Temperature influences on the performance of biodiesel phononic crystal sensor

In the present work, a 1D phononic crystal model is introduced as a biodiesel sensor. The proposed sensor can sense and distinguish between different biodiesel fuels with high performance. The sensor structured from a defect layer filled with biodiesel in the middle of a 1D binary structure as the configuration [(Al/epoxy)biodiesel[(Al/epoxy)]. Using the transfer matrix method, the reflection spectrum of the biodiesel sensor is calculated. The influences of different biodiesel fuels on the resonant peaks are compared and discussed. Also, different temperatures in the range from 20 °C to 50 °C were considered to show the sensor performance under temperature effects. Based on our results, the presented sensor can be considered as a selective sensor between different biodiesel fuels (e.g. Methyl Soy Ester, Oxidized Soy Ester, Ethyl Soy Ester, Certified D-2 and Methyl Laurate) and more sensitive to their temperature changes. For example, at temperature 30 °C the sensitivity and quality factor of the sensor reached the values of about 265 Hz °C−1 and 55.12 for Methyl Soy Ester, 261.5 Hz °C−1 and 55.8 for Oxidized Soy Ester, 260.9 Hz °C−1 and 56.2 for Ethyl Soy Ester, 291.7 Hz °C−1 and 58.3 for Certified D-2, 361.6 Hz °C−1 and 56.3 for Methyl Laurate, respectively.


Introduction
Phononic crystals (PnCs) are artificial composites can inhibit the propagation of mechanical waves at certain frequencies known as phononic band gaps [1][2][3][4][5][6]. Within which elastic and acoustic waves are fully suppressed. PnCs constructed by a periodic arrangement of two or more materials in 1D, 2D or 3D structures [7][8][9][10][11]. In recent years, many interesting applications based on these materials have introduced such as filters, multiplexing systems, heat isolation devices, energy harvesting structures, waveguides, etc [12][13][14][15][16][17][18][19][20][21][22]. One of the fundamental applications of PnCs is the possibility of achieving acoustic sensors [7,23]. A H Aly et al introduced theoretically the idea of using 1D defective PnC as a sensor design based on the intensity of the transmitted (resonant) modes [6]. Where the type of unknown liquid can be specified by the transmission intensity of the resonant mode. A 1D multi-layered PnC sensor structure was introduced experimentally by S Villa-Arango et al [20]. G Sharma et al Designed theoretically a Si-SiO 2 phoxonic crystal for simultaneous sensing of liquids by electromagnetic and mechanical waves. They investigated using the transfer matrix method the possibility of sensing biodiesel in a binary mixture of diesel and biodiesel [24]. Also, a 1D PnC structure works as a biodiesel sensor has been presented theoretically by A Mehaney [25]. This biodiesel sensor has drawbacks such as large size (5 cm) and its inability to recognize the type of the biodiesel fuel. Moreover, R Lucklum et al introduced a different type of PnC sensors in the form of a sandwiched PnC between two stacked layers on the top of surface acoustic wave device [26]. Based on the previous success of PnCs in the field of biosensors. PnC sensors must play a bigger role in detecting and measuring biofuels physical properties due to its important applications. Biodiesel has many benefits like renewable, non-toxic and biodegradable [27][28][29][30][31][32][33][34]. Therefore, a PnC sensor platform must be applied for recognizing and measuring the physical properties of the different biodiesel fuels.
In this paper, we study theoretically the effects of temperature changes on the performance of biodiesel PnC sensor. A 1D binary PnC model can distinguish between different biodiesel fuels under temperature considerations. Different biodiesel fuels such as Methyl Soy Ester, Oxidized Soy Ester, Ethyl Soy Ester, Certified D-2 and Methyl Laurate in a defect layer of the PnC are considered. The effects of these fuels on the resonant peaks will be studied and compared. The quality factor and sensitivity of the sensor will be calculated and tabulated for all fuels at different temperatures.

Theoretical analysis and method of calculations
Depending on the transfer matrix method (TMM) we study the acoustic waves interactions with the PnC structure [35,36]. Assume a 1D binary PnC constructed from n unit cells as shown in figure 1. Each unit cell has two different layers from aluminum and epoxy signified by the symbols ( X, Y), where the lattice constant is = + a a a . 1 2 The acoustic wave motion through the 1D PnC structure can be written in the following equation: )and x denote the density, displacement, stress, external force and the position coordinates, respectively.
The equation of motion will be recast to a dimensionless form by introducing the following dimensionless local coordinates: where a 1 is the thickness mean value of material A (j = 1). The general harmonic dimensionless solution for equation (1) will be as follow: (¯) / z j are the dimensionless thickness of each layer, w is the angular frequency, c j L is the longitudinal acoustic wave velocity and A1, A2 are unknown coefficients to be determined. After conducting a lot of mathematical calculations that can be found in [35,36] the relation between these two state vectors is given in the following form where ¢ T j are2 2transfer matrix. The reflection coefficient of an acoustic wave propagates normally in the x-direction through a PnC composed of N layers and bonded between two semi-infinite materials can be written as follow:  and =´a 1 10 m. 2 3 The acoustic properties (density and speed of sound) of Al and epoxy are listed in table 1. In what follows, we will calculate the reflection spectra of the defective PnC for the different biodiesel fuels at different temperatures In figure 2, the reflection spectrum of the incident acoustic wave is plotted as a function of the nondimensional frequency (w p a 2 c l / ). Where w is the angular frequency of the incident acoustic wave and c L is the speed of sound in epoxy. As shown in figure 2, the perfect PnC has a wide phononic band gap in the normalized frequency range 0.3 −1.5. Such band gap will be the effective region where the localized modes will be generated. Various experimental models similar to our design were introduced in many literatures [25,37,38]. Therefore, the proposed sensor design can be applied and fabricated easily.

Sensor design
The proposed design of the biodiesel PnC sensor is shown in figure 3. The sensor structure is the perfect PnC with a defect layer with a thickness =´a 1.5 10 m d 3 in between as the configuration [(Al/epoxy)defect(Al/ epoxy)]. We consider the defect layer is filled with different biodiesel fuels. Inserting a defect layer inside the perfect PnC results in generating resonant peaks, each peak frequency represents the type of the biodiesel fuel. Firstly, water was taken as a reference or standard material for all measurements.
Secondly, the defect layer is filled with different biodiesel fuels, e.g. Methyl Soy Ester, Oxidized Soy Ester, Ethyl Soy Ester, Certified D-2 and Methyl Laurate, respectively. The acoustic properties of these fuels can be found in [30].
As shown in figure 4, the biodiesel fuels restore the incident acoustic energy and excite a transmitted resonant peak inside the phono0nic band gap. Each peak is related directly to the acoustic properties of each fuel. The resonant peak frequencies of Methyl Soy Ester, Oxidized Soy Ester, Ethyl Soy Ester, Certified D-2 and  Hz, respectively. Also, it is observed that as the sound speed decreases, the resonant peaks shift towards lower frequencies. Therefore, the presented sensor can produce a specific resonant peak for any biodiesel oil at specified frequency value.

Analysis of sensor performance.
The sensitivity of sensors can be calculated according to the following relation [39,40]: is the resonant frequency shift, f r and f w is the resonant peak frequency of each biodiesel oil and water, respectively. x is the change of the input parameter to be measured (sound speed or temperature). Also, the quality factor (Q-factor) can be calculated by the following relation [39,40]: where, FWHM is the full width at half maximum of the resonant peak. The normalized frequency, resonant peak frequency, sensitivity and Q-factor are listed in table 2 for each biodiesel oil at 20°C. For selectivity measurements, x is considered as the change in the sound speed of each fuel.
From table 2, we can see that as the sound speed decreases, the sensitivity value increases. Also, the Q-value is almost constant for the five oils due to the constancy of the FWHM value of each oil.

Temperature effects on the reflection spectrum and sensor performance
In this part, we pay more attention to studying the effects of temperature on the reflection spectrum and on the resonant peak of each biodiesel oil. Such study will be applied to the performance analysis of the defective PnC sensor as well. In      As shown in these figures, the resonant peak position for each oil shifts towards lower frequencies with increasing temperatures. The resonant peak broadening and displacement are related directly to the sound speed and density value each oil.
The normalized frequency, frequency, sensitivity and Q-factor values for each oil at different temperatures are listed in tables 3 and 4. The temperature value of 20°C was taken as the reference value for measurements. As seen in table 4, the temperature has a great effect on the performance parameters of the biodiesel sensor for all   oils. For example, the sensitivity increases to the value 265 Hz°C −1 with increasing the temperature from 20°C to 30°C for the Methyl Soy Ester. This means that each 1°C causes the resonant peak to move about 26.5 Hz which is considered a high value for temperature effects on the biodiesel sensor and better than other previous liquid sensors [37,38]. Also, the sensor has high Q-factor values for all oils and each value increases with increasing temperatures. Thus, we can conclude that the proposed biodiesel sensor besides its selective property between the different biofuels, it is also very sensitive to biodiesel temperature changes.

Conclusions
In conclusion, we studied the performance of a 1D defective PnC biodiesel sensor under temperature effects in this paper. The proposed model sensor can be considered as selective and temperature-sensitive for different biodiesel fuels e.g. Methyl Soy Ester, Oxidized Soy Ester, Ethyl Soy Ester, Certified D-2 and Methyl Laurate. Also, the resonant frequency of each fuel is plotted and analyzed at temperature changes. Moreover, the sensor showed high sensitivity and quality factor for all fuels.