A horn shaped Langevin ultrasonic transducer used in a single axis levitator was characterized to better understand the role of the acoustic profile in establishing stable traps. The method of characterization included acoustic beam profiling performed by raster scanning an ultrasonic microphone as well as finite element analysis of the horn and its interface with the surrounding air volume. The results of the model are in good agreement with measurements and demonstrate the validity of the approach for both near and far field analyses. Our results show that this style of transducer produces a strong acoustic beam with a total divergence angle of 10°, a near-field point close to the transducer surface and a virtual sound source. These are desirable characteristics for a sound source used for acoustic trapping experiments.

Acoustic levitation and trapping techniques are useful for non-contact handling,1 positioning,2 suspending of micro-particles,3 and cell trapping.4 Recent advances using holographic imaging of sound waves5 and phased arrays6–8 demonstrate the outstanding potential to use sound to control, by holding or imparting momentum to, objects a few millimeters in size. The ability to manipulate objects with shaped sound fields opens a variety of opportunities to use sound as a technique for contact free sample delivery and manipulation. Contact-free methods of handling samples have many advantages including (i) reducing the potential for contamination, (ii) avoiding unwanted chemical processes and nucleation associated with contacting container walls, and (iii) eliminating laser distortion and x-ray background scattering in experiments where beams are used to examine samples.9–12 Acoustic levitation methods are particularly appealing because they require no special physical characteristics of the sample, as is the case for optical13 and electrostatic14 trapping, and can therefore be extended to a variety of samples.

A single-axis acoustic levitator (SAL) is perhaps the simplest acoustic device used to suspend spherical objects several millimetres in diameter in a gaseous atmosphere (air).9–11 A SAL typically consists of either an ultrasonic emitter-reflector or emitter-emitter geometry to create an acoustic standing wave. Solid or liquid objects experience a restorative force that constrains them near the nodal points of the standing wave. However, levitated objects exhibit spontaneous oscillations15,16 in both the transverse and longitudinal directions of the resonator. Our research shows [unpublished results] that the amplitude of these oscillations increases as the size of the trapped object decreases and therefore set a lower limit on the size of objects that can be trapped and consequently limit the range of applications for acoustic trapping, e.g., beam based experiments that require or benefit from using small sample volumes and good sample stability.

In the Gor'kov theory,17 the restorative force is given as F=U where U is the acoustic radiation potential. It is clear from this equation that the shape and gradient of the acoustic field within the resonator, determined by the resonator mode, are critical in creating a stable acoustic trap. Optimization of the resonator mode then requires accurate simulations that are verified by measurements. In particular, an accurate characterization must show agreement for a single transducer/emitter in the near field, which is the geometry many SALs employ. We also note that constructive and destructive interference of counter propagating waves obscures the near field region in both of the above geometries. As we discuss below, the transducer style used in this work has a shorter near field region and emphasizes the need to characterize the far field as well. It is the purpose of this letter to develop a robust methodology for characterizing the sound field of a single ultrasonic transducer used in a SAL in both the near and far fields. We show good agreement between our model and experiment for the near and far field regions using finite element analysis with COMSOL Multiphysics platform and discuss the critical role the vibrational mode of the transducer plays in determining the sound field. Our work provides a set of tools for modeling and optimization of more advanced transducer designs as well as the evaluation of different transducer materials. We also comment on the advantages of using a horn shaped Langevin ultrasonic transducer compared to piston style sound sources.

While several papers provide detailed simulations of the resonator mode in a SAL,15,18–21 there are few supporting experimental measurements.21,22 A simple and robust way to measure the acoustic pressure distribution is by scanning the sound field with an ultrasonic microphone.22,23 The amplitude of the sound field is then recorded as a function of position with respect to the emitter. A variety of methods have been employed to calculate the sound field within an ultrasonic resonator including finite element analysis and matrix methods using Rayleigh integral.24,25 Matrix methods, while simple, intuitive, and rather elegant, fail at capturing the essential role the vibrational mode of the emitter plays in determining the shape of the sound field. Below, we discuss the role of the vibrational mode as it pertains to the horn shaped transducer used in this work.

The SAL used in our experiments was purchased from Materials Development, Inc. (MDI). The MDI SAL™ uses a unique two-emitter design with horn shaped Langevin transducers constructed from 6063-T6 series aluminum alloy, Fig. 1. In this study, a single transducer is reported, though we have found that both transducers perform similarly. The SAL operates at a resonant frequency of 22.3 kHz and the radiating surface is 67 mm in diameter. The horn shape is a unique design that can produce sound pressure levels in excess of 160 dB and is strong enough to hold a variety of sample densities from water droplets massing 14 mg to lead shots, ∼140 mg.

FIG. 1.

(a) Experimental setup for measuring the acoustic pressure field. Each block represents a translation stage in the (x,y,z) directions; y is in/out of the page. (b) Axis symmetric model geometry for calculating the acoustic pressure field.

FIG. 1.

(a) Experimental setup for measuring the acoustic pressure field. Each block represents a translation stage in the (x,y,z) directions; y is in/out of the page. (b) Axis symmetric model geometry for calculating the acoustic pressure field.

Close modal

The acoustic field was characterized using a high precision microphone (Brüel & Kjær, type 4138, Naerum, Denmark) mounted on an XYZ motorized stage assembly, Fig. 1. Here, we define the XY-plane as the plane transverse to the propagation, Z, direction. The microphone was raster scanned in 1 mm increments over a total distance of 100 mm in the XY-plane and then moved in Z by 7.5 mm and repeated. A single coordinate was chosen to measure before and after each XY plane to monitor any long-term drift in the acoustic intensity. Sound absorbing foam was placed around the experimental area to prevent secondary reflections from reaching the microphone.

Our numerical model was developed in COMSOL Multiphysics to describe the oscillation mode of the transducer and the relationship between the transducer and the surrounding air volume. The structural deformation of the horn was determined with the following approximations: (1) the piezoelectric wafers were directly bonded to the aluminum horn and (2) the multiple wafers in the piezo stack were simplified as a single wafer. By modeling the interface as a continuum, the horn motion can be strongly coupled to the piezoelectric strain. The coupling relationship between the piezoelectric material strain and the applied voltage goes as E=V and ·D=ρv, where E is the electric field, D is the electric displacement, V is the applied voltage, and ρv is the electric charge concentration.26 To match the acoustic field amplitude in the model to that measured in the experiment, the voltage across the piezoelectric wafer was variably adjusted until the maximum measured and simulated pressure fields were in close agreement.

The air domain was modeled as isothermal and obeying the ideal gas law, PT=nRT/V, where V is the volume of the gas, n is moles, R is the universal gas constant, and T is the temperature. PT is the total pressure of the gas and represents the sum of the static pressure and the dynamic pressure. Noting that n is the ratio of the total mass of the gas (m) to the molar mass of the gas (M) and using the specific gas constant R*=R/M, the ideal gas law becomes

ρ=PTR*T,
(1)

where the density is ρ=m/V. Therefore, the gas density varies proportionately with the pressure.27 

The horn/air interface was modeled as a fluid/structure boundary so that the hydraulic action of the horn motion produced an accurate response from (or on) the air volume.28 This coupling also captures the structural response from the air pressing back on the horn face. Mathematically, this relationship can be described as

n·1ρpd=n·2ut2,
(2)
Phorn=pdn,
(3)

where u is the horn face position, n is the surface normal, pd is the dynamic pressure, and Phorn is the load in force per unit area experienced by the horn.

A frequency domain solver was used to solve the linearized system of equations. The frequency used for the study, 22.3 kHz, is the same as the experiment. The solver produced solutions for the horn deformation and stress, piezoelectric material stress and strain, and the acoustic pressure distribution.

Assuming that heat losses are minimal and energy is conserved from the piezo stack to the surface of the transducer, the oscillation amplitude goes as zM/EI, where M is the bending moment, E is Young's modulus, and I is the area moment of inertial. Assuming that the material is isotropic leaves E constant, while moving radially outward across the surface of the transducer results in an increase in M and a decrease in I proportional to the thinning of the horn material. Thus, the oscillation amplitude increases in proportion to M/I, see Fig. 2(b). The primary mode of oscillation is a buckle-type mode where the majority of the motion (60 μm) is nearest the outer annular region of the horn and the on-axis region moves ∼10 μm with opposite phase. Taking into account the area differences as well as the difference in oscillation amplitude, we estimate that over 90% of the total acoustic energy is radiated by the outer annular region of the horn. High-speed (100 kHz rate) video, not shown here, confirmed the results of the simulation; the outer annular region of the horn was observed to flex by over 50 μm.

FIG. 2.

(a) The numerical simulation of the acoustic pressure field generated by a single horn shaped transducer. The white dashed lines mark the phase front generated by the flexing region of the horn shaped transducer. Panel (b) shows the deformation of the horn-shaped transducer. The left and right sides separated by the dashed-dotted line are the extremes of the horn motion separated by one-half period. The thin black line is the profile of the relaxed transducer.

FIG. 2.

(a) The numerical simulation of the acoustic pressure field generated by a single horn shaped transducer. The white dashed lines mark the phase front generated by the flexing region of the horn shaped transducer. Panel (b) shows the deformation of the horn-shaped transducer. The left and right sides separated by the dashed-dotted line are the extremes of the horn motion separated by one-half period. The thin black line is the profile of the relaxed transducer.

Close modal

The scanned acoustic intensity is shown in Figs. 3(b), 3(e), 3(h), and 3(k) for the XY planes 1.5, 3, 6, and 10.5 cm away from the transducer surface. The corresponding simulations are shown in Figs. 3(c), 3(f), 3(i), and 3(l) with lineouts comparing the two sets shown in Figs. 3(a), 3(d), 3(g), and 3(j). From these measurements and simulations, we extracted the peak on-axis intensity as a function of Z, shown in Fig. 4(a). Using the maximum dynamic pressure, the difference in density, Eq. (1), between the simulation and experiment is less than 5%. We also extracted the divergence of the central and first annular regions shown in Fig. 4(b). Overall, the agreement between the simulations and experiment is good; the model accurately reproduces the shape of the acoustic profile in both the near and far fields as well as the intensity and divergence behaviors. Figure 3(c) shows that the largest disagreement still lies within the near field region. We expect that errors within the model such as not including material inhomogeneity, machining tolerances, and internal defects in the material fully account for the discrepancies shown here.

FIG. 3.

Acoustic intensity plots of planes with (top to bottom row) 1.5 cm, 3 cm, 6 cm, and 10.5 cm distance away from the transducer surface. Panels (a), (d), (g), and (j) are lineouts directly comparing the acoustic field between experimental measurements and simulation results. Lines are simulation and symbols are from measurement. Panels (b), (e), (h), and (k) are from experimental measurements and panels (c), (f), (i), and (l) are from numerical simulations.

FIG. 3.

Acoustic intensity plots of planes with (top to bottom row) 1.5 cm, 3 cm, 6 cm, and 10.5 cm distance away from the transducer surface. Panels (a), (d), (g), and (j) are lineouts directly comparing the acoustic field between experimental measurements and simulation results. Lines are simulation and symbols are from measurement. Panels (b), (e), (h), and (k) are from experimental measurements and panels (c), (f), (i), and (l) are from numerical simulations.

Close modal
FIG. 4.

Panel (a) shows the peak on-axis acoustic intensity as a function of distance from the surface of the transducer. The red line is from simulation and blue symbol from experiment. The near field point, or natural focus, is marked with a vertical dashed line. Panel (b) shows the full width at half maximum (FWHM) of the pressure field as a function of distance from the surface of the transducer for the central peak, red, and first annular ring, blue. Data symbols are from experiment and the solid line is from simulation. The dashed lines are linear fits to the experimental data and show that the virtual sources are 20 mm in back of the transducer.

FIG. 4.

Panel (a) shows the peak on-axis acoustic intensity as a function of distance from the surface of the transducer. The red line is from simulation and blue symbol from experiment. The near field point, or natural focus, is marked with a vertical dashed line. Panel (b) shows the full width at half maximum (FWHM) of the pressure field as a function of distance from the surface of the transducer for the central peak, red, and first annular ring, blue. Data symbols are from experiment and the solid line is from simulation. The dashed lines are linear fits to the experimental data and show that the virtual sources are 20 mm in back of the transducer.

Close modal

We briefly compare the results of the horn shaped transducer to those expected for a circular piston style emitter.29 Circular piston style transducers are well documented in the literature and obey simple analytical formulas, which we use for comparison to the horn shaped transducer used here. From Fig. 4(a), we determine that the natural focus, the point that roughly determines the transition from the near field to the far field, occurs 30 mm from the surface of the transducer. For the case of a circular piston style transducer, N = (D2 − λ2)/4λ, where N is the natural focus, D is the diameter of the transducer, and λ is the wavelength of the acoustic wave. For a piston style transducer with a 67 mm diameter, N = 69 mm, about twice that of the horn shaped transducer. We also measure the opening half angle to half intensity point of the pressure field to be 5°. Again, for a piston style transducer θhalf = arcsin(0.7*lambda/D) = 9.3°. We also note that Fig. 4(b) implies that the sound source lies 20 mm in back of the transducer face implying a virtual sound source.

Evidently, the horn shaped transducer has the net effect of moving the natural focus inward while pushing the sound source back to approximately Z = −20 mm, thus producing a less divergent acoustic beam in the far field. This is equivalent to using a curved emitting surface but is accomplished with a horn design that has larger amplitude oscillations at larger radial values across the surface of the horn and is nearly stationary near the acoustic axis. We also note that similar results were obtained by using a radiating plate transducer in Ref. 25 in an emitter-reflector resonator with both curved and flat reflectors. It was found that the radiating plate resulted in better axial (radial) confinement of the acoustic energy in the resonator. Here, we note that the oscillation amplitude of the horn is much smaller than λ; thus, the observed focusing arises from diffraction effects rather than the geometric distortion of the horn surface. From a Fresnel-Huygens perspective, a horn style transducer reduces the oscillation amplitude of the emitting elements near the central region of the acoustic axis leaving a dominant ring shaped group of emitters around the outer edge of the transducer. This is clearly shown with the white circular arcs in Fig. 2(a). The effect in the far field is a flatter phase front with the acoustic energy concentrated on axis. These are highly desirable characteristics for an acoustic source used in a SAL since it allows the resonator gap to be shorter while still trapping objects in the less structured far field region of the emitter. As a practical matter, it has been shown that the fatigue life of a flat front horn significantly exceeds that of a curved radiator in which large hoop stresses at the circumference cause the formation of radial cracks [unpublished research].

In conclusion, we have shown that including the vibrational mode of an ultrasonic sound source is a critical component to accurately model both near- and far-field behaviors. Our model includes both the mechanical properties of the sound source and its coupling to the surrounding air volume. For the case of a horn shaped transducer, we find that over 90% of the acoustic energy is produced by an outer annular region from a buckle-type oscillation mode. This produces a beam with collimation properties similar to a piston style transducer driven at twice the ultrasonic frequency and with a near field point similar to a piston style transducer driven at half the ultrasonic frequency. The agreement between the experiment and the model will be critical in formulating an understanding of spontaneous oscillations and improving trap stability in SALs.

This material is based upon work supported by Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-06CH11357. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.

1.
V.
Vandaele
,
P.
Lambert
, and
A.
Delchambre
,
Precis. Eng.
29
,
491
505
(
2005
).
2.
3.
H.
Hertz
,
J. Appl. Phys.
78
,
4845
4849
(
1995
).
4.
J.
Nilsson
,
M.
Evander
,
B.
Hammarström
, and
T.
Laurell
,
Anal. Chim. Acta
649
,
141
157
(
2009
).
5.
K.
Melde
,
A. G.
Mark
,
T.
Qiu
, and
P.
Fischer
,
Nature
537
,
518
522
(
2016
).
6.
A.
Marzo
,
S. A.
Seah
,
B. W.
Drinkwater
,
D. R.
Sahoo
,
B.
Long
, and
S.
Subramanian
,
Nat. Commun.
6
,
8661
(
2015
).
7.
C. E. M.
Demore
,
P. M.
Dahl
,
Z.
Yang
,
P.
Glynne-Jones
,
A.
Melzer
,
S.
Cochran
,
M. P.
MacDonald
, and
G. C.
Spalding
,
Phys. Rev. Lett.
112
,
174302
(
2014
).
8.
D.
Foresti
and
D.
Poulikakos
,
Phys. Rev. Lett.
112
,
024301
(
2014
).
9.
J.
Weber
,
C.
Rey
,
J.
Neuefeind
, and
C.
Benmore
,
Rev. Sci. Instrum.
80
,
083904
(
2009
).
10.
C. J.
Benmore
and
J. K. R.
Weber
,
Phys. Rev. X
1
,
011004
(
2011
).
11.
R. J. K.
Weber
,
C. J.
Benmore
,
S. K.
Tumber
,
A. N.
Tailor
,
C. A.
Rey
,
L. S.
Taylor
, and
S. R.
Byrn
,
Eur. Biophys. J.
41
,
397
403
(
2012
).
12.
S.
Tsujino
and
T.
Tomizaki
,
Sci. Rep.
6
,
25558
(
2016
).
13.
O. M.
Maragò
,
P. H.
Jones
,
P. G.
Gucciardi
,
G.
Volpe
, and
A. C.
Ferrari
,
Nat. Nanotechnol.
8
,
807
819
(
2013
).
14.
H. L.
Bethlem
,
G.
Berden
,
F. M.
Crompvoets
,
R. T.
Jongma
,
A. J.
Van Roij
, and
G.
Meijer
,
Nature
406
,
491
494
(
2000
).
15.
N.
Pérez
,
M. A. B.
Andrade
,
R.
Canetti
, and
J. C.
Adamowski
,
J. Appl. Phys.
116
,
184903
(
2014
).
16.
D.
Foresti
,
M.
Nabavi
, and
D.
Poulikakos
,
J. Fluid Mech.
709
,
581
592
(
2012
).
17.
L. P.
Gor'kov
,
Sov. Phys.-Dokl.
6
,
773
776
(
1962
).
18.
K.
Suthar
,
C. J.
Benmore
,
P.
Den Hartog
,
A.
Tamalonis
, and
R.
Weber
, in
IEEE International Ultrasonics Symposium
,
Chicago IL
,
2014
, pp.
467
470
.
19.
S.
Baer
,
M. A.
Andrade
,
C.
Esen
,
J. C.
Adamowski
,
G.
Schweiger
, and
A.
Ostendorf
,
Rev. Sci. Instrum.
82
,
105111
(
2011
).
20.
M. A.
Andrade
,
N.
Pérez
, and
J. C.
Adamowski
,
J. Acoust. Soc. Am.
136
,
1518
1529
(
2014
).
21.
T.
Kozuka
,
K.
Yasui
,
T.
Tuziuti
,
A.
Towata
, and
Y.
Iida
,
Jpn. J. Appl. Phys., Part 1
47
,
4336
(
2008
).
22.
A.
Stindt
,
M.
Andrade
,
M.
Albrecht
,
J.
Adamowski
,
U.
Panne
, and
J.
Riedel
,
Rev. Sci. Instrum.
85
,
015110
(
2014
).
23.
Y.
Abe
,
D.
Hyuga
,
S.
Yamada
, and
K.
Aoki
,
Ann. N. Y. Acad. Sci.
1077
,
49
62
(
2006
).
24.
M. A.
Andrade
,
N.
Perez
,
F.
Buiochi
, and
J. C.
Adamowski
,
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
58
,
1674
1683
(
2011
).
25.
M. A.
Andrade
,
F.
Buiochi
, and
J. C.
Adamowski
,
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
57
,
469
479
(
2010
).
26.
A. F.
Bower
,
Applied Mechanics of Solids
(
CRC Press
,
2009
).
27.
M. J.
Moran
and
H. N.
Shapiro
,
Fundamentals of Engineering Thermodynamics
, 4th ed. (
Wiley
,
2000
).
28.
D. T.
Blackstock
,
Fundamentals of Physical Acoustics
(
John Wiley & Sons
,
2000
).
29.
E. A.
Ginzel
,
Ultrasonic Inspection 2-Training for Nondestructive Testing
(
Prometheus Press, Canada
,
1985
).