 # A PHOTOACOUSTIC METHOD OF DETRMINING THE THERMAL

```Molecular and Quantum Acoustics vol. 28 (2007)
195
A PHOTOACOUSTIC METHOD OF DETRMINING THE THERMAL
DIFFUSIVITY WITH A REFERENCE SAMPLE
Jacek MAZUR, Barbara PUSTELNY
Institute of Physics, Silesian University of Technology,
Krzywoustego 2, 44-100 Gliwice, Poland
This paper presents proposition of thermal wave method of measuring the thermal
diffusivity by means of photoacoustic detection. Described procedure is based on
determination of the phase-shift between light source and registered acoustic
signal. A reference material is used for elimination of the measurement set-up
contribution to phase. As there is no obligatory standard for measurement of
thermal diffusivity by means of thermal wave method the paper suggests a method
suitable for standardization.
1. INTRODUCTION
Thermal diffusivity is one of those physical quantities which determine the thermal
properties of materials. It is defined as the quotient of thermal conductivity and the product
of the specific heat c and the density 
=

m2
, []=
.
c
s
(1)
Thermal diffusivity is applied to describe the properties of materials which are connected with
the dynamics of temperature changes caused by time-dependent heat sources. It is, therefore,
an essential material parameter informing about the rate of heat losses.
2. MEASUREMENT METHOD
The oldest method of the measuring of the thermal diffusivity is the so-called Ångström
method, formulated by Ångström in 1863 . This method is based on periodical heating of
one end of a thermally insulated rod and recording the temperature in several points along its
length. The thermal diffusivity is determined by measuring of the time-lag with witch the
maximum value of the temperature reaches the subsequent points of the sample. As the
temperature must be measured at various points of the sample along the direction of the heat
flow, this method is not applicable for technical purposes, particularly when the tested sample
is a thin plate (a few millimeters or even less).
196
Mazur J., Pustelny B.
The method presented in this paper refers to the way of taking measurements put
forward by Ångström, without the necessity of measuring the temperature at various points of
the sample, but only on its surfaces. A flux of heat is supplied to one of the surface of the
tested sample (e.g. by the absorption of light). It ought to vary periodically with the frequency
f of modulation of the absorbed light. It can be shown that in the case of semi-infinite sample
the spatial distribution of the temperature component with the frequency f may be expressed
by the formula:
  
T  z , t ; f =T 0A exp −
 
f
f
z exp 2 f t−i
z


where z denotes the depth (z = 0 for the illuminated surface), T0
,
(2)
is the undisturbed
temperature of medium. Due to the similarity to waves such a disturbance is called a thermal
wave and the method has been called thermal wave method.
As indicated by the formula (2), the temperature changes decrease with the depth .
This property is characterized by the thermal path of diffusion:
th=


f
.
(3)
Samples of thickness l may be classified by comparing them to the thermal path of
diffusion as thermally thin (l < th) or thermally thick (l > th) . On one hand the thermal
path of diffusion depends on the frequency of modulation of the flux of energy supplied to the
sample, but on the other hand it depends also on the thermal diffusivity of the material of the
sample. This property allows to work out a measurement method, where the temperature of
the sample surfaces is measured as function of the modulation frequency.
The harmonic components of the temperature of the illuminated surface and the reverse
surface are described respectively by the formulas
T F  f =
I cosh l  s
,
  s sinhl  s 
(4)
T B  f =
I
1
,
  s sinh l  s 
(5)
where I is the amplitude of the absorbed flux of energy,  s=1i as , a s=

f

.
=−1
th
The formulas presented above are simplified, assuming that the thermal conductivity of the
medium surrounding the sample is significantly smaller than the thermal conductivity of the
sample itself.
Typically the frequency of modulation of the flux of energy varies within the range of
fractions of single Hz to several kHz. It restricts the possibility of measuring of the
temperature directly, e.g. by means of thermocouples. Therefore, indirect measurements have
Molecular and Quantum Acoustics vol. 28 (2007)
197
to be applied, e.g. ones making use of the photoacoustic effect. This method utilizes detection
of the acoustic signal resulting from pressure changes in the so-called photoacoustic cell
attached to the surface of the sample. It is to be mentioned that such measurements are
effective in the case of the frequency of transmission of the acoustic detector.
An analysis of the way of recording of the photoacoustic signal has led to the
conclusion that it is most feasible to measure the signal from the side opposite to the
illumination. According to  such a signal can be expressed by the relation:
S B=
CB
1
,
 s sinh l  s 
(6)
where CB is the corresponding factor of proportionality.
Fig. 1. Diagram of a typical measurement system.
The acoustic signal generated by the absorption of light is detected by means of a
microphone. It is measured with lock-in amplifier set to the frequency of modulation of the
source of light.
Such measurement system permits to measure the amplitude and phase shift of the
signal with respect to the signal of the generator which controls the source of light. The
amplitude and the phase shift of the signal described by the formula (6) are expressed by
following two equations:
∣S B∣=∣C B∣


2 f
1
,
 cosh 2 a s l−cos 2 a s l
  S B =−arctg


tg  as l

−
.
tgha s l  2
(7)
(8)
198
Mazur J., Pustelny B.
2.1. LINEAR APPROXIMATION
In order to simplify the procedure of data processing and the analysis of the accuracy of
measurements following linear approximation of expression (8) can be applied as suggested
by Calderon et al. 
  S B ≃−
l2
3
.
f−

4
(9)
This approximation represents correctly (with an error not exceeding 1.2%) the relationship
given by formula (8) concerning frequencies not exceeding 2.5 fc (fc denotes the frequency at
which the thickness of the sample is equal to the thermal path of diffusion).
3. THE INFLUENCE OF THE MEASUREMENT SYSTEM TO REGISTERED SIGNAL
Formula (6) describes the signal for an ideal measurement set-up, which only changes
the measured quantities into digits. In real set-up the signal coming from the detector
undergoes deformations. The are two sources of these deformations.
3.1. THE SOURCE OF MODULATED LIGHT
The electronic system which modulates the light is controlled by the generator of the
reference signal of the lock-in amplifier. Measurement of the generated light by means of the
photodetector indicates that the amplitude of the intensity of light decreases with the
increasing frequency. A phase shift depending on the frequency also occurs. Temperature
changes caused by the illumination are directly proportional to the intensity of the light
absorbed in the sample. Thus, it may be concluded that the source of light affects the output
signal due to the changes of the amplitude depending on the frequency (graduating factor
CZS( f )) and effects an additional phase shift ZS( f ).
3.2. AMPLIFIERS AND INPUT FILTERS OF THE LOCK-IN AMPLIFIER
Similarly as in former case, the influence of amplifiers and the high-pass filter situated
at the input step of lock-in amplifier. May be taken into account by applying the graduating
factor CWF( f ) and the additional phase shift WF( f ).
4. ELIMINATION OF THE INFLUENCE OF THE MEASUREMENT SYSTEM BY
REFERENCE MEASUREMENTS
Deformation of the signal caused by the source of light and the path of detection are
independent of the measured sample.
Taking into account deformations described above, the formula (7) and (8) take the
following form:
∣SB∣=C ZS C WF ∣C B∣


2 f
1
,
cosh 2 a s l−cos  2 a s l 
(10)
Molecular and Quantum Acoustics vol. 28 (2007)
 S B =−arctg


199

tg  as l

−  ZS  WF .
tgha s l
2
(11)
In order to eliminate the influence of measurement system to the signal, the real signal for
another (reference) sample must be measured. In this case is
 S B ref =  S B ref  ZS  WF .

(12)
Compilation of formulas (8), (11) and (13) permits to eliminate the phase shifts resulting from
the measurement system:
 S B − 
 S B ref    S Bref = 
  S B −[ 
 S B ref −  S B ref ] (13)
  S B = 
The correction function indicated by the square brackets can be determined once by
measurement with wide-ranged and small-step of frequency, followed by adequate
interpolations in subsequent data analysis. The main advantage of this approach is
simplification of measurement procedure. If the reference sample is thermally thin over the
entire range of measurements the simplified formula (9) can be used and the expression (13)
can be reduced to the simple linear relation:
 S B − 
 S B ref ≃−



l 2ref
l2
−
f .
  ref
(14)
5. EXEMPLARY RESULTS OF MEASUREMENTS
At the presented arrangement the test measurements were carried out. As a reference
sample a copper plate 255 m thick with purity of 99,98% was used. The uncertainty of its
thickness was assumed to be 10 m. As no standard samples were available, high-purity metal
plates were used. Details concerning these samples as well as the results of measurements are
presented in Table 1.
Signals for the reference tested samples were measured by lock-in amplifier SR830,
which ensure measurements of the phase with resolution 0.008° and dynamic reserve
exceeding 100 dB. Examples of the dependency of corrected phase on frequency and the
matched linear dependences are shown in Fig.2. Corrections have been effected in compliance
with the formula (13).
The analysis of the data quoted in Table 1 displays differences between the determined
values and the values quoted in literature (in the case of the samples 1-5 values are
underrated). The results of the tests of determining of thermal diffusivity basing on the data
resulting from the formula (6) have been presented in Fig.3. These results confirm the
observed trends and may by used to correct them.
200
Mazur J., Pustelny B.
Fig. 2. Corrected phase of the signal measured for copper samples with a thickness of 515mm
(+) and 1020mm (●).
Tab. 1 Results of test measurements.
Materiał
l

u()
lit
(purity, %)
m
cm2s-1
cm2s-1
cm2s-1
Ti (99.7)
255
0.0926
0.0082
0.093
Ni (99.98)
510
0.206
0.010
0.229
Si (?)*
515
0.695
0.038
0.8
Cu (99.98)
515
1.046
0.075
1.16
Cu (99.999)
1020
1.052
0.048
1.16
Ag (99.9)**
255
1.92
0.32
1.74
* The degree of the purity of the applied silicon sample and possible doping are not known.
** Sample reflecting the light, the thickness of which is comparable to the reference sample. Measurements
within the frequency range 120 – 1050 Hz.
The accuracy of measurements has been assessed taking into account the uncertainty of
measurements  of the sample thickness, the standard uncertainty resulting from the linear
regression and the uncertainty of the parameters of the reference sample. The value of the
standard uncertainty depends on the scatter of the points of measurement in relation to the
matched straight line. The scatter of the points of the measurement depends on the ratio of the
recorded signal to the noise occurring in the system. The quantity of the measured signal is
Molecular and Quantum Acoustics vol. 28 (2007)
201
proportional to power of the applied source of light, as well as to the light absorption
coefficient. Moreover, the amplitude of the signal decreases more or less exponentially with
square root of modulation frequency. Adequate measurements can minimize the uncertainty
of the measurements. Besides the determined value of the thermal diffusivity Table 1 also
provides its uncertainty.
Fig. 3. Difference between the establishment value (a) and the value determined by means of
the regression method (areg) as the function of areg for samples 250-1000 mm thick and the
reference sample aref = 1.16 cm2s-1 (Cu), l = 250mm.
6. CONCLUSIONS
As indicated by the measurements which have been carried out, the suggested method
may serve as a basis for the standardization of measurements of the thermal diffusivity
making use of thermal wave method. This method is feasible in the case of small samples
assuming a one-dimensional flow of the heat flux and an effective absorption of light by the
material. Reflective samples ought to the be blackened. An advantage of this method is its
simplicity and non-destructive character. Its drawback is the relatively high uncertainty of the
measurements, due to the simplification of the procedure of measurements. The simplicity of
this procedure constitutes a requirement of the standardization of measurements.
202
Mazur J., Pustelny B.
This work is supported as part of project Elaboration of System for Measurement of Thermal
Diffusivity by Thermal Wave Methods Multi-Year Programme PW-004 “Development of
innovativeness systems of manufacturing and maintenance 2004-2008”, established by a
resolution of the Council of Ministers of the Republic of Poland
REFERENCES
1. A. J. Ångström, Phil. Mag. 25, 130 (1863).
2. A. Rosencweig, A. Gersho, J. Appl. Phys. 47/1, 67 (1976).
3. J. Mazur, Theoretical analysis of propagation and photodeflection detection of thermal
waves in layered structures – Ph.D. thesis, Silesian University of Technology, Gliwice
1998 (in Polish).
4. A. Calderon, et al., J. Appl. Phys. 84/11, 6327 (1998).
5. Wyrażanie niepewności pomiaru. Przewodnik (Guide to the Expression of Uncertainty in
Measurement), Główny Urząd Miar 1999 (in Polish).
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