Theoretical optimisation of a novel gas sensor using periodically closed resonators

Abstract

This study investigates using the phononic crystal with periodically closed resonators as a greenhouse gas sensor. The transfer matrix and green methods are used to investigate the dispersion relation theoretically and numerically. A linear acoustic design is proposed, and the waveguides are filled with gas samples. At the center of the structure, a defect resonator is used to excite an acoustic resonant peak inside the phononic bandgap. The localized acoustic peak is shifted to higher frequencies by increasing the acoustic speed and decreasing the density of gas samples. The sensitivity, transmittance of the resonant peak, bandwidth, and figure of merit are calculated at different geometrical conditions to select the optimum dimensions. The proposed closed resonator gas sensor records a sensitivity of 4.1 Hz m⁻¹ s, a figure of merit of 332 m⁻¹ s, a quality factor of 113,962, and a detection limit of 0.0003 m s⁻¹. As a result of its high performance and simplicity, the proposed design can significantly contribute to gas sensors and bio-sensing applications.

Introduction

Periodic structures in optic and acoustic systems have emerged as an active field of scientific research in the last decades in different applications, such as sensors^1,2,3,4,5, filters⁶, energy harvesting⁷, and smart windows⁸. Moreover, different equipment, buildings, and bridges are disturbed by different vibration sources and elastic waves in their surrounding environment. So, the fantastic mechanical and physical properties of phononic crystals (PnCs) remain a fascinating research domain due to their ability to control the wave propagation of elastic wave or vibration sources⁹. PnCs are artificial structures of unit cells with periodic Young’s modulus and mass density.

The Bragg or phononic band gap (Pn-BG), which represents the range of frequencies in which the real component of wavenumbers does not coincide in dispersion curves, is the main competitive advantage of PnC. The Pn-BGs prevent the transfer of energy through the PnCs because of the multiple destructive interferences due to the repetitive change in the acoustic impedance of the structure¹⁰. Furthermore, by introducing a defect layer or unit cell with different geometric, elastic properties or geometric properties into a PnC, a narrow defect mode appears within the PnBG due to the localization of energy¹¹.

The acoustic resonant frequencies in fluid-filled cavities are commonly used for calculating the properties of gases or liquid substances¹². Cylindrical or rectangular resonance devices can be constructed from glass or stainless steel as a stiff material. The shift in the resonant wavelength or frequency confined in a fluid-filled layer or cavity is highly influenced by changes in the sound speed of the fluid. In this connection, PnCs have engaged severe attention in various sensing attempts^13,14. Recently, numerous scientific research has been done on using defective one-dimensional PnCs (1D-PnCs) with periodic materials in sensing applications^15,16,17.

Taha et al.¹⁸ designed a 1D-PnC sensor to detect the presence of NaCl in a water sample. The structure of this model is composed of two mirror PnCs separated by a cavity. In 2022, Zaki et al.¹⁹ suggested a PnC sensor based on a 2D-materials. The structure consists of four periods of MoS₂/PtSe₂ PnCs containing a defect cavity at the center. Such structures of periodic materials need relatively high techniques and quality to be fabricated. Wang et al.²⁰ demonstrated a locally resonant BG at low longitudinal elastic frequencies in harmonic oscillators connected to a slender beam in a quasi-1D configuration.

This paper investigates a novel gas sensor using periodically closed resonators. This novel study provides new information about using periodically closed resonators as greenhouse gas sensors. Furthermore, this study ensures that PnC sensors with closed resonators perform better than PnC sensors with periodic tubes¹¹. A new generation of tube-adapted sensors is an urgent requirement in industry, medicine, and biology because cylindrical structures are the most common shapes in nature that are used for fluid (liquids and gas) transport. The proposed sensor can be used as a factory chimney and continuously detect the concentration of hazardous gases’ path through it. Besides, the structure can transport gases that we need to characterize simultaneously. This sensor can be manufactured more easily than nano-sensors that require high-quality and expensive techniques.

Sensor array and modeling assumptions

In Fig. 1, a unit cell of a closed resonator is repeated for N periods as a 1D-PnC. Each unit cell has four geometrical parameters, including the cross-section (S₁) of the primary waveguide, the length (d₁) of the primary waveguide, the cross-section (S₂) of the finite waveguide, and the height (d₂) of the finite waveguide. The proposed sensor is made up of two PnCs of N unit cells separated by a defect unit cell with the same geometry of the primary waveguide but different geometry of the finite waveguide.

Figure 1

Defected 1D-PnC composed of closed resonators.

Using the transfer matrix method

The incident wave may be deemed a plane wave for sufficiently large wavelengths. The unimodular acoustic transfer matrix method (UATMM) is used to investigate the interaction between acoustic waves and structures^21,22,23,24. For example, the following matrix can represent each unit cell:

$${M}_{i}=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{cc}1& 0\\ {y}_{D}& 1\end{array}\right]\left[\begin{array}{cc}A& B\\ C& D\end{array}\right],$$

(1)

where $A=\cos\left(k\frac{{d}_{i}}{2}\right), B=j {Z}_{i} \sin\left(k\frac{{d}_{i}}{2}\right), C=\frac{j}{{Z}_{i}}\sin\left(k\frac{{d}_{i}}{2}\right),$ $D=A, k=\omega /c$ is an acronym for the wave number. $\rho$ points to the density. $c$ is the speed of sound. ${Z}_{i}$ refers to the impedance of the acoustic waves:

$${Z}_{i}=\frac{\rho c}{{{\text{S}}}_{{\text{i}}}}.$$

(2)

In this closed resonator, the propagation of the acoustic wave at its end is zero. So, the admittance (${y}_{D})$ of the structure of the incident wave is²²:

$${y}_{D}=j \frac{j}{{Z}_{D}} tan\left(k{d}_{D}\right).$$

(3)

The following Bloch’s equation describes the relation of the acoustic wave dispersion of a unit cell of the proposed periodic structure²⁵:

$${\text{cos}}\left(Kd\right)={\text{cos}}\left(k{d}_{1}\right)- \frac{M}{2}{\text{sin}}\left(k{d}_{1}\right){\text{tan}}\left(k{d}_{2}\right),$$

(4)

where $K$ and k are the Bloch and wave vectors, respectively, $d= {d}_{1}+{d}_{2}$, $M=\frac{{S}_{2}}{{S}_{1}}$. The transmittance (T) is as follows:

$$t=\frac{2{\mathrm{\varnothing }}_{1}}{\left({A}_{11}+{A}_{12}{\mathrm{\varnothing }}_{1}\right){\mathrm{\varnothing }}_{1}+\left({A}_{21}+{A}_{22}{\mathrm{\varnothing }}_{1}\right)},$$

(5)

$$T\left(\mathrm{\%}\right)=100*{\left|t\right|}^{2}.$$

(6)

Using Green method

The Green method will be used to check if the results of UATMM are correct. At the end of each closed branch resonator, acoustic velocity (u) is zero. The closed branch resonator is grafted along the horizontal tube periodically. The set of interface spaces of all connections of the finite guides is reduced to M = (0) as a single interface²⁶. The response function of an inverse interface of a unit cell is^26,27:

$${g}^{-1}\left(\mathrm{0,0}\right)=-2{F}_{1}+{g}_{R}^{-1}\left(\mathrm{0,0}\right),$$

(7)

where ${g}_{R}^{-1}\left(\mathrm{0,0}\right)$ is the Green’s surface function of the closed-branched tube according to the conditions of the boundary. The dispersion relation of the infinite periodic waveguide can be written as:

$${\text{cos}}\left(\mathit{Kd}\right)={\text{cos}}\left(k{d}_{1}\right)- \frac{1}{2}\frac{{z}_{1}}{\upomega }{\text{sin}}\left(k{d}_{1}\right){g}_{R}^{-1}\left(\mathrm{0,0}\right),$$

(8)

For closed resonators, ${g}_{R}^{-1}\left(\mathrm{0,0}\right)$ can be written as:

$${g}_{R}^{-1}\left(\mathrm{0,0}\right)=- \frac{{F}_{1}{S}_{2}}{{C}_{2}}=\upomega {y}_{2}{\text{tan}}\left(k{d}_{2}\right),$$

(9)

From Eqs. (8) and (9), the dispersion relation can be written as:

$${\text{cos}}\left(Kd\right)={\text{cos}}\left(k{d}_{1}\right)- \frac{M}{2}{\text{sin}}\left(k{d}_{1}\right){\text{tan}}\left(k{d}_{2}\right),$$

(10)

which is precisely the exact dispersion relation obtained by UATMM in Eq. (4).

Ethics declarations

This article does not contain any studies involving animals or human participants performed by any authors.

Results

The proposed sensor consists of (${M}^{N}{M}_{D}{M}^{N}$) with N = 10. The initial geometrical conditions are selected as S₁ = 1 m², S_D = 0.73 m², d₂ = 0.15 m, d₁ = 0.6 m, d_D = 0.45 m, and S₂ = 0.75 m². The transmittance of the acoustic wave through the proposed novel gas sensor using periodically closed resonators has been studied as clearly in Fig. 2. By plotting the transmittance spectrum without the defect resonator, a Pn-BG extended from 1378.9 to 1429.1 Hz, with an intensity of 0%. The transmittance before and after the bandgap (width and intensity of ripples) depends on the constructive and destructive interferences of reflected waves at each interface due to the multiple Bragg scattering. The Bloch vector (K) within this range of frequencies is complex. So, the waves are evanescent. Real(K) is used to investigate changes in the phase of the pass band propagated wave. The black and red spectra in Fig. 2 clearly show that Bloch wavenumber dispersion and the Pn-BG coincide.

Figure 2

The band structure (red spectrum), the transmittance spectra of the closed resonator system without defect (black spectrum), and with an air cavity (blue spectrum) at d₁ = 0.6 m, S_D = 0.73 m², d₂ = 0.15 m, S₁ = 1 m², d_D = 0.45 m, S₂ = 0.75 m², and N = 10.

Inserting a defect-closed resonator at the center of the design causes the excitation of a sharp resonant peak with a minimal bandwidth inside the Pn-BG at 1398.16 Hz using an air sample.

In Fig. 3, the excited peaks and the Pn-BG shift toward higher frequencies as the speed of sound in the sample increases and its density decreases (Table 1). This behavior is known as the blueshift of the peak. To see the peak dependence on the type of gas sample, we change the sample from air to N₂, NH₃, and CH₄. By replacing the air sample with N₂, NH₃, and CH₄, the position of the excited peak is changed from 1398.16 to 1422.61 Hz, 1752.8 Hz, and 1813.94 Hz, respectively. The intensities of the excited peaks are very high (99.9%) due to the high localization of acoustic waves inside the defect-closed resonator. The dependence of the excited peaks on the acoustic velocity can be explained according to the standing wave equation:

Figure 3

The transmittance spectra of the closed resonator system using different gas samples at d₁ = 0.6 m, N = 10, d₂ = 0.15 m, S₁ = 1 m², d_D = 0.45 m, S_D = 0.73 m², and S₂ = 0.75 m².

Table 1 Acoustic properties of gas samples¹⁶.

$$2d=\frac{n c}{{f}_{R}},$$

(11)

where d is the length of the defect-closed resonator, n is an integer, C is acoustic velocity, and ${f}_{R}$ is the frequency of the excited peak.

Any detector’s sensitivity (S) is calculated by the rate of change of the peak’s frequency ($\Delta {f}_{R}$) and acoustic speed (Hz m⁻¹ s) as the following equation²⁸:

$$S =\frac{\Delta {f}_{R}}{\Delta C}=\left|\frac{{f}_{CH4}-{f}_{air}}{{C}_{CH4}-{C}_{air}}\right|.$$

(12)

Besides, the FoM is expressed as²⁹:

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

(13)

where FWHM is the bandwidth of the defect peak. Also, the Q-factor and LoD can be calculated as follows^28,29:

$$\begin{array}{l}Q-factor=\frac{{f}_{R}}{FWHM}\end{array},$$

(14)

$$LoD=\frac{{f}_{R}}{20 S Q}.$$

(15)

As expected, increasing the length of the defect-closed resonator ${d}_{D}$ does not reflect the position of Pn-BG, as evident in Fig. 4A. This expectation was built on the fact that the Pn-BG depends only on the potential (acoustical and geometrical) contrast between the layers in each unit cell, not the defect cell. Increasing the ${d}_{D}$ only reflects on the shape of the Pn-BG edges, as apparent in Fig. 4B. In Fig. 4B, we selected some values of ${d}_{D}$ that makes ${f}_{R}$ in the middle of the Pn-BG because the central resonance is highly responsive to slight changes in the sample and has the lowest FWHM. On the other hand, increasing the ${d}_{D}$ significantly impact the position (Eq. 11) and shape of the resonant peak. Increasing the length of the defect-closed resonator shifts the resonant peak to lower frequencies until it goes out from Pn-BG, another peak comes from the right, and so on. Besides, it was observed that the peak shift ($\Delta {f}_{R}$) seems constant. This independence of peak shift on the length of the defect-closed resonator may be considered an advantage because it gives flexibility in selecting a suitable length with the same peak shift (same sensitivity), as explicit in Fig. 4C. The resonant peak frequency for air and CH₄ samples slightly changes, and $\Delta {f}_{R}$ seems to be constant with increasing d_D. The resonant peak shift is expected to increase d_D due to the increasing interaction between acoustic waves and sample molecules, as in multilayer PnCs^29,30. This difference between the effect of increasing the defect length inside multilayers and a lateral defect resonator is that increasing the defect length inside multilayers increases the path that waves will travel. As a result, the interaction between the incident wave and the sample inside the defect increases until a saturation occurs. So, the resonant peak shift increases with the defect length inside multilayers. But in our case, the defect resonator is lateral, and any increase in its length isn’t in the path of the incident acoustic waves. Besides, the impedance of the defect closed resonator doesn’t depend on the length (Eq. 2). For these reasons, the resonant peak shift seems to be constant.

Figure 4

(A) transmittance intensity versus frequency as a function of the length of d_D using CH₄ sample, (B) transmittance intensity versus frequency at selected values of d_D for air (black lines) and CH₄ (red lines) samples, and (C) resonant peak positions for air (black lines) and CH₄ (red lines) samples at different lengths of d_D at d₁= 0.6 m, N = 10, d₂= 0.15 m, S₂= 0.75 m², S₁ = 1 m², and S_D = 0.73 m².

In Fig. 5 A–C, sensitivity, transmittance intensity, FWHM of defect mode for air sample, FoM, Q-factor, and LoD as a function of the length of d_D are calculated. In Fig. 5A, the sensitivity and transmittance intensity are obtained as a function of the length of d_D. As the sensitivity is directly proportional to the resonant peak shift (Eq. 12), with changing the d_D from 0.08 to 0.32 m, 0.57 m, 0.81 m, 1.06 m, and 1.3 m, the sensitivity slightly changes. The air sample is used to investigate the intensity of peaks as an indicator. The peak’s transmittance ranges from 96.9 to 99.6% for all studied lengths. Figure 5B shows the FWHM and FoM as a function of the original lengths of d_D. The lowest FWHM (0.064 Hz) and highest FoM (63.7 m⁻¹ s) are recorded at a length of 1.3 m. As apparent in Fig. 5C, the highest Q-factor (21,867) and lowest LoD (8 × 10^–4 m s⁻¹) are recorded at a length of 1.3 m. So, the length of d_D = 1.3 m will be used in the following studies.

Figure 5

(A) resonant peak position for air (black lines) and CH₄ (red lines) samples, (B) sensitivity and transmittance intensity, and (C) FWHM of defect mode for air sample and FoM as a function of the length of d_D at N = 10, d₂ = 0.15 m, d₁ = 0.6 m, S₂ = 0.75 m², S₁ = 1 m², and S_D = 0.73 m².

Similar to the effect of length ${d}_{D}$ on the PnBG, increasing the cross-section of the defect-closed resonator ${S}_{D}$ does not reflect on the position of Pn-BG, as clear in Fig. 6A. Changing the ${S}_{D}$ only reflects on the shape of the Pn-BG edges, as clear in Fig. 6B. However, increasing the ${S}_{D}$ significantly impact the position of the resonant peak. By increasing the ${S}_{D}$, the resonant peak is shifted to higher frequencies. As clear in Fig. 6C, by increasing the ${S}_{D}$ from 0.05 to 1.25 m², the $\Delta {f}_{R}$ slightly increases from 412.98 to 418.66 Hz. As the impedance is inversely proportional to the cross-sectional area (Eq. 2), increasing S_D decreases the impedance of the defect-closed resonator. So, the interaction between the acoustic wave and the sample inside the defect-closed resonator increases. So, the resonant peak position for air and CH₄ samples slightly increases with increasing S_D, and the $\Delta {f}_{R}$ slightly increases.

Figure 6

(A) transmittance intensity versus frequency as a function of the cross-section S_D using CH₄ sample, (B) transmittance intensity versus frequency at selected values of S_D for air (black lines) and CH₄ (red lines) samples, and (C) resonant peak positions for air (black lines) and CH₄ (red lines) samples at different cross-section S_D at N = 10, d₂ = 0.15 m, d₁ = 0.6 m, S₂ = 0.75 m², S₁ = 1 m², and d_D= 1.3 m.

Next, to evaluate the effect of cross-section S_D, the sensor’s performance is studied by changing S_D from 0.05 m² to 0.25 m², 0.50 m², 0.75 m², 1.00 m², and 1.25 m², leaving other geometrical conditions unchanged. With increasing S_D from 0.05 m² to 1.25 m², a slight increase in the sensitivity from 4.05 to 4.11 Hz m⁻¹ s can be observed in Fig. 7A. The peak’s transmittance ranges from 98.2 to 99.7% with increasing S_D. In Fig. 7B, the lowest FWHM (0.062 Hz) and highest FoM (65.7 m⁻¹ s) are recorded at a cross-section S_D of 0.75 m². As explicit in Fig. 7C, the highest Q-factor (22,522) and lowest LoD (7.6 × 10^–4 m s⁻¹) are recorded at a cross-section S_D of 0.75 m². The only reason why this enhancement at the cross-section S_D of 0.75 m² is because at this S_D of 0.75 m², the resonance is very close to the middle of Pn-BG, and the FWHM is very small at the center of the Pn-BG relative to at the edges. As a result, the S_D of 0.75 m² is optimum.

Figure 7

(A) sensitivity and transmittance intensity, (B) FWHM of defect mode for air sample and FoM, and (C) Q-factor and LoD as a function of the cross-section of S_D at N = 10, d₂ = 0.15 m, d₁ = 0.6 m, S₁ = 1 m², d_D = 1.3 m, and S₂ = 0.75 m².

Figure 8 shows that the Pn-BG and resonant peak frequencies for the CH₄ sample don’t change with increasing N from 6 to 14 periods, and the resonant peak shift remains constant. Increasing N can be considered a double-edged sword. Increasing N from 6 to 10 periods enhances the Bragg-scattering. As a result, the edges of the Pn-BG become sharper, and the bandwidth of the resonant peak decreases. On the other hand, in periods higher than 10, the reflectance increases, and the transmittance decreases.

Figure 8

The transmittance spectra using different N = 10 at d₂ = 0.15 m, d₁ = 0.6 m, d_D = 1.3 m, S_D = 0.75 m², S₁ = 1 m², and S₂ = 0.75 m².

In Fig. 9 A–C, sensitivity, transmittance intensity, FWHM of defect mode for air sample, FoM, Q-factor, and LoD are calculated as a function of the number of periods (N). From Fig. 9A, the sensitivity and transmittance intensity are obtained as a function of the number of periods. With changing the number of periods from 6 to 8 periods, 10 periods, 12 periods, and 14 periods, the sensitivity records the same value (4.09 Hz m⁻¹ s). When the number of periods increases from 6 to 10 periods, the peak’s transmittance slightly decreases from 100 to 98%. By increasing the number of periods above 10 periods, the peak’s transmittance strongly decreases. Figure 9B shows the FWHM and FoM versus the number of periods. FWHM strongly decreases with increasing periods from 6 to 10 periods, and FoM slightly increases with increasing periods from 6 to 10 periods due to their inverse relationship (Eq. 13). Then, FWHM slightly decreases, but FoM strongly increases. As clear in Fig. 9C, the Q-factor slightly increases with increasing N from 6 to 10 periods, and LoD strongly decreases with increasing periods from 6 to 10. Then, the Q-factor strongly increases, and LoD slightly decreases. N of 12 periods will be used in the following studies to ensure high FoM and Q-factor with acceptable transmittance.

Figure 9

(A) sensitivity and transmittance intensity, (B) FWHM of defect mode for air sample and FoM, and (C) Q-factor and LoD as a function of N at S_D = 0.75 m², d₂ = 0.15 m, d₁ = 0.6 m, S₁ = 1 m², d_D = 1.3 m, and S₂ = 0.75 m².

Figure 10A clears the transmittance intensity versus frequency as a function of the length d₁ using the CH₄ sample. In the present study, the length of d₁ changes only from 0.23 to 0.96 m because increasing the unit cell length may be a disadvantage of the proposed model. The Pn-BG is shifted to lower frequencies with increasing the length d₁. The resonant peak shift slightly changed and recorded the highest shift at the length of 0.59 m, according to Fig. 10B. As shown in Fig. 11A, the sensitivity slightly increases with the length of d₁. Besides, the transmittance ranges from 99.7 to 100%. The FWHM in Fig. 11B strongly depends on the position of peaks inside Pn-BG and ranges from 0.0148 Hz to 0.0268 Hz. FoM and Q-factor are inversely proportional to the FWHM. According to Eqs. (13) and (14), both have behavior opposite to the FWHM. In Fig. 11C, the LoD has the same behavior as FWHM according to Eq. (15). The length of 0.96 m records the best FWHM, FoM, Q-factor, and LoD so that it will be the optimum.

Figure 10

(A) transmittance intensity versus frequency as a function of the length d₁ using CH₄ sample, and (B) resonant peak positions for air (black lines) and CH₄ (red lines) samples at different length d₁ at N = 12, S_D = 0.75 m², d₂ = 0.15 m, S₂ = 0.75 m², S₁ = 1 m², and d_D = 1.3 m.

Figure 11

(A) sensitivity and transmittance intensity, (B) FWHM of defect mode for air sample and FoM, and (C) Q-factor and LoD as a function of length d₁ at N = 12 periods, d₁ = 0.6 m, d_D = 1.3 m, d₂ = 0.15 m, S₁ = 1 m², S_D = 0.75 m², and S₂ = 0.75 m².

The length of d₂ varies from 0.2 to 1.0 m to study its effect on the transmittance spectra. By increasing the length of d₂, the Pn-BG and peaks are red-shifted, as clear in Fig. 12A. The lengths of 0.276 m, 0.395 m, 0.517 m, 0.641 m, 0.765 m, and 0.885 m are selected to study the model’s performance at them. Unfortunately, when the transmittance spectra for air and CH₄ are plotted at the length of 0.276 m, an undesired peak (P₂) is found between the peaks of air and CH₄ (P₁ and P₃), as clear in Fig. 12B. The same effects were observed at other lengths. So, a length of 0.15 m will be optimum.

Figure 12

(A) transmittance intensity versus frequency as a function of the length d₂ using CH₄ sample, and (B) transmittance intensity versus frequency at selected values of d₂ for air (black lines) and CH₄ (red lines) samples at N = 12, S_D = 0.75 m², d₁ = 0.96 m, S₂ = 0.75 m², S₁ = 1 m², and d_D = 1.3 m.

Then, the transmittance spectra of the closed resonator system are studied in Fig. 13A using the selected geometrical parameters. By changing the air sample with N₂, NH₃, and CH₄, the position of the excited peak changed from 1408.57 Hz, 1433.21 Hz, 1765.84 Hz, and 1827.44 Hz, respectively. The intensities of the excited peaks are very high (99.9%) due to the high localization of acoustic waves inside the defect-closed resonator at the selected geometrical parameters. The resonant peak position versus acoustic speeds using different gas samples at optimum conditions is clear in Fig. 13B. Ar ($\rho = 1.661 kg/{m}^{3}$ and C = 319 m/s) and O² ($\rho = 1.314 kg/{m}^{3}$ and C = 326 m/s) samples are added in Fig. 13B to ensure the linearity of the sensor for different samples.

Figure 13

(A) The transmittance spectra of the closed resonator system, and (B) the resonant peak position versus acoustic speeds using different gas samples at optimum conditions at N=12, d₁=0.96 m, S₁ =1 m², d₂=0.15 m, S₂=0.75 m², S_D =0.75 m², and d_D=1.3 m.

The following relation (Eq. 16) describes the linearity of the sensor with an average sensitivity of 4.07 Hz m⁻¹ s:

$${f}_{R} = 4.107 C - 0.02049.$$

(16)

In Table 2, compared with other designs, the proposed closed system has achieved outstanding performance with a high sensitivity of 4.1 Hz m⁻¹ s, a high FoM of 332 m⁻¹ s, a very outstanding Q-factor of 113,962, and a small LoD of 0.0002 m s⁻¹. Even though many previous studies with complicated structures and materials achieved better outcomes, most of them couldn’t achieve linearity (linear peak shift or constant sensitivity) as in our model. For example, Zaki et al.³¹ proposed a defective 1D-Pn-BG based on a high-sensitivity fano resonance, but the linearity was missed. Aliqab et al.³² suggested a sensor to detect sulfuric acid concentration using 1D-PnC. Their model recorded a good sensitivity, but the linearity between the peak shift and the acoustic speed was missed. Zaky et al.¹¹ studied the ability to use the periodic cross-section of phononic tubes as gas sensors. This structure of periodic cross-section of phononic tubes recorded limited sensitivity (S) of 2.5495 Hz s m⁻¹, limited Q-factor of 4077, and limited FoM of 9.16 s m⁻¹.

Table 2 Comparison study.

Conclusion

The acoustic wave is better localized in the closed resonator by designing a phononic crystal with periodically closed resonators as a greenhouse gas sensor. This acoustic wave localization changes the peak position with any change in the acoustic properties of the analyte. Therefore, the proposed phononic crystal with periodically closed resonators as a greenhouse gas sensor is a good choice with many features as the following:

• 1. Compared with other designs, this sensor is more straightforward to fabricate than nano-dimensional structures. It is also cheaper than other sensors, making them more cost-effective for industrial applications.

• 2. This sensor can be a part of the factory chimney and continuously detect the concentration of hazardous gases’ path through it.

• 3. No recovery time is required.

• 4. High linearity.

• 5. The proposed closed resonator gas sensor records a high sensitivity of 4.1 Hz m⁻¹ s, a high FoM of 332 m⁻¹ s, a high Q-factor of 113,962, and a small LoD of 0.0002 m s⁻¹.