As the operating principle of the GC-SAW gas sensor mentioned in Figure 1, the heated gases from the GC column are condensed onto the surface of the SAW detector at a low temperature as a liquefied layer. Target gases are “adsorbed” onto the surface of the SAW device. Due to the mass loading and viscous property of the condensate itself, the SAW propagation along the piezoelectric substrate is perturbed, causing a velocity shift and attenuation change. The velocity shift induces the oscillation frequency change, which is used for target gas evaluation.

In this section, the condensation-adsorption effect is analyzed theoretically by solving the piezoelectric medium equations of motion and surface effective permittivity method. Considering there is an isotropic and piezoelectric medium occupying a half-space (x₃ ≤ 0) with interdigital transducer (IDT) about the plane (x₃ = 0) and a liquefied condensate adsorbed above (0 ≤ x₃ ≤ h), as schematically illustrated in Figure 2, the dynamic equations of linear piezoelectricity in a half-space piezoelectric substrate takes the following forms in this coordinate system mentioned in Figure 2: (1) ρ ∂ 2 u i ∂ t 2 − c ijkl E ∂ 2 u k ∂ x j ∂ x l − e ijk ∂ 2 φ ∂ x j ∂ x k = 0 e jkl ∂ 2 u k ∂ x j ∂ x l − ε jk ∂ 2 φ ∂ x j ∂ x k = 0 ( i , j , k , l = 1 , 2 , 3 )where x₁, x₂, x₃ denote the Cartesian coordinates x, y, z respectively, E_ijk is the Levi-civita symbol. We denote by u_i the mechanical displacements and by φ the electric potential. c_ijkl, e_kij, and ε_ij stand for the elastic, piezoelectric and dielectric constants, and ρ for the mass density, respectively. The summation convention for repeated tensor indices and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with the index are adopted. The indices i, j, k, and l range from 1 to 3. With the compressed matrix notation [19], the material constants c_ijkl and e_ijk in Equation (1) can be represented by matrices c_pq and e_ip, with the convention that p, q =1, 2, 3, …, 6. Similarly, the strain tensor S_ij and the stress tensor T_ij can be represented by S_p and T_q. The first three terms in the left-hand side of Equation (1) are the equation of the motion related to the inertia, the elasticity and piezoelectricity [20].

First, we assume a general solution of Equation (1), the particle displacement and electrical potential, are in the forms of: (2) u i ( x 1 , x 3 ) = A i   exp { j ω t − jk s ( x 1 + α x 3 ) } , i = 1 , 2 , 3 ; φ ( x 1 , x 3 ) = A 4   exp { j ω t − j k s ( x 1 + α x 3 ) } where k_s and ω are the wave number in the x₁ direction and the angular frequency, respectively. α is a decay constant along the x₃ direction. A_i (i =1, 2, 3) and A₄ are wave amplitudes. Substitution of Equation (2) into Equation (1) leads to four linear algebraic equations (Cristoffel equations) for A_i and A₄. Then, for nontrivial solutions of A_i and/or A₄, the determinant of the coefficient matrix of the linear algebraic equations must vanish, and this leads to a polynomial equation of degree eight for α. The coefficients of this polynomial equation are generally complex. To ensure the decrease in the displacement u_i and the potential φ into the substrate, the generally complex constant α must have a negative imaginary part. Thus, we select four eigenvectors with negative imaginary part denoted by α_n (n = 1, 2, 3, 4), and the corresponding eigenvectors by A i ( n ) = [ A 1 ( n )     A 2 ( n )     A 3 ( n )     A 4 ( n ) ], n = 1, 2, 3, 4. Thus, the general wave solution to Equation (1) in the form of Equation (2) can be written as: (3) u i ( x 1 , x 3 ) = ∑ n = 1 4 A i ( n ) C n   exp { − jk s ( x 1 + a n x 3 ) } ,     i = 1 , 2 , 3 φ ( x 1 , x 3 ) = ∑ n = 1 4 A 4 ( n ) C n   exp { − jk s ( x 1 + a n x 3 ) }where C_n (n = 1, 2, 3, 4) are the weight factors, and can be determined by the boundary condition.

According to Equation (3) and the piezoelectric constitutive equations, the stress T_3j (j = 1, 2, 3) and the electrical displacement component along x₃ direction (D₃) can be obtained in the space of x₃ < 0.

Referring to the Navier-Stokes equations and assuming that the liquefied condensate is a dielectric liquid, the dynamic equation in the liquid layer (0 < x₃ < h) takes the following form in this coordinate system mentioned in Figure 2 [21]: (4) ρ ∂ 2 u i ∂ 2 t = c ijkl ∂ 2 u k ∂ x l ∂ x k − 2 3 μ j ω ∂ ∂ x i ( ∇ • u ) + μ j ω ∇ 2 u i + μ j ω ∂ ∂ x i ( ∇ • u ) , i , j , k , l = 1 , 2 , 3where ∇ 2 = ( ∂ 2 / ∂ x 1 2 + ∂ 2 / ∂ x 2 2 + ∂ 2 / ∂ x 3 2 ) is Laplacian, ∇ • u is the divergence of displacement and μ is the viscous coefficient. The divergence of displacement is equal zero (∇ • u = 0) for incompressible liquid and for incompressible and ideal liquid, the divergence of displacement and viscous coefficients are also equal zero respectively (∇ • u = 0, μ =0).

First, we assume a general solution of Equation (4), the particle displacement, is in the form: (5) u l i ( x 1 , x 3 ) = A i l   exp [ j ω t − j ( k x 1 + a l k x 3 ) ] , i = 1 , 2 , 3where the superscript l on u^l_i and A^l_i differentiates the corresponding parameters from those in Equation (1). Substitution of Equation (5) into Equation (4) leads to three linear algebraic equations (Cristoffel equations) for A^l_i. Then, for nontrivial solutions of A_i, the determinant of the coefficient matrix of the linear algebraic equations must vanish, and this leads to a polynomial equation of degree six for α^l. All of the generally complex constants α^l are available because of the limit liquid layer and are denoted by α^l_n (n = 1, 2, 3), and the corresponding eigenvectors by A i l ( n ) = [ A 1 l ( n ) A 2 l ( n ) A 3 l ( n ) ], n = 1, 2, 3. Thus, the general wave solution to Equation (4) in the form of Equation (5) can be written as: (6) u i l ( x 1 , x 3 ) = ∑ n = 1 6 A i l ( n ) C l n   exp [ − j ( k x 1 + α l n k x 3 ) ] , i = 1 , 2 , 3

According to Equation (6) and the constitutive equations, the stress T^l_3i (j = 1, 2, 3) can be obtained in the space of (0 < x₃ < h) as: (7) T 31 l = μ ( ∂ v 1 ∂ x 3 + ∂ v 3 ∂ x 1 ) T 32 l = μ ( ∂ v 2 ∂ x 3 + ∂ v 3 ∂ x 2 ) T 33 l = ρ v 2 ( ∂ u l 1 ∂ x 1 + ∂ u l 2 ∂ x 2 + ∂ u l 3 ∂ x 3 ) − 2 3 μ ( ∂ v 1 ∂ x 1 + ∂ v 2 ∂ x 2 + ∂ v 3 ∂ x 3 ) + 2 μ ∂ v 3 ∂ x 3where v₁, v₂ and v₃ are the velocity components along x₁, x₂ and x₃ directions, respectively, ρ and v are the density and velocity of the liquid respectively.

Additionally, φ^l and φ^g were introduced to represent the Laplace potential in the space of 0 < x₃ < h and in the space of x₃ > h, respectively: (8) φ l = φ l 1   exp ( − jk x 1 − k x 3 ) + φ l 2   exp ( − jk x 1 + k x 3 ) , 0 < x 3 < h φ g = φ g 1   exp ( − jk x 1 − k x 3 ) , x 3 > hand they should meet the Laplace equation, which means: ∇ 2 φ l = 0 , ∇ 2 φ g = 0

The corresponding electrical displacement component along x₃ direction, D^l₃, can be written as: (9) D 3 l = − jk [ j ε l φ l 1   exp ( − jk x 1 − k x 3 ) − j ε l φ l 2   exp ( − jk x 1 + k x 3 ) ] , 0 < x 3 < h D 3 g = − jk [ j ε 0 φ g 1   exp ( − jk x 1 − k x 3 ) ] , x 3 > hwhere ε₀ and ε_l are the dielectric constant in the liquid and vacuum respectively.

The solutions of the motion equations should satisfy both the mechanical boundary condition and the electrical boundary condition respectively. The mechanical boundary condition and electrical boundary condition at boundary between the piezoelectric substrate and liquid layer (x₃ = 0) and boundary between liquid layer and vacuum (x₃ = h) as schematically illustrated in Figure 2 are: (10) { T 3 j − T 3 j l = 0 u j − u j l = 0 φ − φ l = 0 D 3 − D 3 l = σ ( x 1 ) , j = 1 , 2 , 3 } , x 3 = 0 { T 3 j l = 0 φ l − φ g = 0 D 3 l − D 3 g = 0 , j = 1 , 2 , 3 } , x 3 = h

There are 13 equations in Equation (10), and they can be used to determine 13 unknown terms according to the analyses above (C_n, (n = 1, 2, 3, 4); C^l_n, (n = 1–6);, φ^l₁, φ^l₂, φ^g₁). Usually, after solving the Equation (10), the 13 unknown terms ((C_n, (n = 1, 2, 3, 4); C^l_n, (n = 1–6); φ^l₁, φ^l₂, φ^g₁)) can be determined. However, due to the inhomogeneity of Equation (10), it is difficult to extract directly the unknown terms. Thus, based on the surface-effective-permittivity method, it is easy to describe the acoustic field in the piezoelectric substrate. The surface effective permittivity constant ε_s(s) of the piezoelectric substrate is represented as [22]: (11) ε s ( s ) = σ ¯ ( s ) 2 π f | s | φ ¯ ( s , x 3 ) | x 3 = 0

Here, φ̄(s, x₃) is the electrical potential distribution in slowness domain of the piezoelectric substrate; s is the slowness, and s = 1/V_s, the V_s is the SAW phase velocity. According to Equation (10), φ(s, x₃) denoted by σ can be deduced, and hence the ε_s(s) is obtained by substituting into Equation (11).

Usually, the effective permittivity constant is determined when the material and cut-direction of the crystal substrate, direction of acoustic wave propagation, and normalized thickness of the liquid layer are defined. Using Equations (8–11), the ε_s(s) on ST-X quartz with Euler angle of (0°, 132.75°, 0°) was obtained as shown in Figure 3. In this case, an ideal liquefied condensate with zero viscosity coefficient on the surface of the quartz, the condensate normalized thickness kd (kd = k × h, k is the wave number, and h is the condensate thickness) of 0.2, and operation frequency of 500 MHz were used. The real line and dotted line described the real part and imaginary part of the ε_s(s), respectively. From the picture, there are several discontinuous points, representing different acoustic wave modes. The Rayleigh wave mode is observed at lower acoustic wave velocity (speeds of ∼3.18 × 10⁻⁴ s/m). There is a zero point and pole point pair in the regime of Rayleigh wave mode with the imaginary part being zero. Where, |ε_s(S_mR)| → ∞, at pole point of S_mR, represents the slowness of Rayleigh wave propagating along the metallic surface with no attenuation. ε_s(S_oR) = 0 is observed at zero point of S_oR, and it describes the Rayleigh wave propagating along the free surface with no attenuation. According to the zero and pole points, the phase velocity V_f and V_m in case of free surface and metallic surface coated with condensate can be determined, respectively. To simplify the design procedure, the SAW phase velocity, V_s, can be considered as (V_f + V_m)/2.

Considering the liquid-layer viscosity, the change of the trend of the real part of ε(s) is similar to the above result. However, the imaginary part of ε(s) is non-zero but negative at the zero point and pole point of Re (ε(s)), which denotes the energy attenuation, which in turn was caused by the liquid viscosity.

The piezoelectric material for the SAW sensors analyzed herein is ST-X quartz. Figure 4(a) illustrates the relationships among the velocity shift, operation frequency of sensor, various viscosity coefficient, μ, of liquefied condensate (μ = 0, μ = 0.01 Pa·s, μ = 0.1 Pa·s), on the condition that the condensate coating with normalized thickness, kd, of 0.6, the density of the condensate ρ of 850 kg/m³, and sensing area of 1.5 mm². The higher the operation frequency of the sensor is employed, the larger the velocity shift that will be achieved for the viscous condensate coating with given normalized thickness. This means that a larger sensor response will be achieved when a higher operation frequency is applied. Additionally, the velocity shift increases with an increment of the condensate viscosity coefficient μ, as shown in Figure 4(a).

Additionally, the relationship between the phase velocity shift and normalized thickness, kd, of condensate is described in case where the Formaldehyde is assumed as the condensate as shown in Figure 4(b). The parameters of density of 820 kg/m³, operation frequency of 500 MHz, sensing area of 1.5 mm², and viscosity coefficient assumed as 0.9 × 10⁻³ Pa·s, considering its hydrophilicity, are used to calculate the velocity shift. The velocity shift increases as the normalized thickness of condensate increases, and the velocity shift is approximately proportional to the normalized thickness shift.

Thus, according to the frequency stability description as below, the threshold detection mass of the gas sensor can be deduced by: (12) Δ m = ρ s v 0 2 × 3 N b k c × 2 π × f 0 2

Here, v₀ is the unperturbed SAW phase velocity, f₀ is the operation frequency of the sensor, N_b is the baseline noise of the sensor depending on the frequency stability of the oscillator; s is the sensing area, k_c is considered as a scaling constant, which is the quotient of SAW velocity shift Δv caused by the condensate coating and change of normalized condensate thickness Δ(kd). Thus, under given adsorption efficiency, K, depending on the target specie itself and the experiment condition, the threshold detection limit of the sensor can be evaluated as: (13) m DL = Δ m K = ρ s v 0 2 × 3 N b k c × 2 π × f 0 2 × K

From Equation (13), it is obvious that the detection limit is determined by the baseline noise, operation frequency, detection efficiency, and SAW velocity shift caused by the condensate coating. The higher operation frequency, lower baseline noise (superior frequency stability), and larger sensor frequency response (higher k_c value) applied, the lower detection limit would be. Using the sensor for formaldehyde detection as an example, the relationship between the detection limit and adsorption efficiency (K) is shown in Table 1, in the case of density of 820 kg/m³, baseline noise of 30 Hz, operation frequency of 500 MHz, sensing area of 1.5 mm², and viscosity coefficient assumed as 0.9 × 10⁻³ Pa·s. The k_c value can be obtained by using Equations (1–11) as Figure 4(b). From Table 1, we can find that the detection limit depends mainly on the adsorption efficiency. When increasing the adsorption efficiency, lower detection limit was obtained.

From Equation (15), a higher Q value and lower insertion loss of the SAW device will effectively improve the short-term frequency stability of the oscillator [26]. Here, the dual-port resonator structure with three IDTs is used for the SAW device due to its high Q value, low insertion loss, and enough condensation area for gas sensing, as shown in Figure 5(a). By adjusting the distance between one IDT to another, the distance between IDT to grating, and the ratio of IDT period and grating period, only one resonance mode in the frequency bandwidth was expect to be realized. ST-X quartz was chosen as substrate due to its excellent temperature stability. The SAW velocity on ST-X quartz substrate with 300 nm Al IDTs metallization was 3,158 m/s. The operation frequency of the SAW resonator is specifically given by 500 MHz, thus, the wavelength λ of the SAW resonator is given by 6.3 μm. Two dual-port resonator patterns were fabricated on a same ST-X quartz wafer by standard photolithography technique. Each resonator consists of three transducers (IDT1 and IDT3 act as the launching transducers, and IDT2 is the readout transducer) and adjacent shorted grating reflector distributed in each side of the IDT1 and IDT3; the electrode number of the IDT1, IDT2, and IDT3 were 60, 120, and 60, respectively, and the electrode number in shorted grating reflector was set to 400. Spacing between the reflector and IDT was 1.25λ (λ is the wavelength). Acoustic aperture of 250λ and spacing between IDTs of 22.25λ were used to accommodate enough condense area for gas sensing. The fabricated SAW device was shown in Figure 5(a). The frequency response S₂₁ of the fabricated device was measured with a network analyzer as shown in Figure 5(b). A low insertion loss of 7.6 dB and high Q value of 2,300 were obtained.

Generally, oscillation occurs at the central frequency in the pass band of the SAW devices. However, the central frequency point is not always the frequency point with the lowest insertion loss because of technical issues in the device fabrication. To improve the short-term frequency stability of the oscillator, the oscillation in frequency point with the lowest insertion loss of the SAW devices is advised here by adjusting the phase shifter.

Then, using the fabricated SAW devices as feedback, a 500 MHz SAW oscillator is implemented with an oscillator circuit composed of discrete elements (amplifier, phase shifter, mixer and LPF, etc.) on a printed circuit board (PCB) as shown in Figure 6(b). The working principle of the oscillator was as depicted in Figure 6(a). The output of the amplifier is mixed in order to obtain a differential frequency in the MHz range. This technique allows us to reduce the influence of the thermal expansion of the substrate and to use simple low-frequency counters. The output of the oscillator is connected to a programmable frequency counter which can be used for an analog indicator or a chart recorder.

An experiment was performed to evaluate the frequency stability of the fabricated SAW oscillator using the programmable frequency counter under constant testing temperature (25 °C) and 70% RH. To demonstrate the short term frequency stability (seconds), a partial differentiation is used. It shows the frequency shift per second. The testing condition is 25 °C temperature. Then, the oscillation is modulated at the frequency point with the lowest insertion loss by a strategic phase modulation, in which a low pass filter acts as the phase shifter, and the phase modulation is accomplished by adjusting the inductor or capacitor values. The measured short-term frequency stability (frequency shift in seconds) is shown in Figure 7(b). Excellent short-term frequency stability of ∼±3 Hz per second is obtained, much better than that of the previous oscillator with oscillation at the operation frequency point, in which, weak short-term frequency stability of ±15 Hz were observed as shown in Figure 7(a).