Design of phononic crystal using open resonators as harmful gases sensor

Scientific Reports volume 13, Article number: 9346 (2023) Cite this article

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This paper investigates the ability to use a finite one-dimensional phononic crystal composed of branched open resonators with a horizontal defect to detect the concentration of harmful gases such as CO2. This research investigates the impact of periodic open resonators, defect duct at the center of the structure, and geometrical parameters such as cross-sections and length of the primary waveguide and resonators on the model's performance. As far as we know, this research is unique in the sensing field. Furthermore, these simulations show that the investigated finite one-dimensional phononic crystal composed of branched open resonators with a horizontal defect is a promising sensor.

Massively producing pollutants in the air has threatened human health, the environment, and global biological ecosystems in recent years1,2. So, detecting harmful gases to human health, such as CO2, NO2, NH3, etc., piqued people's interest in protecting humans and the environment3,4,5,6. As a result, numerous optical studies have been done on detecting toxic gaseous using two-dimension nanostructured materials, such as porous materials5,7 and graphene8,9. In addition, fluorescent, chemical, electrochemical, photonic crystal, and mass-sensitive are common gas sensors10,11,12,13.

Phononic crystals (PnCs) are periodic artificial materials14,15,16. PnCs have sparked considerable interest in various biosensing and chemical applications. PnCs can confine acoustic or elastic waves by creating stop frequency bands or phononic bandgaps (PnBGs) to propagate through them17,18. Acoustic properties of materials, such as viscosity, density, speed of sound, elastic moduli, etc., can be probed by propagating the acoustic wave inside19. One-dimensional PnC (1D-PnC) sensors are resonant detectors. The main operating concept of 1D-PnC sensors is the multiple Bragg scattering of acoustic waves at each interface between two mediums with different acoustic impedance to produce a standing wave. The frequency of the PnBG depends on the traveling wave's acoustic speed and the structure's geometrical dimensions. Most 1D-PnC sensors are based on breaking the periodicity at the center of the structure, resulting in a resonant peak inside the PnBG. Adding this defect at the center of the structure confines a specific frequency called resonant frequency.

In traditional PnCs, continuity of flux and pressure are considered along the main direction of propagation. Recently, locally resonant elements have attracted attention in the field of periodic structures. However, lateral elements or resonators that depend on the change of pressure or flux stability in other paths can be added. These lateral elements can be closed or open ducts. In 2008, El Boudouti et al.20 proposed a structure of a slender tube with lateral ducts. The presence of lateral tubes causes the formation of stop bands in the transmittance spectrum. In 2020, Antraoui et al. designed a periodic structure composed of a main duct with open resonators. But utilizing these structures with lateral resonators in gas sensing applications is still lacking.

Recently, gas sensors using PnCs attracted attention due to their advantages. For example, gas sensors using PnCs do not require a recovery time. Besides, as PnC doesn't contain any electronic component, gas sensors using PnCs can give good measurements in complex environments such as in an explosive environment21. Furthermore, the low cost and ease of fabrication of PnC sensors are good advantages22.

As far as we know, this research is unique in the gas sensing field. Using branched open resonators enhanced the sensor's performance. Furthermore, these simulations show that the investigated finite one-dimensional phononic crystal composed of branched open resonators with a horizontal defect is a promising sensor. Furthermore, the proposed PnC sensor with branched open resonators can be easily fabricated using low-cost conventional materials.

In Fig. 1, a schematic of the 1D-PnC composed of branched open resonators is proposed. The main guide has a cross-section S1 and a thickness d1. The branched open resonators have cross-section S2 and height d2. The proposed 1D-PnC comprises branched-open resonators sensor, and a defect guide sandwiched between two PnCs. The structure will be filled with gas samples containing different concentrations of CO2. The plane wave theory can be used for stationary samples inside the sensor, and the effects of temperature gradients, higher-order modes, and viscosity effects are neglected23.

Schematic of the 1D-PnC composed of branched open resonators.

The theoretical method used to study the response of the proposed periodic branched open resonators to the incident acoustic waves is called the transfer matrix method (TMM) as the following23,24,25,26,27,28,29,30:

where \(A_{i} = \cos \left( {k\frac{{d_{i} }}{2}} \right), B_{i} = j Z_{i} sin\left( {k\frac{{d_{i} }}{2}} \right), C_{i} = \frac{j}{{Z_{i} }}sin\left( {k\frac{{d_{i} }}{2}} \right), D_{i}\) = \(A_{i}\), \(k = {\raise0.7ex\hbox{$\omega $} \!\mathord{\left/ {\vphantom {\omega c}}\right.\kern-0pt} \!\lower0.7ex\hbox{$c$}}\) is the wave number, \(\rho\) is the density, \(Z_{i} = {\raise0.7ex\hbox{${\rho c}$} \!\mathord{\left/ {\vphantom {{\rho c} {{\text{S}}_{{\text{i}}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{\text{S}}_{{\text{i}}} }$}}\) is the impedance of each period of the proposed branched open resonators, and \(c\) is the acoustic speed. The acoustic pressure at the end of the opened lateral chimney is approximately zero, and the acoustic admittance of the acoustic wave (\(y_{R} )\) is calculated as:

For the defect cell:

where \(A_{d} = \cos \left( {k\frac{{d_{d} }}{2}} \right), B_{d} = j Z_{d} sin\left( {k\frac{{d_{d} }}{2}} \right), C_{d} = \frac{j}{{Z_{d} }}sin\left( {k\frac{{d_{d} }}{2}} \right), D_{d}\) = \(A_{d}\), and \(Z_{d} = {\raise0.7ex\hbox{${\rho c}$} \!\mathord{\left/ {\vphantom {{\rho c} {{\text{S}}_{{\text{d}}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{\text{S}}_{{\text{d}}} }$}}\).

Bloch's theorem is used to plot the dispersion relation of the elementary cell of the 1D-PnC composed of branched open resonators23:

where \(K\) is the Bloch vector,\(d = d_{1} + d_{2}\), \(M = \frac{{S_{2} }}{{S_{1} }}\), k is the wave vector. The transmission and transmittance of the 1D-PnC composed of branched open resonators are calculated as the following:

As an initial condition, the geometrical parameters of the main guide and open resonators of the proposed sensors will be N = 10, d1 = 0.6 m, d2 = 0.15 m, dd = 0.3 m, S1 = 1 m2, S2 = 0.75 m2, and Sd = S1 m2. Table 1 shows the acoustic properties of an air sample at different concentrations of CO2. The gradient of the density of the sample from low to high and acoustic speed from high to low with the increase of the CO2 concentration ensures that both density and acoustic speed can be considered an indicator of the concentration of CO2.

The transmittance (red spectra) and dispersion relation (blue spectra) curves versus frequency of the proposed 1D-PnC composed of branched open resonators without defect are plotted and coincided using TMM and Bloch's theorem in Fig. 2A. In the frequency range of concern, two PhBGs extend from 1429.2 to 1478.1 Hz and from 1950.6 to 2000.6 Hz. The proposed 1D-PnC sensor composed of branched open resonators has the ability to make the PnBG due to the periodic change in the impedance and admittance of propagated acoustic waves inside the structure. By adding a horizontal defect tube sandwiched between two identical 1D-PnCs, a specific frequency of the incident acoustic wave is localized, making a defect peak inside the PnBG. This peak is very sensitive to any change in the mechanical properties of the medium inside the tubes. Considering an additional defect tube with dd = 0.3 m at the middle of the design and the other geometrical parameters having the same initial values, a resonant peak appears at the center of each PnBG, as clear in Fig. 2B.

(A) The dispersion relation (blue line), the transmittance of the 1D-PnC composed of branched open resonators without defect cell (red line) using air sample (exceed in CO2 = 0%), and (B) the transmittance with a defect (blue spectrum) using air sample with different CO2 concentrations.

Any change in the density or acoustic speed of the gas sample due to the change in the CO2 concentration will result in a transmittance spectrum and cause a wavelength shift to the resonant peaks and PnBGs, as clear in Fig. 3. The defect peak is redshifted to lower frequencies by increasing the concentration of CO2 from 1975.95 Hz (at 0% of CO2) to 1872.83 Hz (at 20% of CO2), 1772.02 Hz (at 40% of CO2), 1672.36 Hz (at 60% of CO2), 1612.45 Hz (at 80% of CO2), and 1575.00 Hz (at 100% of CO2).

The transmittance of the 1D-PnC sensor composed of branched open resonators with a defect cell using different concentrations of CO2.

The sensitivity, figure of merit (FoM), quality factor (Q), and detection limit (LoD) of the harmful gas's sensor are used to examine the efficacy of the sensor and can be defined as follows,

where \(\Delta f_{R}\) is the value of the resonant frequency shift with changing the acoustic speed by (\(\Delta c\)), and FWHM is the peak bandwidth. Sensitivity is the change in the position of the defect peak relative to the acoustic speed relative to the pure air sample as a reference. Q denotes the resonator's energy loss and is expressed as the ratio of the frequency of the defect peak to the FWHM. The sensor's ability to discover the alteration in the resonance frequency is represented by FoM32. LoD denotes the slightest change in the sample that can be detected.

Figure 4A–C shows the S, FWHM, T, FoM, Q, and LoD versus the thickness of dd. Figure 4A clears the sensitivity and FWHM versus the incident frequency for the proposed 1D-PnC sensor composed of branched open resonators with a defect cell at different dd values to select the best value that gives the highest performance. The sensitivity is measured for the proposed sensor at different thicknesses of dd of 0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, and 0.6 m. In Fig. 4A, the sensitivity is slightly reduced from 5.82 Hz m−1 s to 5.79 Hz m−1 s, 5.76 Hz m−1 s, 5.73 Hz m−1 s, 5.71 Hz m−1 s, and 5.69 Hz m−1 s with the increase of dd.

(A) S and FWHM, (B) transmittance and FoM, and (C) Q and LoD versus the thickness of dd.

Sharp defect peaks with 100% intensity at resonant frequencies of 1996.94 Hz, 1986.81 Hz, 1975.95 Hz, 1966.27 Hz, 1958.48 Hz, and 1952.73 Hz for air sample and frequencies of 1591.73 Hz, 1583.67 Hz, 1575.00 Hz, 1567.29 Hz, 1561.08 Hz, and 1556.49 Hz for CO2 sample at thicknesses of 0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, and 0.6 m, respectively. The right axis of Fig. 4A clears the variations in the FWHM of the resonant peak with dd. At dd = 0.3 m, the FWHM has the lowest value of 0.14 Hz. As a result of the behavior of FWHM, the FoM, and Q have the highest values at the same thickness, according to Eqs. (8) and (9). On the other hand, the LoD has a minor performance at dd = 0.3 m. dd = 0.3 m will be the optimum value. This thickness achieved high performance because the resonant peak is located at the center of the PnBG.

The reliability of the 1D-PnC sensor composed of branched open resonators is investigated by studying the impact of the cross-section of Sd on S, FWHM, T, FoM, Q, and LoD at different concentrations of CO2, as shown in Fig. 5A–C. The defect peak and PnBG exhibit a redshift to lower frequencies as the cross-section of Sd gradually increases. The S decreases from 5.77 to 5.74 Hz m−1 s as the cross-section of Sd increases from 0.9 to 1.4 m2. However, the FWHM gradually increases with the cross-section of Sd. Besides, the T of the resonant peak records the highest intensity of (100%). Hence, the FoM and Q gradually decrease, and LoD gradually increases. Depending on the results in Fig. 5A–C, the cross-section of Sd = 1 m2 will be used in the following studies.

(A) S and FWHM, (B) transmittance and FoM, and (C) Q and LoD versus the cross-section of Sd.

As d1 increases from 0.59 m to 0.60 m, 0.61 m, and 0.63 m, the peak of the air sample is redshifted from 2001.36 Hz to 1975.95 Hz, 1950.14 Hz, and 1897.80 Hz, and the peak of the CO2 sample is redshifted from 1595.26 Hz to 1575.00 Hz, 1554.42 Hz, and 1512.70 Hz. In Fig. 6A, the sensitivity decreases linearly with increasing d1. On the other hand, FWHM gradually increases with increasing d1. The transmittance records intensity above 99.9% for thickness d1 higher than 0.59 m, as clear in Fig. 6B,C. Besides, FoM and Q gradually decrease, and LoD gradually increases with increasing d1. Therefore, a thickness of 0.59 m will be optimum.

(A) S and FWHM, (B) transmittance and FoM, and (C) Q and LoD versus the thickness of d1.

Figure 7A clears the sensitivity and FWHM versus the incident frequency for the proposed 1D-PnC sensor composed of branched open resonators with a defect cell at different values of d2 to select the best value that gives the highest performance. The sensitivity is measured for the proposed sensor at different thicknesses of d2 of 0.148 m, 0.149 m, 0.15 m, and 0.152 m. In Fig. 7A, the sensitivity is increased from 4.30 Hz m−1 s to 5.84 Hz m−1 s with the increase of d2 from 0.148 m to 0.149 m. Then, sensitivity slightly decreases to 5.83 Hz m−1 s with the increase of d2 to 0.150 m. After that, sensitivity is significantly reduced to 4.29 Hz m−1 s with the increase of d2 to 0.152 m. At d2 = 0.150 m, the FWHM has the lowest value of 0.068 Hz. The T of the resonant peak changes from 99.24% to 93.26%, 94.15%, and 99.76% by changing the thickness of d2 from 0.148 m to 0.149 m, 0.15 m, and 0.152 m. As a result of the behavior of FWHM and sensitivity, the FoM and Q have the highest values at the same thickness, according to Eqs. (8) and (9) and Fig. 7B,C. On the other hand, the LoD has the smallest performance at d2 = 0.150 m. d2 = 0.150 m will be the optimum value.

(A) S and FWHM, (B) transmittance and FoM, and (C) Q and LoD versus the thickness of d2.

Figure 8A–C shows the variations in S, FWHM, T, FoM, Q, and LoD with cross-sections S2. The defect peak and PnBG exhibit a redshift to lower frequencies as the cross-section of S2 gradually increases. The S gradually decreases from 5.84 to 5.83 Hz m−1 s as the cross-section of Sd increases from 0.71 to 0.85 m2. Also, the FWHM gradually decreases with the increase of the cross-section of S2 for all selected values of cross-sections except at 0.79 m2 and 85 m2. At these values (0.79 m2 and 85 m2), the FWHM records a small increase. The T of the resonant peak changes from 94.97% to 97.18%, 94.15%, 95.07%, 77.00%, 94.46%, 90.5%, and 66.48% by changing the cross-section of S2 from 0.71 m2 to 0.73 m2, 0.75 m2, 0.77 m2, 0.79 m2, 0.81 m2, 0.83 m2, and 0.85 m2. FoM changes from 64.68 m−1 s to 75.45 m−1 s, 85.42 m−1 s, 97.58 m−1 s, 89.94 m−1 s, 130.54 m−1 s, 140.90 m−1 s, and 109.83 m−1 s by changing the cross-section of S2 from 0.71 m2 to 0.73 m2, 0.75 m2, 0.77 m2, 0.79 m2, 0.81 m2, 0.83 m2 and 0.85 m2. Besides, Q changes from 22,183.31 to 25,879.92, 29,298.19, 33,472.38, 30,849.29, 44,775.59, 48,326.07, and 37,670.87 by changing the cross-section of S2 from 0.71 m2 to 0.73 m2, 0.75 m2, 0.77 m2, 0.79 m2, 0.81 m2, 0.83 m2, and 0.85 m2. On the other hand, LoD changes from 8 × 10–4 m s−1 to 7 × 10–4 6 × 10–4 m s−1, 5 × 10–4 m s−1, 6 × 10–4 m s−1, 4 × 10–4 m s−1, 4 × 10–4 m s−1 and 5 × 10–4 m s−1 by changing the cross-section of S2 from 0.71 m2 to 0.73 m2, 0.75 m2, 0.77 m2, 0.79 m2, 0.81 m2, 0.83 m2 and 0.85 m2. As a result, 0.83 m2 will be the optimum cross-section.

(A) S and FWHM, (B) transmittance and FoM, and (C) Q and LoD versus the cross-section of S2.

At selected conditions, the defect peak is redshifted to lower frequencies by increasing the concentration of CO2 from 1999.02 Hz (at 0% of CO2) to 1894.7 Hz (at 20% of CO2), 1792.71 Hz (at 40% of CO2), 1691.89 Hz (at 60% of CO2), 1631.27 Hz (at 80% of CO2), and 1593.39 Hz (at 100% of CO2), as clear in Fig. 9A. This redshift of the PnBG and resonant peak to lower frequencies is due to the direct proportionality between the acoustic speed of the sample and the resonant frequency according to the standing wave equation:

where d and n are the thickness and an integer, respectively. In Fig. 9B, the acoustic speed and resonant frequency versus the concentration of CO2 are plotted. An empirical equation between the resonant frequency (\(f_{R}\)) and the concentration of CO2 (\(C_{CO2}\)) was established using the quadric fitting as the following relation:

(A) The transmittance at selected conditions using air samples with different CO2 concentrations, and (B) acoustic speed and resonant frequency versus the concentration of CO2.

By fitting the simulated data, by knowing the resonant frequency, the CO2 concentration can be predicted according to the following equation:

This study proposed a branched open resonator sensor with a defect guide sandwiched between two PnCs. The structural properties and geometrical parameters of the 1D-PnC sensor composed of branched open resonators were thoroughly optimized. The above simulation studies indicate that the suggested 1D-PnC composed of branched open resonators can effectively detect the concentration of CO2 with a sensitivity of 5.8 Hz m−1 s, FoM of 140 m−1.s, Q of 5 × 104, and LoD of 4 × 10–4. Using branched open resonators enhanced the sensor's performance, according to Table 2. As a result, the suggested design could be useful in different sensing and filtering devices.

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The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the Project Number: IFP22UQU4350068DSR160.

TH-PPM Group, Physics Department, Faculty of Science, Beni-Suef University, Beni Suef, 62521, Egypt

Zaky A. Zaky & Arafa H. Aly

Department of Physics, College of Applied Sciences, Umm Al-Qura University, Mecca, Saudi Arabia

M. A. Mohaseb

Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, Russia, 620002

Ahmed S. Hendy

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Z.A.Z. invented the original idea of the study, implemented the computer code, performed the numerical simulations, analyzed the data, wrote and revised the main manuscript text. M.A.M. discussed the results and analyzed the data. A.S.H. discussed the results and co-wrote the revised manuscript. A.H.A. discussed the results and analyzed the data. Finally, all Authors developed the final manuscript.

Correspondence to Zaky A. Zaky or M. A. Mohaseb.

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Zaky, Z.A., Mohaseb, M.A., Hendy, A.S. et al. Design of phononic crystal using open resonators as harmful gases sensor. Sci Rep 13, 9346 (2023). https://doi.org/10.1038/s41598-023-36216-y

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Received: 11 March 2023

Accepted: 31 May 2023

Published: 08 June 2023

DOI: https://doi.org/10.1038/s41598-023-36216-y

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