# Formation of inverse Chladni patterns in liquids at microscale: roles of acoustic radiation and streaming-induced drag forces

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## Abstract

While Chladni patterns in air over vibrating plates at macroscale have been well studied, inverse Chladni patterns in water at microscale have recently been reported. The underlying physics for the focusing of microparticles on the vibrating interface, however, is still unclear. In this paper, we present a quantitative three-dimensional study on the acoustophoretic motion of microparticles on a clamped vibrating circular plate in contact with water with emphasis on the roles of acoustic radiation and streaming-induced drag forces. The numerical simulations show good comparisons with experimental observations and basic theory. While we provide clear demonstrations of three-dimensional particle size-dependent microparticle trajectories in vibrating plate systems, we show that acoustic radiation forces are crucial for the formation of inverse Chladni patterns in liquids on both out-of-plane and in-plane microparticle movements. For out-of-plane microparticle acoustophoresis, out-of-plane acoustic radiation forces are the main driving force in the near-field, which prevent out-of-plane acoustic streaming vortices from dragging particles away from the vibrating interface. For in-plane acoustophoresis on the vibrating interface, acoustic streaming is not the only mechanism that carries microparticles to the vibrating antinodes forming inverse Chladni patterns: In-plane acoustic radiation forces could have a greater contribution. To facilitate the design of lab-on-a-chip devices for a wide range of applications, the effects of many key parameters, including the plate radius *R* and thickness *h* and the fluid viscosity *μ*, on the microparticle acoustophoresis are discussed, which show that the threshold in-plane and out-of-plane particle sizes balanced from the acoustic radiation and streaming-induced drag forces scale linearly with *R* and \(\sqrt \mu\), but inversely with \(\sqrt h\).

## Keywords

Chladni patterns Acoustic streaming Acoustic radiation force Acoustofluidics Vibrating plates## 1 Introduction

Arranging particles and cells into desired patterns for lab-on-a-chip biological applications using ultrasonic fields, i.e. acoustophoresis, by means of bulk and surface acoustic wave techniques, have attracted increasing interest in recent years (Bruus et al. 2011; Friend and Yeo 2011). When an ultrasonic standing/travelling wave is established in a micro-channel containing an aqueous suspension of particles, two main forces act on the particles: the acoustic radiation force and the streaming-induced drag force. In most bulk and surface micro-acoustofluidic manipulation devices, the latter is generally considered to be a disturbance because it places a practical lower limit on the particle size that can be manipulated by the former (Wiklund et al. 2012; Drinkwater 2016). Nevertheless, acoustic streaming flows have been applied to play an active role in the functioning of such systems. (Hammarstrom et al. 2012, 2014; Yazdi and Ardekani 2012; Antfolk et al. 2014; Devendran et al. 2014; Ohlin et al. 2015; Cheung et al. 2014; Huang et al. 2014; Patel et al. 2014; Destgeer et al. 2016; Rogers and Neild 2011; Tang and Hu 2015; Leibacher et al. 2015; Agrawal et al. 2013, 2015).

The ability to use ultrasonic fields for manipulation of particles and fluids has a long history which can date back to many eminent scientists including Chladni (1787), Faraday (1831), Kundt and Lehmann (1874), Rayleigh (1883), King (1934), Gorkov (1962). As early as 1787, the German physicist Chladni (1787) observed that randomly distributed sand particles on a vibrating metal plate could group along the nodal lines forming a wide variety of symmetrical patterns. The various patterns formed at different modes of resonance were called Chladni figures. Chladni also reported that fine particles would move in the opposite direction, to the antinodes, which was further studied by Faraday (1831), who found that it was due to air currents in the vicinity of the plate, i.e. acoustic streaming. The latter phenomenon was revisited by Van Gerner et al. (2010, 2011) who showed that it will always occur when the acceleration of the resonating plate is lower than gravity acceleration. Zhou et al. (2016) recently proposed an approach which is able to control the motion of multiple objects simultaneously and independently on a Chladni plate.

Recently, Vuillermet et al. (2016) demonstrated that it is possible to form two-dimensional inverse Chladni patterns on a vibrating circular plate in water at microscale, which extended an earlier work from Dorrestijn et al. (2007), who showed formation of one-dimensional (1D) Chladni patterns on a vibrating cantilever submerged in water, where microparticles and nanoparticles were found to move to the antinodes and nodes of the vibrating interface, respectively. Both works have depicted the two-dimensional streaming field in the near-field and emphasized the effects of in-plane streaming flow on the collections of particles at vibrating antinodes or nodes. Practical manipulation on vibrating plates, however, is three-dimensional (3D) including out-of-plane and in-plane manipulation, and interestingly, in such systems, little work has been done on the impact of acoustic radiation forces, the main engine for particle and cell manipulation in other acoustofluidic manipulation devices. Unlike microparticle acoustophoresis in bulk and surface standing wave devices that have been well studied (Barnkob et al. 2012; Muller et al. 2012, 2013; Lei et al. 2014; Hahn et al. 2015; Nama et al. 2015; Oberti et al. 2009), the literature is lacking a quantitative analysis of microparticle acoustophoresis over vibrating plate systems.

## 2 Numerical method

*t*is time, \(\varvec{u}\) is the fluid velocity,

*p*is the pressure and

*μ*and

*μ*

_{ b }are, respectively, the dynamic and bulk viscosity coefficients of the fluid.

In this work, only the boundary-driven streaming field was solved because an evanescent wave field is established (see below) such that the overall streaming field is dominated by the boundary-driven streaming. Moreover, as the inner streaming vortices are confined only at the thin viscous boundary layer [thickness of \(\delta_{v} \approx 0.6\) µm at 1 MHz in water (Bruus 2012)], for numerical efficiency, we solved only the 3D outer streaming fields using Nyborg’s limiting velocity method (Nyborg 1958; Lee and Wang 1989) as those published previously (Lei et al. 2013, 2014, 2016). Although the inner streaming fields were not computed in this work, they can, of course, be known from the limiting velocity field.

## 3 Numerical model, results and discussion

*R*= 800 µm and thickness

*h*= 5.9 µm was firstly considered, which has a same size to the one used in Vuillermet et al.’s experiments (2016). Our model is slightly different to the device in Vuillermet et al.’s experiments. It can be seen from Fig. 1 that our model shows a vibrating clamped plate in a free space while the side boundaries of Vuillermet et al.’s device have sound reflections, which may result in acoustic pressure antinodes at the plate boundaries. More model parameters are found in Table 1. The model configuration is shown in Fig. 3a, where a cylindrical fluid-channel-only model was considered. Cartesian (\(x, y, z\)) and cylindrical (\(r, \theta , z\)) coordinates were used for the convenience of calculations. The finite element package COMSOL 5.2 (COMSOL Multiphysics 2015) was used to solve all equations. The modelled final particle (radius of 30 µm) positions driven by the main forces including acoustic radiation forces, streaming-induced drag forces and buoyancy forces at two vibrating modes are shown in Fig. 2a. It can be seen that the inverse Chladni patterns the microparticles form compare well with Vuillermet et al.’s (2016) experimental observations. In the following, we will show step by step why microparticles are gathered to the vibrating antinodes forming inverse Chladni patterns and the contributions of various driving forces on the acoustophoretic motion of microparticles at various sizes.

Model parameters

Parameter | Symbol | Value | Unit |
---|---|---|---|

Model domain | \(\pi R^{2} \times h\) | \(\pi \times 0.8^{2} \times 0.725\) | mm |

Density of plate | \(\rho\) | 2320 | kg m |

Plate Poisson’s ratio | \(\upsilon\) | 0.22 | |

Plate Young’s modulus | \(E\) | 169 | GPa |

Sound speed in plate | \(u\) | 55 | m s |

Particle density | \(\rho_{p}\) | 1050 | kg m |

Sound speed in particle | \(c_{p}\) | 1960 | m s |

Density of water | \(\rho_{f}\) | 1000 | kg m |

Sound speed in water | \(c_{f}\) | 1480 | m s |

It is noteworthy that we have previously applied a fluid-channel-only model to study the 3D transducer-plane streaming fields in bulk acoustofluidic manipulation devices (Lei 2015), where the excitation of transducer was approximated by a Gaussian distribution of boundary vibration. The fluid-channel-only model applied in this work has more merits because we can easily write down the displacement equation when the circular plate vibrates at a resonant mode (see Eq. (6) below), and thus, there is no need to make an approximation on the boundary vibrations as we did in the previous models (Lei et al. 2013, 2016).

### 3.1 Resonant frequencies

The modelled resonant frequencies (Hz) of first eight modes for various loads

Modes | No load | Air | Water |
---|---|---|---|

(0, 1) | 37,772 | 37,270 | – |

(1, 1) | 78,447 | 77,562 | 16,320 |

(2, 1) | 128,390 | 127,330 | 33,660 |

(0, 2) | 146,330 | 145,400 | 35,122 |

(3, 1) | 187,340 | 186,100 | 55,884 |

(1, 2) | 223,070 | 221,900 | 65,745 |

(4, 1) | 255,100 | 253,660 | 83,448 |

(2, 2) | 309,350 | 307,930 | 102,580 |

The computations were performed on a Lenovo Y50 running Windows 8 (64-bit) equipped with 16 GB RAM and Intel(R) Core(TM) i7-4710HQ processor of clock frequency 2.5 GHz. The mesh constitution was chosen based on the method described in a previous work (Lei et al. 2013), which chooses the mesh size to obtain steady solutions, i.e. ensuring further refining of mesh does not change the solution significantly. This model resulted in 131,521 mesh elements, a peak RAM usage of 4.96 GB (at the acoustic step), and a running time of about 4 h for solving the steps described between Sects. 3.2 and 3.6 below.

### 3.2 First-order acoustic fields

*Pressure Acoustics, Frequency Domain*’ interface, which solves the harmonic, linearized acoustic problem, taking the form,

*ω*is the angular frequency and

*c*is the speed of sound in the fluid. The acoustic fields in the model regime were created by a harmonic vibration of the bottom edge (i.e. the plate) coupled with radiation boundary conditions on all other edges. For comparison, we also tried adding perfect matching layers around the cylindrical domain to absorb all outgoing waves and found that the differences on all the modelled quantities between these two methods are within 3%. To give a clear presentation of results, we show here the results modelled form radiation boundary conditions.

*m*and \(\alpha_{mn}\) is the

*n*th zero of \(J_{m} \left( \cdot \right)\). The results presented in this paper were obtained at a vibration amplitude of 0.4 µm unless otherwise stated. The vibration amplitude has a limited effect on the shape of microparticle trajectories as both the acoustic radiation force and streaming-induced drag force scale with the square of the vibration amplitude (more discussions can be found below).

*λ*is the acoustic wavelength,

*f*

_{ r }is the resonant frequency and

*u*

_{ l }is the speed of sound in the liquid.

### 3.3 Acoustic radiation forces

*λ*in the limit \(r_{0} /\lambda \ll 1\) (where

*r*

_{0}is the radius of the particle) in an inviscid fluid in an arbitrary sound field. (Gorkov 1962) When a particle moves close to the vibrating plate, the acoustic radiation forces may oscillate weakly with a decrease in distance to the plate due to the multiple-scattering interaction and wall interference, while the force magnitudes will not be significantly affected (Wang and Dual 2012).

### 3.4 Acoustic streaming fields

*u*

_{ L }and

*v*

_{ L }are the

*x*- and

*y*-components of the limiting velocity field,

*u*

_{1},

*v*

_{1}and

*w*

_{1}are the

*x*-,

*y*- and

*z*-components of the acoustic velocity vector \(\varvec{u}_{{\mathbf{1}}}\), \(\text{Re} \left\{ \cdot \right\}\) denotes the real part of a complex value and \(*\) is the complex conjugate.

*Creeping Flow*’ interface was used to model the acoustic streaming field, which solves

The limiting velocity field and the 3D acoustic streaming fields are shown in Fig. 5. It can be seen that, similar to the distribution of in-plane acoustic radiation forces, the limiting velocities (i.e. the in-plane acoustic streaming velocity field) converge at the acoustic pressure antinodes from all directions leading to acoustic streaming vortices on out-of-planes perpendicular to the vibrating interface as those plotted in Fig. 5b, c, where, in order to give a clear demonstration of the 3D acoustic streaming fields, only the acoustic streaming vortices above one acoustic pressure antinode are plotted.

### 3.5 Acoustic streaming-induced drag forces

*χ*or

*γ*, respectively, which can be expressed as (Happel 1965)

*A*= 9/16,

*B*= 1/8,

*C*= 45/256 and

*D*= 1/16.

*w*= 0.4 µm, the maximum out-of-plane streaming-induced drag force is larger than the buoyancy force on a particle with a radius of 30 µm. With an increase in vibration amplitude, we can expect even larger acoustic streaming-induced drag forces while the buoyancy forces remain the same. Therefore, it might be reasonable to say that introducing only the streaming effects is not enough to explain the sedimentation of microparticles, especially for those with

*r*

_{0}< 30 µm, where the differences between the out-of-plane streaming-induced drag forces and the buoyancy forces are even larger, as plotted in Fig. 6d, because the former and the latter scale with the particle radius and particle volume, respectively.

### 3.6 Microparticle trajectories

*m*

_{ p }is the particle mass, \(\varvec{F}_{\varvec{B}}\) is the buoyancy, \(\varvec{F}_{\varvec{G}}\) is the gravity and

*g*is the gravity acceleration. In this work, it is assumed that all the forces, including acoustic radiation, streaming-induced drag and buoyancy forces, act on the centre of spherical particles (otherwise, integration of forces over the particle surface would be needed when the particles are close to the boundaries). It is noteworthy that, in addition to these main driving forces, a particle–particle interaction force was used in this model. The particle–particle interaction force can be expressed as

*k*

_{ s }is the spring constant, \(\varvec{r}_{\varvec{i}}\) is the position vector of the

*i*th particle, and

*r*

_{ e }is the equilibrium position between particles. In this model,

*k*

_{ s }= 2.5 × 10

^{−4}N/m for polystyrene beads (Jensenius and Zocchi 1997) and

*r*

_{ e }was set as 2

*r*

_{0}to avoid all particles being concentrated to a single point.

*Particle Tracing for Fluid Flow*’ interface was used to solve Eq. (12) to model the particle trajectories. The shape of the trajectories is independent of the pressure amplitude since both the acoustic radiation forces and steaming-induced drag forces scale with the square of pressure; results are presented here for an excitation amplitude of

*w*= 0.4 µm. An array of tracer particles (given the properties of polystyrene beads of radius 30 µm) are seeded at \(t = 0\). Acoustic radiation forces, streaming-induced drag forces and buoyancy forces act on the particles, resulting in the motion shown in Fig. 7. It can be seen that, in the considered model regime, particles with a radius of 30 µm first move towards the vibrating interface driven by the predominant out-of-plane forces and are then carried to their closest acoustic pressure antinodes by in-plane forces, resulting in

*spider*-

*like*trajectories and inverse Chladni patterns on the vibrating interface within seconds. Generally, particles closer to the vibrating interface take less time to settle for stronger driving forces. Particles with smaller sizes take longer to locate at the acoustic pressure antinodes for smaller driving forces and will follow the out-of-plane streaming vortices leading to acoustic streaming-dominated trajectories close to those shown in Fig. 5b, c while

*r*

_{0}< 6.9 µm (see explanations below and videos in the Supplemental material).

*Out-of-plane acoustophoresis.*A single particle out-of-plane acoustophoresis is directly acted upon by the acoustic radiation force, the buoyancy force and the acoustic streaming-induced drag force. The equation of motion for a spherical particle of out-of-plane velocity \(\varvec{v}^{{\varvec{out}}}\) above an acoustic pressure antinode is then

*w*= 0.4 µm, the threshold particle size \(r_{0}^{out} \approx 6.9\) µm. Considering the wall-effect-correction for the streaming-induced drag forces, \(r_{0}^{out} \approx 9.1\) µm. This threshold out-of-plane particle size may slightly vary with the vibration amplitude as \(\varvec{F}_{\varvec{B}} + \varvec{F}_{\varvec{G}}\) are independent of the vibration amplitude, while \(\varvec{F}_{\varvec{d}}^{{\varvec{out}}}\) and \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{out}}}\) scale with the square of the vibration amplitude. As shown in Fig. 4c, the buoyancy force is approximately 1/20 of \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{out}}}\) at \(w = 0.4\) µm on the vibrating interface. With an increase in vibration amplitude, the contribution of buoyancy force will be even smaller on the microparticle acoustophoresis in the near-field. To calculate the limit value of \(r_{0}^{out}\), we can set

*In-plane microparticle acoustophoresis.*For the in-plane microparticle acoustophoresis, it is acted upon by the acoustic radiation force and the streaming-induced drag force. Similar to the analyses above, the equation of motion for a spherical particle of in-plane velocity \(\varvec{v}^{{\varvec{in}}}\) is then

As shown in Figs. 4b and 6b, both the in-plane acoustic radiation force, \(\varvec{F}_{{\varvec{ac}}}^{{\varvec{in}}}\), and the streaming-induced drag force, \(\varvec{F}_{\varvec{d}}^{{\varvec{in}}}\), move microparticles to the acoustic pressure antinodes (see also the inset in Fig. 8b). To evaluate the contributions of these two forces on the in-plane microparticle acoustophoresis, we compare their average values over the plate interface because considering the maximum force only may not be accurate. Since both of these in-plane forces point to the acoustic pressure antinodes, they jointly contribute to the focusing of microparticles to the acoustic pressure antinodes provided that the particle sizes are big enough to avoid being driven away from the vibrating interface by out-of-plane acoustic streaming vortices (as discussed in the previous step), which could provide evidence for the much larger particle velocities measured in experiments when compared with the predicted streaming velocities as shown in Vuillermet et al. (2016) work.

## 4 Effects of key parameters on microparticle acoustophoresis

Having demonstrated the acoustophoresis of microparticles at various sizes for a particular plate (thickness of 5.9 µm and radius of 0.8 mm), in this section, we investigate the effects of many key parameters, including the plate radius and thickness and the fluid viscosity, on the performance of microparticle acoustophoresis in order to facilitate device design for a wide range of applications.

*Effects of fluid viscosity.*On the one hand, it can be seen from Eq. (8) that the magnitudes of limiting velocities (i.e. the strength of the outer streaming velocities) are independent of the fluid viscosity even though viscosity is the initial cause of acoustic streaming flows. Thus, with a change in fluid viscosity, the streaming-induced drag force, \(\varvec{F}_{\varvec{d}}\), scales linearly with \(\mu\), while \(\varvec{F}_{{\varvec{ac}}}\) will remain the same. From Eq. (21), the following relationships are established,

Therefore, to eliminate the ‘side effect’ of streaming flows on the microparticle manipulation, and we can conclude that lowering the fluid viscosity is a viable way to augment the weight of acoustic radiation force on microparticle acoustophoresis.

*Effects of plate thickness and radius.*To investigate the effects of plate thickness (\(h\)) and radius (\(R\)) on the microparticle acoustophoresis, we considered a series of

*h*and

*R*ranging from 2 to 14 µm and 0.3 to 1.4 mm, respectively. When one parameter was studied, the other parameter was kept the same. For each case, following the whole numerical procedure described in the sections above, we calculated the threshold in-plane and out-of-plane particle sizes, which are shown in Fig. 9. It can be seen that these two threshold particle sizes have similar variation tendencies: they grow with the increase in

*R*and fall with the rise of

*h*.

*Compare with basic theory.*Turning to the theoretical aspect, as seen from Eq. (21), to determine how these two threshold particle sizes change with the many key parameters, we only need to figure out how the force ratio on the right-hand side varies with these parameters. If we define \(\varvec{v}^{{\varvec{rad}}}\) as the contribution of acoustic radiation force on the particle velocity, considering Eqs. (15) and (19), we have

*h*, the angular frequency for an unloaded case for each \(\left( {m, n} \right)\) mode follows (Leissa 1993)

*ρ*is the plate density and

*υ*is the plate Poisson’s ratio. Considering the surrounding water, for a given \(\left( {m, n} \right)\) mode, the angular frequency is reduced to

*r*

_{0}

^{ in }using Eq. (27) and those obtained from our model for various \(R\) and \(h\) are shown in Fig. 10. It can be seen that the modelled \(r_{0}^{in}\) compare reasonably well with the calculated values under the 1D standing wave approximation. The differences between the calculated values and those modelled may be attributed to the reason that, compared to an approximated 1D standing wave, the acoustic field in the near-field is a more complex pattern. However, due to the complexity of the problem, the good comparisons between our model and the calculated values indicate that the approximated 1D standing wave may have captured the main features of (4, 1) mode and our model can be applied to study the basic physics of microparticle acoustophoresis on vibrating plate systems for even more complex vibrating modes.

## 5 Mode switching

For this specific model, both the in-plane acoustic radiation force and streaming-induced drag force diverge from the vibrating nodes and converge at the vibrating antinodes, so when switching from one mode (e.g. mode 1 in Fig. 11) to the other orthogonal mode (e.g. mode 2 in Fig. 11) a particle tends to move from the vibrating antinode of the former to its closest antinode of the latter either clockwise or anticlockwise depending on the initial position of the particle (assuming the initial position of the particle slightly shifts from the vibrating antinode). The potential underlying mechanism for the circular manipulation of a single particle is schematically shown in Fig. 11. It can be seen that, for each mode switching, the particle can move by an angle, \(\theta = \pi /2m\), while its distance to the centre of the circular membrane will remain the same. To complete a full circle of manipulation, 4 \(m\) times of mode switching are required. This method is different from the mode switching proposed by Glynne-Jones et al. (2010) who showed that beads can be brought to any arbitrary point between the half and quarter-wave nodes when rapidly switching back and forth between half and quarter wavelength frequencies in bulk acoustofluidic devices.

## 6 Conclusions

We have investigated the 3D acoustophoretic motion of microparticles due to acoustic radiation, acoustic streaming, gravity and buoyancy over a clamped vibrating circular plate in contact with water. The underlying physics of microparticle acoustophoresis over vibrating plates has been studied in detail. Previous predominant analyses have emphasized the in-plane acoustic streaming flows on the formation of inverse Chladni patterns, which, according to this study, may not be complete. For in-plane microparticle acoustophoresis, both the in-plane acoustic radiation forces and the in-plane streaming-induced drag forces were shown to drive microparticles to their closest vibrating antinodes. For out-of-plane microparticle acoustophoresis above vibrating antinodes, in addition to the buoyancy forces, one has to consider the acoustic radiation forces in the near-field, which prevent the out-of-plane streaming vortices from dragging microparticles away from the vibrating interface.

Based on the high efficiency of this numerical model, the threshold in-plane and out-of-plane particle sizes balanced from the acoustic radiation and streaming-induced drag force under all vibrating modes can be readily obtained. An important next step is to achieve a direct experimental verification of numerical modelling. Given a successful experimental verification, this 3D model could be extended to include the thermoviscous effects (Muller and Bruus 2014) to obtain more accurate results, but it would be very computationally expensive. According to a study by Rednikov and Sadhal (2011), the thermoviscous effects can increase the streaming velocities by 18% for water at 20 °C which, thus, will shift the threshold particle sizes.

The good comparisons between our modelling and experiments and basic theories indicate that our numerical model could be used together with high-precision experiments as a better research tool to study the many yet unsolved problems. For example, modelling suggests that mode switching between two adjacent frequencies may be used for circumferential manipulation of a single particle or a pair of particles, which might provide routes for the study of particle–particle and particle–wall interactions in acoustofluidics.

While we have shown here 3D particle size-dependent acoustophoresis over an ultrathin circular plate in water, we believe that this strategy could be applied to analyse 3D acoustophoretic motion of microparticles in other vibrating plate systems regardless of fluid medium and thickness, shape and material of plates. One particular application would be acoustophoretic handling of sub-micrometre particles, such as small cells, bacteria and viruses, whose movements are usually dominated by acoustic streaming flows. From the modelled results and the general scaling law given in Eq. (27), we can conclude that increasing plate thickness, decreasing the plate diameter and lowering the viscosity of the liquid are probably the most viable way to conduct such manipulation.

The above-mentioned applications demonstrate that our numerical model is timely and has a huge potential on studies of basic physical aspects of microparticle acoustophoresis in vibrating plate systems and the design of lab-on-a-chip devices.

## Notes

### Acknowledgements

This work is supported by the EPSRC/University of Southampton Doctoral Prize Fellowship (EP/N509747/1). The authors gratefully acknowledge helpful discussions with Prof M. Hill and Dr P. Glynne-Jones. Models used to generate the simulation data supporting this study are openly available from the University of Southampton repository at http://dx.doi.org/10.5258/SOTON/404258.

## Supplementary material

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