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Results in Physics 18 (2020) 103166
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Results in Physics
journal homepage: www.elsevier.com/locate/rinp
Design and simulation of the new ultrasonic atomizer using silicon-based
with one step resonator
Yu-Lin Songa,b, , Lavanya Bandib
Department of Bioinformatics and Medical Engineering, Asia University, Taichung 413, Taiwan
Department of Computer Science and Information Engineering, Asia University, Taichung 413, Taiwan
Ultrasonic device
Computer-aided design
COMSOL multi-physics
Space claim
Surface wave
The design and simulation of a new generation Fourier amplifier were investigated in this study. The design and
characterization of 500 kHz micro fabricated silicon-based ultrasonic atomization is presented for the concept of
uniform and fine droplets. Ultrasonic atomizer nozzle is composed of a silicon-based resonator consisting of the
Fourier-Horn., drive section and one step part. The atomization of liquid droplets is verified by using the direct
modeling CAD software Space Claim and to construct a single and three Fourier-Horn Amplifier model. At the
same time, the multi-section geometric model is completed by the array geometry function of the CAE numerical
software COMSOL and the physical characteristics of the Fourier amplifier designed by the finite element
analysis module. The simulation study states that the onset amplitude of single Horn ultrasonic atomizer produces to 6.8 μm droplets with a designated frequency of 492 kHz. The types and basic principles of mechanical
amplifiers (Horn) and the reason for choosing and the designing of required Fourier amplifiers are explained.
The simulation data fit well with experimental data. Micro-droplets are steady and uniformly formed after the
liquid feeding rate is optimized. This newly designed ultrasonic atomizer facilitates the development of capillary
surface-wave resonance at a designated frequency and easy to form atomization of a liquid drop.
There are many uses for uniform and fine droplets, for example, it can be used as a precursor for making
micro-sized drops, and can be used as a coating to form extremely fine surfaces; it can increase the surface of
droplets, increase evaporation speed and even fuel rate; can be suspended Drugs enter the lungs directly and are
absorbed into the blood, treatment of inner ear diseases, nose diseases, and even eye cares can use this single
Horn atomizer with a tube.
Uniform and small droplets can be achieved by atomizing the liquid,
and the atomization methods can be roughly divided into three
methods: pressure, pressure-assisted (Pneumatic), and ultrasonic atomization [1]. The pressure atomizer is mainly atomized by pushing the
liquid into the nozzle with high pressure. The atomized particles are
usually large and uneven. The gas-assisted atomizer uses the kinetic
energy of the gas to impact the fluid to produce larger shear. The shear
stress makes the liquid reach a finer atomizing particle size. The ultrasonic atomizer mainly relies on ultrasonic waves to generate capillary waves or cavitation on the liquid surface to atomize the liquid, and
the atomization process is kept at a low temperature and low atomization. Flow rate, atomized particles are smaller and more uniform.
According to the literature [2,3], when the atomized particle size is less
than 10 µm, 90 nm particles can be produced through pyrolysis, and the
requirements of drug particle size (3 ~ 5 µm), so the main purpose of
this research is to design and manufacture high-frequency atomizing
amplifiers. At low frequencies, a general shaker is sufficient to meet
atomization requirements, but the bandwidth of the oscillator is limited
and cannot meet the needs of high frequencies. Ultrasonic amplifiers
must be used. In 1963, Eisner [4] designed the amplifier in a completely
different way. First, the vibration mode of the amplifier was determined, and then the shape of the amplifier was determined. Eisner
expanded the mode using a Fourier series, and the amplifier shape can
be calculated through the equation of the amplifier's motion. A class
amplifier is called a Fourier amplifier.
By 20th century Ultrasonic atomizer has many practical applications and successfully applied in biomedical engineering such as food
industries, chemical coating, and miniature carriers in pharmaceutical
and for the fabrication of printed electronics and sensors [5]. Propagations of ultrasonic surface‐wave are of great interest in the middle of
Corresponding author.
E-mail address: [email protected] (Y.-L. Song).
Received 29 March 2020; Received in revised form 11 May 2020; Accepted 13 May 2020
Available online 25 May 2020
2211-3797/ © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
Results in Physics 18 (2020) 103166
Y.-L. Song and L. Bandi
Design of Fourier Horn
the 20th century, as it has enabled atomization of incompressible frictionless fluid and has often known in a compact, atomized liquid droplets [6]. The control of the atomized droplet process and the characterization of the vaporized droplet size are important issues to obtain
good physical properties and repeatability. In 2018, M L Hakim [7]
states that the spreading factor decreases with the increasing of surface
roughness at 140 degreesC due to incomplete wetting of droplet on the
surface, the maximum spreading factor is relatively independent to the
change of surface roughness. In 2019, Wang and Junlei proposed a
cross-coupled dual beam structure [8] with various incoming directions. The upper and bottom piezoelectric beams can generate a maximum power output of 6.77 µW and 56.64 µW.
This research focuses on the development of a new generation of
ultrasonic atomizers. The main function is to serve as an efficient drugcovered component device. During this time, the optimized atomizer
structure will be designed by COMSOL software modeling. It meets the
required particle size and particle size, and then the micro-electromechanical process technology [9] is used to complete this component
mechanical device to design a new generation of an ultrasonic atomizer.
The advantages of the early design of the ultrasonic atomizer design
and the general commercial atomization are mainly that the atomization particle size generated is quite uniform and mostly 7 μm (80%),
while the traditional commercial atomizers are 20 μm to 150 μm, the
distribution is quite uneven. And the energy required is high, about
23.5 W. The Fourier Ultrasonic Atomizer of this study only needs
0.65 W to produce very uniform particles. However, although there are
so many advantages in the practical application, when used clinically
device exhibits tolerance. It is mainly the size of the nozzle head, which
causes a lot of drug loss and inconvenience of the device when it is used
for drug coating [10]. Because of the above-mentioned shortcomings,
this research will develop a full-fledged Fourier ultrasonic atomizer,
minimizing the components and devices for drug delivery, and making
its nozzle size only 100 μm × 100 μm (the previous design nozzle size is
1.06 mm × 1.25 mm, it is easy to produce wetting throughout the
nozzle and cannot be materialized.) The design shape is shown in Fig. 1.
This newly added half-wavelength silicon tube greatly reduces the
original wetting at the tip end, which can improve its working performance.
The amplifier is a 3D vibrating body. When it is free to vibrate, the
input and output ends are not stressed, and the amplitude of the output
end (Tip end) is greater than the input end (Base end). Since a 3D vibrating body cannot be directly analyzed using a mathematical model,
the following conditions must be assumed to simplify the original 3D
model into one dimension:
(1) The amplifier material is uniform; the purpose is to make the entire
amplifier have a uniform density.
(2) The vibration on the cross-section of the center axis of the amplifier
is the same. Here, the center axis is located on the x-axis of the
Cartesian coordinate (the axis represents the normal vector of the
cross-section). This condition means that the cross-section vibration
is independent of y, z.
(3) Displacement and velocity are continuous in the amplifier.
Equation of motion
From the above conditions, the equation of motion of the amplifier
can be simplified as:
1 ∂
∂2u¯ (x , t )
[σxx A (x )] = ρs
A (x ) ∂x
∂t 2
Where A(x): cross-sectional area of the amplifier,σxx: stress,ρs : amplifier
density,ū(x,t): longitudinal amplitude.
When the amplifier is in harmonic vibration, the displacement can
be expressed as:
From Hook’s law [11]:
σxx = E¯
Putting formula (2) into formula (1), we get
∂ 2ū
1 dA ∂ū
1 ∂ 2ū
= −2 2
A dx ∂x
c0 ∂t
∂x 2
c0 =
E / ρs
E : Young's coefficient [12].
When that the amplifier is in harmonic vibration, the displacement
can be expressed [13] as
Design and production of amplifier
Using the characteristics of the mechanical amplifier, the vibration
displacement generated at the input end is amplified at the output end,
and the liquid at the output end is atomized. This chapter first introduces the types and basic principles of mechanical amplifiers (Horn)
and explains the reasons for choosing Fourier amplifiers. Second, the
Fourier amplifiers required for the experiment are designed and simulated by the finite element method (FEM).
u (x , t ) = us (x ) eiωt
Substituting Eq. (4) into (3) gives
d 2u
dx 2
1 dA dus
= β¯2us = 0
A dx dx
β =
Dimensionless formula (5) gives the following formula:
u'' + u'
+ Ω̄2U = 0
Based on the 1-D model for longitudinal particle displacement u x of
the horn with varying cross section area A(x) at coordinate x in Eq. (6).
Since A(x) is finite, d [lnA(x )] ~ A′ (x )
Where, X =
Fig. 1. Space Claim constructed Fourier amplifier.
A (x )
Results in Physics 18 (2020) 103166
Y.-L. Song and L. Bandi
Ω = βl are the length of the amplifier,
U ' = dU and the Eq. (6) is recombined to obtain X = us
α1 =
1 −
(M − 1)[3α52 − 4(4 − γ 2)] U0
d ln A
U ′ ′ + Ω̄2U
α2 =
(M + 1)(9 − γ 2) U0
1 −
α3 = − (M − 1)(α52 + γ 2) U0
Horn boundary conditions
α4 =
(1) If there is no stress at both ends of the amplifier, then
U ' (0) = 0
U ' (1) = 0
(M + 1)(γ 2 − 1) U0
32 α5
¯ −1 U0
γ= =
Where, α52 =
Substituting the above Eqs. (23)–(25)
into Eq. (7), the calculation can be obtained:
A (X ) = A (0) e ∫0
(2) The amplitude ratio at both ends is − M̄ , M̄ which the amplitude
amplification factor is. The negative sign is because the vibration
phases at both ends are 180 degrees apart.
f (α52, γ , X ) d (cosπX )
f (α52, γ , X )
U (0) = U0
U (1) = −MU0
(3) If the change in area is to be limited, dA/ dx ≠ ∞ use this relationship and Eqs. (7)–(9) to know
U '' (1) = −Ω U (1)
Fourier Horn simulation discussion
After determining the values, the shape of the Fourier amplifier can
be calculated. Under the premise of constant thickness, the three-dimensional structure of the amplifier can be obtained. The finite element
method is used to establish a grid [18] for simulation.
The required vibration frequency of the amplifier designed in this
research is about 300 kHz to 1 MHz. It can be known from the literature
[19] that when the ultrasonic wave is transmitted through the amplifier, the maximum particle velocity, the ultrasonic wave transmission
speed, and the maximum strain of the amplifier have the following
X →1
theL' Hospital rule [14], we get
U '' + Ω U
U ''' + Ω U '
lnA = − lim
= − lim
X → 0 dX
X →0
X →0
U ''
In Eq. (14), since it is finite [15], it is necessary to satisfy
U (0) = 0
vm =Kεm c
U ''' (1) = 0
μ (3 − γ 2) − 2(γ 2 + 2α52)cosπX + 3μ (γ 2 − 1)cos2 πX + 4α52cos3 πX
(4) Make the area change rate of the input and output terminals
zero, lim dX lnA = 0 and lim dX lnA = 0 . Then formula (7) and using
X →0
¯ + 1)/(M
¯ − 1) [11,17] from the above derivation, the bigμ¯ = (M
gest difference between the Fourier amplifier and other types of amplifiers is that the form of vibration is determined first, and then the
shape is calculated. The shape of the other types of amplifiers is determined after the shape is calculated.
Here X = 0 is the big-endian position and X = 1 little-endian.
U '' (0) = −Ω U (0)
(4 − γ 2)(γ 2 + α52) − μ (9 − γ 2)(γ 2 − 1)cosπX − (16 − γ 2) α52cos2 πX
Where K is constant that is only related to the shape of the amplifier.And εm c are related to the material of the amplifier.
Under the same amplifier shape, it is known that silicon has a higher
value than ordinary metals and εm c can withstand higher frequency
Therefore, this design uses silicon as the amplifier substrate. This
Fourier amplifier is divided into two parts:
Determine the appearance of the Fourier amplifier
If the amplitude is expanded using the Fourier series to:
U (X ) =
∑ αn cos[(n − 1) πX ]
1. The driving part (Piezoelectric Actuator) of piezoelectric material
2. Fourier amplifier with half-wavelength design at double magnification
Due to the characteristics of the cosine function, Eqs. (8), (9), (15),
and (16) can be satisfied by themselves, and there are four remaining
conditions [16], so undetermined coefficients N = 4 can be obtained
atαn . But in order to increase the flexibility of the appearance design,
we add one more variable, that is N = 5. Substituting (17) into four
boundary conditions (10)–(13) gives:
Its structure is shown in Fig. 1. There is a first-rate channel in the
middle. The channel that mainly provides liquid from the input end to
the output end, can also be used as a cooling device. The large and small
ends ratio of area can be reduced [20].
α1 + α2 + α3 + α4 + α5 = U0
COMSOL simulation
α1 − α2 + α3 − α4 + α5 = −MU0
π 2α2 + 4π 2α3 + 9π 2α4 + 16π 2α5 = Ω U0
¯ 2U0
π 2α2 − 4π 2α3 + 9π 2α4 − 16π 2α5 = MΩ
PZT and monocrystalline silicon are both anisotropic materials, the
material parameters must be set according to the lattice direction of the
material parameters in the simulation to obtain the true results. This
simulation uses COMSOL to set the coordinate axis according to the
simulated material orientation settings. Different coordinate systems,
and then setting these materials in these coordinate systems, you can
α5 are optional parameters. From formulas (18) to (21), the following formulas can be solved:
Results in Physics 18 (2020) 103166
Y.-L. Song and L. Bandi
give a material that can simulate non-isotropic materials with different
orientations, and when changing amplifiers with different magnifications, in the past, it was necessary to continuously modify the geometry
through CAD software. Then import CAE software to have a way to
simulate. COMSOL can use the array method. This method not only
arrays the geometry, but also sets the material along with the set material and boundary conditions [17,21]. Amplifiers with different
magnifications can be adjusted quickly, which can greatly reduce the
simulation time.
The finite element method COMSOL 3D simulation program is used
to seek the resonance frequency of pure longitudinal vibration
(Longitudinal Vibration). In addition to the amplifier itself and PZT,
there is also a driving section at one end of the magnification, which is
mainly used to connect the PZT piezoelectric plates on the left and right
sides. It is a device that converts input energy into mechanical energy.
The simulation program calculates the shape of the amplifier following
the previous calculation, establishes a grid, performs amplifier simulation, and adjusts the better sum obtained previously to make the simulation frequency and amplification meet the design requirements.
After a series of simulations to find twice the amplification, seek to
meet design requirements as well. Taking 500 kHz as an example, the
frequency of the single-mode longitudinal vibration mode in Fig. 2 is
491.95 kHz and the magnification is 2.013. Secondly, find the size of
the PZT sheet, get the length of this PZT sheet according to the design
frequency, and then continuously adjust the length to make the PZT
sheet’s pure longitudinal vibration modal frequency and the single-cell
amplifier frequency the same.
The same method is used to make the driving section plus PZT
longitudinal vibration frequency equal to the frequency of the singlesegment amplifier. The above is to determine the size of the siliconbased amplifier. Finally, after combining the amplifier, driving section,
and PZT, find the amplifier section line position and add Upper fixed
end. The simulation substrate is a silicon wafer. The longitudinal vibration is in the direction of the wafer (1 1 0). Si anisotropic element is
used [22]. The piezoelectric plate should use PZT-5H. Finally, the size
of each unit of the Fourier amplifier is shown in Fig. 3.
ANSYS Mechanical finite element analysis software is utilized to
Fig. 3. Single 500 kHz Fourier Horn longitudinal vibration mode.
reproduce PC models of structures, hardware, or machine parts for
breaking down quality, durability, versatility, temperature distribution,
electromagnetism, fluid flow, and different traits [23]. Fig. 3a explains
the Fourier Horn longitudinal vibration at different position.
The mode of longitudinal vibration is also shown and the displacement of single 500 kHz in longitudinal vibrational mode is shown
in Fig. 3a. To verify that the designed amplifier can generate this mode
when the voltage is applied, through COMSOL's frequency sweep
function, ground on one side and the other side of the applied voltage
PZT [24]. The analog frequency range is 450 kHz ~ 550 kHz. The
impedance analysis diagram of this section is shown in Fig. 3b. From
the results obtained by the pattern, the lowest impedance value is the
frequency of the desired operating mode.
Simulation discussion for 3Horn-1Step
The above method is to complete the design and simulation of a
Fig. 2. 500 kHz Fourier's pure longitudinal vibration mode.
Results in Physics 18 (2020) 103166
Y.-L. Song and L. Bandi
Fig. 3a. Single 500 kHz Fourier Horn longitudinal vibration mode and displacement with X-axis, Y-axis, and Z-axis at 492 kHz.
The longitudinal vibration mode of 3-Horn 1-Step with vibrational
frequency of 493 kHz is shown in Fig. 4a. And the displacement of 3section 500 kHz in longitudinal vibrational mode, the impedance analysis diagram of this section is also shown in Fig. 4a. From the results
obtained by the pattern, the lowest impedance value is the frequency of
the desired operating mode. 3-Horn atomizer consume high frequency
compared to single Horn ultrasonic atomizer. Two types of spray nozzles are recommended in industries: air-driven sprays, which are produced by atomizer nozzles (present research), and pressure sprays
which are produced by pressure nozzles [27,28].
This research shows that single Horn design will be longitudinal
wave mode, which is called working mode. Previous mode and Last
mode of frequency vibration does not belong to longitudinal wave
mode. Using the same process can get the design size of the frequency,
the results are shown in Table 1 for 500 kHz simulation results.
Fourier Horn measurement results
Fig. 3b. 1-section 500 kHz Fourier Horn analog impedance analysis diagram.
Fourier Horn electrical characteristics analysis
single double-time Fourier amplifier. There are two ways to design a
higher magnification amplifier:
The One-Step Fourier Horn atomizer obtained by the micro-electromechanical process is shown in Fig. 5. The size of the one-step device
and the three-stage One-Step Fourier Horn atomizer are 27.5 mm and
67.5 mm, respectively. The impedance results obtained by putting the
(HP_4294A_16047A Test Fixture) are shown in Fig. 6. Comparing the
results with the results of the COMSOL simulation (Fig. 3b), it can be
found that the data are quite consistent, and it can be judged that if the
longitudinal vibration [29] (Longitudinal Vibration) is between 490
and 492 kHz, the simulation and actual atomizer results can be seen
Anastomosis. In the meantime, add the signal and energy amplifier to
this atomizer, find its resonance atomization frequency, and get a good
resonance frequency of 491 kHz. It can be known from this that the
simulation system matches the real finished product quite well and is
reliable [30].
In the supporting materials S1 shows the simulation results of single
horn with one step atomizer. The S1 present resonated frequency at
492 kHz. The S2 was shown that simulation results of 3-Horn with one
step atomizer resonate at 493 kHz. The atomization experimental results of a single Horn with one step are shown in S3 and 3-Horn with
one step are shown in S4 supporting materials.
(1) Like the design of the double magnification process, after determining the magnification, find a better sum and then engage in
three-dimensional simulation to determine the size of the amplifier.
However, when this magnification is large, the length of the big end
will be longer than the amplifier longitudinal length [25]. In this
case, the pure longitudinal vibration mode cannot be accurately
obtained during the simulation process, and the cross-section vibrations are related, i.e. the assumptions of the original derivation
of the amplifier are no longer applicable.
(2) As shown in Fig. 4, three separate Fourier amplifiers are connected
in series into one, and the amplification effect of the amplifier can
be known from the design. The magnification obtained by this design can be changed to 3 sections to obtain 8 times magnification
Fourier Horn atomization analysis
Based on the research and development results of this research, the
atomizing droplet size of this atomizer is determined by the resonance
frequency of the atomizer. Compared with the previous research journals [17,22], we can know that λ = [8πT /(ρ f02 )]1/3 it is multiplied by
0.34 to estimate its droplet size to be 6.8 μm.
Particle size analysis instrument (Malvern Particle Sizer) obtained
three distribution indicators of particle size analysis: MMD: 7.77 μm
Fig. 4. 3-section 500 kHz Fourier Horn longitudinal vibration mode.
Results in Physics 18 (2020) 103166
Y.-L. Song and L. Bandi
Fig. 4a. 3-section 500 kHz Fourier Horn longitudinal vibration mode at resonated frequency with displacement and analog impedance analysis diagram.
Table 1
Simulation results of 500 kHz by varying no. of Horns.
No. of Horns
Previous mode
Working mode
Last mode
VMD: 7.48 μm and SMD: 7.48 μm, as shown in Fig. 7. Note that MMD,
VMD and SMD are inset designated droplet mass mean diameter, volume droplet means diameters and surface area droplet mean diameters, respectively. The distribution phase is concentrated at 7 μm. It
is in good agreement with the theoretical predictions. The multiple
One-Step design is mainly because when there is no such designs before, often because the viscosity of the precursor and the silicon wafer
atomize its front surface contact, it is easy to form water droplets,
which causes the resonance frequency of the atomizer to shift, causing
stop atomization [31]. When using the One-Step design, the contact
area is relatively small (0.1 mm × 0.1 mm ≪ 1.06 mm × 1.25 mm).
Therefore, reducing its contact area can avoid the atomization interruption caused by the accumulation of liquid beads at the front of the
atomizer. In addition, it can take advantage of the needle-like characteristics of the front end, which can bring the atomized liquid beads
into the tiny holes. For example, when engaged in otitis media drug
delivery, this nebulizer can first enter the middle ear cavity in the
eardrum. The medicine is atomized and sent to achieve an effective
drug treatment mode.
Fig. 6. 1-section 500 kHz Fourier Horn impedance analysis chart.
verification, the novel ultrasonic atomizer using single horn can be
obtained, which will have the advantages and effectiveness of its previous design. The simulation study states that the onset amplitude of
single horn ultrasonic atomizer is 6.8um with a designated frequency of
491 kHz. The simulated droplet size is in good agreement with that
predicted by the capillary wave atomization machine. This finding represents the ultrasonic atomizer designed in this new study can make a
section atomizer atomization (previously only three or four sections
Based on previous discussions, simulation results and experimental
Fig. 5. 1-section and 3-section 500 kHz Fourier Horn finished products.
Results in Physics 18 (2020) 103166
Y.-L. Song and L. Bandi
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Fig. 7. 1-section 500 kHz Fourier Horn droplet distribution.
could be atomized). Furthermore, the atomization time can be extended
without interruption, and it can be known that the novel atomizer has
more extensive applications in the delivery of medicines.
CRediT authorship contribution statement
Yu-Lin Song: Conceptualization, Methodology, Software,
Investigation, Writing - review & editing, Validation, Supervision.
Lavanya Bandi: Data curation, Writing - original draft, Software,
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this paper.
This work was supported in part by the Ministry of Science and
Technology, Taiwan, through grant MOST 106-2221-E-468-015. The
authors are grateful to the Prof. Y. F. Chou with the Department of
mechanical engineering, National Taiwan University, Taipei, Taiwan,
for his technical support and valuable suggestion in numerical simulation.
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://
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