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Japanese Journal of Applied Physics 52 (2013) 100201
http://dx.doi.org/10.7567/JJAP.52.100201

Behavior of Ultrasonically Levitated Object above Reflector HoleHiroki Tanaka1 , Yuji Wada2 , Yosuke Mizuno1 , and Kentaro Nakamura1
1

Precision and Intelligence Laboratory, Tokyo Institute of Technology, Yokohama 226-8503, Japan
Faculty of Science and Technology, Seikei University, Musashino, Tokyo 180-8633, Japan
E-mail: htanaka@sonic.pi.titech.ac.jp
2

Received April 30, 2013; accepted July 25, 2013; published online September 9, 2013
Using an axis-symmetric model comprising a flat reflector with a hole and a piston vibrator, we simulate the radiation force acting on a small object
levitated near the hole, and predict that the position of the object shifts from the center line of the hole. We also experimentally investigate thebehavior of a levitated polystyrene sphere approaching the hole. As predicted, the sphere avoids approaching the hole with some distance,
depending on the reflector/vibrator interval and the hole diameter. We believe that this effect is useful in discriminating the droplets to bedispensed from those that should not be dispensed. # 2013 The Japan Society of Applied Physics

In recent years, considerable attention has been paid to
the noncontact transport of small objects and droplets
as an important technology in pharmaceutical industries, and new material science/engineering. Compared withvarious methods exploiting air pressure1) and magnetic/electric ﬁelds,2,3) ultrasonic-levitation-based methods havea number of advantages such as cost-eﬃciency, noiselessoperation, and applicability to nonelectric/magnetic objects.4,5) We have been aiming at implementing all theprocedures of the noncontact transport of objects in airusing ultrasonic waves, which consist of injection, linear,and circular transport, direction switching, physical/chemical analysis, and ejection. When the objects to be transported
are droplets, mixing and dispensing are also included in
the procedures. As well as the linear transport,6) circulartransport and direction switching,7) ejection, and droplet
mixing, droplet dispensing has also been demonstrated sofar.8) Other research groups have also reported the properties
of ultrasonically levitated droplets,9,10) but they aimed atchemical analysis rather than noncontact transportation.
In our previous dispensing experiment,8) we showed thatby employing a half-cylindrical (not ﬂat) reﬂector and a stephorn driven with a bolt-clamped Langevin transducer (BLT),a single droplet, levitated above a hole of the reﬂector by
adjusting the reﬂector/horn interval, can fall through thehole into a well with no inﬂuence of the residual acoustic
ﬁeld when the vibration is stopped. To achieve our next
objective, i.e., the dispensing of a droplet that has been
ultrasonically transported for some distance, the acoustic
ﬁeld distribution above the hole needs to be analyzed indetail. Without the hole, the cross section of the experimental setup can be directly used as a model for the ﬁniteelement analysis (FEA). However, with the hole, threedimensional analysis is required, which is practicallydiﬃcult because of its large amount of calculation.
In this study, by analyzing the acoustic ﬁeld distribution
above a hole using an axis-symmetric model composed of a
ﬂat reﬂector and a piston vibrator, we predict that a levitatedobject avoids approaching the center line of the holeaccording to the interval between the reﬂector and thevibrator. Next, to experimentally observe this phenomenon,
we investigate the behavior of a levitated object (polystyrenesphere) when it approaches the hole. The object is levitatedbelow the vibrator, the position of which is ﬁxed, and theposition of the reﬂector is changed. The object predictably
avoids approaching the hole with some distance, whichdepends on the interval between the reﬂector and thevibrator as well as the hole diameter. This phenomenon ispotentially applicable for discriminating the droplets to bedispensed from those that should not be dispensed.
In the noncontact ultrasonic levitation, objects are trappedat the nodal points of the pressure ﬁeld generated in air
between a vibrator and a reﬂector. The objects are levitatedby the acoustic radiation force Fr , which is calculated as11)
Fr ¼ V fDrhek i  ð1  Þrhep ig;

ð1Þ

where V is the volume of the levitated object,  is the ratioof the compressibility of the object to that of the air, and D isgiven byD¼

3ðs  0 Þ;
2s þ 0ð2Þ

where s and 0 represent the densities of the object and theair, respectively. hep i and hek i are the temporally averaged
potential and kinetic energies, respectively, which are
expressed as11)
jpj2 =2;
ð3Þ
20 c2
 jr’j2 =2;
ð4Þ
hek i ¼ 0
2
where p is the acoustic pressure and ’ is the velocity
potential given by
hep i ¼p ¼ 0@’:@tð5Þ

Thus, the distribution of the acoustic radiation force can becalculated using the temporally averaged acoustic pressure
distribution. Although the theory above is derived under
the assumption of k  a  1 (k, wave number; a, objectradius),11) we used this theory in the simulation below
for an object with k  a ¼ 0:5, expecting moderate analyticalerrors.
We simulated the acoustic ﬁeld distribution above a hole
employing an axis-symmetric model composed of a 1-mmthick ﬂat reﬂector, an end surface of a piston vibrator with
30 mm diameter, and air media between the reﬂector and thevibrator, as shown in Fig. 1. A sinusoidal vibration at 27.4
kHz (corresponding to the acoustic wavelength of 12.4 mm
in air; note that this value is valid only for free space andthat the resonance wavelength was 12.9 mm in this model
without a hole) with the zero-to-peak velocity of 0.75 m/s

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Jpn. J. Appl. Phys. 52 (2013) 100201

H. Tanaka et al.

Fig. 3. (Color online) Calculated distribution of the acoustic radiation

force.
Fig. 1. (Color online) Axis-symmetric model used in the simulation.

Fig. 2. (Color online) Acoustic pressure distribution calculated by FEA.

was applied to the end surface of the vibrator. An absorbing
boundary condition was used to terminate the air media
in the FEA. Figure 2 shows the calculated acoustic ﬁeld
distribution when the hole diameter was set to 7.0 mm andthe reﬂector/vibrator interval was set to 13 mm, where thered and blue regions indicate the areas with high and low
temporally averaged acoustic pressures (absolute values),
respectively. Then, the distribution of the acoustic radiation
force was calculated using Eqs. (1)–(5), as shown as vectors
in Fig. 3. In this calculation, each parameter was assumed tobe:12) V , 4:2  109 m3 , , 3:8  105 , s , 1:1  103 kg/m3 ,0 , 1.2 kg/m3 , and c, 340 m/s. The vectors converged on thetwo nodal lines where an object is levitated. The vectors
are almost parallel to the symmetrical axis, but with small
components perpendicular to it. The two areas in air indicate
the regions where the force acts in the directions away fromand toward the symmetrical axis. Therefore, the object istrapped at the boundaries between the two regions in thenodal lines, A and B; here, we deﬁned the distance fromthe symmetrical axis to the trapping point as the avoiding
distance. Figure 4 shows the avoiding distance dependence
on the hole diameter for the reﬂector/vibrator intervals of
13, 14, and 15 mm (in the lower nodal line; B in Fig. 3).
With a 13 mm interval, the avoiding distance was calculated
to be larger than the hole radius, when the diameter was>3 mm. With 14 and 15 mm intervals, the distance wasexpected to be larger than the hole radius when the diameter
was >12 mm. These simulation results indicate that thetrajectories of the levitated objects can be moderately controlled (above the hole for dispensing, or not) by adjusting
the reﬂector/vibrator interval and the hole diameter.

Fig. 4. (Color online) Calculated dependence of the avoiding distance on
the hole diameter. The interval between the reﬂector and the vibrator was13, 14, and 15 mm.

Next, we experimentally observed this phenomenon using
the experimental setup depicted in Fig. 5(a). A polystyrene
sphere with 2 mm diameter and 3 g weight was levitatedat the lower nodal line between an aluminum reﬂector andthe head of a step horn, and was moved close to a hole of
the reﬂector at the speed of 20 mm/s. The driving frequencyof the BLT was 27.4 kHz, and the current and voltage wereadjusted so that the end of the horn vibrates at the zeroto-peak velocity of 0.75 m/s. As shown in Fig. 5(b), when
the reﬂector/vibrator interval was 13 mm and the holediameter was 7 mm, the sphere avoided approaching thehole. Figure 6 shows the measured avoiding distancedependence on the hole diameter for the reﬂector/vibrator
intervals of 13, 14, and 15 mm. Irrespective of the holediameter, the avoiding distance was larger than the holeradius with a 13 mm interval, but the distance was almost
equal to or shorter than the hole radius with 14 and 15 mm
intervals. When the hole diameter was larger than 10 mm,
with 14 and 15 mm intervals, the sphere was not levitatedowing to the weak acoustic radiation force (note that when
the interval was 12 or 16 mm, the sphere was not stably
levitated even without a hole). These measurement results
basically agree with the trends that we predicted through theFEA simulation. The discrepancy in the avoiding distance
between experiment and simulation seems to be caused by
the acoustic ﬁeld distribution disturbed by the sphere itself13)
along with the value of k  a, which is not suﬃciently small
compared with 1.11)
In conclusion, ﬁrst, we analyzed the acoustic ﬁeld distribution above a hole using an axis-symmetric model

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H. Tanaka et al.

(a)

(b)
Fig. 5. (Color online) (a) Schematic setup and procedure of the experiment. (b) Photographs of the levitated polystyrene sphere avoiding approaching thehole.

should not be dispensed. The inﬂuence of the acoustic ﬁeld
generated with this technique on the phenomena such as
deformation, evaporation, and cavitation of the droplets also
needs to be studied further.
Acknowledgements The authors are grateful to Mr. Ryohei Nakamura andMr. Neisei Hayashi, Tokyo Institute of Technology, Japan, for their helpful
discussions and comments. This work was partially supported by a Grant-in-Aidfor Scientiﬁc Research (Challenging Exploratory Research, 23656163) from theJapan Society for the Promotion of Science (JSPS).

1) S. Konishi, M. Harada, Y. Ogami, Y. Daiho, Y. Mita, and H. Fujita: ETFA

6th Int. Conf., 1997, p. 232.

Fig. 6. (Color online) Measured dependence of the avoiding distance on2) S. C. Mukhopadhyay, J. Donaldson, G. Sengupta, S. Yamada, C.

the hole diameter. The interval between the reﬂector and the vibrator was13, 14, and 15 mm.

3) F. Hellman, E. M. Gyorgy, D. W. Johnson, H. M. O’Bryan, and R. C.

Chakraborty, and D. Kacprzak: IEEE Trans. Magn. 39 (2003) 3220.
Sherwood: J. Appl. Phys. 63 (1988) 447.

composed of a ﬂat reﬂector and a BLT-based vibrator. Then,
we predicted that a levitated object avoids approaching
the hole depending on the reﬂector/vibrator interval. We
also experimentally investigated the behavior of a levitatedpolystyrene sphere when it was moved close to the hole. Thesphere avoided approaching the hole with a certain avoiding
distance, which depends on the reﬂector/vibrator interval
and the hole diameter. We believe that this phenomenon canbe exploited in developing noncontact ultrasonic dispensing
systems in the near future by adding a new function of
discriminating the droplets to be dispensed from those that

4) R. R. Whymark: Ultrasonics 13 (1975) 251.
5) E. G. Lierke: Acustica 82 (1996) 220.
6) D. Koyama and K. Nakamura: IEEE Trans. Ultrason. Ferroelectr. Freq.

Control 57 (2010) 1152.
7) D. Koyama and K. Nakamura: IEEE Trans. Ultrason. Ferroelectr. Freq.

Control 57 (2010) 1434.
8) H. Tanaka, Y. Wada, Y. Mizuno, and K. Nakamura: presented at 4th Int.

Congr. Ultrasonics, 2013, paper P0243.
9) A. L. Yarin, G. Brenn, and D. Rensink: Int. J. Heat Fluid Flow 23 (2002)

471.
10) J. Sloth, S. Kiil, A. D. Jensen, S. K. Andersen, K. Jrgensen, H. Schiﬀter,and G. Lee: Chem. Eng. Sci. 61 (2006) 2701.

11) L. P. Gor’kov: Sov. Phys. Dokl. 6 (1962) 773.
12) National Astronomical Observatory of Japan: Chronological Science

Tables (Maruzen, Tokyo, 1993) [in Japanese].
13) Y. Kagawa and T. Murai: Proc. IEEE Ultrasonics Symp., 1985, p. 572.

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