Theoretical design of porous phononic crystal sensor for detecting CO2 pollutions in air

Theoretical design of porous phononic crystal sensor for detecting CO2 pollutions in air

Physica E 124 (2020) 114353 Contents lists available at ScienceDirect Physica E: Low-dimensional Systems and Nanostructures journal homepage: http:/...

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Physica E 124 (2020) 114353

Contents lists available at ScienceDirect

Physica E: Low-dimensional Systems and Nanostructures journal homepage:

Theoretical design of porous phononic crystal sensor for detecting CO2 pollutions in air Ahmed Mehaney *, Ashour M. Ahmed Physics Department, Faculty of Science, Beni-Suef University, Beni-Suef, 62514, Egypt



Keywords: Phononic crystal Gas sensor CO2 concentration Transmission mode Peak intensity Density

Phononic crystal sensor is a novel technology for sensing applications with high performance. The present work proposes theoretically a design of gas (CO2) sensor based on a one-dimensional (1D) porous silicon (PSi) pho­ nonic crystal (PnC) sandwiched between two thin rubber layers. The transfer matrix method (TMM) was used for the numerical modeling of the acoustic waves spectra through the 1DPSi-PnC sensor structure. The results showed that a resonant mode was created inside the transmission spectrum as a result of the presence of the twosided rubber layers. Also, the position of the resonant mode was invariant with changing CO2 concentration, temperature, and pressure. On the contrary, the intensity of the transmitted mode is very sensitive to any change in these parameters. With increasing the CO2 concentration (from 0% to 90%) and pressure (from 2 atm to 6 atm), the intensity of the resonant mode are significantly increased. While, with increasing temperature (from 20 � C to 200 � C), the intensity of the resonant mode is decreased. These results are correlated directly to the density of the CO2/air mixture. Therefore, the proposed 1DPSi-PnC sensor can measure CO2 pollutions in the sur­ rounding air over a wide range of concentration, temperature, and pressure values. The merits of a gas sensor based on porous materials and PnC structure are numerous. For example, the ease of fabrication, working under tough conditions, and its capability to sense CO2 pollutions from the surrounding air directly. Also, the proposed sensor can be developed as a monitor for many gases in industrial and biomedical applications. Moreover, porous materials enable the proposed design to be compatible with other fluidic com­ ponents specifically liquids. Thereby, allowing the proposed gas sensor to be replicated for various fluidic sensing applications.

1. Introduction Phononic crystals (PnCs) are artificial periodic structures consist of two or more materials different in their mass density, elasticity, and sound speed [1]. PnCs can be fabricated in one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) structures [2]. The reflection and transmission of acoustic waves occur at the interfaces of the constituents. Hence, the wave interference occurs at each layer interface. The periodic arrangement leads to destructive and construc­ tive interferences dependent on the frequency of the incident acoustic waves. PnCs exhibit stop bands for waves at certain frequencies called phononic band gaps (PnBGs) due to the complete destructive interfer­ ence [3,4]. Within these frequency regions, the acoustic waves cannot propagate through the structure. The condition of the PnBG is derived by analogy to Bragg condition. This band gap is similar to what happens for electromagnetic waves in periodic structures of photonic crystals [5–7].

The physics of photonic crystals and PnCs focus on the properties of optical and acoustic waves, respectively [8,9]. The unique characteris­ tics of photonic and PnCs offer wonderful applications based on different designs [10–12]. Recently, sensor technology is considered a very important research field for many modern technology applications. PnCs attracted scien­ tists’ attention in the sensors field because they are considered flexible, precise, and reliable materials with long life expectancy cloaking [13–16]. Most of the previous investigations of PnCs focused on liquid sensors. For example, Zubtsov et al. determined the concentration of ethanol in gasoline using 2D PnCs having a resonant cavity [17]. The result showed a strong correlation between gasoline properties and the frequency of the maximum transmission modes. Samar et al. presented a 1D PnC based on Si/SiO2 multilayer with a defect cavity for alcohol detection [18]. Feiyan et al. demonstrated experimentally that a slotted PnC plate (SPCP) can act as a liquid sensor with high sensitivity [19].

* Corresponding author. E-mail address: [email protected] (A. Mehaney). Received 5 May 2020; Received in revised form 19 June 2020; Accepted 6 July 2020 Available online 11 July 2020 1386-9477/© 2020 Elsevier B.V. All rights reserved.

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Fig. 1. A schematic diagram of a porous PnC gas sensor consists of [rubber(PSi1/PSi2/PSi3)2rubber].

The acoustic properties of many liquids are determined by observing the transmission spectra of the SPCP system. Shehatah et al. introduced a 1D PnC model with a defect layer using the transfer matrix method as a biodiesel sensor [20]. This sensor can distinguish between different biodiesel fuels in the range of temperatures from 20 � C to 50 � C with high performance. Manzhu et al. presented theoretically and experi­ mentally an acoustic liquid sensor based on a PnC model consists of a steel plate with an array of holes filled with liquid [1]. The position of the resonant peak has been proven to be highly sensitive to the sound speed of the liquid fill the holes. Beside liquid sensors, other types of new sensors were introduced by PnCs as well. Nagaty et al. analyzed the propagation of acoustic waves in a 1D PnC contains a piezomagnetic material as a defect layer for detecting the magnetic field and external pressure [21]. Mehaney studied theoretically a PnC sensor as a neutron detector over a wide fluence range of neutron irradiations [22]. On the other side, gas sensing has strong applications in many fields such as industry, food, medicine, agriculture, and environment. Unfor­ tunately, the idea of using a PnC design as a gas sensor is still very limited. We expect a PnC design as a gas sensor to impose promised results. Wherein, the acoustic properties of fluids can ultimately change higher than optical, magnetic, or electrical properties. Today, detection of the level of carbon dioxide in atmospheric air is very important, especially in the environmental protection area [23]. Burning fossil fuel makes a high rate of air pollutions and global warming as a result of the increment of CO2 in the atmosphere. More­ over, CO2 is emitted from cars and many factories, which increase the environmental pollution. Few previous researches were focused on PnCs as a gas sensor. For example, Ahmet et al. investigated numerically and experimentally CO2 sensors based on a 1D surface acoustic PnC [24,25]. Therefore, the development of CO2 gas becomes inevitable. Many types of materials are used in the design of PnCs including polymers, metals, semiconductors, liquids, and dielectrics. The latest studies are mostly focused on the integration of porous silicon (PSi) layers in a novel sensor design in order to improve its sensitivity [13,26]. PSi has perfect morphological, electrical, and mechanical characteris­ tics. Also, the PSi material has a high surface-to-volume ratio and its compatibility with modern silicon microelectronics fabrication tech­ nologies [27]. PSi has been commonly used in modern applications especially in sensor devices [28]. The mechanical properties of the PSi layers such as Young’s modulus, speed of sound, and density can be controlled by changing the porosity (fractional volume of voids) ratio and the filling medium. Electrochemical etching of high-quality crystalline Si wafer in the ethanol-hydrofluoric solution can generate porous silicon multilayers PnCs [29]. The electrical current-driven etching breakdowns Si–Si bonds as pores penetrate the volume of the wafer. The porosity and thickness are controlled by the current density and etching time [30]. This method is easy and inexpensive. As a result of these advantages, PSi is considered a very promised material for the fabrication of a PnC sensor device. The PSi-PnCs have

many advantages such as very light-weighted and ease of fabrication with low manufacturing cost over other ordinary materials. Also, the tunability of the mechanical properties of PSi has been used for hyper­ sonic acoustic structures [31,32]. Based on the above characteristics of PSi and PnCs, in this study, we aim to combine these benefits of porous materials side by side with the pioneering role of PnCs in the field of sensors to design a 1D PnC gas sensor. The present work introduces a new design of CO2 sensor based on a porous silicon multilayer PnC. Two soft rubber layers are bonded at the two sides of the 1DPSi-PnC to obtain resonant modes. The effects of many parameters such as temperature and pressure on the sensitivity of the PnC CO2 sensor are investigated and discussed. The transfer matrix method is applied for calculations of the transmission spectra of the PnC models. 2. Design of 1DPSi-PnC gas sensor The proposed configuration of the CO2 sensor is a ternary multilayer 1DPSi-PnC sandwiched between two rubber layers as shown in Fig. 1. So, the sensor structure will be as the configuration, [rubber(PSi1/PSi2/ PSi3)2rubber]. We used porous materials in the design of the PnC gas sensor rather than other conventional solid material due to some rea­ sons. First, it has high elastic constants despite its light-weight. Sec­ ondly, the porous materials are considered as hollow media, so it can be used for other types of fluids specifically liquids by injecting them in the pores and compute their effective sound speed and mass density. Thereby we added the rubber layers at both sides of the PnC to introduce the resonant peaks and at the same time to prevent the leakage of liquids in the case of using PnCs as a liquid sensor. The layers PSi1, PSi2, and PSi3 were chosen with the thicknesses d1 ¼ 6 μm, d2 ¼ 3 μm, and d3 ¼ 1.5 μm and porosities 20%, 40%, and 60%,

Fig. 2. Density and speed of sound of CO2-air mixture. 2

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Where, U is the displacement potential and vj is the acoustic wave speed in each layer. The subscript j ¼ 1, 2, and 3 are identifying each layer of the PSi layers. The displacement component can be written in the following form, vx ¼ δU=δx:


The general solution of equation (1) for each layer (let’s say j) can be formed as a superposition of the transmitted and reflected traveling waves with harmonic time dependence as follows, � � þi ​ kj ​ x U ¼ Aþ þ A j ​ e i ​ kj ​ x e i ​ ω ​ t (3) j ​ e pffiffiffiffiffiffi Where, i (¼ 1) is a complex number, kj ¼ 2π ​ f=vj is the wave­ number. ω is the angular frequency and f is the frequency of the incident wave. The first and second term in equation (3) represents the transmitted and reflected waves through the j layer, respectively. The two co­ efficients Aþ j and Aj are the amplitudes of the transmitted and reflected

Fig. 3. Dependence of the density and sound speed of PSi layer on the porosity ratio.

waves, respectively. The boundary conditions of TMM imply the continuity of the displacement and stress at each layer interface. The acoustic impedance Zj of the layer j depends on the mass density ρj and the acoustic speed vj according to the equation,

respectively. These porosities were chosen based on many previous theoretical and experimental studies [33,34]. Each rubber layer has a thickness value of 0.5 μm. The mass density and longitudinal sound speed of rubber are 1300 kg/m3 and 22.87 m/s, respectively [13]. The total thickness of this sensor is in the mesoscopic scale, PSi materials were extremely fabricated with such size [35]. Also, the operating fre­ quency of the sensor in all calculations is located in the ultrasonic range.

Zj ¼ ρj ​ vj

According to equation (4), for each porous layer, the impedance is depending on the porosity of each layer. By using the analysis of the TMM, the wave matrix through the j layer with thickness dj is written as, � � i​k d 0 e j​ j​ Mj ¼ (5) i ​ kj ​ dj ​ : 0 e

3. Theoretical model The acoustic waves propagation through a solid material (e.g. bulk Si) can be either transversal or longitudinal. In fluids (gas and liquid), the propagating wave is mostly a longitudinal wave. So, in this paper we consider only acoustic waves propagate through the PnCs [36]. The transfer matrix method (TMM) is used to estimate the acoustic/elastic wave propagation in PnCs. This method can calculate exactly the elas­ ticity solutions in one-dimensional multilayer structures in the form of the transmission-reflection frequency spectra. For a plane time-harmonic wave incident normally from left to right on a 1D multilayer structure as shown in Fig. 1, the wave equation is given by the next equation, 2


r U ¼ vj U


The wave matrix at the interface between two successive layers (j, jþ1) is given as, 3 2 Zjþ1 þ ​ Zj Zjþ1 ​ Zj 7 6 Zjþ1 Zjþ1 16 7 Mj;jþ1 ¼ 6 (6) 7: 2 4 Zjþ1 ​ Zj Zjþ1 þ ​ Zj 5 Zjþ1 Zjþ1 If we consider a structure formed from N layers, the relationship between the incident displacement U0 and the transmitted displacement UN is written as follows,


Fig. 4. Transmission spectra of the ternary PnC gas sensor structure (a) (PSi1/PSi2/PSi3)2 and (b) [rubber(PSi1/PSi2/PSi3)2rubber], both are surrounded by air. 3

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Fig. 5. The resonant peak of the PnC CO2 sensor at (a) CO2 concentration in air is 50%, (b) CO2 concentration in air ranging from 0% to 90% and (c) Resonant peak intensity versus different CO2 concentrations at room temperature (20 � C).

U0 ¼ M ​ UN

mixtures as [40],


Here, M is the accumulative transfer matrix of the total multilayer structure. Finally, the transmission coefficient of the 1D PnC structure is calculated by using the following relation, T ¼ j1=M11 j2 :

ρm ¼ ρ1 f 1 þ ρ2 f 2 ;


where ρm , ρ1 and ρ2 are the densities of the mixture, air, and CO2, respectively. f 1 and f 2 are the volume fraction of air and CO2, respectively. For a binary mixture, the relationship between the longitudinal sound speed and the composition of the gaseous mixture (air and CO2) can be calculated using the following equation [41], ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffi sffi� R Cp1 ð1 X12 Þ þ ​ Cp2 ​ X12 c¼ T: (10) m1 ð1 X12 Þ þ m2 ​ X12 Cv1 ð1 X12 Þ þ ​ Cv2 ​ X12


Where, M11 is the first element of the accumulative matrix M. More details about the TMM and its derivation can be found in many previous articles [37–39]. 4. Results and discussion

Where, T is the temperature of the gas in Kelvin, R is the gas constant 8.314 J/mol-K, X12 is the partial pressure of component 1 in component 2, m1 and m2 are the molecular weights of the two components in the mixture. Cp1, Cp2, Cv1, and Cv2 are specific heat capacity at constant pressure and volume in the two components, respectively. Fig. 2 gives the density and sound speed as a function of the CO2/air mixture. The density of CO2/air mixture increases with increasing CO2 concentration because the density of CO2 (1.98 kg/m3) is higher than

4.1. Air and CO2 mixture calculations In the beginning, all the below calculations are taken at the sea level altitude and atmospheric pressure. Suppose a mixture of air and CO2 with a total volume of 100 m3. At temperature 20 � C, the density of air and CO2 is 1.225 and 1.98 kg/m3, respectively. The density of the air and CO2 mixture can be calculated according to the well-known rule of 4

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the air (1.225 kg/m3). Oppositely, the speed of sound decreases in the CO2/air mixture with increasing CO2 concentrations because the speed of sound in air (343 m/s) is higher than in CO2 (271.5 m/s).

experimentally based on similar previous studies. The PnBG was observed experimentally in PnCs based on multilayer PSi in many pre­ vious reports. Goller at al. verified experimentally the transmission of the acoustic wave in Fibonacci one-dimensional hypersonic PSi-PnC [42]. Andrews et al. observed the hypersonic PnBG experimentally in PSi superlattices [44]. Zorayda et al. studied the localization of acoustic modes in periodic PSi structures with cavity modes in the gigahertz range [37]. Thomas et al. fabricated PSi-PnC as an acoustic filter at hypersonic frequencies [31]. Arriaga et al. successfully measured experimentally the acoustic Bloch oscillations in PSi structures [33]. Moreover, porous silicon was fabricated experimentally as sensor de­ signs for ethanol [48], humidity [49], glucose [50], DNA hybridization [51], and antibodies [52]. Herein, by bonding two rubber layers at the two opposite sides; [rubber(PSi1/PSi2/PSi3)2rubber], the resonant transmission mode (maximum transmission) appeared in the transmission spectrum as seen in Fig. 4(b). The low elastic constants of the rubber besides the high elastic constant of Si introduce a strong locally resonant mode in the transmission spectrum of the PSi-PnC. This resonant mode appeared at the frequency of 3.57 � 107 Hz with an intensity of about 49.5%. Similar to our design, Wang et al. investigated the local resonance in a 1D ternary periodic PnC [53]. Transmission modes allow the confining of acoustic energy in the PnC designs [54]. The resonant mode acts as the cornerstone of the proposed sensor measurements. The position and intensity of the resonant mode depend on the acoustic properties of the surrounding medium. Hence, its position and intensity will be calculated in the next sections to determine the effect of CO2 concentration, tem­ perature, and pressure on the transmission spectrum of the [rubber (PSi1/PSi2/PSi3)2rubber].

4.2. Porous Si layer calculations The average density of the PSi layer is given a function of the porosity by the following relation [42], � (11) ρj ¼ ρ0 1 Pj where ρ0 (¼ 2.33 g/cm3) is the density of bulk silicon material and Pj is the porosity in the j-th layer. The effective longitudinal sound speed (v�L Þ in a porous structure can be calculated by Hashin’s expression as [43], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffi 4 þ 5 ​ Pj 1 Pj � � � vLj ¼ ​ vL0 (12) 4 1 þ bK ​ Pj 1 þ bG ​ Pj bK ¼

ð1 þ ν0 Þ ð11 19 ν0 Þ ; bG ¼ 2 ð1 2 ν0 Þ 4 ð1 þ ν0 Þ

ν0 (¼ 0.265) is the Poisson’s ratio of silicon. vL0 (¼ 8322 m/s) is the longitudinal speed of sound in the bulk silicon at zero-porosity. Fig. 3 gives the density and sound speed as a function of the porosity in the air at zero concentration of CO2. As the porosity increases, both density and sound speed of PSi linearly decrease as seen in Fig. 3. This means the impedance of the PSi layer falls rapidly with increasing the porosity because both the density and sound speed decrease in highly porous media [36]. The same results were verified in many experimental studies [33,44].

4.4. Effect of CO2 concentration

4.3. Effect of rubber layers

The wave transmission in PnCs is not only influenced by their to­ pological construction, but also by the material in the surrounding medium [55]. Fig. 5(a c) presents the transmission spectra of the PnC [rubber(PSi1/PSi2/PSi3)2rubber] surrounded by air at different CO2 concentrations from 0% to 90%. The position of the resonant peak is constant for all CO2 concentrations as clear in Fig. 5(a c). This behavior is due to the change in the CO2 concentration has a negligible influence on the effective sound speed and effective mass density of PSi layers for any porosity ratio. But, the intensity of the peak is changed with increasing CO2 concentrations, which can act as an indicator for the CO2 concentration in air as seen in Fig. 5 (b). By comparing the resonant peak intensity in Fig. 4(b) (the CO2 concentration is 0%) with the peak intensity in Fig. 5 (a) (the CO2 concentration is 50%), we see that the

The rubber layers are added on the two sides of the 1DPSi-PnC to generate a resonant mode, these modes can be tuned with changing the gas properties in the surrounding medium. The choice of the rubber material due to its past novel results in locally resonant and resonator structures [13,45]. Also, the rubber layers were chosen with a very small thickness (0.5 μm) to enhance its sensitivity to the smallest changes in the gas properties. Fig. 4 shows the transmission spectra as a function of the frequency for the two PnCs, (PSi1/PSi2/PSi3)2 and rubber(PSi1/PSi2/PSi3)2rubber, both are surrounded by air. The operating frequency of the PnC sensor is in the ultrasonic range. For the PnC structure (PSi1/PSi2/PSi3)2, the transmission spectrum is characterized by a complete stop band gap with good depth (the lowest transmission intensity in the frequency spectrum) which occupies the full frequency range. Hence, the acoustic waves cannot propagate through the entire structure as seen in Fig. 4(a). The high mismatch in the acoustic impedance between the PSi layers results in a high atten­ uation for acoustic waves at their interfaces. The stop band is formed due to the destructive interference between the reflected and incident acoustic waves at the interfaces between every two successive layers dependent on Bragg diffraction law. The high-speed mismatch between pores (air) and elastic solids (Si) produces a large difference in the impedance between the constituent materials. This produces a very wide phononic band gap. Air has a low acoustic impedance compared with Si inclusions which have a high acoustic impedance. This large acoustic impedance mismatch results in producing such a wide band gap. To produce a wide phononic band gap, the high acoustic impedance (Z ¼ ρ v) mismatch between the materials is desired [45,46]. Many previous studies used porous PnCs consist of air hole inclusions in a solid matrix material to obtain a high impedance mismatch [47]. Also, the proposed sensor design can be easily fabricated

Fig. 6. Theoretical density and sound speed of CO2/air mixture versus tem­ perature ranging from 20 � C to 200 � C. The CO2 concentration is taken as 70%. 5

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Fig. 7. (a) Effect of temperature on the resonant peak of the PnC CO2 sensor, (b) Resonant peak intensity versus different temperatures.

peak intensity is decreased with increasing the CO2 concentration. Fig. 5(c) summarizes the intensity of the resonant mode as a function of the CO2 concentration. The intensity of the resonant peak increases linearly with increasing the CO2 concentration. The highest peak in­ tensity at CO2 concentration 90% coincides with the maximum density of the CO2/air mixture (1.9045 kg/m3). All previous PnCs sensors are dependent on observing the change in the position of the resonant mode. From an experimental point of view, the source (transmit transducer) and detector (receive transducer) of the acoustic wave must have very wide frequency ranges. In our work, the position of the resonant peak is invariant and while its intensity changes extremely. So, any source and any detector with a short wide frequency range or a single frequency can be used easily to detect the acoustic signal in our sensor. The intensity of the resonant peak (T %) and the CO2 concentration (C %) are the output signal of the sensor and the input value, respectively. Hence, the sensitivity can be defined by the following equation, S ¼ ΔTð%Þ=ΔCð%Þ

4.5. Effect of temperature The effect of temperatures on the PnC CO2 sensor is illustrated in Fig. 7. The sensor structure [rubber(PSi1/PSi2/PSi3)2rubber] is supposed to be surrounded by air with 70% of CO2 concentration. The sensor may be affected by other surroundings properties besides the property being measured. For example, most sensors are influenced by the environment temperature. The effect of temperature on the volume of the mixture and the volume of each gas is calculated based on the well-known Charles’ Law. This law states that the volume of a given amount of gas is directly proportional to the temperature of the gas at constant pressure as, V1 T2 ¼ V2 T1:


Where, V1 is the initial volume of the mixture at room temperature T1 , while V2 is the final volume of the mixture at a high temperature T2 . Hence, the new density and sound speed at high temperatures can be calculated. Fig. 6 shows the effect of temperature on the density and sound speed of the CO2/air mixture at 70% of CO2 concentration. As temperature increases, the density of the mixture decreases while the sound speed of the mixture increases. Consider the propagation of sound waves through the PSi-PnC at different temperatures from 20 � C to 200 � C. The intensity of the transmitted peak inside the PnBG decreases as the temperature increases (Fig. 7). At the same time, the position of the peak is not affected. By increasing the temperature, the sound speed in PSi increases while the effective density of the PSi decreases (Fig. 6). The resonant peak intensity decreases from the value of 0.559%–0.443% when the temperature increases from 20 � C to 200 � C. The reduction is mainly caused by the decrease in the density of the CO2/air mixture from the value of 1.6317 kg/m3 to the value of 1.0061 kg/m3 (Fig. 6). Also, there is a significant increase in the mixture sound speed (from 292 m/s to 371 m/s). Therefore, the amount of acoustic energy that localizes in the PSi-PnC is decreased as well.


The value of S can be calculated from the slope of the line in Fig. 5(c) which equals the value of 0.12. Based on this figure, the CO2 pollutions in the air can be determined by the resonant peak strength.

4.6. Effect of pressure Finally, we will show the effects of pressure on the PnC CO2 sensor. The physical properties of the gas are varying in a significant way when the pressure of the ambient conditions is not stable. The effect of pressure on the volume of a mixture can be deduced from Boyle’s law. The absolute pressure exerted by a given gas is inversely proportional to the volume it occupies if the temperature

Fig. 8. Theoretical density and sound speed of CO2/air mixture versus pressure ranging from 2 atm to 6 atm. The CO2 concentration in air is taken as 70%. 6

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Fig. 9. (a) Effect of pressure on the resonant peak of the CO2 PnC sensor, (b) Resonant peak intensity versus different pressures.

remains unchanged. This is shown by the following equation, V1 P1 ¼ ​ V2 P2 :

localized resonant mode caused by the rubber layers. Based on this technique, we have introduced a sensor capable of sensing different CO2 concentrations in the surrounding air dependent on the change in the resonant peak intensity. The effects of temperature and pressure on the resonant peak intensity are presented as well. The numerical results showed the excellent perspectives of phononic crystals as a gas sensor. To the best of our knowledge, it is the first time to realize a 1D PnC as a gas sensor based on porous materials. Finally, the proposed sensor can be replicated on other gases and liquids easily.


Where, V1 is the initial volume of the mixture at a pressure P1 , while V2 is the final volume of the mixture at high pressure P2 . So we can calculate the volume and density for each new pressure value. The speed of sound depends on the elasticity and density of the conveying medium. In general, the speed of sound in a gas is given by Newton-Laplace equations as [56], sffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi γ​R​T γ​R​T ​ Bs v¼ ¼ ; (16) ¼ M ρ​V ρ

Declaration of competing interest The authors report no conflict of interest.

Where, R is the universal gas constant, γ is the adiabatic index. M and Bs are the molar mass of and the bulk modulus of the gas, respectively. The sound speed in gases does not affect by pressure at a constant temperature because the pressure is inversely proportional to the gas density according to the general law of gases (P/ρ ¼ constant) [57]. Therefore, density is the only factor that can be changed with pressure changes [58]. In Fig. 8, different pressure values were plotted versus the sound speed and density of the CO2/air mixture (the CO2 concentration is taken as 70%). Different pressure values on the PnC structure are considered (2 atm–6 atm). The results show that the sound speed has a very minor effect on the resonant peak intensity compared to the density of the gas. As shown in Fig. 9, the resonant peak shows higher a trans­ mission intensity with increasing the pressure. This effect is opposite to the effect of temperature on the resonant peak (Fig. 7). The same behavior was observed in our previous work [59]. Also, the position of the peak is not affected by changing the pressure. The pressure reduces the volume of the CO2/air mixture, and hence increases its density. At a pressure value of 1 atm, there is a resonant frequency peak with an intensity of 57.6%. With increasing the pressure to the value 6 atm (the density of the CO2/air mixture increases ac­ cording to Fig. 8), the intensity of the resonant peak increases to 98.1%.

CRediT authorship contribution statement Ahmed Mehaney: Conceptualization, Data curation, Funding acquisition, Investigation, Methodology, Project administration, Re­ sources, Software, Supervision, Validation, Visualization. Ashour M. Ahmed: Formal analysis, Writing - original draft, Writing - review & editing. References [1] R. Lucklum, M. Ke, M. Zubtsov, Two-dimensional phononic crystal sensor based on a cavity mode, Sensor. Actuator. B Chem. 171–172 (2012) 271–277. [2] Y.-Z. Wang, F.-M. Li, K. Kishimoto, Y.-S. Wang, W.-H. Huang, Wave localization in randomly disordered layered three-component phononic crystals with thermal effects, Arch. Appl. Mech. 80 (6) (2009) 629–640. [3] M.I. Hussein, K. Hamza, G.M. Hulbert, R.A. Scott, K. Saitou, Multiobjective evolutionary optimization of periodic layered materials for desired wave dispersion characteristics, Struct. Multidiscip. Optim. 31 (1) (2005) 60–75. [4] A. Khelif, A. Choujaa, S. Benchabane, B. Djafari-Rouhani, V. Laude, Guiding and bending of acoustic waves in highly confined phononic crystal waveguides, Appl. Phys. Lett. 84 (22) (2004) 4400–4402. [5] M. Shaban, A.M. Ahmed, E. Abdel-Rahman, H. Hamdy, Tunability and sensing properties of plasmonic/1D photonic crystal, Sci. Rep. 7 (1) (2017). [6] A.M. Ahmed, M. Shaban, A.H. Aly, Electro-optical tenability properties of defective one-dimensional photonic crystal, Optik 145 (2017) 121–129. [7] A.M. Ahmed, A. Mehaney, Ultra-high sensitive 1D porous silicon photonic crystal sensor based on the coupling of Tamm/Fano resonances in the mid-infrared region, Sci. Rep. 9 (1) (2019). [8] A. Hatef, M.R. Singh, Effect of a magnetic field on a two-dimensional metallic photonic crystal, Phys. Rev. 86 (4) (2012). [9] A. Hatef, M. Singh, Decay of a quantum dot in two-dimensional metallic photonic crystals, Optic Commun. 284 (9) (2011) 2363–2369. [10] J.D. Cox, M.R. Singh, C. Racknor, R. Agarwal, Switching in polaritonic–photonic crystal nanofibers doped with quantum dots, Nano Lett. 11 (12) (2011) 5284–5289.

5. Conclusions In conclusion, this work presents theoretically a porous silicon phononic crystal as a new type of gas sensors (CO2). The idea of sensing CO2 pollutions in the surrounding air is based on the change in the transmitted resonant mode intensity. The expression of the transmitted wave is presented using the transfer matrix method. An exciting inter­ action occurred between the mechanical properties of PSi-PnC and the 7

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