Simulation study of gas sensor using periodic phononic crystal tubes to detect hazardous greenhouse gases

Abstract

Here, we investigate a gas sensor model based on phononic crystals of alternating tubes using the transfer matrix method to detect hazardous greenhouse gases. The effect of the thicknesses and cross-sections of all tubes on the performance of the proposed sensor is studied. The results show that longitudinal acoustic speed is a pivotal parameter rather than the mass density variations of the gas samples on the position of the resonant peaks due to its significant impact on the propagation of the acoustic wave. The suggested sensor can be considered very simple and low-cost because it does not need a complicated process to deposit multilayers of different mechanical properties’ materials.

Introduction

In recent years, there has been a lot of interest in using periodic media in different acoustics^1,2 and optics applications^3–12. Acoustic devices with good performance are vital in biosensing¹³, acoustic communication¹⁴, gas detection¹⁵, etc. The occurrence of sound wave dispersion and forbidden frequency bands in phononic crystals (PnCs) has increased interest in them. Photonic crystals’ (PCs) acoustical counterparts are PnCs^4,16,17. PnC is a periodic geometry developed from the expression of lattice vibration, which refers to phonon. The tubes of PnC are structured in a regular pattern, which allows for spatial manipulation of the acoustic characteristics and promotes the reflection of elastic and acoustic waves, resulting in band gaps. Phononic bandgap (PnBG) is proposed to prevent the propagation of a certain frequency range of elastic and acoustic waves within PnC.

PnBG property offers a novel method of regulating vibration and noise¹⁸. Besides, PnBG can be created using a periodic branched resonator¹⁹. The ratio of acoustic impedances between successive layers, periodicity, physical properties, and the geometry of the materials will determine the depth, location, and bandwidth of the created PnBG. Kushwaha et al.²⁰ investigated the PnCs band structure and studied the acoustic wave control concept. PnCs containing a defect layer have gotten much attention due to the appearance of a very thin defect mode with high transmittance. As a result of the defect mode, the acoustic wave at specific energy can travel through the PnBG. Within the defect layer, the acoustic energy is trapped in the form of a passband, like what’s happening for photons in the defected PC²¹. The defect mode's frequency depends on the defect layer's geometry, position, and nature. The appearance of this resonant mode gives the PnC a chance to be used in different applications such as filters and detectors.

There have been numerous gas detectors developed based on PC and PnCs^22,23. PnCs are well-known for gas and liquid monitoring because elastic characteristics of materials vary dramatically from other properties such as density, resulting in high performance²⁴. Infrared radiation is absorbed and reradiated by hazardous greenhouse gases such as CH₄CO₂, N₂O, and CO₂ generated from the Earth and trap it. As a result, these gases are considered particularly harmful chemicals²⁵. In 2022, Imanian et al.²⁴ presented a gas sensor to detect different types of gases such as CH₄, O₂, NH₃, CO₂, N₂, Ar, CO₂, and C₃H₈ at different concentrations in the air with a sensitivity of 0.69 MHz kg^–1 m³. Zaki and Basyooni²⁵ proposed CO₂, N₂O, and CH₄ gas sensor using the concept of Fano resonance within PnC. Kaya et al.²⁶ introduced an acoustic ring resonator for spoof surface acoustic waves (SSAW) on 1D-PnC. Authors observed high Q-factor peaks with very small bandwidths that are exceptionally qualified for gas sensing applications. Cicek et al.²⁷ numerically and experimentally proposed a gas detector based on SSAWs using 2D-PnC spherical Helmholtz 2D-PnC resonators to detect CO₂ and CH₄ in air.

The proposed gas sensor is based on alternating tubes as a one-dimensional PnC (1D-PnC). This study aims to analyze the acoustic interaction with an expansion chamber as a gas sensor using the transfer matrix method (TMM) to detect hazardous greenhouse gases. The frequency response of the defect mode that depends on the change in the gas sample's acoustic properties, which fills the structure, will be studied. The suggested sensor is simple and low-cost because it does not need a complicated process to deposit multilayers of different acoustic properties’ materials. Besides, the proposed sensor does not need a recovery time. It is simple alternating tubes with different cross-sections in periodic sequences. Also, The suggested structure recorded very high linearity between the change in the resonant peak and the acoustic velocity of the sample.

Basic equations and model design

The intended detector design is depicted in Fig. 1. The unit cell consists of two cavity muffler tubes²⁸ with different thicknesses and cross-sections. This unit cell is repeated for N finite periods to make a PnC crystal. The first tube has a thickness of d₁ and a cross-section of S₁. Also, the second tube has a thickness of d₂ and a cross-section of S₂. The defect tube has a thickness of d_D and a cross-section of S_D. The cross-sectional areas of resonators are designed perpendicularly to the main tube. The gas sample will fill the proposed sensor. The gas samples are studied at standard pressure (1 atm) and room temperature (20 °C). For a symmetrical periodic structure with a defect, there may be two resonance peaks (topological edge state peak and defect mode peak). However, we need only one peak to prevent the overlap between peaks. So, we preferred to use an asymmetrical structure.

Figure 1.

The schematic of the suggested PnC gas sensor with tube sequence of (M₁ M₂)^N M_D ( M₁ M₂)^N.

The Helmholtz equation investigates the propagation of the acoustic wave within the proposed sensor structure as²:

$$\mathrm{\nabla }.\left(\frac{1}{\rho },\mathrm{\nabla },P\right)+\frac{{\omega }^{2}P}{\rho c_L^2}=0$$ 1

where $c_L$ is the longitudinal speed of the acoustic wave through the tub, and P is the acoustic pressure. TMM is a standard method for studying the transmission and reflection of the incident acoustic and optic waves with multitube structures^29–34. The following matrix describes the interplay between the proposed system and acoustic waves.

$$\left|\begin{array}{cc}{A}_{11}\hfill & A_{12}\hfill \\ A_{21}\hfill & A_{22}\hfill \end{array}\right|={\left(M_1,M_2\right)}^N{M}_D{\left(M_1,M_2\right)}^N,$$ 2

$$M_i=\left[\begin{array}{cc}cos(k{d}_i)\hfill & \left(-,\frac{i}{{\varnothing }_i}\right)sin(k{d}_i)\hfill \\ -i{\varnothing }_{i}sin(k{d}_i)\hfill & cos(k{d}_i)\hfill \end{array}\right],i=1,2,N,$$ 3

where $\mathrm{k}$ is the wave vector ($k=\omega /c)$,${\varnothing }_i$ is the reciprocal of the impedance ($Z_i$):

$$Z_i=\rho c/{\mathrm{S}}_{\mathrm{i}}.$$ 4

Chebyshev’s polynomial is used to repeat the unit cell $\left(M_1,M_2\right)$ for N times. The transmittance of acoustic waves through the proposed hazardous gases sensor can be calculated as a function of transmission:

$$t=\frac{2{\varnothing }_1}{\left(A_{11},+,A_{12},{\varnothing }_1\right){\varnothing }_1+\left(A_{21},+,A_{22},{\varnothing }_1\right)}$$ 5

$$T\left(\%\right)=100\ast {\left|t\right|}^2$$ 6

Results and discussion

Pure gases calculations

This section studies the acoustic wave propagation within the proposed hazardous gases sensor structure without and with the defect tube. The initial geometrical conditions of the design will be d₁ = 0.1 m, d₂ = 0.2 m, d_D = 0.15 m, N = 6, S₁ = 0.1 m², S₂ = 0.2 m², S_D = 0.15 m², the acoustic wave will fall on the structure with incident angle ${\theta }_0$= 0°, at atmospheric pressure, at the sea level altitude, and the room temperature. Besides, the gas samples inside tubes are stationary. Table 1 clear the mass density and acoustic speed of gas samples.

Table 1.

Acoustic properties of different gas samples:

Gas sample	Density (kg/m3)	Acoustic speed (m/s)	Reference
SO2	2.279	201	35
CO2	1.8393	267	35
C3H8	1.8102	249	35
Ar	1.661	319	35
O2	1.314	326	35
Air	1.2047	343	35
N2	1.165	349	24
NH3	0.7069	430	35
CH4	0.659	445	35

For a perfect PnC (M₁M₂)^N (M₁M₂)^N without the defect tube, an ideal PnBG is extended from 460 to 670 Hz, as evident in Fig. 2A. Within this range, the propagation of acoustic waves is forbidden. Besides the plotting of the transmittance spectra of acoustic waves, the dispersion relation is studied according to Bloch’s theorem as the following¹⁹:

$$\cos \left(K,d\right)=\cos \left(k,d_1\right)\cos \left(k,d_2\right)-\frac{1}{2}\left[m,+,\frac{1}{m}\right]\sin \left(k,d_1\right)\sin \left(k,d_2\right)$$ 7

where $d$ is the thickness of the elementary unit cell, $K$ is the Bloch wave propagation vector, k is the wave vector, and $m=\frac{S_2}{S_1}$. The real part of $K$ is used to study the change in the propagating phase of waves in a pass band. Figure 2A shows that the PnBG of the transmittance spectra coincides with the band structure of the real part of the Bloch wavenumber.

Figure 2.

(A) The transmittance spectra of the proposed gas sensor without defect (left axis) and the band structure of the real part of the Bloch wavenumber (right axis) using air samples, and (B) the defect peaks of different gas samples by using defect.

As straightforward in Fig. 2B, due to the existence of a defect tube with a cross-section of 0.15 m² and thickness of 0.15 m, a sharp resonant peak to the change in the nature of the sample of gas inside the tube of the proposed sensor appeared inside the PnBG due to the localization of acoustic waves in the defect layer. With the change of the gas sample from SO₂ to C₃H₈, CO₂, Ar, O₂, Air, N₂, NH₃, and CH₄, the PnBG extended from 265 to 392 Hz (SO₂ gas sample), from 326 to 486 Hz (C₃H₈ gas sample), from 350 to 517 Hz (CO₂ gas sample), from 418 to 621 Hz (Ar gas sample), from 430 to 635 Hz (O₂ gas sample), from 449 to 671 Hz (Air sample), from 460 to 681 Hz (N₂ gas sample), from 563 to 842 Hz (NH₃ gas sample), and from 582 to 873 Hz (CH₄ gas sample), respectively. Besides, the defect mode is generated at 333.6 Hz, 413.2 Hz, 443.1 Hz, 529.4 Hz, 541.1 Hz, 569.3 Hz, 579.2 Hz, 713.7 Hz, and 738.6 Hz, respectively.

Although the peak shift depends on both the mass density and acoustic speed of each sample, the sensitivity will be calculated as a function of mass density during the optimization process^24,35,36. The sensitivity (S) of the investigated gas PnC sensor is represented as the ratio between the peak shift for each gas sample and the change in mass density ($\mathrm{\Delta }{\rho }_{\mathit{gas}}$). The figure of merit (FoM) is the ratio between S and the bandwidth (FWHM) of the resonant mode. The quality factor (Q-factor) is the ratio between the position of the resonant mode ($f_R)$ and the bandwidth. The limit of detection (LoD) is the lowest mass density change that can be reliably detected. S, FoM, Q-factor, and LoD can be calculated as the following to evaluate the performance of the selected conditions:

$$S=\frac{\mathrm{\Delta }{f}_R}{\mathrm{\Delta }{\rho }_{\mathit{gas}}}=\left|\frac{f_{\mathit{air}}-f_{CH4}}{{\rho }_{\mathit{air}}-{\rho }_{CH4}}\right|,$$ 8

$$FoM=\frac{S}{\mathit{FWHM}},$$ 9

$$\begin{array}{c}Q=\frac{f_R}{\mathit{FWHM}}\hfill \end{array},$$ 10

$$LoD=\frac{f_R}{20SQ}.$$ 11

In the following optimization process, the change between the resonant peak of air and pure CH₄ samples will be used for sensitivity calculations. The FWHM will be calculated as an indicator for the resonant peak of air samples.

Figure 3 shows the sensitivity, transmittance of the resonant mode of air samples, FWHM of the resonant mode of air samples, FoM, Q-factor, and LoD of the proposed sensor at different defect tube thickness considerations from 0.15 to 0.25 m. Outside the scope of this range of defect tube thickness, the resonant peaks overlap (Supplementary Data: Sup. 1). At a frequency range of 0 Hz < f < 1000 Hz, the PnBG and the resonant peak are shifted to lower frequencies (red-shift), and the sensitivity is decreased from 310 to 271  Hz m³ kg⁻¹ with the increase of the thickness of the defect tube, as evident in Fig. 3A. The resonant peaks for which the transmittance varies between 89 and 94% are shifted to lower frequencies³⁷ from 738.56 Hz (at d_D = 0.15 m), 711.13 Hz (at d_D = 0.175 m), 686.16 Hz (at d_D = 0.2 m), 663.93 Hz (at d_D = 0.225 m), and 644.47 Hz (at d_D = 0.25 m). The transmittance of the sample resonant peaks slightly increases with the defect tube thickness increase due to the acoustic wave confinement inside the sensor tube. As explicit in Fig. 3B, a change in the thickness of the defect tube leads to a change in the FWHM of the transmitted peaks. The FWHM increases from 0.25 to 0.50 Hz with the increase of the thickness of the defect tube from 0.15 to 0.25 m. As the FoM, Q-factor, and LoD are dependent parameters; their behaviors can be predicted according to Eqs. (9, 10, 11), as apparent in Fig. 3A–C. Therefore, a thickness of 0.15 m will be the optimum value.

Figure 3.

(A) sensitivity between air and pure CH₄ samples and transmittance of the resonant mode of air samples, (B) FWHM of the resonant mode of air samples and FoM, and (C) Q-factor and LoD of the proposed gas sensor at different values of defect tube thickness.

Figure 4A represents the decrease in sensitivity of the proposed detector with the increase of the first tube thickness of the unit cell. Figure 4A shows the slight change in the transmittance of the resonant mode of air samples at different values of the first tube thickness of the unit cell. When the first tube thickness of the unit cell changes from 0.075 to 0.2 m, the FWHM decreases from 0.55 to 0.11 Hz, as evident in Fig. 4B. As a result of the behavior of sensitivity and FWHM, FoM and Q-factor record the highest value at a thickness of 0.15 m, then seems to be constant, as apparent in Fig. 4B,C. On the other hand, the LoD decreases to the minimum value at a thickness of 0.15 m and seems to become constant at higher thicknesses, as evident in Fig. 4C. Therefore, the thickness of 0.075 m is better because the proposed sensor recorded the most heightened sensitivity.

Figure 4.

(A) sensitivity between air and pure CH₄ samples and transmittance of the resonant mode of air samples, (B) FWHM of the resonant mode of air samples and FoM, and (C) Q-factor and LoD of the proposed gas sensor at different values of first tube thickness of the unit cell.

Figure 5 shows the performance of the proposed sensor at different values of the second tube thickness considerations from 0.1 to 0.2 m. At a frequency range of 0 Hz < f < 1500 Hz, the sensitivity is decreased from 456 to 332 Hz m³ kg⁻¹ with the increase of the thickness of the second tube, as clear in Fig. 5A. The position of resonant peaks was found at 1085.86 Hz (at d_D = 0.1 m), 992.01 Hz (at d_D = 0.125 m), 914.89 Hz (at d_D = 0.15 m), 848.57 Hz (at d_D = 0.175 m), and 790.07 Hz (at d_D = 0.2 m). The transmittance of the resonant peaks at these thicknesses slightly varies between 90 and 93%, as clear in Fig. 5A. As straightforward in Fig. 5B, a change in the thickness of the second tube leads to a change in the FWHM of the transmitted dip. The FWHM decreases from 0.61 to 0.37 Hz with the increase of the thickness of the second tube from 0.1 to 0.15 m. After that, the FWHM gradually increases to 0.55 Hz. As the FoM, Q-factor, and LoD are dependent parameters on sensitivity and FWHM according to Eqs. (9, 10, 11), their behaviors depend on sensitivity and FWHM, as clear in Fig. 3B,C. As the highest sensitivity is at the thickness of 0.1 nm, this thickness of the second tube will be optimum.

Figure 5.

(A) sensitivity between air and pure CH₄ samples and transmittance of the resonant mode of air samples, (B) FWHM of the resonant mode of air samples and FoM, and (C) Q-factor and LoD of the proposed gas sensor at different values of second tube thickness of the unit cell.

Figure 6A represents the slight change in sensitivity of the proposed detector with the change in the cross-sections of the defect tube. Figure 6A shows that the transmittance of the resonant mode of air samples drops quickly with the increase of the cross-sections of the defect tube. The transmittance decreases from 92.5 to 55.2% with the rise of the cross-sections of the defect tube from 0.15 to 0.3 m². When the cross-section of the defect tube changes from 0.15 to 0.25 m², the FWHM increases from 0.61 to 0.67 Hz, as clear in Fig. 6B. Then, the FWHM decreases to 0.65 Hz at a cross-section of 0.3 m². As a result of the behavior of sensitivity and FWHM, FoM and Q-factor record the lowest value of 683 m³/kg and 1627 at a thickness of 0.25 m, then decreases to 705 m³/kg and 1678 at cross-section of 0.3 m, as clear in Fig. 6B,C respectively. On the other hand, the LoD increases to a maximum value at a cross-section of 0.25 m² and seems constant at higher cross-sections, as clear in Fig. 4C. The cross-section of 0.15 m² is the best because the proposed sensor recorded the lowest FWHM, highest FoM, highest transmittance, highest Q-factor, and lowest LoD.

Figure 6.

(A) sensitivity between air and pure CH₄ samples and transmittance of the resonant mode of air samples, (B) FWHM of the resonant mode of air samples and FoM, and (C) Q-factor and LoD of the proposed gas sensor at different values of cross-sections of the defect tube.

Figure 7A shows the sensitivity when the cross-section of the first tube of the unit cell is S₁ = 0.09 m², S₁ = 0.1 m², S₁ = 0.11 m², and S₁ = 0.125 m². As the cross-section of the first tube gradually increases, it can be seen that the resonant peak undergoes a blue shift to higher frequencies, and the sensitivity increases gradually from 448 to 476 Hz m³ kg⁻¹. This behavior is because the cross-section of the first tube of the unit cell is large, the impedance of the proposed structure to the acoustic wave decreases, and the PnBG containing the defect peak is blue-shifted to higher frequencies, resulting in an increase in sensitivity. Above the cross-section of 0.125 m², the resonant peak goes out from the PnBG. Besides, the transmittance of the peaks increases from 87.7 to 97.7% with the increase of the cross-section of the first tube of the unit cell from 0.09 to 0.125 m², as clear in Fig. 7A. As regards the impact of the increase of the cross-section of the first tube of the unit cell on the FWHM, the FWHM increases from 0.24 to 3.75 Hz with the increase of the cross-section of the first tube of the unit cell from 0.09 to 0.125 m², as clear in Fig. 7B. In Fig. 7B,C, the FoM and Q-factor decrease with the increase of the cross-section of the first tube. For LoD, it increases with the increase of the cross-section of the first tube, as clear in Fig. 7C. The cross-section of 0.125 m² is optimum.

Figure 7.

(A) sensitivity between air and pure CH₄ samples and transmittance of the resonant mode of air samples, (B) FWHM of the resonant mode of air samples and FoM, and (C) Q-factor and LoD of the proposed gas sensor at different cross-sections of the first tube of the unit cell.

Similarly, Fig. 8A gives the impact of the cross-section of the second tube of the unit cell on sensitivity at S₂ = 0.2 m², S₂ = 0.22 m², S₂ = 0.24 m², and S₂ = 0.28 m². As demonstrated, the sensitivity is slightly decreased from 475.5 to 451.1 Hz m³ kg⁻¹ with the increase of the cross-section of the second tube. Figure 8A gives the relation between the transmittance of the resonant peaks and the cross-section of the second tube. Clearly, the transmittance is slightly affected by the change in the cross-section of the second tube from 0.20 to 0.28 m². In contrast, the FWHM strongly decreases from 3.75 to 0.18 Hz with increasing the cross-section of the second tube from 0.20 to 0.28 m², as clear in Fig. 8B. Figure 8B gives the FoM of the proposed sensor as a function of the cross-section of the second tube. The highest and lowest values of FoM are at S₂ = 0.28 m² and 0.20 m², respectively. Similar to Fig. 8B, the Q-factor changes dramatically with the cross-section of the second tube from 0.20 to 0.28 m², as clear in Fig. 8C. In the case of the LoD, with the increase of the cross-section of the second tube from 0.20 to 0.28 m², the LoD dramatically decreases from 4 × 10^–4 to 2 × 10^–5 $\mathrm{kg}{\mathrm{m}}^{-3}$, as clear in Fig. 8C.

Figure 8.

(A) sensitivity between air and pure CH₄ samples and transmittance of the resonant mode of air samples, (B) FWHM of the resonant mode of air samples and FoM, and (C) Q-factor and LoD of the proposed gas sensor at different cross-sections of the second tube of the unit cell.

Figure 9A demonstrates the sensitivity when the numbers of unit cells are N = 6, N = 7, N = 8, N = 9, and N = 10. As the number of unit cells increases, the resonant peak undergoes a slight blue shift to higher frequencies, and the sensitivity increases modestly from 475.5 to 476.5 Hz m³ kg⁻¹. The resonant peak for N smaller than 5 goes out from the PnBG. In Fig. 9A, the transmittance of the peaks seems to be constant (98.5%) with the increase in the number of unit cells. As regards the impact of the unit cell number on the FWHM, the FWHM decreases from 3.75 to 0.29 Hz with the increase of the unit cell number from N = 6 to N = 10, as clear in Fig. 9B. In Fig. 9B,C, the FoM and Q-factor increase with the increase of the unit cell number. For LoD, it decreases with the increase of the unit cell number, as clear in Fig. 9C.

Figure 9.

(A) sensitivity between air and pure CH₄ samples and transmittance of the resonant mode of air samples, (B) FWHM of the resonant mode of air samples and FoM, and (C) Q-factor and LoD of the proposed gas sensor at different numbers of unit cells.

Even though there is no contrast in the mass density of the two tubes of each unit cell (both will be filled with the same gas sample), their acoustic impedances are different due to the change in cross sections according to Eq. (4). Due to the periodicity of the acoustic impedance, the PnBG was formed as clear in Fig. 2A. By adding a defect tube with a different cross-section, the periodicity of the acoustic impedance was broken, and the acoustic waves were localization inside the defect tube, as clear in Fig. 2B. The position of the PnBG and PnC resonant mode depends on the amount of energy localized in the defect tube, which depends on the acoustic properties of the gas filling it. So, by knowing the position of the localized state position within the PnBG, an unknown gas can be detected because each gas has a distinguished localized state position.

In general, PnBG and resonant peak were red-shifted with the increase of thickness (to lower frequencies) according to the standing wave equation:

$$2d=\frac{\mathit{nc}}{f},$$ 12

where d is the thickness, n is an integer, c is the acoustic velocity, and f is the frequency. According to Eq. (12), the frequency is inversely proportional to the thickness of the tubes. Besides, the sensitivity of the proposed sensor decreases with the increase of the thickness of any tube. This negative thickness behavior may be related to the acoustic path and the attenuation of the wave within the tube.

For the cross-section of the tubes, according to Eq. (4), the acoustic impedance is inversely proportional to the cross-section of tubes, and the interaction between the acoustic wave and the gas sample increases with the decrease of the cross-section. Although, we noticed that the sensitivity sometimes increased with the increase of the cross-section and other times decreased. For both the first and second tubes of the unit cell, we found that the proposed sensor recorded high sensitivity when the cross-section of the first tube has a broad cross-section and the second tube has a small cross-section. For the defect tube cross-section, the sensitivity increases with the cross-section because this increase in the defect cross-section increased the contraction between the defect layer and the next layer to it (the first layer of the unit cell). When we tried to increase or decrease the thicknesses or cross-sections from the selected ranges, the resonant peaks overlapped or went out from the PnBG.

Figure 10A shows the transmittance of different gas samples at the selected conditions. As clear in Fig. 10B,C, the acoustic velocity is a significant parameter rather than the mass density variations of the gas samples on the position of the resonant peaks due to its significant impact on the propagation of the acoustic wave (Eq. 1). On the other hand, the change in the mass density of the gas samples is tiny, as evident in Table 1. As a result, the acoustic velocity is an excellent choice to evaluate the performance of the proposed sensor because the mass density does not have a linear relation with the position of the resonant peaks as occurred in the case of the acoustic velocity. So, we will re-calculate the sensitivity of the sensor at the optimum conditions as the ratio between the resonant peak shift and the change in the acoustic velocity as the following:

$$S=\frac{\mathrm{\Delta }{f}_R}{\mathrm{\Delta }{C}_{\mathit{gas}}}=\left|\frac{f_{\mathit{air}}-f_{CH4}}{C_{\mathit{air}}-C_{CH4}}\right|,$$ 13

Figure 10.

(A) the transmittance of different gas samples at the selected conditions, (B) the defect peak position versus the mass density for each gas sample, and (C) the defect peak position versus the acoustic velocity for each gas sample.

The linear relation between the position of the resonant peak and the acoustic velocity of the gas sample can be described as the following:

$$f_R=2.5495c-0.0096,R^2=1$$ 14

According to Eq. (14), the sensitivity of the proposed sensor is constant (2.5495 Hz s m⁻¹) for all gas samples.

Concentration of CO₂ in air calculations

To calculate the mass density of a mixture of air and CO₂ as a function of the concentration of CO₂, the following equation will be used³⁸:

$${\rho }_{\mathit{mix}}={{\rho }_{CO2}{\alpha }_{CO2}+\rho }_{\mathit{air}}(1-{\alpha }_{CO2}),$$ 15

where ${\rho }_{\mathit{mix}}$ is the density of the mixture, ${\rho }_{\mathit{air}}$ is the density of the air, ${\rho }_{CO2}$ is the density of the CO₂, ${\alpha }_{\mathit{air}}$ is the volume fraction of air and ${\alpha }_{CO2}$ is the volume fraction of CO₂. The following relation calculates the longitudinal sound speed of the mixture of air and CO₂ as a function of the temperature (at T = 20 K), the ratio of CO₂ in the air (${\alpha }_{CO2}$), the molecular weights of air ($m_{\mathit{air}}$) and CO₂ ($m_{CO2}$), and specific heat capacity at constant pressure ($C_{p1}$ and $C_{p2}$) and volume ($C_{v1}$ and $C_{v2}$)³⁸:

$$C_{\mathit{mix}}=\sqrt{\left(\frac{R}{m_{\mathit{air}}\left(1,-,{\alpha }_{CO2}\right)+m_{CO2}{\alpha }_{CO2}}\right)\left(\frac{C_{p1}\left(1,-,{\alpha }_{CO2}\right)+C_{p2}{\alpha }_{CO2}}{C_{v1}\left(1,-,{\alpha }_{CO2}\right)+C_{v2}{\alpha }_{CO2}}\right)T},$$ 16

where R is the gas constant 8.314 J/mol-K. By changing the ratio of CO₂ in the air from 0 to 20%, 40%, 60%, 80%, and 100%, the longitudinal sound speed of the mixture changes from 343 m/s to 325.1 m/s, 307.6 m/s, 290.3 m/s, 279.9 m/s, and 273.4 m/s, respectively.

Figure 11A clears the mixture's mass density and the mixture's longitudinal sound speed versus the concentration of CO₂ in the air. With increasing the ratio of CO₂ in the air from 0 to 20%, 40%, 60%, 80%, and 100%, the position of the resonant peak is shifted from 874.5 Hz to 828.8 Hz, 784.2 Hz, 740.1 Hz, 713.6 Hz, and 697.0 Hz, as clear in Fig. 11B. the proposed sensor recorded high stability (fixed sensitivity) for all concentrations, as clear in Fig. 11C.

Figure 11.

(A) the mass density of the mixture and longitudinal sound speed of the mixture versus the concentration of CO₂, (B) the transmittance of different ratios of CO₂ in the air at the optimum conditions, and (C) the sensitivity versus the acoustic velocity at different concentrations.

The main advantage of the proposed sensor is:

1. The structure is very simple and low-cost to be fabricated compared to many other models because it does not need a complicated process to deposit a multi-tube of different mechanical properties’ materials. It is simple alternating tubes with different cross-sections in periodic sequences.

2. The proposed sensor does not need a recovery time.

3. The suggested structure recorded very high linearity between the change in the resonant peak and the acoustic velocity of the sample.

Conclusion

In summary, the proposed PnC gas sensor depends on the contrast in the acoustic velocities between gas samples with tube sequence of (M₁M₂)^N M_D (M₁M₂)^N. The transmittance spectra of the interaction between the incident acoustic wave and the proposed PnC structure were studied using the TMM. All geometrical parameters are optimized to achieve the highest performance. The proposed model can monitor many samples of gases depending on the physio-chemical variations in the structure due to the interaction between the gas sample and acoustic wave inside the system. Our sensor recorded sensitivity, Q-factor, and FoM of 2.5495 Hz s m⁻¹, 4077, and 9.16 s m⁻¹. The main merits of the proposed sensor are the simplicity of fabrication and linearity.

Author contributions

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. S.A. supervised this work. E.I.Z. supervised this work. A.H.A. discussed the results and supervised this work. All Authors developed the final manuscript.

Data availability

All data generated or analysed during this study are included in this published article [and its supplementary information file]. The MATLAB code that support the findings of this study should be addressed to Zaky A. Zaky.

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-022-26079-0.