- Email: [email protected]

S0167-7322(19)33122-8

DOI:

https://doi.org/10.1016/j.molliq.2019.111878

Reference:

MOLLIQ 111878

To appear in:

Journal of Molecular Liquids

Received Date: 2 June 2019 Revised Date:

26 August 2019

Accepted Date: 3 October 2019

Please cite this article as: H. Gharibi, A. Khaligh, A. Bahrami, H.B. Ghavifekr, A very high sensitive interferometric phononic crystal liquid sensor, Journal of Molecular Liquids (2019), doi: https:// doi.org/10.1016/j.molliq.2019.111878. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

A Very High Sensitive Interferometric Phononic Crystal Liquid Sensor

Hamed Gharibi1, Aynaz Khaligh1, Ali Bahrami1* and Habib Badri Ghavifekr2

1

Optoelectronics and Nanophotonic Research Lab. (ONRL), Faculty of Electrical Engineering, Sahand University of Technology, Tabriz, Iran 2

Microelectronics Research Lab., Faculty of Electrical Engineering, Sahand University of Technology, Tabriz, Iran

*Corresponding author. A. Bahrami; Tel.: +98 33459419; Fax: +98 33459419 E-mail address: [email protected] Address: Optoelectronics and Nanophotonic Research Lab. (ONRL), Faculty of Electrical Engineering, Sahand University of Technology, Tabriz, Iran

1

1

Abstract

2

In this work, a novel structure has been proposed in order to design a highly sensitive phononic-

3

crystals-based sensor for sensing different water-ethanol mixtures. The proposed structure

4

consists of the steel background and water-filled holes as inclusions. Having a resonant mode

5

inside the band gap is very important for wave propagation inside the structure and performance

6

analysis of the sensor. The defect mode has been created by utilizing a linear radii-decreased

7

hollow cylinder. Also, the proposed structure includes two unequal pathways to use the

8

interferometric advantages of the asymmetric Mach-Zehnder interferometers. In this work, the

9

frequency change arises by changing the material concentration of defects area which is a water-

10

ethanol mixture. The proposed phononic crystal-based sensor displays this frequency change

11

well by varying the sound’s speed of the propagating wave inside the structure. Finally, the

12

sensor was optimized for a sensing step of 0.2% concentration of the analyzed mixture.

13

Keywords: Phononic crystals; Liquid sensor; Mach-Zehnder interference; Irreducible brillouin

14

zone; Finite element method

2

1

1. Introduction

2

Phononic crystals (PnCs) are artificial periodic homogeneous structures which allow

3

phonon vibrating in lattice constant. In recent decades, phononic crystals have introduced a large

4

variety of applications, such as sensing [1], filtering [2], demultiplexing [3], etc. In 1979,

5

Narayanamurti et al. were focused on limiting the wave propagation at high-frequencies [4]. The

6

main feature of phononic crystals is controlling and manipulating of acoustic wave inside the

7

structure. Using the wave vector-frequency relation and frequency analysis of these structures, a

8

unique curve is obtained, called band structure, which is the basis of the desired structure's

9

behavior versus input waves. Due to the difference in the elastic and acoustic properties of

10

materials used, their band structure may comprise a region in which waves within the frequency

11

range are not allowed to propagate inside the structure [5]. In Bragg's scattering conditions,

12

oscillation velocity of the phonon reaches the lowest value, and the propagation of any frequency

13

will be forbidden in this region [6].

14

Lucklum et al. have previously investigated the phononic crystals sensor which was able

15

to evaluate the water and 1-propanol properties [7]. They used a steel background with air

16

cavities and then placed the model in a glass container of water. The shift in frequency was due

17

to the density and speed of sound’s variations. In a different work, Oseev et al. have presented a

18

structure with a linear defect which led to estimate general properties of the gasoline [8]. In

19

2018, Nagaty et al. introduced a one-dimensional structure including multilayer aluminum and

20

epoxy. The defect region was based on a piezomagnetic material called Terfenol-D. This

21

material changes the resonance mode’s location in the dispersion curve by varying its

22

piezomagnetic properties which were affected by the variable external applied pressure [9]. In

3

1

2016, a phononic sensor has been introduced that senses only some strict percentages of liquids

2

such as Hexanol-n-Propanol, etc [10].

3

In this work, a two-dimensional structure will be introduced with a background of steel

4

and the water-filled holes. In the proposed structure a line defect as the waveguide was

5

implemented based on the radius change of inclusions. The defect composed of hollow cylinders

6

is designed as a line waveguide with two unequal branches. The purpose was to create a

7

destructive interference in the transmission process. The linear resonance mode was obtained in

8

the band-gap, which is very suitable for our sensor operation. The water-ethanol mixture

9

considered as the target material by taking linear values of the density and speed of sound

10

relations. Compare to other works in the same field, here we’ve presented a sensor which is

11

suitable for specify the target liquid material and sensing very small amounts of that target

12

material in the mixture. The main advantage of the proposed structure is that the mentioned

13

mixture can be sensed even with 0.2% differences in 0-100% sensitivity range. This is due to

14

interference properties of asymmetric Mach-Zehnder interferometer. In this work, A new idea of

15

using an asymmetric Mach-Zehnder channel to eliminate additional transmissions is proposed.

16

While the substrate material in some PnC-based sensor applications was made of mercury-like

17

materials which are highly toxic, the material that is suggested in this work is non-toxic and

18

carefully selected with consideration of possibility for fabrication.

19

2. Target Sample

20

Different concentration percentages of water and ethanol mixtures were chosen from valid

21

values as shown in Fig. 1. Input data of this practice have been experimentally obtained by

22

Salman et al. [1]. It is obvious that concentration variations of ethanol in the mixture will change

23

the acoustic parameters of the material, including the density and speed of sound. 4

1

Fig. 1. Density and speed of sound vs different concentrations of ethanol in the water-ethanol

2

mixtures

3

As can be seen in Fig. 1, x-axis is consisted of different concentration percentages (from 0-

4

100%). Note that, the state of water and ethanol is polar and non-polar, respectively. Density-

5

concentration relation is assumed to be linear [11]. However, in order to have a linear relation

6

between the concentration and speed of sound, these values have divided into the range of 0-10%

7

and 13.3-100%. In some cases, the same speed of sound can be observed for two different values

8

of concentrations; For example, 5% and 19.2% concentrations are showing a same speed at 1575

9

m/s. Therefore, designing a perfect structure that can properly distinguish these similar points is

10

going to be a challenge in this work.

11

3. Design Procedure

12

Generally, the speed of sound plays a dominant role in phononic crystal sensor structures.

13

In this work, a two-dimensional phononic crystal structure is constructed by a square matrix of

5

1

water holes and steel background. The filling fraction of the structure has been chosen as

2

ff = π r 2 / a 2 = 0.325 , which r = 9mm is the scatterer radius and a is the lattice constant. This

3

will help to design a structure that can be fabricated more easily. By doing such, our work area

4

will fall into sonic range, which adds an acoustic wave to the structure by using a piezoelectric

5

transducer. Physical parameters of constituent materials are provided in Table 1. Table 1. Physical parameters of constituent materials of structure

6

Material

Mass density (kg/m3)

Longitudinal and Transvers sound velocity (m/s)

Steel

7780

5825-3230

Water

1000

1490

7

Generally, transverse waves can be seen in concrete or rigid mediums. So, waves which are

8

propagating through a solid material can be longitudinal or transverse, but those waves that are

9

propagating in fluids are mostly longitudinal waves [6]. It should be noted that shear waves are

10

exist in viscous fluids. However, water-ethanol mixture is a diluted fluid. Thus, only longitudinal

11

waves can propagate in the mixture.

12

To begin with, we are going to analyze the smallest part of the structure called unit cell. Its

13

irreducible Brillouin zone (IBZ) is investigated by using the finite element method (FEM) as

14

shown in Fig. 2.

6

1

Fig. 2. Schematic view of a perfect phononic crystal unit cell and irreducible brillouin zone of the proposed structure

2

3

Figure 2 shows a unit-cell which is the smallest repeating part of the whole structure. The

4

values of geometry’s main parameters including lattice constant and inclusion radius, are 28mm

5

and 9mm, respectively.

6

As part of modal analysis, The Helmholtz equation is applied for given frequency domains.

7

Harmonic diversity of fields and sources are given by e i ωt . The governing equations and

8

boundary conditions are formulated using the general potential of pressure, called dispersed

9

field. In the presence of a pressure background and a stress-dispersion field, the general pressure

10

relationship can be introduced as following:

Ptotal = P + Pb

11 12

(1)

where Pb is background pressure and P is scattering pressure. In order to solve acoustic pressure equations, we have:

7

k eq 2 1 ∇ * − = Qm ρc (∇pt − qd ) ρc

(2)

k eq 2 = (ω / c ) − k z 2

(3)

2

1

where

ρ is density, c is speed of sound, keq includes both wave numbers and out-of-plane

2

wave numbers, Qm is monopole domain source, qd is dipole domain source, p t is total pressure

3

and ω is angular frequency. The Bloch-Floquet method is applied on a two-dimensional

4

structure for parametric sweep of wave vectors k x and k y on IBZ spans [5]. So the band

5

structure diagram is achieved as shown in Fig. 3.

6

Fig. 3. Dispersion curve of the proposed 2D phononic crystal structure without defect

7

In periodic structures, the most important phenomenon is band gap. The band gap in PnC is

8

a frequency range that no acoustic wave is allowed to propagate. 8

1

Figure 3 shows the band structure which includes allowed bands as well as a wide band

2

gap region. This band gap in PnC represents the forbidden frequency range where acoustic

3

waves cannot be propagated through the structure.

4

A supercell was designed with a hollow cylinder as defect rod contains two concentric

5

circles. The inner circle with a radius of rh2 = 2mm is a soft polymer, and the outer part with a

6

radius of rh1 = 3mm contains a mixture of water and ethanol at 20˚C. The soft polymer has been

7

applied to fit the acoustic properties with the host material. We obtained the speed of sound and

8

mass density of the mixture from the following equations [8]: v = ∑ i =1u i .v i N

(4)

ρ = ∑ i =1 x i . ρi

(5)

N

9

where u i is the ratio of sample for speed of sound, x i is the ratio of sample for density

10

voluminosity, v i and ρi are the speed of sound and density of water-ethanol mixture,

11

respectively. Physical parameters of defect cylinder constituent materials are given by Table 2. Table 2. Physical parameters of constituent materials of defect rods

12

Mass density (kg/m3)

Longitudinal sound velocity (m/s)

Soft polymer

960

1000

Ethanol

790

1156

Material

13 14

The supercell analysis was applied on ΓΧ direction of IBZ and defect modes were obtained in the band gap which can be seen in Fig. 4.

9

1 2

Fig. 4. Dispersion curve of the proposed 2D phononic crystal sensor structure, showing resonance mode in the gap center in presence of defect

3

As shown in Fig. 4, the band gap engineering is very important for sensing operation. The

4

frequency range of the band gap region completely depends on geometric parameters such as

5

lattice constant and holes’ diameters. Defect in the structure is applied in such a way that

6

resonance mode is produced for every concentration of ethanol in water-ethanol mixture.

7

It is noteworthy that modes which are emerging close to the lower and upper edge of the

8

band gap, get distorted during transmission. Such modes are not considered as good propagating

9

modes. So, their frequencies are not selected for propagation in the waveguide structure. While

10

propagating through waveguide, they enter to the structure and are broadcasting in the crystal;

11

because they are close to the allowed modes of the crystal. This means that they are considered 10

1

as allowed frequencies that are not in our favor. In general, those resonance modes that are closer

2

to the center of the band gap, can better concentrate and confine in the waveguide channel.

3 4

The defect in the structure is applied in such a way that resonance mode is produced for every concentration of ethanol in water-ethanol mixture.

5

Finally, a general structure was introduced including line defects as the waveguide

6

branches. Three-dimensional schematic of the proposed phononic crystal interferometric sensor

7

is depicted in Fig. 5. The holes were filled with water and the defects were filled with sample

8

material. In fact, the main purpose of applying two waveguide branches is to allow destructive

9

interferences to remove excess frequency peaks. Therefore, one of the waveguide channels is

10

closer to the input waveguide. So, the path traversed by two branches of wave will be different in

11

each channel. This causes a phase shift between waves and they will reach to the output

12

waveguide with a time delay results in interference phenomenon.

11

1

Fig. 5. Schematic of the proposed phononic crystal interferometric sensor using asymmetric

2

Mach-Zehnder structure

3

The periodic boundary conditions were applied in y-direction. The transducer creates

4

propagating waves along x-direction. In order to obstruct reflections of scattering wave in spans

5

of the structure, an absorber part (perfectly matched layer) set at end of the structure. Output

6

results including frequency peaks will be presented and discussed in next section.

7

4. Results and Discussion

8

4.1. Spectral Response

9

Here, transitions of resonance modes in the band gap will be described. Using Bloch-

10

Floquet theorem, a supercell with a defect inside, is simulated. In the unit cell, wave vectors k x

11

and k y are introduced for two directions of IBZ. The supercell analysis is only applied on ΓΧ

12

direction of IBZ. As explained in section 2, the target sample is the water-ethanol mixture. We

13

intend to present the sensing capability of proposed structure. We will show that it’s able to

14

detect a complete range of concentration percentages from water to ethanol with steps of 0.2%.

15

For this purpose, the total region of concentrations is divided to six regions including 5-20%, 20-

16

35%, 35-50%, 50-65%, 65-80% and 80-95%. It is obvious that the spectral separation of output

17

signals for these six regions can assure us that different regions can be individually sensed.

18

Therefore, in first step, we will show the ability of structure to distinct between frequency work

19

regions. Resonant modes have been created in phononic band gap by filling the target cylinders

20

with water-ethanol mixture for all six percentage regions. The phononic band structure of

12

1

proposed sensor including six resonance modes for 5-95% ethanol concentrations in the mixture

2

(with steps of 15%) is presented in Fig. 6.

3

Fig. 6. Phononic Band Structure including six resonance modes for 5-95% ethanol

4

concentrations in the mixture (with concentration step of 15%)

5

It provides complete linear and distinctive modes for different frequency ranges. Reffering

6

the band structure diagram, neither of concentration percentages have frequency interference

7

with each other. Therefore, the sensor can detect any concentration percentages. The oscillation

8

boundary fixing on the water and sample solid-liquid contact is assumed as [12]: 1 na .( ∇ρ ) = an

ρ

9 10

Here, n a is the unit vector direction,

(6)

ρ is the mass density of the sample and an is the

displacement on the spans of the structure.

13

So, for computing elastic properties, the solid region ( Ω s ) can be evaluated as follows

1 2

[12]:

∇σ = −ρω2u

(7)

3

where σ is the Cauchy stress tensor, u is the displacement vector, ρ is density and ω is the

4

angular frequency. Body forces are ignored. The region of band gap which contains all of these

5

resonance modes corresponding to the border points of six mentioned concentrations is called the

6

total working region (WR). For a better understanding, the large-scale of the total working region

7

diagram is depicted in Fig. 7.

8

Fig. 7. Defect modes related to concentrations (5-95% with step of 15%) of working regions

9

border points (WR1 to WR6) 14

1

Different modes are achieved for 5-95% ethanol concentrations of the mixture. These

2

modes don't have a frequency overlap or interference with each other as shown in Figs. 6 and 7.

3

As a result, inside the total working region, 6 separate working regions corresponding to

4

concentrations with a step of 15% are formed that have distinct frequencies. This is considered as

5

an important point for the sensor application. Therefore, we expect that the final ideal

6

transmission-frequency curve for working regions WR1 to WR6 (for six concentration ranges) to

7

be as close as possible to what is shown in Fig. 8.

8 9

Fig. 8. Ideal expected transmission-frequency curve for total working region including six WRs with different frequency regions

10

The working frequencies of each region are derived from the Fig. 7. Here each region

11

corresponds to a 15%-width concentration range. Due to achieved results, we can assure that

12

there is a frequency distinction for WRs with a step of 15%. We put our focus on analyzing the

13

working region WR6 as the most difficult WR to sense. We will analyze WR6 (for 5-20% 15

1

concentration) with steps of 0.2%. The difficulty of this region is due to existence of

2

concentrations with close frequencies (according to Fig. 1). Transmission of the five other

3

working regions can be easily achieved as they show a linear behavior of their elastic properties

4

depicted in Fig. 1.

5

Transmission spectrum of the proposed structure for different molar values of WR6 (5-

6

20% with the step of 0.2%) has been plotted in Fig. 9. Figure 9 is divided into two parts because

7

of the high number of plots. The central frequency of each transmission has been varied by

8

changing the molar ratio of the mixture. Location of the resonance modes within the band gap of

9

the dispersion curve and their intensity are changed with different concentration values. The

10

input probe was placed to the input of structure. This can be done by using the piezoelectric

11

transducer. The output probe is placed at the end of the output channel.

16

(a)

(b) 1

Fig. 9. Transmission spectrum of structure for different molar values in WR6 with 0.2% steps;

2

the figure has been separated into two parts due to the high number of plots

3

Moreover, frequency change is shown in Fig. 9 (the steps are based on 0.2% of ethanol

4

concentration in the water-ethanol mixture). This amount of steps indicates that the sensor can

5

well reveal very low amounts of the water-ethanol mixture. It is also noteworthy that the

6

detection frequency of some different concentrations will be very close together due to the

7

similarity of their acoustic properties. Despite, detection of these concentrations has been truely

8

accomplished in the proposed structure.

9

4.2. Confinement

17

1

Some important parameters were obtained such as quality factor and maximum

2

transmission of each step (0.2%) in WR6. In this section, the wave propagation has been shown

3

in the provided sensor. The wave propagation and pressure transfer in branches of structure is

4

presented in Fig. 10 for two percentages of starting and end points of WR6 (5% and 20%).

(a)

5

(b) Fig. 10. Schematic of pressure transfer through the branches at WR6 (5-20%) for ethanol

6

concentration of (a) 5% and (b) 20% in the mixture

7

As shown in Fig. 10, waves are efficiently confined along waveguide branches which

8

correspond to resonance modes’ frequencies of 5% and 20% concentrations. The best wave

9

confinement is achieved on the central frequency. It is also obvious that the field is initially

10

concentrated on the defect cylinders. The difference of wave propagation seen in Fig. 10 is due

11

to the concentration changes of the mixture (which is led to changing of density and speed of

18

1

sound). The central frequency of maximum transmission must be obtained as shown in Fig. 11.

2

Then, by using equation Q = f c / ∆f the values of quality factor are calculated for each step.

3

Fig. 11. Central frequency and Q-Factor for different molar values of WR6 with 0.2% steps

4

Figure 11 shows the quality factor for different values of concentrations. As can be

5

concluded from Fig. 11, despite the fact that smallest sensible difference of concentration is

6

decreased to 0.2%, the quality factor values are still high for each concentration.

7

4.3. Fabrication Tolerance

8

When it come to production concerns, the proposed structure can be affected by fabrication

9

tolerance. Although different parts of the structure can be affected in the fabrication process, it

10

seems that the variations of hollow cylinders’ radius can have the highest effect on sensing

11

operation. Hence, variations of central peak frequency and quality factor of the output spectrum

12

resulted from change in radius difference of two concentric circles (shown in Fig. 4) will be

13

evaluated. In previous section we introduced rh1 and rh2 as 3mm and 2mm, respectively. In this 19

1

section, we consider that changes may arise in the radius size of defect rods during the

2

fabrication process. In order to analyze practical applications of above-mentioned structure, the

3

effect of changing in defect rods radius during the fabrication process has been considered as the

4

fabrication error tolerance and shown in Fig. 12.

5

Fig. 12. Effects of changing the radius of defect rods during the fabrication process on (a)

6

transmission-frequency diagram, (b) central frequency peak, (c) quality Factor and (d)

7

transmission of structure

8

The transmission-frequency diagram is presented in Fig. 12a. The effect of fabrication

9

tolerance on the central peak frequency, quality factor and transmission of structure are depicted

10

in Figs. 12 (b) to (d). These figures show variations that can happen in main factors if the radius

11

of defect rod is changing due to a fabrication error.

12 13

According to Fig. 12, we assume the outer hollow rods radius with ∆rh error as the fabrication tolerance from theoretical designed parameters which can be expressed as: 20

= rtheory + ∆rh

(8)

1

r

2

where r Final is the radius of each rod that has fabrication error comparing to theory value,

3

and r theory is theory design value of each rod’s radius. It can be understood that the variations of

4

rods' radius in this device have the tolerable effect on the structure performance. It is clear that

5

due to the presence of casualties, the output results will decay in experiment.

6

5. Conclusion

Final

7

A novel structure has been proposed in order to design a highly sensitive phononic-

8

crystals-based sensor for sensing different water-ethanol mixtures. Using the finite element

9

method, different linear frequency shifts were measured for various concentrations of the water-

10

ethanol mixture. The proposed structure includes two unequal pathways to use the

11

interferometric advantages of the asymmetric Mach-Zehnder interferometers. As a result, the

12

occurred phase difference removes the excess frequencies around the transmission peak. This

13

leads to a linear variation in frequency peaks due to the linear change in concentration of water-

14

ethanol mixture.

21

1

References

2

[1] A. Salman, O. A. Kaya, and A. Cicek, Determination of concentration of ethanol in water by

3

a linear waveguide in a 2- dimensional phononic crystal slab, Sensors and Actuators A: Physical

4

208 (2014) 50-55.

5

[2] P. Moradi and A. Bahrami, Design of an optomechanical filter based on solid/solid phoxonic

6

crystals, Journal of Applied Physics 123 (2018) 115113.

7

[3] P. Moradi and A. Bahrami, Three channel GHz-ranged demultiplexer in solid-solid phononic

8

crystals, Chinese Journal of Physics 59 (2019) 291-297.

9

[4] V. Narayanamurti, H. L. Stormer, M. A. Chin, A. C. Gossard, and W. Wiegmann, Selective

10

transmission of high-frequency phonons by a superlattice: the “Dielectric” phonon filter,

11

Physical Review Letters 43 (1979) 2012.

12

[5] A. Khelif and A. Adibi, Phononic Crystals, Springer, Berlin, Germany, 2015.

13

[6] T. Gorishnyy, M. Maldovan, C. Ullal, and E. Thomas, Sound ideas, Physics World 18 (2005)

14

24.

15

[7] M. Zubtsov, R. Lucklum, M. Ke, A. Oseev, R. Grundmann, B. Henning, and U. Hempel, 2D

16

phononic crystal sensor with normal incidence of sound, Sensors and Actuators A: Physical 186

17

(2012) 118-124.

18

[8] A. Oseev, M. Zubtsov, and R. Lucklum, Gasoline properties determination with phononic

19

crystal cavity sensor, Sensors and Actuators B: Chemical 189 (2013) 208-212.

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[9] A. Nagaty, A. Mehaney, and Arafa H. Aly, Acoustic Wave Sensor Based on Piezomagnetic

21

Phononic Crystal, Journal of Superconductivity and Novel Magnetism 31 (2018) 1-5.

22

[10] F. Lucklum and M.J. Vellekoop, 3D Phononic-fluidic cavity sensor for resonance measurements of

23

volumetric fluid properties, IEEE SENSORS (2016) 1-3.

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[11] W. Schaaffs, Molecular Acoustics, Springer-Verlag, Berlin, Germany, 1967. 22

1

[12] G. H. Yoon, J. S. Jensen, and O. Sigmund, Topology optimization of acoustic–structure

2

interaction problems using a mixed finite element formulation, International Journal for

3

Numerical Methods in Engineering 70 (2007) 1049-1075.

23

Highlights: A highly sensitive PnC-based sensor is proposed to distinguish water-ethanol mixtures The proposed structure uses advantages of asymmetric Mach-Zehnder Interferometers Ethanol concentration values of 5% to 95% can be sensed with 0.2% differences

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