A Theoretical Study of Love Wave Sensors Based on ZnO–Glass Layered Structures for Application to Liquid Environments

The propagation of surface acoustic Love modes along ZnO/glass-based structures was modeled and analysed with the goal of designing a sensor able to detect changes in the environmental parameters, such as liquid viscosity changes and minute amounts of mass supported in the viscous liquid medium. Love mode propagation was modeled by numerically solving the system of coupled electro-mechanical field equations and Navier–Stokes equations. The phase and group velocities and the attenuation of the acoustic wave propagating along the 30° tilted c-axis ZnO/glass structure contacting a viscous non-conductive liquid were calculated for different ZnO guiding layer thicknesses, added mass thicknesses, and liquid viscosity and density. The three sensor responses, i.e., the wave phase and group velocity, and attenuation changes are calculated for different environmental parameters and related to the sensor velocity and attenuation sensitivities. The resulted sensitivities to liquid viscosity and added mass were optimized by adjusting the ZnO guiding layer thickness corresponding to a sensitivity peak. The present analysis is valuable for the manufacture and application of the ZnO-glass structure Love wave sensors for the detection of liquid properties, such as viscosity, density and mass anchored to the sensor surface.


Supplementary Materials: A Theoretical Study of Love Wave Sensors Based on ZnO-Glass Layered Structures for Application to Liquid Environments
Cinzia Caliendo and Muhammad Hamidullah

Half-Space/Guiding Layer
A piezoelectric ZnO layer of thickness hgl overlays a glass isotropic half-space, as shown in Figure S1. The space above the layer is occupied by air or vacuum which is assumed to have no mechanical contact with the layer. The ZnO layer has its c-axis 30° tilted with respect to the surface normal. The ZnO stiffness constants cαβ , piezoelectric constants eαβ, and dielectric constants εαβ were rotated by applying the Bond matrix method described in [1]. The coordinate system used through the paper is the following: the x2 axis is parallel to the surface normal, the x3 as axis is parallel to the wave polarization vector, and the x1 axis is parallel to the wave propagation direction. In the quasi-static approximation, the matrix notation for the piezoelectric constitutive equations of the c-axis 30° tilted ZnO is the following: The wave under consideration is assumed to travel in the x1-direction along a surface whose normal is in the x2 direction, and to be polarized parallel to the x3 direction. The only non null particle displacement component is U3, and both U3 and the electric potential Φ are independent of the x3 coordinate: since travelling wave solutions are in the form U3 = U3(x1, x2, t) and Φ = Φ(x1, x2, t), then the Equation (S1) can be rewritten as The equations of motion for the piezoelectric finite thickness layer and for the isotropic half-space are: The assumed solutions to the propagation equations in the substrate (sub), guiding layer (gl) and in the region above the free surface of the guiding layer are the following: where C1, C2, C3, C4, A and B are arbitrary constants, k=ω/v is the wave-number (it is real since the ZnO and glass are lossless materials), v is the Love mode velocity (whose value is in between the shear horizontal bulk acoustic wave velocity in the layer and in the substrate, and , ω = 2πf, f = v/λ, q and β account the variation in depth of the wave amplitude. By substituting Equations (S6-S8) into Equations (S3-S5), two system of equations for the displacement and the potential are obtained, that involve the layer and substrate material parameters. An algebraic equation in β and one in q are obtained by solving the secular equations for the layer and for the substrate. From the two algebraic equations, only q and β values are retained that correspond to a wave displacement that decay to zero with depth below the x2 = 0 plane, and that varies sinusoidally into the layer. By substituting the Equations (S6-S8) into the boundary and continuity conditions, a set of homogeneous equations for the C1, C2, C3, C4, A and B coefficients are obtained with v as the unknown. By setting the determinant of the coefficients equal to zero, real values of v are found for fixed layer thickness and wavelength λ. An optimized numerical procedure was used to find a real velocity value that drives the size of the determinant of the coefficients as close to zero as possible.

Half-Space/Guiding Layer/Liquid
The guiding layer, as well as the half-space, is assumed to be isotropic with the constant c44 numerically equal to the stiffened value calculated in the previous paragraph. A viscous non conductive liquid half-space contacts the upper surface of the layer, as shown in Figure S2: ρl and η are the liquid mass density and viscosity. Figure S2. The half-space/guiding layer/viscous liquid system.
The equations of motion for the three media are the following: The assumed solutions to the Equations (S9-S11) are the following: From Equation (S15) the wave dispersion equation is obtained: the system of two equations, the real and imaginary parts of the dispersion equation, were numerically solved by using the Levenberg-Marquardt-Fletcher method implemented within a Matlab routine, and the real and imaginary parts of the Love wave velocity were then calculated, and .

Half-Space/Guiding Layer/Mass Layer/Liquid
An added mass layer (am) of thickness ham is supposed to cover the guiding layer surface as shown in Figure S3. The guiding layer, the mass layer and the half-space are assumed to be isotropic. Figure S3. The half-space/guiding layer/mass layer/viscous liquid system. The equations of motion for the four media are the following: = + , x2 > 0 (S16) = + , 0 < x2 < -hgl (S17) = + , -hgl <x2 < -ham (S18) =  + , for x2 < -H (S19) The assumed solutions to the Equations (S16-S19) are the following: The particle displacement and the traction components of stress must be continuous across the substrate/guiding layer, guiding layer/mass layer, and mass layer/liquid interfaces. When the Equation (S20-S23) are substituted into the boundary and continuity conditions, a set of six homogeneous algebraic equations are obtained in the six coefficients , , , , , and : a non trivial solution of this equations system exists if the determinant of the coefficients vanishes. The determinant is: