1 Introduction

Thickness-shear mode (TSM) quartz crystal resonator (QCR) is an acoustic-wave-based sensor, which has been commonly used as a gravimetric tool to monitor thin film deposition and a versatile tool to characterize the mechanical behaviors of the biological/chemical materials bounded onto the QCR surface^[1-4]. Resonant frequency and electrical impedance are two basic properties of major interest for the complete circuit analyses of a QCR. The effect of the surface loadings on QCR frequencies is often described through a positive equivalent mass, considering that the attached materials are rigid with the sizes much smaller than the QCR thickness^[5]. However, the over layer is not always attached rigidly, and the characteristic scale of the surface loadings gradually becomes comparable to the QCR thickness with the miniaturization of resonators. Therefore, it is required to develop proper two-dimensional models to describe the effect of the geometrical configurations of the surface materials/structures on the dynamic behavior of compound QCR systems. Liu et al.^[6] established a theoretical model of a TSM quartz crystal plate, carrying an array of micro-beams (MBs) with their bottoms fixed to the top surface of the plate, in which the beams were modelled with the Euler-Bernoulli theory for beam bending and the bending moment effect at the MBs/QCR interface was not considered. In order to clarify the effect of the bending moment on the frequency shift, Zhang et al.^[7] put a rigid beam/QCR model forward, where the couple-stress was introduced into the governing equations of the QCR to balance the distributed moments produced by the surface MBs at the interface. Kong et al.^[8-9] and Kong and Hu^[10] further performed a series of works based on this two-dimensional model considering the couple-stress, studied the corresponding frequency-shifts of an Euler-Bernoulli beam/QCR model and a Timoshenko beam/QCR model, compared the obtained results with each other, and detailedly discussed the effect of the strain-gradients of the surface micro-beams on the frequency-shift of the compounded system. Xie et al.^[11] established a liquid-solid coupled dynamic model to study the vibration characteristics of the compounded system immersed in liquid. However, all these researches focus on the influence of the intrinsic characteristics of the surface substance-units on the QCR frequency shift, and the study on the influence of the surface materials/structures on the electric admittance of a compound QCR system has not been found. Therefore, an electrically forced vibration analysis of the resonator admittance is still needed, whose results can be practicably verified in subsequent experiments.

Nowadays, many techniques have been employed to develop MB arrays^[12-14]. One of the potential methods is photolithography using photoresists, such as epoxy resin (SU-8), for high aspect ratio and three-dimensional (3D) submicron lithography. SU-8 has been proven as an ideal viscous polymer, which can be spun or spread onto the QCR surface as a thin overlayer^[15]. This SU-8 overlayer then can be used to pattern high aspect ratio (≥20) structures^[16].

In this paper, the coupling thickness-shear vibrations of a quartz crystal resonator loaded by an array of viscoelastic MBs is discussed. The admittance spectra is studied when a QCR surface is covered by an array of SU-8 MB units. The effects of both the inertia and constraint of the surface materials on the admittance spectra and the shear deformation on the Timoshenko-beam admittance spectra are detailedly investigated.

2 Displacement and stress of AT-cut QCR in TSM vibrations

Figure 1 schematically shows an AT-cut quartz crystal resonator covered with an array of cylindrical MB viscoelastic material units.

For the thickness-shear vibrations of the QCR, the governing equations^[4] are

(1)

where T₂₁ is the stress component, u₁ is the displacement component, and D₂ is the electric displacement. E₂ stands for the electric field, and φ denotes the electric potential. ρ_Q and η_Q stand for the mass density and the viscosity of the quartz, respectively. c₆₆, e₂₆, and ε₂₂ denote the elastic, piezoelectric, and dielectric constants, respectively. Since the crystal plate is under a TSM harmonic vibration, the time-harmonic factor e^iωt will be dropped in the following content for simplicity, where i stands for the imaginary unit, and ω=2πf stands for the driving frequency. We can get the general solution for Eq. (1) by

(2)

where B₁, B₂, B₃, and B₄ are constants to be determined, and

(3)

3 Analysis of micro-beam vibrations from Timoshenko-beam theory and Euler-beam theory

The following discussion is presented based on the work of Kong et al.^[9]. When the above QCR is excited into TSM vibrations, the attached MBs in Fig. 1 will present flexural motions. The deflection and angular rotation of each beam are denoted by w(x, t) and θ(x, t), respectively. The bending moment and shear force are denoted by M and T, respectively.

According to the Timoshenko-beam theory^[17], the equations of motion of the beam shown in Fig. 2 are

(4)

for the rotation and

(5)

for the translation in the direction of the y-axis. Eliminating θ from Eqs. (4) and (5), we obtain the following required equation:

(6)

where E, ρ, I, A, G, and κ denote the elastic modulus, the mass density, the moment of inertia, the cross sectional area, the shear modulus, and the shear coefficient of the MBs, respectively. The complex shear modulus G=G’ + iG” is obtained from Ref. ^[15]. The corresponding elastic modulus E is derived from E=2(1 + ν)G.

The boundary conditions at the tops of the MBs are

(7)

The boundary conditions on the beam/QCR interface are

(8)

In the above equations, W is the non-dimensional general solution of Eq. (6), and the expressions of W, M, T, and θ are extracted directly from Ref. ^[9].

In addition, the bottom edge of the QCR is free from any traction. Therefore, we have

(9)

Let the electric potentials on the top and bottom electrodes of the quartz be

(10)

Substituting Eq. (2) and the expressions of W, M, T, and θ into Eqs. (7) to (10), we obtain

(11)

where

(12)

Moreover, we can obtain a similar expression from the Euler-Bernoulli beam theory as follows:

(13)

where

(14)

In order to determine the admittance, define the ratio of current to voltage. Then, an expression for the current is obtained. Tiersten^[18] gave the surface charge over an area S for two-dimensional piezoelectric plates as Q=∫D₂dS. Substitute φ from Eq. (2) into Eq. (1). Then, the displacement can be expressed explicitly in terms of B₃ as D₂=-ε₂₂B₃. Accordingly, the charge on the electrode of area S₀ can be written as Q=-ε₂₂B₃S₀. The current is the time derivative of Q, which means

(15)

Therefore, the admittance of the system is

(16)

4 Numerical results

We consider an AT-cut quartz crystal plate with the following parameters:

(17)

and the SU-8 MB array with the storage shear modulus G'=1.66 × 10⁹ Pa, the loss shear modulus G"=6 × 10⁷ Pa, the Poisson ratio ν=0.22, and the mass density ρ=1 200 kg·m^-3[15]. The MB distribution density N is set to be 1 × 10¹² units·m^-2 in the analysis. It is readily obtained that the fundamental frequency of the uncoated QCR is f₀=10.035 5 MHz. For convenience in the analysis, we make a non-dimensionalization on the real part of the admittance Y (=Y₁ + iY₂) to produce a non-dimensional quantity Y₁^*(=lg (Y₁/Y₀)), where Y₀ (=0.022 8 S) is the corresponding real admittance amplitude of the uncoated QCR.

Figure 3 presents the non-dimensional real admittance spectra of the compound MBs/QCR system versus the normalized frequency f/f₀ upon different MB lengths (3.5 μm≤L≤6.0 μm), while the radius R of the MBs is set to be 0.5 μm. The real admittance spectrum of the uncoated QCR is also shown for reference. It follows from the left figure of Fig. 3 that there always exists an admittance peak near f₀ for every admittance profile, e.g., Points J_i(i=1, 2, 3, 4), implying the arrival of a natural frequency of the QCR/MB compound system. When L increases from 0.0 μm to 4.5 μm, the admittance peak moves left. When L=5.5 μm, the admittance peak is at the right-side of f₀, and continues to move left with a further increase in the MB length. Obviously, there should exist such a special MB length L^* between 4.5 μm and 5.5 μm that the corresponding non-dimensional admittance spectra possess two admittance peaks, respectively, located at the two sides of f₀. A detailed calculation shows L^*=5.0 μm, whose real admittance spectra near f₀ are shown in the right figure of Fig. 3.

The reason for incurring the above phenomenon will be analyzed as follows. The vibration mode of the MBs is calculated in Fig. 4. It can be found that the MBs are under the first-order vibration mode when L=3.5 μm and 4.5 μm and the second-order vibration mode when L=5.5 μm and 6.5 μm. However, two vibration modes exist near f₀ when L^*=5.0 μm. One is the first-order vibration mode at Point J₅ (see (1) in Fig. 4), and the other is the second-order vibration mode at Point J₆ (see (2) in Fig. 4). We note from a detailed calculation that the fundamental frequency of the MBs when L^*=5.0 μm is just identical to the fundamental frequency f₀ of the QCR, which indicates the resonance and mode jump of the MBs between the first-order vibration mode and the second-order vibration mode. We introduce a factor as follows:

which stands for the ratio of the interface shear force T|_x=0 to the QCR surface displacement u₁|_x2=h. Figure 5 shows the dependence of Γ upon the driving frequency near f₀ for L (L=3.5 μm, 4.5 μm, 5.0 μm, 5.5 μm, 6.5 μm). From Fig. 5, we can see

As illustrated in Ref. ^[8], Γ >0 means that T|_x=0 is with the same phase as u₁|_x2=h, i.e., the interface shear force produces a pushed-forward effect on the QCR vibration, resulting in an increase in the system inertia. The system resonant frequency reduces or produces a left frequency-shift F_S. Moreover, a larger positive Γ yields a larger left frequency-shift. Therefore, F_S(J₅) > F_S(J₂) > F_S(J₁). Moreover, negative Γ means that T|_x=0 is in an opposite phase to u₁|_x2=h. Under this situation, the interface shear force produces a dragged-back effect or a constraint effect on the QCR vibration to induce the increase or right frequency-shift in the system resonant frequency. A smaller negative Γ produces a stronger constraint effect to induce a larger right frequency-shift. Thus, F_S(J₆) > F_S(J₃) > F_S(J₄).

The above analysis is focused on the discussion for the case of the MBs with the fundamental frequency of the MBs close to f₀. From the analysis, we can see that the MB length is within the region (3.5 μm, 6.0 μm). In the following content, we will similarly present a brief discussion for the case of the MBs with the second-order natural frequency close to f₀. Figure 6 presents the non-dimensional real admittance spectra of the compound MB/QCR system upon different MB lengths (11.0 μm≤L≤14.0 μm). There exists an admittance peak near f₀ for every admittance profile, i.e., Points J_i (i=7, 8, 9, 10). When L increases from 11.0 μm to 12.0 μm, the admittance peak moves left. When L=13.0 μm, the admittance peak is at the right-side of f₀, and continues to move left with a further increase in the MB length. Similarly, we obtain a special MB length L^*=12.5 μm, whose admittance spectra near f₀ is shown in the right figure of Fig. 6.

The vibration modes of the MBs are calculated in Fig. 7, whose tendencies are the same with Fig. 4. We also note from the calculation that the second natural frequency of the MBs with L^*=12.5 μm is just identical to f₀, which indicates a resonance and mode jump between the second-order vibration mode and the third-order vibration mode. Figure 8 shows the dependence of Γ upon the driving frequency near f₀ for L (L=11.0 μm, 12.0 μm, 12.5 μm, 13.0 μm, 14.0 μm). It follows from Fig. 8 that Γ (J₁₁) >Γ (J₈) >Γ (J₇) >0 and Γ (J₁₂) < Γ(J₉) < Γ (J₁₀) < 0, which indicates the left frequency-shift sequence F_S(J₁₁)>F_S(J₈)>F_S(J₇) and the right frequency-shift sequence F_S(J₁₂)>F_S(J₉)>F_S(J₁₀).

Figure 9 shows two normalized real admittance spectra obtained from the Timoshenko-beam theory and the Euler-beam theory. In order to show the effect of the shear deformation on the admittance spectra more evidently, we focus on the discussion of the MBs whose length is larger than 12.5 μm, while the beam radius is still set to be 0.5 μm. The corresponding vibration modes are presented in Fig. 10. It can be obtained from Figs. 9 and 10 that the effect of the shear deformation possesses the following characteristics:

(i) For each MB length, the evident effect of the shear deformation arises at the natural frequency of the MB/QCR system, e.g., 10.091 MHz for L=13.5 μm and 9.821 MHz for L=20 μm;

(ii) When a natural frequency of the MBs is identical to f₀, the effect of the shear deformation becomes the most evident at the admittance peaks, e.g., 9.540 MHz and 10.524 MHz for L^*=12.5 μm, where the second natural frequency of the MBs is close to f₀, and 9.645 MHz and 10.368 MHz for L^*=21.0 μm, where the third natural frequency of the MBs is close to f₀;

(iii) The higher order vibration mode of the MBs corresponds to a larger effect of shear deformation at the jumping transition, e.g., when L^*=12.5 μm, the second-order vibration mode is with the frequency deviation 0.084 MHz at 9.540 MHz, the third-order vibration mode is with the frequency deviation 0.147 MHz at 10.524 MHz; when L^*=21.0 μm, the third-order vibration mode is with the frequency deviation 0.071 MHz at 9.645 MHz, and the fourth-order vibration mode is with the frequency deviation 0.119 MHz at 10.369 MHz.

5 Conclusions

A two-dimensional model is exhibited for the influence of a covered array of MB viscoelastic material units on the impedance and dynamics of a compound QCR system by introducing the effects of the shear deformation. The dependence of the admittance spectrum upon the MB length is investigated. It is revealed that both the inertia effect and the constraint effect of the MBs produce competitive influence on the resonant frequency and admittance of the compound QCR system. The numerical results of the Timoshenko-beam model are further compared with those of the Euler-beam model. The deviations of the admittance spectra induced by the shear deformation are detailedly discussed. The deviations are found to be evident around the admittance peak (s) and reach the maximum when a natural frequency of the MBs is identical to the fundamental frequency of the QCR. Moreover, the higher order vibration mode of the MBs corresponds to a larger deviation at the resonance. This research lays the theoretical foundation for using the electric admittance of an MB array loaded QCR to characterize the physical and geometric parameters of MBs.