance amplitude, ω = 2πf, f is the resonance frequency, n denotes number of antinodes along x in a mode shape [n = 3 for the 3rd mode in Fig. 4(a)], and L is the beam length. Therefore, the maximum acceleration that a microbead could experience when attaching to the device surface is am= 2 [in Fig. 4(b)]. We assume the microbead attachment is supported by van der Waals force, which is calculated as29 

Fv=Ahd12hs21+2ac2hsd,
(3)

where hs is the Lennard-Jones separation distance, ac is the contact area, Ah is the Hamaker constant, and d is the microbead diameter. Ah is estimated as30 

Ah34kBTε1ε3ε1+ε3ε2ε3ε2+ε3+3hυe82×(n12n32)(n22n32)(n12+n32)1/2(n22+n32)1/2[(n12+n32)1/2+(n22+n32)1/2],
(4)

where kB is the Boltzmann constant, T is room temperature, h is the Planck’s constant, υe is the main electronic absorption frequency, ni and ɛi are the refractive indices and dielectric permittivity values of SiC, SiO2, and water. ac is predicted by the Derjaguin-Muller-Toporov (DMT) model [as shown in Fig. 4(c)],31 

ac|P=0=(πWAd2)1/3/(2K),
(5)

where WA=Ah12πhs2 is the thermodynamic work of adhesion and K=431υ12E1+1υ22E21 is the composite Young’s modulus. Maximum acceleration force (Fam, blue) and van der Waals force (Fv, red) as functions of resonance amplitude (A) and microbead diameter (d) are plotted in Fig. 4(d). Using 3.62 μm microbeads as an example [the yellow plane in Fig. 4(d), also see Fig. 4(e)] and considering A could easily be >25 nm in the experiment, we can reasonably assume that the microbeads frequently bounce with the beam since the van der Waals adhesion cannot sustain particle attachment. One clear evidence is that when we intentionally increase the driving amplitude, the microbeads appear “floating” above the device (the microbeads are in and out of focus under a 50× objective). Therefore, we perceive that the formation of inverse micro-Chladni figures in the beam experiments is dominated by the top layer of inner streaming (up to bottom layer of the outer streaming) pointing from node to antinode.

Similarly, to explain the formation of the 2D microscale standard Chladni figures and microparticle patterning in the trampoline experiments, we take the “\”-shaped diagonal mode (2nd in Fig. 3) as an example. Streaming flow velocity profile in Fig. 5(a) is calculated from a simplified “center-clamped beam” model vibrating at its torsional mode to mimic the trampoline’s “\” mode from its cross-sectional view [indicated by the white dashed line, Fig. 5(a) bottom panel]. Such a simplification accelerates computations and evades the super-demanding direct simulations of the realistic 3D system.

FIG. 5.

2D standard Chladni pattern formation. (a) Streaming flow induced by a 50 × 50 μm2 square trampoline resonator vibrating at its “\”-shaped diagonal flexural mode at 0.5 MHz. A cross-sectional view of the inner streaming is highlighted. (b) Schematic illustration of the vibrational acceleration a microbead experiences within single vibration cycle. Lateral component (a||) always points toward nearest node. (c) Model illustration of Stokes drag force (lateral component, Fd||¯) exerting on the microbead. (d) 3D view of time-averaged acceleration force (lateral component, Fa||¯) vs. Fd||¯ as functions of microbead diameter (d) and vibration amplitude (A) at x = 12.5 μm [red dashed line in (a)]. (e) 2D view of Fa||¯vs.Fd||¯ at d = 3.62 μm.

FIG. 5.

2D standard Chladni pattern formation. (a) Streaming flow induced by a 50 × 50 μm2 square trampoline resonator vibrating at its “\”-shaped diagonal flexural mode at 0.5 MHz. A cross-sectional view of the inner streaming is highlighted. (b) Schematic illustration of the vibrational acceleration a microbead experiences within single vibration cycle. Lateral component (a||) always points toward nearest node. (c) Model illustration of Stokes drag force (lateral component, Fd||¯) exerting on the microbead. (d) 3D view of time-averaged acceleration force (lateral component, Fa||¯) vs. Fd||¯ as functions of microbead diameter (d) and vibration amplitude (A) at x = 12.5 μm [red dashed line in (a)]. (e) 2D view of Fa||¯vs.Fd||¯ at d = 3.62 μm.

Close modal

Considering that the vibrational amplitude of the trampoline resonator would be much smaller than the doubly-clamped beam (due to the smaller dimensions and clamping conditions at four tethers), we assume that the van der Waals force can sufficiently sustain the microbead attachment on the device surface. This assumption is also supported by observing (under the microscope) the microbeads “rolling” on the trampoline surface rather than “floating” above the device. In this case, we need to implement careful analysis to calculate the lateral force component that dominates the microbead movement. As illustrated in Fig. 5(b), when a microbead attaches to and vibrates along with the device, we can decompose the lateral component (with approximation based on the fact that the vibration amplitude is much smaller than the vibration wavelength) of vibrational acceleration (a||) as

a||adydx=A2ω2kπLsin2(ωt)sinkπxLcoskπxL.
(6)

Within one vibration cycle, a|| always points toward the nearest node. This restoring term, if averaged within one vibrational cycle, can be expressed as

Fa||¯=m1T0Ta||dt=14mA2ω2kπLsin2kπxL.
(7)

On the other hand, the microbead also experiences streaming flow [illustrated in Fig. 5(c)]. The total lateral component of the Stokes drag force can be calculated by an integral of vortex velocity (uvortex||) over the microbead diameter (d):

Fd||¯=1T0T3πη0dufluid||dhdt=3πη0duvortex||dh.
(8)

Furthermore, it can be rewritten as

Fd||¯=3πηuvortex||¯d.
(9)

We set x = 12.5 μm [red dashed line in Fig. 5(a)], meaning a microbead is positioned at the middle point between the node and antinode, and both Fa||¯ (blue) and Fd||¯ (red) can be calculated as functions of vibrational amplitude (A) and particle diameter (d), as shown in Fig. 5(d). We use 3.62 μm microbeads as an example [the yellow plane in Fig. 5(d), also see Fig. 5(e)], and we clearly see that the acceleration force always overrides the Stokes drag force, which agrees with our experimental observations that the microbeads are always aggregated at the nodal line/circles. The orders of magnitude smaller forces in Fig. 5(d), in contrast to those in Fig. 4(d), also help explain the relatively much longer pattern formation time in the trampoline experiments (∼40 s) than that in the beam experiments (∼2 s).

Table I summaries the new contributions in this study regarding the creation of genuinely microscale Chladni figures in liquid beyond previous works in the literature. We first compare our findings with Ref. 24, in which 1D Chladni patterns were observed using much larger cantilevers (560 × 100 μm2). First, the observations of inverse Chladni patterns in our beam experiments agree with the analysis in Ref. 24 that boundary streaming dominates the particle movement and pattern formation. However, as predicted by Ref. 24, particle size-dependent pattern inversion would happen at a “critical diameter” (CD ≈ 2.8δ, where δ=2ν/ω is only frequency dependent), i.e., inverse Chladni figures form when d > CD, while standard Chladni patterns form when d < CD. This infers that in our beam experiments, standard Chladni figures would appear when d < CD ≈ 3 and 1.5 μm for the 1st and 3rd flexural modes. However, we have consistently observed inverse Chladni figures for d = 1–8 μm. Second, the 2D standard Chladni figures in our trampoline experiments are demonstrated for the first time, which cannot be explained by the theory in Ref. 24, but have been well explained by the modeling results in this work (in Fig. 5). This demonstrates a new mechanism for micro-Chladni figures and patterning in liquid, where particle movement is dominated by vibrational acceleration rather than boundary streaming flow. We also compare our results with another work25 in which inverse Chladni figures were demonstrated in liquid using 1.6 mm-diameter Si circular membrane resonators. First, we have demonstrated 2D Chladni figures and patterning in liquid at the genuine microscale (down to 50 μm) with device surface areas and volumes 3–4 orders of magnitude smaller than those in Ref. 25. Second, our microbeads (1 to 8 μm-diameter) are comparable to biological cells in size and are much smaller than those (70 μm-diameter) in Ref. 25. The scaling in both device and microparticle dimensions has very important implications for future biological applications. Third, only 2D inverse Chladni figures were reported in Ref. 25, which were dominated by boundary streaming. In addition to that, we have demonstrated 2D standard Chladni figures and patterning in liquid, which is dominated by a different mechanism, i.e., vibrational acceleration. While acoustic radiation effect has also been reported to dominate the particle motion to the “pressure nodes” in the acoustofluidic devices,32,33 here we have found that it plays a much less important role as compared to the vibrational acceleration in the formation of 2D standard micro-Chladni figures and patterning of microparticles (see Sec. VI in supplementary material).

TABLE I.

State-of-the-art micro-Chladni figures and patterning of microparticles in liquid.

DevicesDevice dimensionsMicrobead diameter(s)Observed Chladni figuresPatterning mechanism
This work SiC doubly-clamped beam 100 × 10 × 0.4 μm3 (Area: 1 × 103μm2; Volume: 4 × 102μm30.96 μm
1.70 μm
3.62 μm
7.75 μ
1D Inverse Boundary streaming 
 SiC trampoline 50 × 50 × 0.2 μm3 (Area: 2.5 × 103μm2; Volume: 5 × 102μm32D Standard Vibrational acceleration 
Dorrestijn et al.a Si cantilevers 560 × 100 × 7 μm3 (Area: 5.6 × 104μm2; Volume: 3.9 × 105μm34 μ1D Inverse Boundary streaming 
0.5 μ1D Standard Boundary streaming 
Vuillermet et al.b Si circular membrane 1.6 mm × 5.9 μm (Area: 2 × 106μm2; Volume: 1.2 × 107μm370 μ2D Inverse Boundary streaming 
DevicesDevice dimensionsMicrobead diameter(s)Observed Chladni figuresPatterning mechanism
This work SiC doubly-clamped beam 100 × 10 × 0.4 μm3 (Area: 1 × 103μm2; Volume: 4 × 102μm30.96 μm
1.70 μm
3.62 μm
7.75 μ
1D Inverse Boundary streaming 
 SiC trampoline 50 × 50 × 0.2 μm3 (Area: 2.5 × 103μm2; Volume: 5 × 102μm32D Standard Vibrational acceleration 
Dorrestijn et al.a Si cantilevers 560 × 100 × 7 μm3 (Area: 5.6 × 104μm2; Volume: 3.9 × 105μm34 μ1D Inverse Boundary streaming 
0.5 μ1D Standard Boundary streaming 
Vuillermet et al.b Si circular membrane 1.6 mm × 5.9 μm (Area: 2 × 106μm2; Volume: 1.2 × 107μm370 μ2D Inverse Boundary streaming 
a

Reference 24.

b

Reference 25.

In summary, we have experimentally demonstrated both 1D and 2D micro-Chladni figures at sub-100 μm scale and exploited them for micromechanical patterning of microparticles in both 1D and 2D fashions in an aqueous environment. Silica microbeads (1–8 μm in diameter) are manipulated into 1D inverse Chladni patterns when dispersed on top of resonating doubly-clamped beams, while into 2D standard Chladni figures when dispersed on top of resonating trampolines. We have developed a FEM model that can visualize streaming flow over a finite, oscillating boundary, and we have explained the new experimental observations by careful analysis of governing forces experienced by the microbeads. We find that boundary streaming induced Stokes drag dominates the inverse Chladni pattern formation in the beam experiments, while the vibrational acceleration directs the standard Chladni pattern formation in the trampoline experiments. We have also achieved dynamic patterning, switching, and erasing of 2D microparticle patterns by controlling the excitation frequency of the multimode microresonators. These results not only lead to better understanding of micro-Chladni figure creation in liquid but also may facilitate potential biological applications. Patterned micro/nanoscale biological objects (e.g., cells and biomolecules) could enable functional interfaces and building blocks for biosensors, tissue engineering, etc. Meanwhile, these genuinely microscale Chladni figures hold promise for capturing and trapping of the micro/nanoparticles via microscale streaming flows above the device surfaces, which may enhance the sensing modalities and performance. Furthermore, combining frequency-shift based mass sensing and the ability to precisely position biological species (cells, bacteria, viruses, etc.) at the fluid-resonator interface via Chladni figures, multimode MEMS/NEMS resonators may also offer hybrid microdevice technologies for inertial sensing and imaging34 of topologically complex, deformable biological objects in liquid. In addition, non-invasive and versatile manipulation of cells in biosolutions may foster a variety of biophysical studies, such as probing cellular properties and controlling cell-cell interactions.

See supplementary material for details about device fabrication, experimental system, modeling, and supporting movies.

We acknowledge financial support from the National Science Foundation (Nos. ECCS-1408494, EFMA-1641099) and the Case School of Engineering, technical support from the Swagelok Center for Surface Analysis of Materials (SCSAM) and the Materials for Opto/Electronics Research & Education (MORE) Center at Case Western Reserve University (CWRU). P.X.-L.F. thanks Professor Ron Lifshitz for his inspirational discussion on the classical Chladni figures years ago at Caltech Red Door Café.

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Supplementary Material