Compressional acoustic wave generation in microdroplets of water in
contact with quartz crystal resonators
L. McKenna
a
, M.I. Newton
a+
, G. McHale
a
, R. Lucklum
b
and J. Schroeder
b
a
Department of Chemistry and Physics, The Nottingham Trent University,
Clifton Lane, Nottingham NG11 8NS, UK
b
Institute for Micro- and Sensor Systems, Faculty of Electrical Engineering and
Information Technology, Otto-von-Guericke-University Magdeburg, P.O.B 4120, D-39016
Magdeburg, Germany
Abstract
Resonating quartz crystals can be used for sensing liquid properties by completely
immersing one side of the crystal in a bulk liquid. The in-plane shearing motion of the
crystal generates shear waves which are damped by a viscous liquid. Thus only a thin layer
of fluid characterised by the penetration depth of the acoustic wave is sensed by a thickness
shear mode resonator. Previous studies have shown that the finite lateral extent of the
crystal results in the generation of compressional waves, which may cause deviations from
the theoretical behavior predicted by a one-dimensional model. In this work, we report on a
simultaneous optical and acoustic wave investigation of the quartz crystal resonator
response to sessile microdroplets of water, which only wet a localized portion of the
surface. The relationship between initial change in frequency and distance from the center
of the crystal has been measured for the compressional wave generation regions of the
0
crystal using 2μl and 5μl droplets. For these volumes the initial heights do not represent
integer multiples of a half of the acoustic wavelength and so are not expected to initially
produce compressional wave resonance. A systematic study of the acoustic response to
evaporating microdroplets of water has then been recorded for droplets deposited in the
compressional wave generation regions of the crystals whilst simultaneously recording the
top and side views by videomicroscopy. The data is compared to theoretically expected
values of droplet height for constructive acoustic interference. Results are highly
reproducible and there is good correlation between theory and experiment.
PACS 43.35 Bf
Keywords: quartz crystal resonator, QCM, acoustic wave, contact angle, evaporation.
+
author to whom correspondence should be addressed;
electronic mail: michael.newton@ntu.ac.uk
telephone: +44 115 8483365
fax: +44 115 9486636
1
Introduction
Quartz crystal resonators (QCR) are fabricated by adhering thin electrodes onto the
opposing surfaces of AT-cut quartz crystals. Application of an alternating electric field
causes the crystal to oscillate in a shearing motion between the electrodes. Changes in the
surface mass of the crystal are manifested as a change of the oscillating frequency. Thus the
QCR technique lends itself to applications investigating small mass changes at the
interface, and the device is termed a quartz crystal microbalance (QCM).
The change of frequency, Δf, of a QCM device varies directly with mass changes,
Δm, of a thin solid film covering the surface by the relation
qq
o
A
mf
f
μρ
Δ−
=Δ
2
2
(1)
where f
o
is the initial resonant frequency of the quartz crystal, A is the active area, defined
by the electrode overlap,
ρ
q
is the density of quartz and
μ
q
is the shear modulus for AT-cut
quartz.
A rigid mass, such as a glassy polymer, will oscillate as a unit at the same, reduced,
frequency as the QCM. A non-rigid mass, such as water, in contact with the device does not
oscillate as a whole in sympathy with the crystal
1,2
. The shear wave disturbance decreases
as an exponentially damped cosine function and a boundary layer is conveyed with the
surface during the oscillation. The interface between the crystal and adjacent medium is at
2
an antinode. The length where the shear wave amplitude reduces to e
-1
of the original
maximum oscillation amplitude, is defined as the penetration depth,
δ
,
2
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
Lo
L
f
ρπ
η
δ
(2)
where
η
L
and
ρ
L
are the viscosity and density of the fluid, respectively. The penetration
depth for water contacting a crystal operating at 5MHz is approximately 0.25μm. The water
within the decay length is effectively attached to the crystal and can be viewed as a mass
load which decreases the operating frequency of the device. The change in operating
frequency of the QCM for operation with a thick layer of a viscous Newtonian fluid fully
covering and immersing one of the electrodes is
qq
LL
o
ff
μπρ
ηρ
2
3
−=Δ (3)
The frequency response of a real QCM with its finite lateral dimensions to loading is
further complicated because the mass sensitivity across the crystal is not linear. As a
consequence of the finite size of the electrodes, the translational disturbance amplitude, and
hence sensitivity, is greatest at the center and decreases towards the perimeter in an
approximately Gaussian fashion
3,4
.
3
Solving the Navier-Stokes equation and the equation of continuity with the
assumption of a non-uniform shear flow, as required by the finite electrode size, results in
flow normal to the surface
3,5
; the out of plane displacement is around two orders of
magnitude smaller than the in-plane displacement. The longitudinal acoustic pressure wave
has a wavelength that is dependent upon the acoustic properties of the medium and device
frequency. If the path length through the medium is an integer number of half wavelengths,
and dissipation of the wave energy into the medium is not too great, a stationary wave may
be formed in the liquid cavity. The longitudinal waves may travel a significant distance
through the fluid without experiencing great loss. Subsequent reflection at the free liquid
surface can result in constructive interference of acoustic waves, it is often termed
compressional wave resonance; previous work
6
has used cavities of 8mm and 16mm in
order to obtain a sharp resonance. Compressional waves may recombine with the shear
wave component to influence the apparent mass loading on the crystal and induce a
significant change in the crystal’s oscillating frequency; in addition to the frequency shift
there is also a resistance increase
6
. Detection of compressional waves has been reported
previously for fluids enclosed between a resonator and a hard, flat reflecting surface
5-7
;
passing the reflector through the fluid results in successive resonator frequency
discontinuities which are associated with acoustic interference within the fluid. The free
surface of a liquid is also known to act as a reflector of compressional waves. It is therefore
common for experiments using quartz crystal resonators with one face immersed fully in a
fluid to incorporate a non-plane reflector in the cell design to avoid the problem of
compressional wave resonance.
4
