A Multi-Parameter Decoupling Method with a Lamb Wave Sensor for Improving the Selectivity of Label-Free Liquid Detection ^†

Abstract

In this paper, a liquid multi-parameter decoupling method with only one Lamb wave sensor is presented. In a Lamb wave sensor, antisymmetric modes (A₀₁ mode for low frequency, A₀₃ mode for high frequency) and symmetric modes (S₀ mode) are used to detect multiple parameters of a liquid, such as its density, sound velocity, and viscosity. We found they can play very different roles in the detections. For example, the A₀₁ mode is very sensitive to the liquid's density but the A₀₃ mode is sensitive to the sound velocity. Here, the A₀ mode is used to identify the density of the detected liquid and with this density value we obtained the viscosity by the amplitude shifts of the S₀ mode. This could be a way to distinguish an unknown liquid with high sensitivity or to solve the problem of selectivity of label-free detection on biosensors.

1. Introduction

Acoustic sensors have been widely used in chemical/biological fields with the label-free detection method to detect the mass changes on the sensor surface [1–17]. However, the selectivity of this method is often poor due to the absorption of non-target molecules, which are difficult to distinguish using only one-parameter sensors [18,19]. This has been a common problem of all kinds of label-free biosensors. The detection of multiple parameters using a multi-mode acoustic sensor significantly improves the label-free detection method. Acoustic waves which travel in a medium can have multiple modes and this character has been successfully used in a combined detection of density and viscosity for high viscosity solutions [20]. However, for low viscosity solutions like aqueous electrolytic solutions or bio liquids, it is still a challenge [9,20–24].

The micro Lamb wave sensor is a powerful tool for liquid detection because it is easier to get multi-mode vibration, and it performs with high sensitivity and low attenuation [25–29]. The two well-known basic modes, the antisymmetric mode (A₀ mode) and the symmetric mode (S₀ mode), have already been used for solving the problem of temperature compensation on a chip by the authors of [26,30]. The A₀ mode has been used to measure the concentration of bio/chemical liquids, such as the concentration of methanol for direct methanol cell applications [18]. However, it shows uncertain frequency shifts direction with the concentration changes of the liquid [18,31]. In fact, the parameters of a liquid, like the density, acoustic sound velocity and viscosity, will work together for a mode. So it's hard to decouple them with only one mode.

In this work, both the antisymmetric mode and the symmetric modes are used to decouple the liquid physical parameters. The characters of A₀₃ mode, with a wavelength of about one-third of A₀₁ mode (low frequency of A₀ mode), are investigated with the liquid loading for the first time. In order to discuss the response of Lamb wave sensor to the liquid loading, two types of experiments were set up. The first experiment was used to measure the frequency shifts with the same type of solutions with different concentrations loaded on the sensors' surface, such as the solutions of NaCl. It shows that the characters of the modes are very different. Although both A₀₁ and A₀₃ modes belong to A₀ mode, their frequency shifts showed an opposite behavior with the increase of the concentration of the loading solutions. The second experiment was used to measure the frequency changes with different types of solutions and different concentrations, such as solutions of KCl, NaBr and KBr. Although the tested objects are two different kinds of solutions with different concentrations, the frequency of A₀₁ mode shows almost has no movement and A₀₃ mode frequency shows a great difference. The frequency shifts of S₀ mode for these solutions are not apparent. These will make it possible to be only sensitive to the liquid viscosity except for the environmental temperature. This essay attempts to decouple the functions of A₀₁, A₀₃ and S₀ modes to obtain the basic physical parameters (density and sound velocity, viscosity) with different liquid loading responses. From the values which have been reported from literature [32] we can also determine the types of the solution. This work provides a selective, sensitive method for the measurement of physical parameters to an unknown solution. It can be useful for label-free biosensors.

2. Working Principle

Multi-mode wave, such as A₀₁ mode, S₀ mode and their harmonic ones (A₀₃ mode, for example) can be excited in Lamb wave sensors and detected directly by using a pair of inter-digital transducers (IDTs) [33] located on the surface of the piezoelectric layer [34,35]. In liquid sensing, the liquid and IDT will be located on the two opposite sides of the membrane. For the A₀₁ mode and the A₀₃ mode, when the Lamb wave phase velocity is less than the liquid sound velocity, there are evanescent waves produced around the membrane-liquid interface. The phase velocity of the Lamb waves within the evanescent penetration field will be the same with the phase velocity inside the membrane. Taking account of the bending stiffness and the in-plane tension of the plate (B_i) for the A₀₁ mode and the A₀₃ mode, the phase velocity will be [31,36]:

$$c_{P_i}=\sqrt{B_i/(M+m_{Li})}$$

(1)

where i denotes 1 or 3 for the A_0i mode, B_i reflects the influences of the bending stiffness and the in-plane tension of the plate for the A_0i mode, M is the mass per unit area of the plate, the effective mass m_Li in the so-called evanescent penetration depth δ_Ei equals:

$$m_{Li}={\rho }_L{\delta }_{Ei}$$

(2)

In which, ρ_L is the detected liquid density. The evanescent penetration depth δ_Ei obeys the following equation [37–39]:

$${\delta }_{Ei}={\lambda }_i/(2\pi \sqrt{1-{(c_{Pi}/c_L)}^2})$$

(3)

where λ_i and c_Pi denote the Lamb wave wavelength and phase velocity respectively in each mode. c_L is the bulk acoustic velocity of the detected liquid. c_Pi is determined by the frequency (f_i) via c_Pi = f_i λ_i.

By substituting the Equations (2) and (3) into the Equation (1), we will get the following formula:

$${\rho }_L{\lambda }_i/(2\pi \sqrt{1-{(c_{Pi}/c_L)}^2})=B_i/c_{Pi}^2-M$$

(4)

In this formula, the constants B_i, λ_i, and M are independent of the liquid type; λ_i is determined by the structure of the device. With the implicit Equation (4), the density and sound velocity of the liquid on the sensor can be decoupled when the frequency response of the A₀₁ mode and the A₀₃ mode are obtained by the experiments.

In the measurement, the relative frequency shifts (Δf/f) will be used frequently. By taking the term c_Pi = f_i λ_i into Equation (1) and deriving of the equation, the value of Δf/f) for the A₀₁ mode and the A₀₃ mode will be:

$$\mathrm{\Delta }{f}_i/f_i=-\mathrm{\Delta }({\rho }_L{\delta }_{Ei})/2M=-({\delta }_{Ei}\mathrm{\Delta }{\rho }_L+{\rho }_L\mathrm{\Delta }{\delta }_{Ei})/2M$$

(5)

Obviously, the density and the sound velocity are the two factors affecting the relative frequency shifts. When the phase velocity is far less than the liquid sound velocity (c_P « c_L), such as the A₀₁ mode, the values of δ_E and Δδ_E approach λ/2π and 0, respectively. Then, in this situation the variations of the liquid sound velocity will have an almost negligible influence on the relative frequency shift. When the phase velocity is close to the liquid sound velocity, such as the A₀₃ mode, the value 1 – (c_P/c_L)² approaches 0. The influence of the terms δ_E and c_L cannot be neglected any more. The liquid density and sound velocity will simultaneously affect the frequency shifts. The value of Δf/f can be positive or negative which depends on the sum of the value of ρ_LΔδ_E and the value of δ_EΔρ_L. Anyway, if we combine these two cases (A₀₁ mode, A₀₃ mode) simultaneously, we can get the values ρ_L and c_L with high sensitivity.

For the S₀ mode, when the wavelength of the Lamb mode is larger than the thickness of the plate, the phase velocity (c_p) for the principal symmetric mode can be simplified as [40]:

$$c_p=c_t{\left[4\left(1-\frac{c_t^2}{c_d^2}\right)\right]}^{\frac{1}{2}}\left[1+\frac{I}{2}\frac{{\rho }_L{c}_L^2}{{\rho }_m{c}_t^2}\frac{\omega d}{2c_L}\left(\frac{1}{4\left(1-c_t^2/c_d^2\right)}-\frac{c_t^2}{c_d^2}\right)\right]$$

(6)

where c_d and c_t are the velocities of longitudinal (dilatational) and transverse (shear) waves of the membrane, c_L is the liquid sound velocity, ρ_m and ρ_L are the density of the solid membrane and liquid respectively, d is the membrane thickness, I denotes the square root of −1.

Evidently, the effect of liquid on the propagation of the S₀ mode does not change the real part of the phase velocity, but adds a very small attenuation of the amplitude [40]. This mode is suitable for the detection of attenuation. The amplitude (A_L) response of an unknown solution can be expressed by:

$$A_L=A_0{e}^{-\alpha {}_{XL}x}$$

(7)

where the attenuation coefficient α_XL is proportional to (ρ_Lη_L)^0.5, with η_L being the viscosity of the liquid. Similarly, the amplitude response (A_W) of water is given by:

$$A_W=A_0{e}^{-\alpha {}_{XW}x}$$

(8)

In engineering, the insertion loss (dB) is widely used. The values of A_L and A_W can be transformed into values in dB scale which are denoted by A_LdB and A_WdB, respectively. Therefore, the attenuation difference (ΔA_dB) between an unknown solution and water can be expressed in dB scale as follows:

$$\mathrm{\Delta }{A}_{dB}=A_{\mathit{\text{LdB}}}-A_{\mathit{\text{WdB}}}={\gamma }_1{\left({\rho }_L{\eta }_L\right)}^{1/2}+{\gamma }_2$$

(9)

in which both of the slope γ₁ and γ₂ are constant, and the constant γ₂ is determined by the reference liquid (water). As the density and the sound velocity of the solution can be estimated by both the frequency shifts of the A₀₁ mode and the A₀₃ mode, the multi-parameters (ρ_L, c_L, η_L) of the solution can be decoupled simultaneously with these three modes. According to these principles, a series of experiments were set up to investigate the responses of the multi modes of Lamb wave to the loading solutions.

3. Experiments for Liquid Detection

3.1. Experimental Setup

The micro Lamb wave device in Figure 1(a) contains a silicon membrane (length 7.8 mm, thickness ∼12 μm) with a ground layer (Ti/Mo, GND, ∼0.2 μm) and a piezoelectric layer (aluminum nitride, AlN, ∼1.8 μm). Lamb waves are excited and detected directly using bidirectional inter-digital transducers (IDTs) [33] located on the surface of the AlN layer. There are six pairs of fingers on bidirectional IDTs in each exciting and detecting transducer. The period of bidirectional IDT is about 400 μm. As waves are partly reflected at the end of length-limited membrane (∼7.8 mm), the device has strong signal without reflectors on both ends of the membrane. With the layers described as in [34], the mass per unit area of the membrane (M) is about 0.0355 kg/m².

The micro Lamb wave device is packaged directly with the printed circuit board (PCB), as shown in Figure 1(b). The network analyzer (Agilent 4395A), connected with the PCB, is used to excite and receive the acoustic signals. The device is protected with one cover on top of the system. The tube is used to pass the liquid into/out of the chamber, which is sealed up with the polymethyl methacrylate (PMMA) cover under the PCB.

When the device is loaded with air or water, multi-modes can be excited and detected effectively, including the A₀₁ mode, the A₀₃ mode, and the S₀ mode (Figure 2). As the A₀₁ and A₀₃ modes are harmonic, the wavelength of the A₀₃ mode is about one third of the wavelength of the A₀₁ mode. According to the positions of the A₀₁ mode, A₀₃ mode and S₀ mode in the dispersion curves [41,42], the resonant frequencies (f) of these three modes should have the following relation: f_A01 < f_A03 < f_S0. The measured resonant frequency of the A₀₁ and A₀₃ modes for water loading are 0.987 and 9.233 MHz, respectively. With the frequency (f) and the wavelength (λ), the corresponding phase velocities (fλ) of these two modes are 385 m/s and 1,200 m/s respectively. Although the phase velocity of the A₀₁ mode is far less than the sound velocity of water (1,485.5 m/s), it is close to the sound velocity of water for the A₀₃ mode. With the same water loading for the S₀ mode, the measured central resonant frequency is about 20.86 MHz. The corresponding phase velocity of the S₀ mode is about 8,135.4 m/s. Further experiments were conducted to investigate the response of multi-modes to the different solutions.

3.2. Experiments for One Species Solutions (NaCl Solutions) with Different Concentrations

The first experiment was done to measure known solutions with different concentrations (Figure 3), such as NaCl solutions. When water is designated as the reference liquid, the relative frequency shifts (Δf/f = (f_solution − f_water)/f_water) with concentration are different for these three modes (A₀₁ mode, A₀₃ mode, and S₀ mode), where f_solution and f_water are the measured frequencies for the solution and the water, respectively. The frequency of the A₀₁ mode decreases with the concentrations and the A₀₃ mode behaves oppositely. In the case of the A₀₃ mode of NaCl solutions measured in Figure 3, the absolute value ρΔδ_E is bigger than the absolute value δ_EΔρ (

Table 1. Based on the measured frequency shifts of NaCl solutions measured in Figure 3, comparison of the value ρΔδ_E and the value δ_EΔρ for the A₀₃ mode.

Concentration (%)	Density (103 kg/m3)	cP (m/s)	δE (μm)	δEΔρ	ρΔδE
0	0.9981	1,200.30	35.12	0	0
0.99	1.0052	1,201.71	34.70	0.25	−0.42
5.67	1.0389	1,206.55	32.81	1.34	−2.39
9.09	1.0640	1,209.03	31.58	2.08	−3.77
13.79	1.0995	1,211.59	30.01	3.04	−5.62
16.64	1.1212	1,212.69	29.13	3.59	−6.71
19.33	1.1425	1,213.53	28.40	4.10	−7.67

). This causes the positive frequency shifts of A₀₃ mode in NaCl solutions measurements. In any case, when the frequency shift is positive, this means the Lamb wave phase velocity is close to the liquid sound velocity. In this case, this phenomenon can be used to decouple the liquid sound velocity.

Compared with the frequency shifts measured with the A₀₁ mode and A₀₃ mode, the frequency of the S₀ mode shows negligible shift with different concentrations (Figure 3).

3.3. Experiments for Three Different Unknown Species Solutions

The second experiment was done to measure the modes' responses to three different species of aqueous electrolytic solutions with different concentrations (

Table 2. Relative frequency shifts (Δf/f) for the A₀₁ mode and the A₀₃ mode in the measurements of three kinds of unknown solutions. In each kind solution, the concentration increases with the serial number.

	Solution A						Solution B		Solution C
No. of solutions	1	2	3	4	5	6	7	8	9	10
Concentration (%)	0.99	6.05	9.97	16.03	19.96	23.95	9.35	18.07	8.98	17.32
Δf/f (A01 mode) (−1 × 104 ppm)	0.236	1.09	1.846	2.886	3.654	4.434	2.243	4.474	2.216	4.356
Δf/f (A03 mode) (104 ppm)	0.027	0.216	0.357	0.373	0.325	0.287	−0.649	−1.422	−0.876	−1.848

). We will check the possibility of the method to decouple the density and the acoustic sound velocity of these solutions with the measurements of A₀₁ mode and A₀₃ mode. Three different species of aqueous electrolytic solutions are prepared, they are listed as solution A, solution B, and solution C respectively, as it is shown in Table 2. From No. 1 to No. 6 solutions, they are the solution A. No. 7 and No. 8 solutions are the solution B. No. 9 and No. 10 are solution C. In each kind of solution, the concentration increases with the serial number. The S₀ mode is still insensitive to the solution changes compared with the measurement in water, and its values are not listed here.

In the case of solution A, the relative frequency shifts Δf/f of A₀₁ mode and A₀₃ mode have similar properties with the solution of NaCl measured in Figure 3. But for the solution B and solution C, the relative frequency shifts of A₀₃ mode is negative. As it is indicated in Equation (5), the absolute value ρΔδ_E of solution B and solution C measured in Table 2 should be smaller than the absolute value δ_EΔρ.

Even for different kinds of solutions, such as the No. 6, No. 8, and No. 10 solutions, the values of Δf/f of the A₀₁ mode are almost the same, but the values of Δf/f of the A₀₃ mode are apparently different. This means only one antisymmetric mode measurement, such as A₀₁ mode, can not determine the solution with high selectivity. In order to specify the liquid with high selectivity, it is better to decouple the liquid density and the liquid sound velocity with the measured relative frequency shifts of A₀₁ mode and A₀₃ mode. These will be analyzed in following section.

3.4. Viscosity Measurements with the S₀ Mode

The responses of the S₀ mode to the solutions of NaBr are similar to the ones of NaCl (Figure 3), the central frequency does not show apparent shifts due to their changing concentrations. However, the changes of the amplitude are related to the liquid's concentrations (Figure 4). It is because the frequency is mainly affected by the liquid density and sound velocity, which has no effects on the real part of the phase velocity of S₀ mode (Equation (6)) [40]. The amplitude of the S₀ mode in NaBr solution (Figure 4) shows that energy losses increase with the concentration, which is caused by the density and the viscosity of the liquid (Equation (7)). This can be used to decouple the density and the viscosity of the solution.

4. Results and Discussions

4.1. To Decouple the Density and the Acoustic Sound Velocity with the A₀₁ Mode and A₀₃ Mode

In order to specify the liquid with high selectivity, we attempted to decouple the liquid density and the liquid sound velocity with the measured frequency shifts of A₀₁ mode and A₀₃ mode. Based on the Equation (4), two steps are undertaken in order to decouple the density and the sound velocity of a liquid as follows:

• Determine the constant B_i by measuring the frequency response of the reference liquid. Here, the water works as the reference liquid, the values of B₁ and B₃ are 14,767.3 N/m and 101,642.8 N/m respectively.

• Get the physical parameters (ρ_L, c_L) for a unknown liquid by measuring c_Pi. All other parameters (B_i, λ_i, M) in Equation (1) have already been calculated or measured.

The first decoupled result using Equation (1) is the NaCl solution with different concentrations, as measured in Figure 3. Comparing the measured density and sound velocity with the values from the literature [32], the two results are close, as shown in Figure 5.

Based on the same process, the density and the sound velocity for other three different species solutions with different concentrations were also decoupled and are shown in Figure 5. Compared with the values from the literature [32], No. 1–No. 6 solutions are KCl solutions, No. 7 and No. 8 solutions are NaBr solutions, and No. 9 and No. 10 solutions are KBr solutions. Aside from decoupled density and sound velocity, the species of the solutions can be identified by comparing the measured values with the already known values [32]. After these density and sound velocity being decoupled using A₀₁ mode and A₀₃ mode, the liquid are determined with higher selectivity compared with only A₀₁ mode measurement.

For Nos. 6, 8, and 10 solutions, the decoupled densities (Figure 5) are almost the same because the relative frequency shifts are very close (Table 2). For these solutions with adjacent density, sound velocities become the main factors affecting the frequency shifts in the A₀₃ mode (Table 2). With the relations of the absolute frequency shifts: Δf/f_No.6 > Δf/f_No.8 > Δf/f_No.10 (Table 2), the decoupled sound velocity of these three solutions have the following relation: c_No.6 > c_No.8 > c_No.10 (Figure 5).

4.2. The Linear Response of (Density × Viscosity)^0.5 Changing with the Item of Amplitude Shifts (ΔA_dB) in S₀ Mode

The item (density × viscosity)^0.5 changes almost linearly with the item of amplitude shifts (ΔA_dB), and the linear fitting coefficient is about 1.02 ± 0.09 (dB/(kg·m⁻²·s^−0.5)), such as the NaBr and NaCl solutions (Figure 6). Beside the density and the viscosity, other parameters, such as the liquid's conductivity, affect the amplitude shifts. This is maybe the reason why the high linearity of (density × viscosity)^0.5 doesn't change with the amplitude shift. However, by analyzing the amplitude shifts (ΔA_dB) of the S₀ mode, the viscosity of the solution can be derived. With the determined density by measuring the frequency response of the A₀₁ and A₀₃ modes, the viscosity will be determined by checking the amplitude response in the S₀ mode.

5. Conclusions

In this study, for the first time to our knowledge, the density, sound velocity, and viscosity of the liquid are obtained simultaneously based on the measurements of the same solution sample volume with a Lamb wave sensor. When the frequency shifts of the A₀₁ and A₀₃ modes are combined, the density and sound velocity are decoupled. This happens because the phase velocity of the A₀₁ mode is far from the sound velocity of the liquid and the phase velocity of the A₀₃ mode is close to the sound velocity of the liquid. Viscosity is obtained by measuring the amplitude of the S₀ mode, which is especially suitable for Newtonian fluids. Effects of the aqueous electrolytic solutions' conductivity on these three modes are not clearly observed. The term (density × viscosity)^0.5 doesn't show high linearity as a function of amplitude shifts.

However, with this multi-parameter detection method, unknown solutions such as aqueous electrolytic solutions can be distinguished successfully and with high selectivity. Results clearly indicate that the multi-modes of a micro Lamb wave sensor are promising in the investigation of molecular thermodynamics, adiabatic compressibility, and molecular label free detection, among others.