Quality factor of circular microdiaphragm with surface effects in contact with liquids

Abstract

This paper reports an investigation of the influences of surface effects and residual stress on the quality factor of a circular microdiaphragm in contact with liquids on one side. Acoustic radiation, as the main source of energy dissipation, can decrease the quality factor of a circular microdiaphragm. An approximate solution for the natural frequency can be obtained based on the Rayleigh–Ritz energy method. The results show that the influence of surface effects on the quality factor is obvious, in particular for thinner microdiaphragms. A stiffened surface decreases the quality factor, while a softened surface increases it. Furthermore, the quality factor increases with increases in the normalized tension parameter k for both stiffened and softened surfaces. If k is greater than 30, the influence of the surface effects on the quality factor must be considered. The results can provide effective guidance for the determination of dimensions and selection of material parameters in designing microdiaphragm resonant sensors.

Keywords

Quality factor microdiaphragm liquid surface effects residual stress

1. Introduction

Resonant microbiochemical sensors have found wide applications in a variety of fields, such as the label-free detection of biological species, gas detection, and medical health care [1 –3]. The chief components of resonant microbiochemical sensors are various transducers. The frequency stability, mass sensitivity, and detection limit of resonant microbiochemical sensors are mainly determined by the quality factor (Q factor) of the transducers [4].

Microcantilevers and microdiaphragms are the major structures of transducers [5, 6]. In gaseous working environments, microcantilevers have already demonstrated their potential for use as transducers of highly sensitive sensors [7, 8]. Nonetheless, for microcantilevers in liquid working environments, external factors, such as viscous drag, can cause a low Q factor [9]. In recent years, researchers found that the Q factor of microdiaphragms is high in both gaseous and liquids environments [10, 11]. Thereby, the Q factor of microdiaphragms has raised the interest of many scientists.

It is known that the Q factor can be decreased by different energy dissipation mechanisms. The main dissipation mechanisms for transducers can be recognized as medium loss, which is caused by acoustic radiation or viscous drag [12], clamping or support loss [13], and bulk loss, incurred by internal friction, phonon–phonon scattering, thermoelastic dissipation, motion of lattice defects, and so on [14]. These dissipation mechanisms contribute unequally to the total energy loss of the system. For microcomponents, research has shown that the surrounding medium loss due to acoustic radiation or viscous drag can cause a low Q factor [15]. Olfatnia et al. [16] found that the bulk loss can be ignored when compared with the medium loss for microcomponents. Considering the medium loss due to acoustic radiation, Wu et al. [17] analyzed the influences of residual stress on the Q factor of a sensor diaphragm in theory. Experiments have been conducted by Blanco-Gomez and Agache [18] to analyze the influences of different dissipation sources, such as flow conditions, vibration amplitude, and acoustic energy dissipation, on the Q factor of square microtransducers. However, the influences of structure dimensions on the Q factor were not taken into consideration in these experiments. Subsequently, it was experimentally found that the Q factor of nanomechanical transducers depends highly on the structure dimensions and does not decrease linearly as a function of viscosity [19]. To investigate the effects of structure dimensions on the Q factor of square nanoplate transducers, experiments have been carried out by Qian et al. [20]. In these experiments, the Q factors of square nanoplates with different thicknesses from 200 nm down to 50 nm and with different lateral dimensions from 60 µm to 28 µm were obtained. The results indicated that the Q factor of square nanoplate transducers changes obviously with decreases in thickness and lateral dimensions. The Q factors of a circular microdiaphragm for different radii were analyzed experimentally and theoretically by Olfatnia et al. [15]. In the theoretical model, since the influences of thickness and stresses on the Q factor were not taken into account, the theoretical value deviated from the experimental data. Thereby, in this paper, surface effects are taken into account to reflect the influences of structure dimensions on the Q factor of a circular microdiaphragm.

For solids at the micro- or nanoscale, owing to the significant ratio of surface to bulk, surface effects play a dominant role in mechanical behavior [21 –24]. To incorporate surface effects, Gurtin and Murdoch [25, 26] developed a surface continuum elasticity theory. Based on this theory, Assadi [27] presented an analytical model of rectangular nanoplates to study the size-dependence of forced vibration. Sahmani and Bahrami [28] studied the free vibration characteristics of circular nanoplates subjected to hydrostatic and electrostatic forces. Subsequently, Zhou et al. [29, 30] developed a theoretical model to investigate the vibrational characteristics of diaphragms with a uniformly distributed mass based on this theory and Kirchhoff’s plate theory. Based on this theory and the higher-order shear deformation plate theory, Gholami and Ansari [31] investigated the influences of initial imperfection, thickness, surface residual stress, and temperature rise on the vibration response of nanoplates in thermal environments.

The coupling vibration of circular plates in contact with liquids was first investigated by Lamb [32] based on the assumed modes approach. Later, Lamb’s model was revised by Amabili and Kwak [33] to study the coupled free vibration of circular plates. In recent years, various extended coupling vibration models have been developed to investigate the vibrations of rectangular and circular plates at the macroscale [34, 35]. Experiments have proved that Lamb’s theoretical vibration model of plates is also applicable to microscale structures [36]. Based on Lamb’s assumption, Wu et al. [37] analyzed the vibration characteristics of microscale rectangular plates in contact with liquids by utilizing the Rayleigh–Ritz energy method.

As mentioned, there are many experimental and theoretical studies regarding the media influences on the Q factor of transducers. Nonetheless, the influences of surface effects were ignored in these studies. The classical theory without surface effects is inadequate for predicting the quality factor of microresonant sensors. In this study, to improve the weakness, a new model is developed to analyze, theoretically, the influence of surface effects and residual stress on the Q factor of microdiaphragms in contact with liquids. The model of microdiaphragms is provided in Section 2. Subsequently, in Section 3, the reference kinetic energy of liquids is obtained, based on the Hankel transform and inverse Hankel transform. In Section 4, an approximate solution of the natural frequency for a circular microdiaphragm in contact with liquids is given by using the Rayleigh–Ritz method. In Section 5, the energy dissipation and Q factor are obtained. In Section 6, numerical results are presented and the effects of surface properties and residual stress on the Q factor of a circular microdiaphragm are discussed accordingly. Finally, the major conclusions of this paper are given. The results of this study can provide effective guidance for the design of microdiaphragm resonant sensors.

2. Theoretical model of circular microdiaphragm

There are many kinds of liquid microbiochemical sensor [38 –43]. Some of them need to contact the liquid medium on one side during operation. Examples are circular micromembrane transducers for real-time label-free biochemical detection in deionized water [39], circular resonant pressure drumhead sensors [40], circular film bulk acoustic resonators [41], and high-sensitivity microdiaphragm fluid-loaded resonating sensors [42]. Based on these practical applications, Figure 1 depicts the structure of a sensor based on a microdiaphragm; a circular microdiaphragm with thickness h and radius a is clamped to a rigid wall, which is in contact with incompressible and inviscid liquids on one side. Moreover, the liquid domain is infinite in both r and z directions. The elasticity modulus and Poisson ratio are E and ν, respectively. The mass density is ρ_p. The surface elasticity constants E_s and corresponding Lamé constants µ_s and λ_s are considered for the top and bottom surfaces, while τ₀ is the residual surface stress. The partially uniformly distributed mass m_d is located at the center of the top surface, and the radius is r₀. Although the model is developed for a circular microdiaphragm, it can be easily extended to square geometries for biochemical sensors [43].

By introducing surface continuum elasticity, the surface effects of the circular microdiaphragm are taken into account [25, 26]. Then, considering the distributed mass, the governing equation is that developed by Zhou et al. [29]. Based on the results of Zhou et al., the motion equation of a circular microdiaphragm without considering surface mass can be written as

D^{*} \nabla^{4} w - 2 τ_{0} \nabla^{2} w + [ρ_{p} h + \frac{m_{d}}{π {r_{0}}^{2}} H (ξ, ξ_{0})] \frac{\partial^{2} w}{\partial t^{2}} - \frac{ρ_{p} h^{3}}{12} \nabla^{2} \overset{\cdot\cdot}{w} = 0 .

(1)

The influence of distributed mass can be regarded as a transverse inertia force described by the Heaviside function H(ξ, ξ₀),

H (ξ, ξ_{0}) = {\begin{matrix} 1 ξ \leq ξ_{0} \\ 0 ξ > ξ_{0} \end{matrix} .

(2)

The dimensionless parameters ξ and ξ₀ are defined as

ξ = \frac{r}{a}, ξ_{0} = \frac{r_{0}}{a} (0 \leq ξ_{0} \leq 1, 0 \leq ξ \leq 1) .

(3)

The Laplace operator $\nabla^{2}$ and biharmonic operator $\nabla^{4}$ are given by polar co-ordinates with respect to r and θ. As the two surfaces have the same surface properties, the effective flexural rigidity D^* [29] can be expressed as

D^{*} = D + \frac{h^{2} μ_{s}}{2} + \frac{h^{2} λ_{s}}{2} - \frac{h^{2} (3 - 4 v) τ_{0}}{6 (1 - ν)},

(4)

where D = Eh³/12(1 −ν²) is the flexural rigidity in classical plate theory.

The solution of equation (1) can be obtained by summing a series of eigenfunctions in each separated mode:

w (r, θ, t) = \sum_{m = 0}^{N_{m}} \sum_{n = 0}^{N_{n}} W_{mn} R_{mn} (r) Θ_{m} (θ) \cdot f (t),

(5)

where f(t) = e^jωt is the time function and R_mn(r) and Θ_m(θ) are the mode shape functions that satisfy the boundary conditions in the r and θ directions, respectively. For a clamped circular plate, R_mn(r) and Θ_m(θ) are given as [44]

R_{mn} (r) = α [J_{m} (λ_{mn} r) - \frac{1}{2} λ_{mn} r^{m} (1 - r^{2}) J_{m + 1} (λ_{mn})],

(6)

Θ_{m} (θ) = \cos m θ,

(7)

in which the coefficient α for a clamped circular plate is

α = {\begin{matrix} 1 & m \neq 0 \\ 1 / 2 & m = 0 \end{matrix},

(8)

and λ_mn is the nth positive root of Bessel function J_m(x).

3. Effect of liquids on circular microdiaphragm

The theoretical analysis is based on the assumption that liquid is incompressible and inviscid. The movement of liquid, considered only to result from the microdiaphragm vibration, is assumed to be irrotational. The liquid movement is described by the velocity potential,

U (r, θ, z, t) = u (r, θ, z) \overset{\cdot}{f} (t)

(9)

where u is the spatial distribution of the velocity potential and $\overset{\cdot}{f} (t)$ is the derivative of f(t) in equation (5) with respect to time. As a consequence of the hypotheses made, u satisfies the Laplace equation,

\nabla^{2} u = 0 in L .

(10)

Equation (10) can be simplified by the separation of variables

u (r, z, θ) = Φ (r, z) \cos m θ .

(11)

Then equation (10) can be rewritten as

\frac{\partial^{2} Φ}{\partial r^{2}} + \frac{1}{r} \frac{\partial Φ}{\partial r} + \frac{\partial^{2} Φ}{\partial z^{2}} - \frac{m^{2}}{r^{2}} Φ = 0 in L .

(12)

The liquid–structure interface boundary conditions for Φ are given by:

{\frac{\partial Φ (r, z)}{\partial z} |}_{z = 0} = 0 on the rigid wall - liquid interface,

(13)

{\frac{\partial Φ (r, z)}{\partial z} |}_{z = 0} = - W_{mn} R_{mn} (r) on the diaphragm - liquid interface .

(14)

To solve equation (12), the Hankel transform and inverse Hankel transform are introduced [31]:

{\bar{Φ}}_{h} (ζ, z) = \int_{0}^{\infty} r Φ (r, z) J_{m} (ζ r) d r,

(15)

Φ (r, z) = \int_{0}^{\infty} ζ {\bar{Φ}}_{h} (ζ, z) J_{m} (ζ r) d ζ .

(16)

By applying the Hankel transform (equation (15)), to the first, second, and fourth terms of equation (12), the equation can be rewritten as

\int_{0}^{\infty} r (\frac{\partial^{2} Φ}{\partial r^{2}} + \frac{1}{r} \frac{\partial Φ}{\partial r} - \frac{m^{2}}{r^{2}} Φ) J_{m} (ζ r) d r = - ζ^{2} \int_{0}^{\infty} r Φ (r, z) J_{m} (ζ r) d r = - ζ^{2} {\bar{Φ}}_{h} (ζ, z) .

(17)

Substituting equation (12) into equation (17) and using the Hankel transform (equation (15)), equation (17) can be expressed as

\int_{0}^{\infty} r (\frac{\partial^{2} Φ}{\partial r^{2}} + \frac{1}{r} \frac{\partial Φ}{\partial r} - \frac{m^{2}}{r^{2}} Φ) J_{m} (ζ r) d r = - \int_{0}^{\infty} r \frac{\partial^{2} Φ}{\partial r^{2}} J_{m} (ζ r) d r = - \frac{d^{2}}{d z^{2}} {\bar{Φ}}_{h} (ζ, z) .

(18)

Then equation (12) is reduced to the ordinary differential equation, taking the form

\frac{\partial^{2} Φ}{\partial z^{2}} - ζ^{2} {\bar{Φ}}_{h} = 0 .

(19)

The solution of equation (19) is expressed as

{\bar{Φ}}_{h} (ζ, z) = B (a ζ) e^{- ζ z},

(20)

where $B (a ζ)$ is a function that must be determined. By substituting equation (20) into the inverse Hankel transform (equation (16)), one can obtain

Φ (r, z) = \int_{0}^{\infty} ζ B (a ζ) e^{- ζ z} J_{m} (ζ r) d ζ

(21)

Substituting equation (21) into equations (13) and (14), the liquid–structure interface boundary conditions can be obtained as

\int_{0}^{\infty} ζ^{2} B (a ζ) J_{m} (ζ r) d ζ = W_{mn} R_{mn} (r), 0 \leq r \leq a

(22)

\int_{0}^{\infty} ζ^{2} B (a ζ) J_{m} (ζ r) d ζ = 0, r \geq a

(23)

The solution of these integral equations (equations (22) and (23)) can be obtained by using the properties of the Hankel transform:

ζ B (a ζ) = \int_{0}^{a} W_{mn} R_{mn} (r) J_{m} (ζ r) r d r .

(24)

Substituting equation (24) into equation (21), the function Φ at the liquid–structure interface can be obtained:

Φ (r, 0) = \int_{0}^{\infty} \int_{0}^{a} W_{mn} R_{mn} (r) J_{m} (ζ r) J_{m} (ζ r) r d r d ζ .

(25)

Based on the assumption of irrotational movement and simple connection of the liquid domain, the reference kinetic energy of liquid can be obtained according to its boundary condition [45],

T_{L} = - \frac{1}{2} ρ_{L} \int_{0}^{2 π} \int_{0}^{a} \frac{\partial Φ (r, 0)}{\partial z} Φ (r, 0) \cos^{2} (m θ) r d r d θ,

(26)

where ρ_L is the density of the liquid.

Combining the liquid–structure interface boundary conditions (equation (14)) and substituting equation (25) into equation (26), the reference kinetic energy of liquid can be written as

T_{L} = \frac{1}{2} ρ_{L} ψ_{m} \int_{0}^{\infty} \int_{0}^{a} w^{2} (r) J_{m} (ζ r) J_{m} (ζ r) r^{2} d r d ζ,

(27)

where the coefficient ψ_m is

ψ_{m} = {\begin{matrix} 2 π m = 0 \\ π m > 0 \end{matrix} .

(28)

The integral in equation (25) can be evaluated numerically.

4. Approximate solution using Rayleigh–Ritz method

To acquire the natural frequency of circular plates vibrating in contact with liquids, the Rayleigh–Ritz method is applied. The potential energy and reference energy of the microdiaphragm can be written as

U_{P} = \frac{D^{*}}{2} \int \int_{S} {{(\frac{\partial^{2} w}{\partial r^{2}} + \frac{\partial w}{r \partial r} + \frac{\partial^{2} w}{r^{2} \partial θ^{2}})}^{2} - 2 (1 - v) [\frac{\partial^{2} w}{\partial r^{2}} (\frac{\partial w}{r \partial r} + \frac{\partial^{2} w}{r^{2} \partial r^{2}}) - {(\frac{\partial^{2} w}{\partial r \partial θ} - \frac{\partial w}{r^{2} \partial θ})}^{2}]} r d r d θ,

(29)

T_{P} = \frac{1}{2} ρ_{P} h \int \int_{S} w^{2} (r, θ) r d r d θ .

(30)

The reference kinetic energy of the distributed mass can be expressed using the Heaviside function:

T_{M} = \frac{1}{2} \int \int_{S} \frac{m_{d}}{π {r_{0}}^{2}} H (ξ, ξ_{0}) w^{2} (r, θ) r d r d θ .

(31)

Based on the hypothesis that wet mode shapes of the microdiaphragm are the same as dry mode shapes, all these energy formulations can be expressed by the dry mode. The total energy of the coupled system can be written as [37]

V = U_{P} - ω^{2} (T_{P} + T_{L} + T_{M}) .

(32)

A series of eigenfunctions can be obtained by minimizing equation (32) with respect to the unknown deflection coefficient W_pq, such that

\frac{\partial V}{\partial W_{pq}} = 0 .

(33)

These eigenfunctions are

\sum_{m = 0}^{N_{m}} \sum_{n = 0}^{N_{n}} {U_{P, mnpq} - ω^{2} (T_{P, mnpq} + T_{L, mnpq} + T_{M, mnpq})} W_{mn} = 0,

(34)

where

\begin{matrix} U_{P, mnpq} = \frac{D^{*}}{2} \int \int_{S} {{\overset{\cdot\cdot}{R}}_{mn} (r) {\overset{\cdot\cdot}{R}}_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) + \frac{1}{r^{2}} {\overset{\cdot}{R}}_{mn} (r) {\overset{\cdot}{R}}_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) \\ + \frac{1}{r^{4}} R_{mn} (r) R_{pq} (r) {\overset{\cdot\cdot}{Θ}}_{m} (θ) {\overset{\cdot\cdot}{Θ}}_{p} (θ) + \frac{1}{r} {\overset{\cdot\cdot}{R}}_{mn} (r) {\overset{\cdot}{R}}_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) + \frac{1}{r} {\overset{\cdot}{R}}_{mn} (r) {\overset{\cdot\cdot}{R}}_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) \\ + \frac{1}{r^{3}} R_{mn} (r) {\overset{\cdot}{R}}_{pq} (r) {\overset{\cdot\cdot}{Θ}}_{m} (θ) Θ_{p} (θ) + \frac{1}{r^{3}} {\overset{\cdot}{R}}_{mn} (r) R_{pq} (r) Θ_{m} (θ) {\overset{\cdot\cdot}{Θ}}_{p} (θ) + \frac{1}{r^{2}} R_{mn} (r) {\overset{\cdot\cdot}{R}}_{pq} (r) {\overset{\cdot\cdot}{Θ}}_{m} (θ) Θ_{p} (θ) \\ + \frac{1}{r^{2}} {\overset{\cdot\cdot}{R}}_{mn} (r) R_{pq} (r) Θ_{m} (θ) {\overset{\cdot\cdot}{Θ}}_{p} (θ) + (1 - v) [\frac{1}{r} {\overset{\cdot\cdot}{R}}_{mn} (r) {\overset{\cdot}{R}}_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) + \frac{1}{r} {\overset{\cdot}{R}}_{mn} (r) {\overset{\cdot\cdot}{R}}_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) \\ + \frac{2}{r^{2}} {\overset{\cdot\cdot}{R}}_{mn} (r) {\overset{\cdot\cdot}{R}}_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) - 2 {\overset{\cdot}{R}}_{mn} (r) {\overset{\cdot}{R}}_{pq} (r) {\overset{\cdot}{Θ}}_{m} (θ) {\overset{\cdot}{Θ}}_{p} (θ) + \frac{2}{r^{2}} {\overset{\cdot}{R}}_{mn} (r) R_{pq} (r) {\overset{\cdot}{Θ}}_{m} (θ) {\overset{\cdot}{Θ}}_{p} (θ)] \\ + {\overset{\cdot}{R}}_{mn} (r) R_{pq} (r) {\overset{\cdot}{Θ}}_{m} (θ) {\overset{\cdot}{Θ}}_{p} (θ) - \frac{2}{r^{4}} R_{mn} (r) R_{pq} (r) {\overset{\cdot}{Θ}}_{m} (θ) {\overset{\cdot}{Θ}}_{p} (θ)} d r d θ, \end{matrix}

(35)

T_{P, mnpq} = \frac{1}{2} ρ_{P} h \int \int_{S} R_{mn} (r) R_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) d r d θ,

(36)

T_{L, mnpq} = \frac{1}{2} ρ_{L} ψ_{m} \int_{0}^{\infty} \int_{0}^{a} R_{mn} (r) R_{pq} (r) J_{m} (ζ r) J_{p} (ζ r) r^{2} d r d ζ,

(37)

T_{M, mnpq} = \frac{1}{2} \frac{m_{d}}{π {r_{0}}^{2}} \int \int_{S} R_{mn} (r) R_{pq} (r) Θ_{m} (θ) Θ_{p} (θ) H (ξ, ξ_{0}) d r d θ .

(38)

Equation (34) can be further simplified in the matrix form:

{[A] - ω^{2} [I]} {W} = 0,

(39)

where [I] is the unit matrix and $[A] = [M]^{- 1} [K]$ .

The elements M_ij of the inertial matrices [M] and the elements K_ij of the stiffness matrices [K] for the vibration system are given by

K_{ij} = U_{P, mnpq},

(40)

M_{ij} = T_{P, mnpq} + T_{L, mnpq} + T_{M, mnpq},

(41)

where the indices i and j are defined as

\begin{matrix} i = l (p - 1) + q \\ j = l (m - 1) + n \\ l \in N^{+} . \end{matrix}

(42)

If there is a non-zero solution of equation (39), the determinant of the coefficient matrix is required to be zero. The explicit form can be expressed as

| \begin{matrix} A_{11} - ω^{2} & A_{12} & A_{13} & . . . & A_{1 N} \\ A_{21} & A_{22} - ω^{2} & A_{23} & . . . & A_{2 N} \\ ⋮ \\ ⋮ \\ A_{N 1} & A_{N 2} & A_{N 3} & . . . & A_{NN} - ω^{2} \end{matrix} | = 0 .

(43)

Then, considering surface effects and distributed mass, the natural frequencies of microdiaphragm in contact with liquids can be determined by solving the characteristic equation about ω². By assuming µ_s = λ_s = τ₀ = 0 and m_d = 0, the new model reduces to the classical plate model.

5. Emission of energy and quality factor

For a microdiaphragm in contact with liquids, the decrease in the Q factor is due to damping. For incompressible and inviscid liquids, the Q factor is mainly determined by acoustic radiation. There are various methods to calculate acoustic radiation. For the convenience of theoretical calculation, the far-field approach based on the Rayleigh surface integral is taken to calculate the acoustic pressure and acoustic intensity. The acoustic power radiation from a circular microdiaphragm vibrating in an infinite baffle is shown in Figure 2.

Let the spherical co-ordinates of the observation point P be r′, φ, and ψ; r′ is the distance between the center of the co-ordinate system and the observation point. The acoustic pressure at observation point P can be obtained by dividing the radiating surface of the microdiaphragm into infinitesimal elements. The co-ordinates of infinitesimal element ds are r and θ. The acoustic pressure generated by one of these infinitesimal sources at observation point P is given by Strutt and Rayleigh [46] as

d p = \frac{jk ρ_{L} c_{L}}{2 π l} V_{mn} (r, θ) d s \cdot e^{j (ω t - k^{*} l)},

(44)

where k^* is the acoustic wave number, k^* = ω/c_L, c_L is the propagation velocity of waves in liquids, and V_mn(r,θ) is the velocity amplitude distribution on the surface of the microdiaphragm. According to equation (5), the velocity amplitude can be expressed as

V_{mn} (r, θ) = j ω W_{mn} R_{mn} (r) Θ_{m} (θ)

(45)

and l is the distance between observation point P and infinitesimal element ds:

l = r' - r \sin φ (\cos ψ \cos θ + \sin ψ \sin θ) .

(46)

The resulting acoustic wave radiated by a circular microdiaphragm can be found from Rayleigh’s integral [46]. Then the total acoustic pressure p in the far-field (r’>>r) can be obtained by integrating equation (44) over the whole surface of the circular microdiaphragm:

p (r, θ, t) = - \frac{k^{*} ρ_{L} c_{L} ω}{2 π l} \int \int_{s} W_{mn} R_{mn} (r) Θ_{m} (θ) e^{j (ω t - k^{*} l)} d s .

(47)

For the far-field approximation, the effect of distance on the amplitude of radiation acoustic pressure can be ignored, while its effect on the phase cannot be ignored [47]. Using the far-field approximation, the distance l in acoustic pressure amplitude can be approximated as r′. Then, substituting equations (45) and (46) into equation (47), the total acoustic pressure p can be expressed as

p = - \frac{k^{*} ρ_{L} c_{L} ω}{2 π r'} W_{mn} e^{j (ω t - k^{*} r')} \int_{0}^{2 π} \int_{0}^{a} R_{mn} (r) Θ_{m} (θ) e^{j k^{*} r \sin φ (\cos ψ \cos θ + \sin ψ \sin θ)} r d r d θ .

(48)

Then its magnitude can be written as

| p | = \frac{k^{*} ρ_{L} c_{L} ω}{2 π r'} W_{mn} | \int_{0}^{2 π} \int_{0}^{a} R_{mn} (r) Θ_{m} (θ) e^{j k^{*} r \sin φ (\cos ψ \cos θ + \sin ψ \sin θ)} r d r d θ | .

(49)

The far-field average acoustic intensity can be expressed as $I = {| p |}^{2} / 2 ρ_{L} c_{L}$ [48]. This is the time-averaged rate at which acoustic energy is propagated per unit area. Integrating the expression over a hemisphere enclosing the vibrating surface, the far-field acoustic power radiation can be given by

W_{p} = \frac{ρ_{L} c_{L}}{8} {(\frac{k^{*} ω}{π})}^{2} {W_{mn}}^{2} \int_{ψ = 0}^{2 π} \int_{ψ = 0}^{π / 2} {| \int_{0}^{2 π} \int_{0}^{a} R_{mn} (r) Θ_{m} (θ) e^{j k^{*} r \sin φ (\cos ψ \cos θ + \sin ψ \sin θ)} r d r d θ |}^{2} \sin φ d φ d ψ .

(50)

It is well known that the Q factor is generally given as a function of the resonance frequency, the damping coefficient, and the effective mass. The Q factor can also be defined as the ratio between stored energy W and dissipated energy W_p per cycle of vibration:

Q = \frac{2 π W}{W_{p} T} = \frac{ω W}{W_{p}},

(51)

where T is the period of vibration. The total kinetic energy of the system contains the kinetic energies of the microdiaphragm, the distributed mass, and the liquids: W = T_P+T_L+T_M. According to the far-field acoustic power radiation equation (equation (50)), equation (51) becomes

Q = \frac{4 π^{2}}{ρ_{L} c_{L} ω {k^{*}}^{2}} \frac{ρ_{P} h {\int \int_{S} w}^{2} (r, θ) r d r d θ + \int \int_{S} \frac{m_{d}}{π {r_{0}}^{2}} H (ξ, ξ_{0}) w^{2} (r, θ) r d r d θ + ρ_{L} ψ_{m} \int_{0}^{\infty} \int_{0}^{a} w^{2} (r) J_{m} (ζ r) J_{m} (ζ r) r^{2} d r d ζ}{\int_{ψ = 0}^{2 π} \int_{φ = 0}^{π / 2} {| \int_{0}^{2 π} \int_{0}^{a} w (r, θ) e^{j k^{*} r \sin φ (\cos ψ \cos θ + \sin ψ \sin θ)} r d r d θ |}^{2} \sin φ d φ d ψ}

(52)

Then, based on equation (52), the Q factor of a circular microdiaphragm, including the influences of surface effects and residual stress can be obtained.

6. Numerical results and discussion

To validate the new model, the results of the reduced present model, in which the mass is equal to zero, are compared with the theoretical and experimental results of Olfatnia et al. [15, 16] in Figure 3. Although the microdiaphragm model of Olfatnia et al. [16] contains a piezoelectric layer, neither piezoelectric effects nor surface effects were taken into account; this leads to differences between the theoretical and experimental results. The geometrical dimensions and material and air parameters are the same as those of Olfatnia et al. [16]:

\begin{matrix} E = 125.9 \times 1 0^{9} N m^{- 2}, \\ ν = 0.24, \\ ρ_{p} = 5548 kg m^{- 3}, \\ h = 3.65 μ m, \\ ρ_{L} = 1.18 kg m^{- 3}, \\ c_{L} = 343 m s^{- 1} . \end{matrix}

For surface layers, the material properties can be obtained by atomistic calculations or experiments. However, accurate values of the surface elasticity constants E_s in [15, 16] are not yet available, owing to a lack of such work on metal electrode nanomaterials and silica. As a result, the surface elasticity constants E_s in the comparison are assumed to be an approximation: E_s = 0.02Eh = 9 × 10³ N m⁻¹ and E_s = 0.04Eh = 1.8 × 10² N m⁻¹. The ratios refer to the value of silver with a stiffened surface [27].

As seen in Figure 3, there are considerable differences between the theoretical and experiment results of Olfatnia et al. [16] based on the classical theory. In our model, surface effects are considered, which makes the theoretical results closer to the experimental results of Olfatnia et al. [16]. Therefore, this comparison can prove that our model based on the surface elasticity theory is reliable in describing the mechanical properties of a microdiaphragm. Moreover, the results of the present model, for both stiffened and softened surfaces, are also compared with the theoretical results of Olfatnia et al. [16], respectively. The geometrical dimensions and material and ethanol liquid parameters are the same as those of Olfatnia et al. [16]:

\begin{matrix} E = 125.9 \times 1 0^{9} N m^{- 2}, \\ ν = 0.24, \\ ρ_{p} = 5548 kg m^{- 3}, \\ h = 3.65 μ m, \\ ρ_{L} = 789 kg m^{- 3}, \\ c_{L} = 1144 m s^{- 1} . \end{matrix}

As illustrated in Figure 4, the results of the reduced present model are consistent with those of Olfatnia et al. [16]. The Q factor increases with increasing radius. Therefore, considering the application range and manufacturing in application, a large Q factor of resonant sensors can be obtained by increasing the radius of the diaphragm. Considering surface effects, there are significant differences between the results of the present model (m_d = 0) and the model of Olfatnia et al. [16]. As seen in Figure 4, the results predicted by the new model for a stiffened surface are smaller than those given by the model of Olfatnia et al. [16], while for a softened surface, the results predicted by the present mode are larger than those given by the model of Olfatnia et al. [16]. This indicates that the Q factor of a microdiaphragm in liquids can be influenced by surface effects. For the diaphragm with a softened surface, the results predicted by the present model are larger than those given by the classical theory [16], which can improve the sensitivity and resolution of sensors. In addition, the large quality factor can eliminate the influence of mechanical coupling on precision and stability to some extent. There are opposite results for the diaphragm with a stiffened surface.

To study the influence of surface effects on the Q factor for both stiffened and softened surfaces, numerical results are generated based on calculated results from two sets of material parameters [49]:

\begin{matrix} E = 5.625 \times 1 0^{10} N m^{- 2}, \\ ν = 0.25, \\ ρ = 3 \times 1 0^{3} kg m^{- 3}, \\ λ_{s} = 7 \times 1 0^{3} N m^{- 1}, \\ μ_{s} = 8 \times 1 0^{3} N m^{- 1}, \\ τ_{0} = 110 N m^{- 1}, \end{matrix}

for material I (stiffened surface), and

\begin{matrix} E = 17.73 \times 1 0^{10} N m^{- 2}, \\ ν = 0.27, \\ ρ = 7 \times 1 0^{3} kg m^{- 3}, \\ λ_{s} = - 8 N m^{- 1}, \\ μ_{s} = 2.5 N m^{- 1}, \\ τ_{0} = 1.7 N m^{- 1}, \end{matrix}

for material II (softened surface). The radius of the microdiaphragm is a = 1 × 10⁻⁴ m. Assuming that the liquid is ethanol, the liquid density is ρ_L = 1.18 kg m⁻³ and the propagation velocity of sound waves in ethanol is c_L = 1144 m s⁻¹. The surface elasticity constant E_s can be expressed by the corresponding surface Lamé constant λ_s:

E_{s} = \frac{λ_{s} (1 + v) (1 - 2 v)}{v}

(53)

When ignoring surface effects, the Q factor of the microdiaphragm is denoted as Q₀. Figure 5(a) shows how the normalized Q factor varies with the thickness of the microdiaphragm for the case of a stiffened surface (material I).

As shown in Figure 5(a), the normalized Q factor increases with decreasing thickness, while in Figure 5(b), for a microdiaphragm with a softened surface (material II), the normalized Q factor decreases with decreasing thickness. Moreover, for both material I and material II, if h < 0.4 × 10⁻⁸ m, the dimensionless Q factor is more obviously size-dependent. As the thickness increases, the Q factor predicted by the present model gradually approach the results of the model without surface effects. Moreover, as discussed for Figure 4, it can also be seen in Figure 5(a) and (b) that the stiffened surface decreases the Q factor, while the softened surface increases it. Therefore, to obtain a large Q factor, the surface mechanical performance should be considered when selecting the diaphragm material. A larger Q factor of resonant sensors can be obtained by selecting a diaphragm material with a softened surface.

According to equation (53), the surface elasticity constants of materials I and II are E_s = 1.75 × 10⁴ N m⁻¹ and E_s = −17.3 N m⁻¹, respectively. It is known that the vibrational behavior of microdiaphragms in sensors will be affected by various factors, such as the change in surface mechanical properties caused by biochemical media, assuming that the surface elasticity constants E_s of materials I and II are changed by biochemical media. For material I, with a stiffened surface, in Figure 5(a), the influence of the elasticity constant E_s on the Q factor is more significant for larger E_s. For material II, with a softened surface, in Figure 5(b), the influence is more significant for smaller E_s. These results show that the surface effects have greater influence on the Q factor of thinner microdiaphragms for both stiffened and softened surfaces. For a stiffened surface, the influence is more significant for larger E_s, and the influence for a softened surface is more significant for smaller E_s.

The effects of residual stress on the Q factor of a sensor diaphragm were investigated by Wu and Zhou [17] without including the effects of surface properties. Considering surface effects, the influences of residual stresses on the Q factor are analyzed and compared with the results of Wu and Zhou [17] in Figure 6. The influences of residual stress on Q factor can be represented by the normalized tension parameter k [17, 50]:

k = a \sqrt{\frac{2 τ_{0}}{D}} .

(54)

The geometrical dimensions and material and liquid parameters are the same as those of Wu [17]:

\begin{matrix} E = 170 \times 1 0^{9} N m^{- 2}, \\ ν = 0.26, \\ ρ_{p} = 2.3 \times 1 0^{3} kg m^{- 3}, \\ a = 1 \times 1 0^{- 3} m, \\ h = 40 \times 1 0^{- 6} m, \\ ρ_{L} = 1 \times 1 0^{3} kg m^{- 3}, \\ c_{L} = 1480 m s^{- 1} . \end{matrix}

Figure 6 shows the variation of Q factor with respect to the normalized tension parameter k for stiffened and softened surfaces. It can be seen from Figure 6 that the Q factor increases with increasing normalized tension parameter k for both stiffened and softened surfaces. The varying tendencies of Q factor are similar to those of Wu and Zhou [17]. For the stiffened surface in Figure 6, the Q factor is obviously smaller than the results of Wu and Zhou [17]. The Q factor of the microdiaphragm with a softened surface is obviously larger than the results of Wu and Zhou [17]. Moreover, compared with the results of Wu and Zhou [17], when k is small, the influence of surface effects on the Q factor is not obvious. When k is greater than 30, the Q factor is significantly affected by surface effects. At this point, the influence of surface effects on Q factor must be considered. The results indicate that surface effects have a significant influence on the Q factor. If k is greater than 30, the influence of the surface effects on the Q factor must be considered. Moreover, residual surface stresses are inevitably generated in the fabrication process, owing to temperature variation and for other reasons, which can greatly affect the mechanical properties and piezoelectric properties of the diaphragm. Therefore, in the design and manufacture of resonant sensors, the influence of residual stress on several performance parameters of sensors should be comprehensively considered.

7. Conclusions

The Q factor of a circular microdiaphragm with surface effects in contact with liquids has been investigated. An approximate solution is obtained using the Rayleigh–Ritz energy method. The influences of surface effects and residual stress on the Q factor of a microdiaphragm are discussed. Results show that the Q factor increases with increasing radius. The Q factor of a microdiaphragm is significantly affected by surface effects and exhibits strong size-dependence. The effect of stiffened and softened surfaces on the Q factor is more significant for thinner microdiaphragms. Moreover, a stiffened surface decreases the Q factor, while a softened surface increases it. The influence of a stiffened surface is more obvious for larger E_s, and the influence of a softened surface is more obvious for smaller E_s. Also, it is observed that the Q factor increases with an increase in the normalized tension parameter k for both stiffened and softened surfaces. If the normalized tension parameter k is greater than 30, the influence of surface effects on the Q factor must be considered.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Found of Shandong Province of China (grant number ZR2017BA001).

ORCID iD

SS Zhou

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