REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 71, NUMBER 7
JULY 2000
Thermal-wave resonator cavity design and measurements
of the thermal diffusivity of liquids
J. A. Balderas-López,a) A. Mandelis,b) and J. A. Garcia
Photothermal and Optoelectronic Diagnostics Laboratories (PODL), Department of Mechanical
and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto,
Ontario M5S 3G8, Canada
~Received 14 December 1999; accepted for publication 24 March 2000!
A liquid-ambient-compatible thermal wave resonant cavity ~TWRC! has been constructed for the
measurement of the thermal diffusivity of liquids. The thermal diffusivities of distilled water,
glycerol, ethylene glycol, and olive oil were determined at room temperature ~25 °C!, with
four-significant-figure precision as follows: (0.144560.0002)31022 cm2/s ~distilled water!;
(0.092260.0002)31022 cm2/s ~glycerol!; (0.091860.0002)31022 cm2/s ~ethylene glycol!; and
(0.088160.0004)31022 cm2/s ~olive oil!. The liquid-state TWRC sensor was found to be highly
sensitive to various mixtures of methanol and salt in distilled water with sensitivity limits 0.5% ~v/v!
and 0.03% ~w/v!, respectively. The use of the TWRC to measure gas evolution from liquids and its
potential for environmental applications has also been demonstrated. © 2000 American Institute of
Physics. @S0034-6748~00!01407-6#
particularly air,8,9 and vapors10 to a high degree of precision.
Although it is possible to carry out thermal diffusivity measurements in the frequency domain with the TWRC, cavity
scans on this device have shown further advantages in terms
of precision and stability.11 This article reports a new liquidstate compatible design of the TWRC, and its application to
the measurement of the thermal diffusivity of liquids. The
sensitivity of this technique is also evaluated by examining
various liquid mixtures. Finally, the potential of the TWRC
as an environmental sensor is demonstrated in detecting dissolved gases in liquids.
I. INTRODUCTION
Matter in the liquid state plays a fundamental role in
science and technology. Its importance is demonstrated by
the variety of scientific disciplines that deal with it ~i.e., the
biomedical, food, and agricultural sciences!. Thus, the design
of experimental techniques to carry out measurements of the
properties of a substance in the liquid state becomes very
relevant. Especially important is the measurement of the
thermophysical properties of liquids because of their widespread use as refrigerants, lubricants, and heat exchangers.
The utility of the photothermal techniques for measuring
thermal properties of materials has been well documented in
the literature.1–7 The basic principle of these techniques consists of measuring the temperature fluctuations in a sample as
a result of the nonradiative deexitation process that takes
place following the absorption of intensity-modulated radiation. The thermal-wave propagation in a material depends on
its thermal diffusivity. Traditionally this thermal property
has been determined from the frequency- and time-domain
behavior of the thermal wave in a fixed volume of the material. The development of the thermal wave resonant cavity
~TWRC!8–10 has introduced the possibility of measuring
thermal properties by monitoring the spatial behavior of the
thermal wave through cavity-length scans, instead of scanning the modulation frequency. The major advantages of
cavity-length scans are the fixed noise bandwidth of the system, which improves the signal-to-noise ratio ~SNR!, as well
as disposing with the requirement for instrumental transferfunction normalization. This kind of device has been successfully utilized to measure the thermal diffusivity of gases,
II. THEORY
According with the mathematical theory of the TWRC,8
it has been shown that the pyroelectric signal from this device, Fig. 1, at a fixed thermal-wave oscillation frequency f
5v/2p is given by
V ~ L, a l , v ! 5Const~ v !
~1!
where V is the voltage signal across the pyroelectric detector
in the open-circuit configuration, L is the cavity length, and
s l is the complex thermal diffusion coefficient, defined by
s l 5 ~ 11i ! Av /2a l .
~2!
Here a l is the thermal diffusivity of the liquid sample. The
interfacial thermal coefficients g jk are defined as
g jk 5
~ 12b jk !
,
~ 11b jk !
~3!
where b jk 5e j /e k is a thermal coupling coefficient, the ratio
of thermal effusivities of media j and k; the subscripts s, p,
and l refer to the thermal-wave source ~a plane metallic light
absorber, such as a copper or aluminum strip; see Fig. 1!, the
a!
On leave from Centro de Investigación en Ciencia Aplicada y Tecnologı́a
Avanzada del IPN, Av. Legaria 694, Col. Irrigación, C.P. 11500, México,
D. F., México.
b!
Electronic mail: mandelis@mie.utoronto.ca
0034-6748/2000/71(7)/2933/5/$17.00
e 2slL
,
12 g ls g lp e 22 s l L
2933
© 2000 American Institute of Physics
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2934
Rev. Sci. Instrum., Vol. 71, No. 7, July 2000
Balderas-López, Mandelis, and Garcia
FIG. 1. Side view of a schematic of the TWRC configuration.
pyroelectric material and the liquid sample, respectively.
The magnitude of the complex expression given in Eq.
~1! can be written as
u V ~ L, a l , v ! u 'Const~ v ! e 2A l L ,
FIG. 2. Schematic representation of the experimental setup. ~a! Cylindrical
thermal-wave emitter head containing the aluminum foil absorber, ~b!, and
the photothermal chamber with the optical fiber, ~c!; ~d! container walls, ~e!
liquid sample filling the TWRC and ~f! dielectric substrate. The bottom
surface of the PVDF was attached to a copper plate. This plate worked as
electric contact and support. A current-modulated laser diode was used.
~4!
1/2
where A l 5( p f / a l ) . To obtain this expression, only the
forward thermal wave has been considered. The remaining
terms form an infinite series of exponential factors, exp
(2nAlL), n52,3,..., constituting coherent thermal-wave
power accumulation in the intracavity region of length L.
They have been neglected compared to n51. This approximation is valid, since the product of the interfacial thermal
coefficients g jk and the exponential term in the denominator
of Eq. ~1! is !1 ~by taking water for reference, with thermal
diffusivity 0.001 45 cm2/s, this term decreases to 0.001 for a
0.03 cm cavity length!. Equation ~4! shows that it is possible
to carry out simple TWRC measurements of the thermal diffusivity of liquids by monitoring the polyvinylidene fluoride
~PVDF! signal as a function of the cavity length, and fitting
the data to a linear equation in a semilog scale. The thermal
diffusivity can be obtained from the slope fitting parameter
Al .
III. INSTRUMENTATION AND EXPERIMENT
The experimental setup used for the thermal diffusivity
measurements of liquids, shown in Fig. 2, consisted of an
infrared ~806 mm! semiconductor laser ~Opto Power Corporation!, operating at powers up to 200 mW. The intensitymodulated laser light was incident on an aluminum foil ~80
mm thick and 1 cm in diameter!. Thermal waves were generated in this foil which was mounted on a micrometer stage.
This stage allowed the cavity length, L, to vary with 10 mm
step resolution. The cross-sectional design of the signal generation head ~a cylinder! is also shown in Fig. 2. The bottom
was sealed hermetically with a highly conducting thin metal
~aluminum! foil acting as an optical-to-thermal power converter, as thermal-wave generator, and as a sealant to prevent
liquids from seeping into the photothermal signal-generation
chamber. The cylindrical module was dipped in various liquids as shown in Fig. 2. Thermal waves conducted across the
liquid interface ~‘‘intracavity region’’!, reached the pyroelectric sensor, which consisted of a PVDF pyroelectric film ~25
mm thickness and 1.5 cm diameter! with metal electrodes
~Ni–Al! on both sides. To avoid vibrations and possible contributions of the piezoelectric response of the PVDF sensor,
its bottom face was attached to a metal ~copper! electrode
with conductive epoxy. The pyroelectric voltage signal generated in the sensor was preamplified ~ITHACO model 1201!
and then sent to a lock-in amplifier ~EG&G model 5210! for
further amplification and demodulation.
For sensitivity analysis of the liquid-state TWRC sensor,
time scans were used to observe the changes in the pyroelectric signal when small amounts of various substances were
added to distilled water. The signal output with the sensor
immersed in distilled water was used as the base line. The
sensitivity analysis procedure was as follows: The container
with the TWRC was filled with 10–13 ml of distilled water.
By maintaining the cavity length constant ~about 200 mm!,
the PVDF signal was recorded as a function of time. As soon
as a steady-state signal was obtained ~about 600 s!, the solute
substance ~methanol or a saturated solution of sodium chloride in distilled water! was added. The experiments were
performed at a modulation frequency of 4.35 Hz. This frequency was chosen because it combined a good SNR and a
satisfactory signal amplitude in our system. The test consisted of establishing the minimum volume necessary to obtain a change in the signal base line. This procedure helped
determine the sensitivity limit of the TWRC device.
The next step was to measure the thermal diffusivity of
various liquid mixtures obtained by cavity length scanning
on the TWRC. For comparison purposes, the thermal diffusivity of some pure substances ~glycerol and ethylene glycol!
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Rev. Sci. Instrum., Vol. 71, No. 7, July 2000
FIG. 3. PVDF signals recorded vs time for the mixtures of methanol and salt
dissolved in distilled water used in this work. The correspondences between
different lines and substances is as follows: ~–h–! Distilled water in distilled water, ~–,–! five drops of methanol in 10 ml of distilled water,
~–L–! ten drops of methanol in 10 ml of distilled water, ~–s–! one drop of
saturated solution of salt in 13 ml of distilled water, and ~–n–! three drops
of saturated solution of salt in 10 ml of distilled water. The arrow indicates,
approximately, the time at which the various substances were added to the
water.
whose thermal diffusivity values have been reported in the
literature were also measured. The thermal diffusivity values
were obtained by fitting the experimental data to the model
described in the theory. Fitting was done by the least-squares
minimization method.
IV. RESULTS AND DISCUSSION
In order to ensure that base line changes were only due
to the addition of a different substance in distilled water, the
following experiment was conducted. Once a base line time
scan for the given volume of distilled water was obtained,
ten drops of the same distilled water were added. Following
this experiment the container was emptied and filled again
with the 10 ml of reference sample for the next time scan.
This time five drops of methanol were added to the distilled
water. The procedure was repeated with ten drops of methanol and three drops of a saturated solution of sodium chloride in distilled water. In order to determine the sensitivity of
the sensor to smaller concentrations of salt solutions, the
experiment was carried out with only one drop of saturated
solution of salt in 13 ml of distilled water. All solutes were
added, drop by drop, by means of a high precision syringe.
The variation on the base line signal due to various concentrations of methanol and salt in distilled water is shown in
Fig. 3. In this figure the line-symbol trace –h– corresponds
to the base line for distilled water and the line-symbol trace
–s– to the mixture obtained by the dilution of one drop of
salt solution in 13 ml of distilled water ~0.03% w/v!. It is
clear that this concentration is close to the sensitivity limit of
the sensor. It was verified that there was no significant variation on the pyroelectric signal following the addition of distilled water in distilled water.
Considering that the volume of ten drops of methanol
~and the same number of drops of the saturated solution of
salt! were approximately 0.1 ml, it was concluded that the
Thermal wave resonant cavity
2935
FIG. 4. Typical results of the pyroelectric signal amplitude vs relative cavity
length for two liquid substances: glycerol ~s! and distilled water ~h!. The
continuous lines correspond to best fits to Eq. ~4!.
sensor has enough sensitivity to detect at least 0.5% ~v/v! of
methanol, or as low as 0.03% ~m/v! of salt in distilled water.
To obtain a saturated solution of salt in distilled water about
35 g of salt in 100 ml of distilled water are required.12
To complete the analysis, the thermal diffusivities of the
mixtures described previously and of some other homogeneous liquid substances ~distilled water, glycerol, ethylene
glycol, and olive oil! were measured by performing a cavitylength scan of the TWRC device. This cavity-length scan
was carried out in 10 mm steps and at a modulation frequency of 4.35 Hz. All the measurements were made at room
temperature ~25 °C!. In Fig. 4 a typical behavior of the amplitude of the pyroelectric signal is shown versus the cavity
length for two substances. The continuous lines are the best
fits to Eq. ~4!. The resulting thermal diffusivity values measured with this technique are summarized in Table I. The
thermal diffusivities reported in this table are averages over,
at least, five measurements. The reported uncertainties therefore constitute the standard deviation. Despite the fact that
TABLE I. Thermal diffusivities obtained using the TWRC technique and
comparison with ~some! known literature values.
Liquid sample
Distilled water
Lake water
Tap water
Salt in dist. water
~0.03%!
Salt in dist. water
~0.1%!
Methanol in dist.
water ~0.5%!
Methanol in dist.
water ~1%!
Glycerol
Ethylene glycol
Olive oil
TWRC technique
a s (31022 cm2/s)
Literature
a s (31022 cm2/s)
Reference
0.144560.0002
0.144660.0004
0.144860.0003
0.144260.0003
0.1456 ~24 °C!
¯
¯
¯
16
¯
¯
¯
0.143960.0004
¯
¯
0.143860.0002
¯
¯
0.142560.0004
¯
¯
0.092260.0002
0.091860.0002
0.088160.0004
0.0929 ~30 °C!
0.0939 ~20 °C!
0.145
0.0799 a
16
17
5
3, 14
a
Calculated from the specific heat and the thermal conductivity for olive oil
given in Ref. 3, and the density for olive oil reported in Ref. 14.
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2936
Rev. Sci. Instrum., Vol. 71, No. 7, July 2000
there are other experimental channels of information ~inphase, quadrature and phase channels! from the lock-in, the
amplitude channel gave a more steady signal, with a better
signal to noise ratio. Moreover, the analytical expression for
measuring the thermal diffusivity from the amplitude is simpler, having only one parameter (A i ), to be determined from
the best fit of Eq. ~4!. The expressions of the other channels
of information ~in-phase, quadrature, and phase! are more
complex and have more parameters to be fitted.
By comparing the thermal diffusivity values measured
for one drop of saturated solution of sodium chloride in 13
ml of distilled water ~0.03% w/v! with that of pure distilled
water ~0.001 442 and 0.001 445 cm2/s, respectively!, the high
precision of the present device is evident. Variations up to
the fourth significant figure are meaningful. To further examine the sensitivity of the TWRC, the thermal diffusivity of
water samples from various sources was determined. These
measurements were made with distilled water, tap water, and
a sample of water from Lake Ontario ~in Toronto!. The measured values of these thermal diffusivities are also reported
in Table I. It is seen that the thermal diffusivity values for the
various types of water overlap with each other within the
reported standard deviations. However, we believe that the
small variations observed in the mean values are significant.
This issue is currently being investigated more fully, using a
novel signal-generation wave form recently introduced in our
laboratory, which suppresses the base line of the thermalwave signal.13
The thermal diffusivity value of olive oil determined
with the TWRC showed significant differences ~;50%!
when compared to reported literature values ~0.000 881 and
0.001 45 cm2/s,5 respectively!. By taking the thermal properties reported for olive oil in Ref. 3 and a density of 0.918
g/cm3 ~Ref. 14!, a thermal diffusivity value of 0.000 799
cm2/s is estimated, in close agreement with the one reported
by using the TWRC technique. It is worth pointing out that
the thermal diffusivity value reported for the olive oil in Ref.
5 was obtained directly, by fitting the photothermal phase
using the standard photopyroelectric configuration.15 The
thermal properties for the olive oil in Ref. 3 were estimated
by using an inverse frequency-domain photopyroelectric
configuration and by fitting the data to approximate expressions for the phase and the amplitude.
Finally, in order to evaluate the potential of the liquid
TWRC sensor for monitoring gases in liquids and its possible applications in environmental analysis, the time evolution of the pyroelectric signal was monitored as a function of
the evolution of CO2 in carbonated water. To do this a can of
commercial carbonated water was opened and some of the
liquid was poured in a beaker flask. The liquid was left to
rest for 5 min in order to allow for the initial profuse evolution of gas bubbles. After this, the container was filled with
this liquid and the PVDF signal was recorded as a function
of time. The result is shown in Fig. 5. In this figure the initial
flat line corresponds to the distilled water base line ~reference!. Upon exposure to the carbonated water, a rapid signal
decrease followed by an approximately linear increase due to
the evolution of the gas was observed. At the end of the gas
evolution a flat base line at the distilled-water level was ob-
Balderas-López, Mandelis, and Garcia
FIG. 5. Evolution of the PVDF signal vs time for carbonated water from a
commercial can. The distilled water base line and the transient due to carbon
dioxide occlusion are shown.
tained again. The increase of the signal with time for the
carbonated water is a measure of the kinetics of gas occlusion from the liquid and the corresponding change on its
thermal properties. These results suggest the possibility of
using the TWRC in assessing content and evolution of gases
~methane, CO2, O2, etc.! from water resources, or other biological systems, with important applications to environmental science.
In conclusion, a novel TWRC design for thermophysical
measurements in the liquid state has been demonstrated. The
device was shown to be capable of measuring thermal diffusivities of liquids, including mixtures and solutions, with
fourth-significant-figure precision. The high sensitivity of the
liquid-state TWRC raises the possibility of monitoring the
quality of a large range of liquids and liquid compounds,
carrying out water pollution assessments based on thermophysical property dependence on gas concentrations ~oxygen, methane! or solid-matter content, as well as measuring
transient responses due to gaseous occlusions or possibly
solid-matter dissolution. The limitations on the current design of this device are associated with the PVDF detector
itself, specifically the temperature range of operation18 ~220
to 80 °C! and the exclusion of aggressive liquids that could
be harmful to the sensor material.
ACKNOWLEDGMENTS
One of the authors ~J.A.B-L.! wishes to thank the Consejo Nacional de Ciencia y Tecnologı́a ~CONACyT! of
Mexico for the partial support of this work trough a PostDoc Grant. The support of the Natural Sciences and Engineering Research Council of Canada ~NSERC! is gratefully
acknowledged.
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Thermal wave resonant cavity
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