Theoretical optimisation of a novel gas sensor using periodically closed resonators

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.


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: where is an acronym for the wave number.ρ points to the density.c is the speed of sound.Z i refers to the impedance of the acoustic waves: 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 is 22 : The following Bloch's equation describes the relation of the acoustic wave dispersion of a unit cell of the proposed periodic structure 25 : where K and k are the Bloch and wave vectors, respectively, d = d 1 + d 2 , M = S 2 S 1 .The transmittance (T) is as follows:

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 interface 26 .The response function of an inverse interface of a unit cell is 26,27 : (1) (3) y D = j j Z D tan(kd D ).
(4) cos(Kd) = cos(kd 1 ) − M 2 sin(kd 1 )tan(kd 2 ), www.nature.com/scientificreports/where g −1 R (0, 0) 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: For closed resonators, g −1 R (0, 0) can be written as: From Eqs. ( 8) and ( 9), the dispersion relation can be written as: 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 S 1 = 1 m 2 , S D = 0.73 m 2 , d 2 = 0.15 m, d 1 = 0.6 m, d D = 0.45 m, and S 2 = 0.75 m 2 .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.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 N 2 , NH 3 , and CH 4 .By replacing the air sample with N 2 , NH 3 , and CH 4 , the position of the excited peak is changed from 1398.16 to 1422.61Hz, 1752.8Hz, and 1813.94Hz, 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: 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.( 8) cos(Kd) = cos(kd 1 ) − 1 2 Besides, the FoM is expressed as 29 : where FWHM is the bandwidth of the defect peak.Also, the Q-factor and LoD can be calculated as follows 28,29 : 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 ( 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 CH 4 samples slightly changes, and 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.www.nature.com/scientificreports/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  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 m 2 , the f R slightly increases from 412.98 to 418.66 Hz.As the impedance is inversely  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 m 2 .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 m 2 .The only reason why this enhancement at the cross-section S D of 0.75 m 2 is because at this S D of 0.75 m 2 , 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 m 2 is optimum.
Figure 8 shows that the Pn-BG and resonant peak frequencies for the CH 4 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 www.nature.com/scientificreports/ 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.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 (4.09Hz 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.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.
The length of d 2 varies from 0.2 to 1.0 m to study its effect on the transmittance spectra.By increasing the length of d 2 , 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 CH 4 are plotted at the length of 0.276 m, an undesired peak (P 2 ) is found between the peaks of air and CH 4 (P 1 and P 3 ), as clear in Fig. 12B.The same effects were observed at other lengths.So, a length of 0.15 m will be optimum.The following relation (Eq.16) describes the linearity of the sensor with an average sensitivity of 4.07 Hz m −1 s: 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 .

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:
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 4 .
Figure 4. (A) transmittance intensity versus frequency as a function of the length of d D using CH 4 sample, (B) transmittance intensity versus frequency at selected values of d D for air (black lines) and CH 4 (red lines) samples, and (C) resonant peak positions for air (black lines) and CH 4 (red lines) samples at different lengths of d D at d 1 = 0.6 m, N = 10, d 2 = 0.15 m, S 2 = 0.75 m 2 , S 1 = 1 m 2 , and S D = 0.73 m 2 .

Figure 5 .
Figure 5. (A) resonant peak position for air (black lines) and CH 4 (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, d 2 = 0.15 m, d 1 = 0.6 m, S 2 = 0.75 m 2 , S 1 = 1 m 2 , and S D = 0.73 m 2 .

Figure 6 .Figure 7 .Figure 8 .
Figure 6.(A) transmittance intensity versus frequency as a function of the cross-section S D using CH 4 sample, (B) transmittance intensity versus frequency at selected values of S D for air (black lines) and CH 4 (red lines) samples, and (C) resonant peak positions for air (black lines) and CH 4 (red lines) samples at different crosssection S D at N = 10, d 2 = 0.15 m, d 1 = 0.6 m, S 2 = 0.75 m 2 , S 1 = 1 m 2 , and d D = 1.3 m.

Figure 9 .
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 m 2 , d 2 = 0.15 m, d 1 = 0.6 m, S 1 = 1 m 2 , d D = 1.3 m, and S 2 = 0.75 m 2 .

Figure 10 .
Figure 10.(A) transmittance intensity versus frequency as a function of the length d 1 using CH 4 sample, and (B) resonant peak positions for air (black lines) and CH 4 (red lines) samples at different length d 1 at N = 12, S D = 0.75 m 2 , d 2 = 0.15 m, S 2 = 0.75 m 2 , S 1 = 1 m 2 , and d D = 1.3 m.

Figure 13 .
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, d 1 =0.96 m, S 1 =1 m 2 , d 2 =0.15 m, S 2 =0.75 m 2 , S D =0.75 m 2 , and d D =1.3 m.