# Numerical study of the coupling layer between transducer and chip in
acoustofluidic devices

William Naundrup Bodé<EMAIL_ADDRESS>Department of Physics, Technical
University of Denmark,
DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark  Henrik Bruus
<EMAIL_ADDRESS>Department of Physics, Technical University of Denmark,
DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark

(28 January 2021)

###### Abstract

We study by numerical simulation in two and three dimensions the coupling
layer between the transducer and the microfluidic chip in ultrasound
acoustofluidic devices. The model includes the transducer with electrodes, the
microfluidic chip with a liquid-filled microchannel, and the coupling layer
between the transducer and the chip. We consider two commonly used coupling
materials, solid epoxy glue and viscous glycerol, as well as two commonly used
device types, glass capillary tubes and silicon-glass chips. We study how
acoustic resonances in ideal devices without a coupling layer is either
sustained or attenuated as a coupling layer of increasing thickness is
inserted. We establish a simple criterion based on the phase of the acoustic
wave for whether a given zero-layer resonance is sustained or attenuated by
the addition of a coupling layer. Finally, we show that by controlling the
thickness and the material, the coupling layer can be used as a design
component for optimal and robust acoustofluidic resonances.

##  I Introduction

The acoustic impedance matching techniques for piezoelectric sensors and
transducers is in general a well-studied field, as exemplified by the recent
review by Rathod.(Rathod2020, ) However, specifically for ultrasound
acoustofluidic devices, the role of the coupling layer (also known as the
carrier or matching layer) between the transducer and the microfluidic chip
remains poorly understood beyond one-dimensional (1D) planar
systems.(Hill2002, ; Glynne-Jones2012, ) Whereas the function of a matching
layer for 1D traveling waves through layered structures, is simply to couple
acoustic energy more efficiently into subsequent layers, its role in resonant
1D acoustofluidic systems is less straightforward. As analyzed by Glynne-
Jones, Boltryk, and Hill,(Glynne-Jones2012, ) the function of the coupling
layer may be more structural, or to isolate the transducer from the fluid
layer. In the present work, we study the more complex case of a fully three-
dimensional (3D) acoustofluidic system.

We present 3D numerical simulations of a piezoelectric transducer coupled to
an acoustofluidic chip through a thin coupling layer consisting of either a
solid glue or a viscous liquid. We consider two commonly used types of
acoustofluidic devices sketched in Fig. 1, glass capillary tubes and silicon-
glass chips. Capillary-tube devices have been applied as acoustic particle
traps relying on a small transducer that actuates a vertical resonant pressure
mode locally in the capillary.Hammarstrom2012 ; Lei2013 ; Mishra2014 ;
Gralinski2014  Silicon-glass devices have been applied for continuous-flow
focusing and separation of particle suspensions relying on bulk actuation of
horizontal resonance modes in embedded microchannels.(Barnkob2010, ;
Magnusson2017, ; Petersson2018, )

Figure 1:  Sketch of the two types of acoustofluidic devices considered in
this study. (a) A glass capillary (beige) mounted on a small piezoelectric
transducer (gray). (b) A silicon-glass chip (black base, beige lid) mounted on
a bulk piezoelectric transducer (gray). Both sketches are drawn to scale.

The paper is organized as follows: In Section II, we present the basic theory,
including governing equations and boundary conditions. In Section III we model
the capillary-tube particle traps and show their dependency on the coupling-
layer thickness. In Section IV a similar analysis is carried out for the
conventional acoustophoresis silicon-glass devices. Finally in Section V, we
present a concluding discussion regarding the criterion established in the two
previous sections for designing acoustofluidic devices, with acoustic
resonance modes that are relatively insensitive to the thickness of the
coupling layer.

##  II Theory and model assumptions

In establishing the numerical model, we follow closely the theory presented by
Skov et al.(Skov2019, ) Our model consists of a lead-zirconate-titanate (PZT)
piezoelectric transducer coupled to an elastic solid, which contains the
fluid-filled microchannel. Theoretically, the system is described by three
continuous fields: the electric potential $\varphi$ in the PZT transducer, the
mechanical displacement field $\bm{u}$ in the elastic solid and the PZT
transducer, and the acoustic pressure $p_{1}$ in the fluid. The coupling layer
is described by $p_{1}$ if it is a liquid, and by $\bm{u}$ if it is a solid.
Due to the linearity of the governing equations, all fields has a harmonic
time dependence $\mathrm{e}^{-\mathrm{i}\omega t}$ with angular frequency
$\omega=2\pi f$ and frequency $f$. Thus, a given field has the structure
$\tilde{f}(\bm{r},t)=f(\bm{r})\>\mathrm{e}^{-\mathrm{i}\omega t}$, and we need
only determine the complex-valued space-dependent amplitude $f(\bm{r})$.

The acoustic pressure field $p_{1}$ is modeled using the effective pressure
acoustic theory by Bach and Bruus,(Bach2018, ) where the viscous boundary
layers are included analytically in the effective boundary conditions. Using
the effective theory, the acoustic pressure $p_{1}$ in a fluid with density
$\rho_{0}$, sound speed $c_{0}$, dynamic viscosity $\eta_{0}$ and bulk
viscosity $\eta^{\mathrm{b}}$ is governed by the Helmholtz equation, and the
acoustic velocity $\bm{v}_{1}$ is proportional to $\bm{\nabla}p_{1}$,

$\displaystyle\nabla^{2}p_{1}$ $\displaystyle=-k_{\mathrm{c}}^{2}p_{1},$
$\displaystyle\bm{v}_{1}$
$\displaystyle=-\mathrm{i}\frac{1-\mathrm{i}\Gamma}{\omega\rho_{0}}\bm{\nabla}p_{1},$
(1a) $\displaystyle\text{with }k_{0}$ $\displaystyle=\frac{\omega}{c_{0}},$
$\displaystyle k_{\mathrm{c}}$
$\displaystyle=\Big{(}1+\frac{\mathrm{i}}{2}\Gamma\Big{)}\>k_{0},$ (1b)
$\displaystyle\text{and }\;\Gamma$
$\displaystyle=\Big{(}\frac{\eta^{\mathrm{b}}}{\eta_{0}}+\frac{4}{3}\Big{)}\frac{\omega\eta_{0}}{\rho_{0}c_{0}^{2}}.$
(1c)

In cases where the fluid coupling layer thickness $\Delta$ is comparable or
smaller than the viscous boundary-layer length scale
$\delta_{\mathrm{visc}}=\sqrt{2\eta_{0}/(\rho_{0}\omega)}$, the effective
theory fails, and the full perturbation model is used instead.(Muller2012, ;
Muller2013, )

The mechanical displacement field $\bm{u}$ is governed by the linear Cauchy
equation involving the stress tensor $\bm{\sigma}$,

$-\rho_{0}\omega^{2}\bm{u}=\bm{\nabla}\cdot\bm{\sigma},$ (2)

The components $\sigma_{ik}$ are related to the strain components
$\frac{1}{2}(\partial_{i}u_{k}+\partial_{k}u_{i})$ by the stiffness tensor
$\bm{C}$, which for linear isotropic or cubic-symmetric elastic materials are
written in the Voigt notation as

$\displaystyle\left(\begin{array}[]{c}\sigma_{xx}\\\ \sigma_{yy}\\\
\sigma_{zz}\\\ \hline\cr\sigma_{yz}\\\ \sigma_{xz}\\\
\sigma_{xy}\end{array}\right)=\left(\begin{array}[]{ccc|ccc}C_{11}&C_{12}&C_{12}&0&0&0\\\
C_{12}&C_{11}&C_{12}&0&0&0\\\ C_{12}&C_{12}&C_{11}&0&0&0\\\ \hline\cr
0&0&0&C_{44}&0&0\\\ 0&0&0&0&C_{44}&0\\\ 0&0&0&0&0&C_{44}\\\
\end{array}\right)\,\left(\begin{array}[]{c}\partial_{x}u_{x}\\\
\partial_{y}u_{y}\\\ \partial_{z}u_{z}\\\
\hline\cr\partial_{y}u_{z}+\partial_{z}u_{y}\\\
\partial_{x}u_{z}+\partial_{z}u_{x}\\\
\partial_{x}u_{y}+\partial_{y}u_{x}\end{array}\right).$

(3)

Mechanical damping is implemented as complex-valued elastic moduli, defined as
$C_{ik}=(1-\mathrm{i}\Gamma_{\mathrm{sl}})c_{ik}$. In the PZT transducer, the
electric potential $\varphi$ is governed by the quasi-static Gauss equation
involving the electric displacement $\bm{D}$,

$\bm{\nabla}\cdot\bm{D}=0.$ (4)

Furthermore in PZT, the complete linear electromechanical coupling relating
the stress and the electric displacement to the strain and the electric field
is given by the Voigt notation as,

$\displaystyle\left(\begin{array}[]{c}\sigma_{xx}\\\ \sigma_{yy}\\\
\sigma_{zz}\\\ \hline\cr\sigma_{yz}\\\ \sigma_{xz}\\\ \sigma_{xy}\\\ \hline\cr
D_{x}\\\ D_{y}\\\
D_{z}\end{array}\right)=\left(\begin{array}[]{ccc|ccc|ccc}C_{11}&C_{12}&C_{13}&0&0&0&0&0&-e_{31}\\\
C_{12}&C_{11}&C_{13}&0&0&0&0&0&-e_{31}\\\
C_{13}&C_{13}&C_{33}&0&0&0&0&0&-e_{33}\\\ \hline\cr
0&0&0&C_{44}&0&0&0&-e_{15}&0\\\ 0&0&0&0&C_{44}&0&-e_{15}&0&0\\\
0&0&0&0&0&C_{66}&0&0&0\\\ \hline\cr 0&0&0&0&e_{15}&0&\epsilon_{11}&0&0\\\
0&0&0&e_{15}&0&0&0&\epsilon_{11}&0\\\
e_{31}&e_{31}&e_{33}&0&0&0&0&0&\epsilon_{33}\end{array}\right)\,\left(\begin{array}[]{c}\partial_{x}u_{x}\\\
\partial_{y}u_{y}\\\ \partial_{z}u_{z}\\\
\hline\cr\partial_{y}u_{z}+\partial_{z}u_{y}\\\
\partial_{x}u_{z}+\partial_{z}u_{x}\\\ \partial_{x}u_{y}+\partial_{y}u_{x}\\\
\hline\cr-\partial_{x}\varphi\\\ -\partial_{y}\varphi\\\
-\partial_{z}\varphi\end{array}\right).$

(5)

###  II.1 Acoustic energy density

Throughout this study, the time- and volume-averaged acoustic energy density
$E_{\mathrm{ac}}$ in the water-filled channel is used as a measure and
indicator of how the acoustic resonances are affected by the coupling layer.
In a fluid volume $V$ the averaged acoustic energy density is given as

$E_{\mathrm{ac}}=\frac{1}{V}\int\Big{(}\frac{1}{4}\rho_{0}\left|\bm{v}_{1}\right|^{2}+\frac{1}{4}\kappa_{0}\left|p_{1}\right|^{2}\Big{)}\>\mathrm{d}V.$
(6)

###  II.2 Boundary conditions

At the fluid-solid interface the boundary conditions are no-slip and
continuous stress, together with zero stress on free surfaces. Introducing the
mechanical displacement velocity
$\bm{v}_{\mathrm{sl}}=-\mathrm{i}\omega\bm{u}$ and a shear wave number
$k_{\mathrm{s}}=(1+\mathrm{i})/\delta_{\mathrm{visc}}$ the effective
continuous velocity and stress boundary conditions become,(Bach2018, )

$\displaystyle\bm{n}\cdot\bm{v}_{1}$
$\displaystyle=\bm{n}\cdot\bm{v}_{\mathrm{sl}}+\frac{\mathrm{i}}{k_{\mathrm{s}}}\bm{\nabla}_{\parallel}\cdot(\bm{v}_{\mathrm{sl}}-\bm{v}_{1}),$
(7a) $\displaystyle\bm{\sigma}\cdot\bm{n}$
$\displaystyle=-p_{1}\bm{n}+\mathrm{i}k_{\mathrm{s}}\eta_{0}\Big{(}\bm{v}_{\mathrm{sl}}+\frac{\mathrm{i}}{\omega\rho_{0}}\bm{\nabla}p_{1}\Big{)},$
(7b)

where the unit vector $\bm{n}$ is the outward surface normal from the solid
domain. In experiments, the electrical signal is driven by a sine-wave
function generator coupled to the transducer electrodes, this is implemented
as a constant potential boundary condition on the electrode-transducer
interface. Furthermore we assume no free charges, implemented as a zero flux
condition on the electric displacement field. In the 3D capillary-tube device,
symmetries are exploited such that the full system can be reduces to one
quarter. The boundary conditions are listed in Table 1. Except for the
symmetry conditions, the same boundary conditions applies for the silicon-
glass device.

Table 1:  List of boundary conditions used in the modeled acoustofluidic systems. The unit vector $\bm{n}$ is the surface outward normal with respect to the solid domain, and $\bm{t}$ is any of the two tangential unit vectors. Domain $\leftarrow$ boundary | Boundary condition
---|---
Solid domain $\leftarrow$ air | $\bm{\sigma}\cdot\bm{n}=\bm{0}$
Fluid domain $\leftarrow$ solid | Eq. (7a)
Solid domain $\leftarrow$ fluid | Eq. (7b)
Fluid domain $\leftarrow$ air | $p_{1}=0$
PZT domain $\leftarrow$ bottom electrode | $\varphi=0$
PZT domain $\leftarrow$ top electrode | $\varphi=V_{0}$
Solid domain $\leftarrow$ symmetry |  $\bm{u}\cdot\bm{n}=0$, $\bm{t}\cdot\bm{\sigma}\cdot\bm{n}=0$
Fluid domain $\leftarrow$ symmetry | $\bm{n}\cdot\bm{\nabla}p_{1}=0$

###  II.3 Unbounded perfectly matched layers

For long systems like the capillary tubes, a no-reflection boundary condition
can by established closer to the origin of the domain by using the perfectly
matched layer (PML) technique, thus reducing the computational domain
substantially. It involves a complex coordinate transformation of the form
$x\to
x+\frac{\mathrm{i}}{\omega}\int^{x}\theta(x^{\prime})\mathrm{d}x^{\prime}$,
such that outgoing waves are attenuated within a distance comparable to the
wavelength. The PML technique requires a choice of damping function $\theta$,
the specific function is adopted from Bermúdez et al.(Bermudez2007, ),

$\theta(x)=\begin{cases}0,&\text{for }x\leq L_{\mathrm{cap}},\\\
\frac{\beta}{L_{\mathrm{pml}}-(x-L_{\mathrm{cap}})}-\frac{\beta}{L_{\mathrm{pml}}},&\text{for
}x>L_{\mathrm{cap}}.\end{cases}$ (8)

The parameters defining $\theta(x)$ is chosen appropriately for a given
system: $L_{\mathrm{cap}}$ is the position of the interface between the
physical capillary tube and the PML domain, $L_{\mathrm{pml}}$ is the length
of the PML domain, and $\beta$ is the damping strength. The axial coordinate
$x$ is complex-valued for $x>L_{\mathrm{cap}}$ inside the PML domain. The
function $\theta$ is classified as a continuous unbounded damping function,
effective in terms of numerical error and reflections at the PML interface
$x=L_{\mathrm{cap}}$.(Bermudez2007, )

###  II.4 Numerical implementation in COMSOL Multiphysics

The numerical model was implemented in the finite element software COMSOL
Multiphysics(Comsol55, ) using ”Weak Form PDE” in the Mathematics module and
closely following Ref. Skov2019, , where further implementation details can be
found. The mesh settings is adopted from Ley and Bruus.(Ley2017, ) The scripts
were computed on a workstation with a 12 core 3.5 GHz CPU processor, and 128
GB ram.

##  III Capillary-tube particle traps

As the first example, we investigate the capillary-tube device widely used as
a versatile acoustic trap in many experimental studies.Hammarstrom2012 ;
Lei2013 ; Mishra2014 ; Gralinski2014  The corresponding model system is
sketched in Fig. 2 indicating the different domains together with the PML
layer and a zoom-in on the coupling layer. The dimensions and materials used
in the numerical model are listed in Tables 2 and 3, respectively. The model
system is similar to the one studied by Ley and Bruus,Ley2017  but now the
model is extended to include a PZT transducer and a coupling layer. Typically
the capillary-tube device is characterized by having a standing half-wave-like
resonance in the vertical direction. This is achieved with a PZT transducer
having a predesigned mode at 5 MHz, also used in preliminary experiments by
the Laurell group at Lund University.

Figure 2:  One quarter of the capillary-tube-based model system with a zoom-in on the coupling layer of thickness $\Delta$. The model system includes a water-filled glass capillary coupled to a PZT transducer with silver electrodes. The top electrode is coupled to a time-harmonic function generator and the bottom electrode is grounded. Table 2:  The length ($L$), width ($W$), and height ($H$) of the glass capillary tube (cap), the channel (ch), the piezoelectric transducer (pzt), and the silver electrodes (el). The curvature of the outer and inner rounded corners are $240~{}\textrm{\textmu{}m}$ and $25~{}\textrm{\textmu{}m}$, respectively. The bottom $H_{\mathrm{gl,bot}}$ and top $H_{\mathrm{gl,top}}$ glass-wall thicknesses are both $140~{}\textrm{\textmu{}m}$. Symbol | Value |  | Symbol | Value
---|---|---|---|---
$L_{\mathrm{cap}}$ | $1573~{}\textrm{\textmu{}m}$ |  | $L_{\mathrm{ch}}$ | $1573~{}\textrm{\textmu{}m}$
$W_{\mathrm{cap}}$ | $2280~{}\textrm{\textmu{}m}$ |  | $W_{\mathrm{ch}}$ | $2000~{}\textrm{\textmu{}m}$
$H_{\mathrm{cap}}$ | $480~{}\textrm{\textmu{}m}$ |  | $H_{\mathrm{ch}}$ | $200~{}\textrm{\textmu{}m}$
$L_{\mathrm{pzt}}$ | $1160~{}\textrm{\textmu{}m}$ |  | $L_{\mathrm{el}}$ | $1160~{}\textrm{\textmu{}m}$
$W_{\mathrm{pzt}}$ | $3350~{}\textrm{\textmu{}m}$ |  | $W_{\mathrm{el}}$ | $3350~{}\textrm{\textmu{}m}$
$H_{\mathrm{pzt}}$ | $400~{}\textrm{\textmu{}m}$ |  | $H_{\mathrm{el}}$ | $9~{}\textrm{\textmu{}m}$
Table 3:  List of parameters used in the numerical simulations. Note that $C_{12}=C_{11}-2C_{44}$ for isotropic materials. Isotropy in the $xy$-plane implies $C_{66}=\frac{1}{2}(C_{11}-C_{12})$ for the PZT. Parameter | Symbol | Value | Unit
---|---|---|---
Water at $25~{}^{\circ}\\!\textrm{C}$ (Muller2014, ; Karlsen2016, )
Mass density | $\rho_{0}$ | $997.05$ |  kg m-3
Speed of sound | $c_{0}$ | $1496.7$ | $\textrm{m}\,\textrm{s}^{-1}$
Compressibility | $\kappa_{0}$ | $447.7$ | $\textrm{TPa}^{-1}$
Dynamic viscosity | $\eta_{0}$ | $0.890$ |  mPa s
Bulk viscosity | $\eta^{\mathrm{b}}$ | $2.485$ |  mPa s
Isotropic Pyrex borosilicate glass (CorningPyrex, )
Mass density | $\rho_{\mathrm{sl}}$ | $2230$ |  kg m-3
Elastic modulus | $c_{11}$ | $69.7$ | GPa
Elastic modulus | $c_{44}$ | $26.2$ | GPa
Mechanical damping coeff. | $\Gamma_{\mathrm{sl}}$ | 0.0004 | -
Isotropic silver(AZOsilver2001, )
Mass density | $\rho_{\mathrm{sl}}$ | $10485$ |  kg m-3
Elastic modulus | $c_{11}$ | $133.9$ | GPa
Elastic modulus | $c_{44}$ | $25.9$ | GPa
Mechanical damping coeff. | $\Gamma_{\mathrm{sl}}$ | 0.0004 | -
Cubic-symmetric silicon (Hopcroft2010, )
Mass density | $\rho_{\mathrm{sl}}$ | $2329$ |  kg m-3
Elastic modulus | $c_{11}$ | $165.7$ | GPa
Elastic modulus | $c_{44}$ | $79.6$ | GPa
Elastic modulus | $c_{12}$ | $63.9$ | GPa
Mechanical damping coeff. | $\Gamma_{\mathrm{sl}}$ | 0.0001 | -
Pz26 PZT ceramic (Ferroperm2017, ; Hahn2015, )
Mass density | $\rho_{\mathrm{sl}}$ | $7700$ |  kg m-3
Elastic modulus | $c_{11}$ | $168$ | GPa
Elastic modulus | $c_{12}$ | $110$ | GPa
Elastic modulus | $c_{13}$ | $99.9$ | GPa
Elastic modulus | $c_{33}$ | $123$ | GPa
Elastic modulus | $c_{44}$ | $30.1$ | GPa
Coupling constant | $e_{15}$ | $9.86$ | C/m2
Coupling constant | $e_{31}$ | $-2.8$ | C/m2
Coupling constant | $e_{33}$ | $14.7$ | C/m2
Electric permittivity | $\varepsilon_{11}$ | $828$ | $\varepsilon_{0}$
Electric permittivity | $\varepsilon_{33}$ | $700$ | $\varepsilon_{0}$
Mechanical damping coeff. | $\Gamma_{\mathrm{sl}}$ | 0.02 | -

###  III.1 The specific perfectly matched layer

We implement no-reflection boundary conditions using a PML layer with
parameter values $\beta=2c_{\mathrm{lo}}^{\mathrm{(gl)}}=11.294$ m/s and
$L_{\mathrm{pml}}=413~{}\textrm{\textmu{}m}$. The superscript refer to the
material, in this case glass (gl), and $c_{\mathrm{lo}}^{\mathrm{(gl)}}$ and
$\lambda_{\mathrm{lo}}^{\mathrm{(gl)}}$ is the longitudinal sound speed and
wavelength, respectively. The numerical error introduced by using the PML is
shown in Fig. 3, in terms of the convergence parameter $C$, which for a given
field solution $g$ is defined by

$C(g)=\sqrt{\frac{\int\left|g-g_{\mathrm{ref}}\right|^{2}\mathrm{d}V}{\int\left|g_{\mathrm{ref}}\right|^{2}\mathrm{d}V}},$
(9)

where $g_{\mathrm{ref}}$ is a reference solution. The integration domain is
taken as the transducer region for $x\leq L_{\mathrm{pzt}}/2$ defined in Fig.
2. In Fig. 3, the error measure $C$ is evaluated at six different geometries
at a fixed frequency $f=3.84\,\textrm{MHz}$ with
$\lambda_{\mathrm{lo}}^{\mathrm{(gl)}}=1.47\,\textrm{mm}$. The reference
solution is taken as
$L_{\mathrm{cap}}=1.35\lambda_{\mathrm{lo}}^{\mathrm{(gl)}}$. We choose the
system length $L_{\mathrm{cap}}=1.07\lambda_{\mathrm{lo}}^{\mathrm{(gl)}}$
such that the maximal numerical error due to the unbounded PML is estimated to
be $C=3\times 10^{-4}$.

Figure 3:  Numerical convergence $C$ for $\bm{u}$, $p_{1}$, and $\varphi$ in
the PML of Fig. 2 at frequency $f_{\mathrm{res}}=3.84$ MHz with wavelength
$\lambda_{\mathrm{lo}}^{\mathrm{(gl)}}=1.47\,\textrm{mm}$, PML length
$L_{\mathrm{pml}}=413\,\textrm{\textmu{}m}$, and the physical system length
$L_{\mathrm{cap}}$ is varied as
$L_{\mathrm{cap}}/\lambda_{\mathrm{lo}}^{\mathrm{(gl)}}=0.53$, 0.67, 0.80,
0.94, 1.07, and 1.21.

###  III.2 Coupling-layer analysis in 3D and 1D

Continuing with the physics studies, we consider two coupling materials, a
viscous mixture of 99 vol.% glycerol and 1 vol.% water, and a solid ED-20
epoxy resin, from now on referred to as glycerol and epoxy. The coupling
material parameters used in the simulations are listed in Table 4.

In practice, the glycerol coupling allows for reuseability of the
acoustofluidic chip and/or the transducer, whereas the epoxy is used to ensure
a well-defined but permanent coupling.Hawkes2001 ; Barnkob2010 ;
Hammarstrom2010 ; Ohlsson2016 ; Bode2020  The effect of the coupling layer is
investigated by calculating the resonances as a function of coupling-layer
material and thickness $\Delta$ using the 3D model. For each coupling-layer
thickness $\Delta$, the average acoustic energy density $E_{\mathrm{ac}}$ in
the water-filled channel is computed as a function of frequency from 3.0 to
4.5 MHz. Resonances are then identified as peaks in the acoustic energy
spectrum $E_{\mathrm{ac}}(f)$. The resonances are illustrated in the scatter
plot of Fig. 4, where the points represent resonances at frequency $f$ for a
coupling layer thickness $\Delta$ in the range from 0 to
$100~{}\textrm{\textmu{}m}$, with point areas proportional to
$E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$, where $E_{\mathrm{ac}}^{0}$ is the
acoustic energy density without a coupling layer. Also the resonances of the
unloaded PZT resonances are plotted to indicate where the transducer is most
active.

Table 4:  List of the coupling layer parameters for glycerol (a 99% v/v glycerol and 1% v/v water mixture) and epoxy at $20~{}^{\circ}\\!\textrm{C}$. The coefficient $C_{12}$ of the epoxy is obtained through the relation $C_{12}=C_{11}-2C_{44}$. Parameter | Symbol | Value | Unit
---|---|---|---
Glycerol (Slie1966, ; Negadi2017, ; Cheng2008, ) |  |  |
Mass density | $\rho_{\mathrm{glc}}$ | $1260.4$ |  kg m-3
Speed of sound | $c_{\mathrm{glc}}$ | $1922.8$ | $\textrm{m}\,\textrm{s}^{-1}$
Compressibility | $\kappa_{\mathrm{glc}}$ | $214.6$ | $\textrm{TPa}^{-1}$
Dynamic viscosity | $\eta_{\mathrm{glc}}$ | $1.137$ |  Pa s
Bulk viscosity | $\eta^{\mathrm{b}}_{\mathrm{glc}}$ | $0.790$ |  Pa s
Epoxy(Perepechko1996, ) |  |  |
Mass density | $\rho_{\mathrm{sl}}$ | $1205$ |  kg m-3
Elastic modulus | $c_{11}$ | $9.583$ | GPa
Elastic modulus | $c_{44}$ | $2.164$ | GPa
Mechanical damping coeff. | $\Gamma_{\mathrm{sl}}$ | $0.01$ | -
Figure 4:  Resonance frequencies in the capillary-tube device, as a function
of increasing coupling-layer thickness $\Delta$ for (a) glycerol and (b)
epoxy. The 3D and 1D model resonance frequencies are plotted as filled and
empty circles, respectively, with an area proportional
$E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$. The dashed lines represents the 1D
resonance frequencies, indicating a sustained or attenuated behavior. The
solid green lines indicate unloaded PZT resonances with a linewidth
proportional to the logarithm of the acoustic energy density in the PZT.

The 3D model reveals a distinct behavior for both glycerol and epoxy coupling
layers: As the coupling layer $\Delta$ increases, one resonance is attenuated
($E_{\mathrm{ac}}$ decreases) and has a large downshift in frequency, whereas
another is sustained ($E_{\mathrm{ac}}$ increases) and has a small downshift
in frequency. This behavior is also observed in an idealized 1D layer model
along the vertical $z$-axis with seven domains: electrode, PZT, electrode,
coupling layer, glass, water, glass of respective thicknesses
$H_{\mathrm{el}}$, $H_{\mathrm{pzt}}$, $H_{\mathrm{el}}$, $\Delta$,
$H_{\mathrm{gl,bot}}$, $H_{\mathrm{ch}}$, and $H_{\mathrm{gl,top}}$. This 1D
model takes into account only $z$-components, $z$-dependencies, densities and
longitudinal sound speeds in the governing equations. The 1D model resonances
are plotted together with the 3D resonances in Fig. 4. Of course, Fig. 4
reveals that the 3D model exhibits more resonances than the 1D model due to
the extended degrees of freedom in the transverse directions and the shear
waves. However, for the two indicated modes in each system, the 1D model
agrees fairly well with the 3D model, which indicates that the attenuated and
sustained resonance effect can be explained by this 1D fluid-like model.

Figure 5:  Plots of normal stress $\sigma_{zz}(z)$ and pressure $p_{1}(z)$ in
the 1D model with an epoxy coupling layer for the three layer thicknesses
$\Delta=0$, 20, and $50~{}\textrm{\textmu{}m}$ for (a) the sustained and (b)
the attenuated zero-layer resonance modes. The profiles are plotted at a phase
where the amplitude is at a maximum. The accumulated phase $\Phi$ is
calculated without a coupling layer at the fluid-solid interface at
$z_{0}=H_{\mathrm{pzt}}+2H_{\mathrm{el}}+H_{\mathrm{gl,bot}}$ (brown arrow).

###  III.3 A design criterion for coupling layers

The sustaining and attenuating behavior of the zero-layer resonances, as the
layer thickness $\Delta$ is changed from 0 to $100~{}\textrm{\textmu{}m}$, is
elucidated by studying the pressure profiles for each of the resonances in the
1D model. For each profile we choose a temporal phase factor
$\mathrm{e}^{-{\mathrm{i}\omega t}}$ that gives the maximum positive amplitude
in the PZT domain.

In Fig. 5(a) the 1D resonance pressure profiles are evaluated at three
different epoxy coupling-layer thicknesses $\Delta=0$, 20, and
$50~{}\textrm{\textmu{}m}$ for the sustained resonance. In Fig. 5(b), the same
is shown for the attenuated resonance. Without the coupling layer, the fluid-
solid interface is located at
$z_{0}=H_{\mathrm{pzt}}+2H_{\mathrm{el}}+H_{\mathrm{gl,bot}}=558\,\textrm{\textmu{}m}$,
and with a coupling layer at $z_{\Delta}=z_{0}+\Delta$. As a result, the value
of the pressure $p_{1}(z_{\Delta})$ is decreasing. In Fig. 5(a), labeled
”sustained”, we have $p_{1}(z_{\Delta})<p_{1}(z_{0})<0$, yielding an increased
magnitude of the pressure as $\Delta$ is increased. In contrast, in Fig. 5(b),
labeled ”attenuated”, we have $0<p_{1}(z_{\Delta})<p_{1}(z_{0})$, yielding a
decreasing magnitude of the pressure for increasing $\Delta$. Clearly, if a
given zero-layer resonance has a negative (positive) value of $p_{1}(z_{0})$
for the specified temporal phase factor, the resonance is sustained
(attenuated). The sign of $p_{1}(z_{0})$ is determined by the accumulated
spatial phase factor $\Phi(z_{0})$ of the fluid-solid value $p_{1}(z_{0})$
relative to the surface value $p_{1}(0)$. In the 1D model, $\Phi(z_{0})$ is
given by the wavenumber $k_{i}$ and layer thickness $H_{i}$ of each layer
($i=$ PZT, electrodes, and glass placed at $z<z_{0}$). Consequently, we arrive
at the criterion,

sustaining coupling if $\displaystyle\Phi(z_{0})>\pi,$ (10a) attenuating
coupling if $\displaystyle\Phi(z_{0})<\pi,$ (10b)
$\displaystyle\text{with}\;\Phi(z_{0})=\sum_{i}k_{i}H_{i}$
$\displaystyle=\sum_{i}\frac{\omega}{c_{\mathrm{lo}}^{\mathrm{(\mathit{i})}}}H_{i}.$
(10c)

Note that this criterion is only valid for $|\Phi(z_{0})|<\frac{3}{2}\pi$. For
the given capillary-tube device, the values of $\Phi(z_{0})$ for the sustained
and the attenuated zero-layer resonance is $3.32=1.06\pi$ and $2.77=0.88\pi$,
respectively. This criterion is one of the main results of the paper. It can
be used to design optimally coupled capillary devices with minimum attenuation
caused by the coupling layer.

###  III.4 Characteristic coupling-layer attenuation thickness
$\bm{\Delta_{0}}$

Based on the 1D model, we derive a semi-analytical estimate for the
characteristic thickness $\Delta_{0}$, at which the acoustic energy is
attenuated for the above-mentioned attenuated zero-layer resonance modes, see
Fig. 6. The pressure solution $p_{1,i}$ to the Helmholtz equation in each
domain $i$, is written as

$p_{1,i}=p_{a,i}\sin(k_{i}z+\phi_{i})\quad\text{for}\quad z\in\Omega_{i}.$
(11)

At the interface between domain $i$ and $i+1$, the acoustic pressure and
velocity must be continuous,

$p_{1,i+1}=p_{1,i},\qquad\frac{1}{\rho_{i+1}}\partial_{z}p_{1,i+1}=\frac{1}{\rho_{i}}\partial_{z}p_{1,i}.$
(12)

This results in an iterative formula for the amplitude $p_{a,i}$ and phase
$\phi_{i}$ with coefficients $\beta_{i+1,i}$, $a_{i+1,i}$, and $b_{i+1,i}$,

$\displaystyle p_{a,i+1}$ $\displaystyle=\beta_{i+1,i}\>p_{a,i},$ (13a)
$\displaystyle\phi_{i+1}$
$\displaystyle=\phi_{i}-k_{i+1}\textstyle\sum^{i}_{j}H_{j}+\arctan(a_{i+1,i}/b_{i+1,i}),$
(13b) $\displaystyle\beta_{i+1,i}$
$\displaystyle=\sqrt{1+\cos^{2}(k_{i}\textstyle\sum^{i}_{j}H_{j}+\phi_{i})(Z_{i+1,i}^{2}-1)},$
(13c) $\displaystyle a_{i+1,i}$
$\displaystyle=\beta_{i+1,i}^{-1}\sin(k_{i}\textstyle\sum^{i}_{j}H_{j}+\phi_{i}),$
(13d) $\displaystyle b_{i+1,i}$
$\displaystyle=\beta_{i+1,i}^{-1}Z_{i+1,i}\cos(k_{i}\textstyle\sum^{i}_{j}H_{j}+\phi_{i}),$
(13e) $\displaystyle Z_{i+1,i}$
$\displaystyle=\frac{\rho_{i+1}c_{\mathrm{lo}}^{\mathrm{(\mathit{i}+1)}}}{\rho_{i}c_{\mathrm{lo}}^{\mathrm{(\mathit{i})}}}.$
(13f)

For a coupling layer (cl) made of either epoxy or glycerol, we have a mismatch
$Z_{\mathrm{cl,pzt}}\ll 1$ in the acoustic impedance, and the pressure
amplitude $p_{1}(z_{0})$ at the interface $z_{0}$ can therefore be
approximated as

$\displaystyle p_{1}(z_{0})$ $\displaystyle\approx
p_{\mathrm{0}}\sqrt{\alpha^{2}+\beta^{2}}\sin(k_{\mathrm{cl}}H_{\mathrm{pzt}})$
(14a)
$\displaystyle\qquad\times\sin(\arctan(\alpha/\beta)+k_{\mathrm{gl}}H_{\mathrm{gl}})+\mathcal{O}(Z_{\mathrm{cl,pzt}}),$
with
$\displaystyle\alpha=Z_{\mathrm{gl,cl}}\sin(k_{\mathrm{cl}}\Delta)\;\text{ and
}\;\beta=\cos(k_{\mathrm{cl}}\Delta).$ (14b)

Figure 6:  The normalized acoustic energy density
$E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$ for the attenuated capillary-tube
resonance versus the normalized coupling-layer thickness $\Delta/\Delta_{0}$
for glycerol and epoxy, simulated in the 3D and in the 1D model. The
exponential function $\mathrm{e}^{-\Delta/\Delta_{0}}$ (black) is inserted as
a guide to the eye.

By further assuming $E_{\mathrm{ac}}\propto p_{1}^{2}(z_{0})$ and
$k^{2}_{\mathrm{cl}}\Delta^{2}\ll 1$, the lowest order functional dependence
in the layer-thickness $\Delta$ becomes

$\displaystyle E_{\mathrm{ac}}$ $\displaystyle\propto
p_{\mathrm{0}}^{2}\sin^{2}(k_{\mathrm{cl}}H_{\mathrm{pzt}})$ (15)
$\displaystyle\qquad\times\Big{[}\cos^{2}(k_{\mathrm{gl}}H_{\mathrm{gl}})-Z_{\mathrm{gl,cl}}k_{\mathrm{cl}}\Delta\sin(2k_{\mathrm{gl}}H_{\mathrm{gl}})\Big{]}.$

By setting $E_{\mathrm{ac}}=0$, we extract the characteristic thickness scale
$\Delta_{0}$, at which the resonance is attenuated

$\Delta_{0}=\frac{Z_{\mathrm{cl,gl}}c_{\mathrm{lo}}^{\mathrm{(cl)}}}{2\omega}\cot\\!\bigg{(}\frac{\omega
H_{\mathrm{gl}}}{c_{\mathrm{lo}}^{\mathrm{(gl)}}}\bigg{)}.$ (16)

In Fig. 6 the normalized acoustic energy density
$E_{\mathrm{ac}}(\Delta)/E_{\mathrm{ac}}^{0}$ is plotted as a function of the
normalized coupling-layer thickness $\Delta/\Delta_{0}$. The acoustic energy
is seen to be attenuated on the length scale $\Delta_{0}$ predicted by Eq.
(16), with $\Delta_{0}=12.1\,\textrm{\textmu{}m}$ and
$24.8\,\textrm{\textmu{}m}$ for glycerol and epoxy, respectively. For both
coupling materials, we observe an approximate exponential decay,
$E_{\mathrm{ac}}(\Delta)/E_{\mathrm{ac}}^{0}=\mathrm{e}^{-\Delta/\Delta_{0}}$.
The figure also show, that the attenuation computed in the 3D model is
captured fairly well by the 1D model, including good quantitative agreement
within 7% between the estimated characteristic length scale $\Delta_{0}$ in 1D
and 3D, respectively.

##  IV Acoustophoretic bulk devices

We now move on to the second type of acoustofluidic devices, namely the bulk
silicon-glass devices used in many lab-on-a-chip applications, as reviewed by
Lenshof, Magnusson and Laurell.(Lenshof2012a, ) As sketched in Fig. 1(b),
these devices consist of a silicon-glass-based acoustofluidic chip coupled to
a bulk PZT transducer. In contrast to the capillary-tube devices, the
manipulation of particles in the silicon-glass chip relies on horizontal half-
wave pressure resonances. Because the pressure half-wave is anti-symmetric
around the vertical center plane of the channel, the symmetric motion actuated
by a usual PZT transducer must be broken. This is normally done geometrically,
by placing the transducer off-center,(Barnkob2010, ; Muller2013, ;
Tahmasebipour2020, ) or by splitting the top electrode of the transducer, and
actuate it by an anti-symmetric voltage actuation.(Moiseyenko2019, ; Bode2020,
) In this work we use the former method, and displace the silicon-glass chip
by $y_{0}=1\,\textrm{mm}$ with respect to the $(xz)$-mirror-plane of the PZT
transducer, see Fig. 7.

###  IV.1 Coupling layer analysis in 2D

It is well known that even with an ideal long straight channel in such
devices, axial variations and acoustic hot spots appears along the
channel.(Augustsson2011, ) However, when studying a region in the channel near
a local maximum in the acoustic field, where the axial gradients are vanishing
small, 2D models describes the acoustic fields very well.(Augustsson2011, ;
Muller2013, ) In this analysis, we therefore study the silicon-glass device in
a 2D model as shown in Fig. 7. As introduced in Section II, the PZT transducer
is a $z$-polarized Meggitt-Pz26 transducer, but now with a thickness of
$1000~{}\textrm{\textmu{}m}$ with a resonance mode near 2 MHz. The materials
and dimensions used in the silicon-glass-device simulations are listed in
Tables 3 and 5, respectively.

Figure 7:  Cross-section in the vertical $yz$-plane of the silicon-glass device in Fig. 1(b) showing the PZT transducer, the coupling layer, and the silicon-glass chip with the microchannel. The chip is displaced from the PZT center plane by $y_{0}=1$ mm. The sketch defines the 2D model. Table 5:  The width ($W$) and height ($H$) of the silicon base (si), the glass cover (gl), the channel (ch), the piezoelectric transducer (pzt), and the silver electrodes (el). Symbol | Value |  | Symbol | Value
---|---|---|---|---
$W_{\mathrm{si}}$ | $2520~{}\textrm{\textmu{}m}$ |  | $H_{\mathrm{si}}$ | $350~{}\textrm{\textmu{}m}$
$W_{\mathrm{gl}}$ | $2520~{}\textrm{\textmu{}m}$ |  | $H_{\mathrm{gl}}$ | $1130~{}\textrm{\textmu{}m}$
$W_{\mathrm{ch}}$ | $377~{}\textrm{\textmu{}m}$ |  | $H_{\mathrm{ch}}$ | $157~{}\textrm{\textmu{}m}$
$W_{\mathrm{pzt}}$ | $12000~{}\textrm{\textmu{}m}$ |  | $H_{\mathrm{pzt}}$ | $982~{}\textrm{\textmu{}m}$
$W_{\mathrm{el}}$ | $12000~{}\textrm{\textmu{}m}$ |  | $H_{\mathrm{el}}$ | $9~{}\textrm{\textmu{}m}$

The coupling layer analysis is analogous to the one in Section III. The
acoustic resonances are located as peaks in the acoustic energy density
spectrum $E_{\mathrm{ac}}(f)$ in the frequency range 1.5 to 2.5 MHz as a
function of coupling material (glycerol or epoxy) and layer thickness $\Delta$
from 0 to 100 µm.

Similar to Fig. 4, we show in Fig. 8(a) a scatter plot, where the points
represent resonances at frequency $f$ for a coupling layer thickness $\Delta$
in the range from 0 to $100~{}\textrm{\textmu{}m}$, with point areas
proportional to $E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$. Multiple resonances are
identified, however, the one at $f_{\mathrm{res}}=1.940~{}\textrm{MHz}$ stands
out with $E_{\mathrm{ac}}^{0}$ being more than 60 times larger than any other
zero-layer peak. The frequency of this resonance is nearly independent of the
coupling material and the layer thickness $\Delta$. However, in Fig. 8(b) we
see a fundamental difference between the two coupling materials: for glycerol
the normalized acoustic energy density $E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$
decreases rapidly to nearly zero at a length scale ${\sim}100\,\textrm{nm}$,
whereas for epoxy $E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$ stays nearly constant
up to $\Delta\sim 1\,\textrm{\textmu{}m}$ followed by a slow drop to 0.75 at
$\Delta=10\,\textrm{\textmu{}m}$ and 0.2 at $\Delta\sim
50\,\textrm{\textmu{}m}$. This behavior may be explained by the geometry of
the acoustics and the fundamental mechanical difference between elastic solids
and viscous fluids. In the silicon-glass device, the direction of the standing
pressure half-wave is orthogonal to the transducer polarization, and to excite
this resonance mode, the transmission of shear-waves from the transducer to
the microchannel is required. However, only a solid coupling layer, and not a
viscous fluid, can support such transmission of shear waves.

Figure 8:  Scatter plots with point areas proportional to
$E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$ of responses to increasing coupling-
layer thickness $\Delta$ simulated in the 2D model of the silicon-glass
device. (a) Resonance frequencies $f_{\mathrm{res}}$. (b) Normalized acoustic
energy density $E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$.

###  IV.2 Dissipation in the glycerol coupling layer

The critical glycerol coupling-layer thickness $\Delta_{\mathrm{crit}}=100$ nm
observed in Fig. 8(c) requires a physical explanation, as this length scales
is far from any of the geometrical sizes or acoustic wavelengths in the
system. The small thickness $\Delta<0.1$ mm of the coupling layer implies
large shear strain rates, and a large amount of viscous dissipation. We assume
that $\Delta_{\mathrm{crit}}$ is the coupling-layer thickness where the time-
averaged viscous dissipation power
$\big{\langle}P_{\mathrm{crit}}^{\mathrm{visc}}\big{\rangle}$ in the glycerol
coupling layer equals the time-averaged acoustic power
$\big{\langle}P_{\mathrm{ac}}\big{\rangle}$ delivered to the half-wave
pressure resonator. For an ideal resonator with height $H_{\mathrm{ch}}$,
average acoustic energy density $E_{\mathrm{ac}}$, and quality factor $Q$,
together with a coupling layer of thickness $\Delta_{\mathrm{crit}}$, width
$W_{\mathrm{glc}}$, and dynamic viscosity $\eta_{\mathrm{glc}}$, we obtain

$\displaystyle\big{\langle}P_{\mathrm{ac}}\big{\rangle}$
$\displaystyle=\big{\langle}P_{\mathrm{crit}}^{\mathrm{visc}}\big{\rangle},$
(17a) $\displaystyle\big{\langle}P_{\mathrm{ac}}\big{\rangle}$
$\displaystyle=\int\big{\langle}p_{1}\bm{v}_{1}\cdot\bm{n}\big{\rangle}\>\mathrm{d}A=\frac{16\pi}{Q}c_{0}E_{\mathrm{ac}}H_{\mathrm{ch}},$
(17b)
$\displaystyle\big{\langle}P_{\mathrm{crit}}^{\mathrm{visc}}\big{\rangle}$
$\displaystyle=\int\big{\langle}\bm{\nabla}\bm{v}_{1}:\bm{\tau}\big{\rangle}\>\mathrm{d}V\approx\frac{4\pi^{2}\eta_{\mathrm{glc}}E_{\mathrm{ac}}}{Q^{2}\rho_{0}\Delta_{\mathrm{crit}}}W_{\mathrm{glc}}.$
(17c)

Solving for $\Delta_{\mathrm{crit}}$, we obtain

$\Delta_{\mathrm{crit}}=\frac{\pi\eta_{\mathrm{glc}}W_{\mathrm{glc}}}{4Q\rho_{0}c_{0}H_{\mathrm{ch}}}.$
(18)

The effect of the surrounding silicon-glass chip is included in the quality
factor $Q=f_{\mathrm{res}}/\Delta f$, found from the resulting full-width
$\Delta f$ at half maximum of the corresponding resonance peak
$E_{\mathrm{ac}}(f)$ at the resonance frequency $f_{\mathrm{res}}$. The
estimate for $\Delta_{\mathrm{crit}}$ is validated numerically by varying the
material and geometrical parameters in (18). The chosen material and geometric
variations are listed in Table 6 together with critical thicknesses and
quality factors. Using the 2D model, the acoustic energy density
$E_{\mathrm{ac}}$ is simulated, and the result is normalized by
$E_{\mathrm{ac}}^{0}$, the value without a coupling layer. In Fig. 9 the
simulated $E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$ is plotted versus the
normalized coupling-layer thickness $\Delta/\Delta_{\mathrm{crit}}$, and for
the wide range of parameters, it is seen that indeed
$E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$ decays on the critical coupling-layer
thickness scale $\Delta_{\mathrm{crit}}$.

Table 6: Critical coupling-layer thickness $\Delta_{\mathrm{crit}}$ and
quality factor $Q$ in nine different system configurations, categorized as
material and geometry variations.

| Material | Geometry
---|---|---
Parameter | $0.05\eta_{\mathrm{glc}}$ | $0.1\eta_{\mathrm{glc}}$ | $0.2\eta_{\mathrm{glc}}$ | $0.5\eta_{\mathrm{glc}}$ | $\eta_{\mathrm{glc}}$ | $2\eta_{\mathrm{glc}}$ | $2\rho_{0}$ | $0.5H_{\mathrm{ch}}$ | $0.5W_{\mathrm{glc}}$
$\Delta_{\mathrm{crit}}$ [nm] | 3.4 | 6.9 | 13.7 | 34.3 | 68.5 | 137.0 | 73.0 | 202.2 | 38.5
$Q$ [-] | 140 | 140 | 140 | 140 | 140 | 140 | 140 | 95 | 125

Figure 9:  Simulated normalized acoustic energy density
$E_{\mathrm{ac}}/E_{\mathrm{ac}}^{0}$ versus the normalized coupling layer
thickness $\Delta/\Delta_{\mathrm{crit}}$ for the nine different system
configurations listed in Table 6.

##  V Conclusion

We have developed a numerical 3D model to study the role of coupling layers in
acoustofluidic devices. The model includes the PZT transducer with electrodes,
the coupling layer, and the acoustofluidic chip with the fluid-filled
microchannel. The model is used to study two well-known types of
acoustofluidic devices: a glass capillary tube and a silicon-glass chip,
classified as vertical and horizontal resonators, respectively, relative to
the polarization axis of the transducer. For each device, a viscous glycerol
and a solid epoxy coupling layer was studied.

For vertical resonators, the capillary-tube device, we have found that for a
given zero-layer resonance, the coupling layer can either result in a
sustaining or attenuating resonance. We have established the criterion (10) to
predict which of the two behaviors will occur, based on a relation involving
the phase of the acoustic wave. For the attenuated waves, we have derived
expression (16) to estimate the characteristic layer thickness $\Delta_{0}$ at
which the resonance is attenuated.

For horizontal resonators, the silicon-glass device, the acoustic resonances
are partially powered by shear-wave transmission through the coupling layer.
Since a fluid cannot sustain such a shear, the glycerol coupling layer works
as a dissipative layer. A critical viscous dissipation thickness
$\Delta_{\mathrm{crit}}$ was presented in Eq. (18), based on scaling arguments
in a 1D two-component model.

The simulation results presented, has led to the formulation of design rules
for choosing an optimal coupling layer between the piezoelectric transducer
and the acoustofluidic device. The design rules involves material parameters,
geometrical parameters, and information about the orientation of the given
acoustic resonance mode relative to the polarization axis of the transducer.
We hope that these rules will prove useful, and that their limitations will be
understood better by experimental validation.

## Acknowledgments

The authors were supported by Independent Research Fund Denmark, Technology
and Production Sciences (Grant No. 8022-00285B).

## References

  * (1) V. T. Rathod, “A review of acoustic impedance matching techniques for piezoelectric sensors and transducers,” Sensors 20(14), 4051 (2020) 10.3390/s20144051.
  * (2) M. Hill, Y. Shen, and J. J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics 40(1-8), 385–392 (2002) 10.1016/S0041-624X(02)00127-0.
  * (3) P. Glynne-Jones, R. J. Boltryk, and M. Hill, “Acoustofluidics 9: Modelling and applications of planar resonant devices for acoustic particle manipulation.,” Lab Chip 12(8), 1417–1426 (2012) 10.1039/c2lc21257a.
  * (4) B. Hammarström, T. Laurell, and J. Nilsson, “Seed particle enabled acoustic trapping of bacteria and nanoparticles in continuous flow systems,” Lab Chip 12, 4296–4304 (2012) 10.1039/C2LC40697G.
  * (5) J. Lei, P. Glynne-Jones, and M. Hill, “Acoustic streaming in the transducer plane in ultrasonic particle manipulation devices,” Lab Chip 13(11), 2133–2143 (2013) 10.1039/c3lc00010a.
  * (6) P. Mishra, M. Hill, and P. Glynne-Jones, “Deformation of red blood cells using acoustic radiation forces,” Biomicrofluidics 8(3), 034109 (2014) 10.1063/1.4882777.
  * (7) I. Gralinski, S. Raymond, T. Alan, and A. Neild, “Continuous flow ultrasonic particle trapping in a glass capillary,” Journal of Applied Physics 115(5), 054505 (2014) 10.1063/1.4863645.
  * (8) R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip 10(5), 563–570 (2010) 10.1039/b920376a.
  * (9) C. Magnusson, P. Augustsson, A. Lenshof, Y. Ceder, T. Laurell, and H. Lilja, “Clinical-scale cell-surface-marker independent acoustic microfluidic enrichment of tumor cells from blood,” Anal. Chem. 89(22), 11954–11961 (2017) 10.1021/acs.analchem.7b01458.
  * (10) K. Petersson, O. Jakobsson, P. Ohlsson, P. Augustsson, S. Scheding, J. Malm, and T. Laurell, “Acoustofluidic hematocrit determination,” Anal. Chim. Acta 1000, 199–204 (2018) 10.1016/j.aca.2017.11.037.
  * (11) N. R. Skov, J. S. Bach, B. G. Winckelmann, and H. Bruus, “3D modeling of acoustofluidics in a liquid-filled cavity including streaming, viscous boundary layers, surrounding solids, and a piezoelectric transducer,” AIMS Mathematics 4, 99–111 (2019) 10.3934/Math.2019.1.99.
  * (12) J. S. Bach and H. Bruus, “Theory of pressure acoustics with viscous boundary layers and streaming in curved elastic cavities,” J. Acoust. Soc. Am. 144, 766–784 (2018) 10.1121/1.5049579.
  * (13) P. B. Muller, R. Barnkob, M. J. H. Jensen, and H. Bruus, “A numerical study of microparticle acoustophoresis driven by acoustic radiation forces and streaming-induced drag forces,” Lab Chip 12, 4617–4627 (2012) 10.1039/C2LC40612H.
  * (14) P. B. Muller, M. Rossi, A. G. Marin, R. Barnkob, P. Augustsson, T. Laurell, C. J. Kähler, and H. Bruus, “Ultrasound-induced acoustophoretic motion of microparticles in three dimensions,” Phys. Rev. E 88(2), 023006 (2013) 10.1103/PhysRevE.88.023006.
  * (15) A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodri´guez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” Journal of Computational Physics 223(2), 469–488 (2007) 10.1016/j.jcp.2006.09.018.
  * (16) COMSOL Multiphysics 5.5 (2019), http://www.comsol.com.
  * (17) M. W. H. Ley and H. Bruus, “Three-dimensional numerical modeling of acoustic trapping in glass capillaries,” Phys. Rev. Applied 8, 024020 (2017) 10.1103/PhysRevApplied.8.024020.
  * (18) P. B. Muller and H. Bruus, “Numerical study of thermoviscous effects in ultrasound-induced acoustic streaming in microchannels,” Phys. Rev. E 90(4), 043016 (2014) 10.1103/PhysRevE.90.043016.
  * (19) J. T. Karlsen, P. Augustsson, and H. Bruus, “Acoustic force density acting on inhomogeneous fluids in acoustic fields,” Phys. Rev. Lett. 117, 114504 (2016) 10.1103/PhysRevLett.117.114504.
  * (20) CORNING, Houghton Park C-8, Corning, NY 14831, USA, _Glass Silicon Constraint Substrates_ , http://www.valleydesign.com/Datasheets/Corning%20Pyrex%207740.pdf, accessed 28 November 2019.
  * (21) AZoMaterials, Charlotte Street, Manchester, UK, _Silver - Applications and Properties of Silver_ , https://www.azom.com/properties.aspx?ArticleID=600, accessed 27 November 2021.
  * (22) M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s modulus of silicon,” J. Microelectromech. Syst 19, 229–238 (2010) 10.1109/JMEMS.2009.2039697.
  * (23) Meggitt A/S, Porthusvej 4, DK-3490 Kvistgaard, Denmark, _Ferroperm Matdat 2017_ , https://www.meggittferroperm.com/materials/, accessed 28 Januar 2021.
  * (24) P. Hahn and J. Dual, “A numerically efficient damping model for acoustic resonances in microfluidic cavities,” Physics of Fluids 27, 062005 (2015) 10.1063/1.4922986.
  * (25) J. J. Hawkes and W. T. Coakley, “Force field particle filter, combining ultrasound standing waves and laminar flow,” Sensor Actuat B-Chem 75(3), 213–222 (2001) 10.1016/S0925-4005(01)00553-6.
  * (26) B. Hammarström, M. Evander, H. Barbeau, M. Bruzelius, J. Larsson, T. Laurell, and J. Nillsson, “Non-contact acoustic cell trapping in disposable glass capillaries,” Lab Chip 10(17), 2251–2257 (2010) 10.1039/c004504g.
  * (27) P. Ohlsson, M. Evander, K. Petersson, L. Mellhammar, A. Lehmusvuori, U. Karhunen, M. Soikkeli, T. Seppa, E. Tuunainen, A. Spangar, P. von Lode, K. Rantakokko-Jalava, G. Otto, S. Scheding, T. Soukka, S. Wittfooth, and T. Laurell, “Integrated acoustic separation, enrichment, and microchip polymerase chain reaction detection of bacteria from blood for rapid sepsis diagnostics,” Analytical Chemistry 88(19), 9403–9411 (2016) 10.1021/acs.analchem.6b00323.
  * (28) W. N. Bodé, L. Jiang, T. Laurell, and H. Bruus, “Microparticle acoustophoresis in aluminum-based acoustofluidic devices with PDMS covers,” Micromachines 11(3), 292 (2020) 10.3390/mi11030292.
  * (29) W. Slie, A. Donfor Jr, and T. Litovitz, “Ultrasonic shear and longitudinal measurements in aqueous glycerol,” The Journal of Chemical Physics 44(10), 3712–3718 (1966).
  * (30) L. Negadi, B. Feddal-Benabed, I. Bahadur, J. Saab, M. Zaoui-Djelloul-Daouadji, D. Ramjugernath, and A. Negadi, “Effect of temperature on density, sound velocity, and their derived properties for the binary systems glycerol with water or alcohols,” The Journal of Chemical Thermodynamics 109, 124–136 (2017) 10.1016/j.jct.2017.01.011.
  * (31) N.-S. Cheng, “Formula for the viscosity of a glycerol-water mixture,” Ind. Eng. Chem. Res. 47(9), 3285–3288 (2008) 10.1021/ie071349z.
  * (32) I. I. Perepechko, V. A. Danilov, and V. V. Nizhegorodov, “Ultrasonic velocity in epoxy resin at temperatures down to 4.2 K,” Mechanics of Composite Materials 32, 316–320 (1996) 10.1007/BF02254743.
  * (33) A. Lenshof, C. Magnusson, and T. Laurell, “Acoustofluidics 8: Applications in acoustophoresis in continuous flow microsystems,” Lab Chip 12, 1210–1223 (2012) 10.1039/c2lc21256k.
  * (34) A. Tahmasebipour, L. Friedrich, M. Begley, H. Bruus, and C. Meinhart, “Toward optimal acoustophoretic microparticle manipulation by exploiting asymmetry,” J. Acoust. Soc. Am. 148(1), 359–373 (2020) 10.1121/10.0001634.
  * (35) R. P. Moiseyenko and H. Bruus, “Whole-system ultrasound resonances as the basis for acoustophoresis in all-polymer microfluidic devices,” Phys. Rev. Applied 11, 014014 (2019) 10.1103/PhysRevApplied.11.014014.
  * (36) P. Augustsson, R. Barnkob, S. T. Wereley, H. Bruus, and T. Laurell, “Automated and temperature-controlled micro-PIV measurements enabling long-term-stable microchannel acoustophoresis characterization,” Lab Chip 11(24), 4152–4164 (2011) 10.1039/c1lc20637k.