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Infrared Physics & Technology xxx (2002) xxx–xxx
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F. Cernuschi *, L. Lorenzoni, P. Bianchi, A. Figari
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CESI, Via Reggio Emilia, 39, 20090 Segrate (MI), Italy
Abstract
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The eﬀects of sample surface treatments on laser
ﬂash thermal diﬀusivity measurements
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The eﬀects of darkening the sample surfaces in thermal diﬀusivity measurements by laser ﬂash method have been
studied both experimentally and theoretically. In particular, by modelling the thermal properties of the blackening
layers, a prediction of the relative error in the thermal diﬀusivity evaluation has been obtained and a good agreement
with the experimental results has been observed. Ó 2002 Published by Elsevier Science B.V.
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Keywords: Thermal diﬀusivity; Surface treatments; Porous and multi-phases materials
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1. Introduction
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The thermal diﬀusivity is one of the most important parameters when heat transfer phenomena
are involved. Nowadays the most worldwide used
technique is the laser ﬂash that is currently considered as a standard for the thermal diﬀusivity
measurements of solid materials [1–3]; this method
consists in heating a sample (typically a disk) by a
short laser pulse and detecting (usually by an infrared detector) the time evolution of the temperature on its rear surface. The absorption of optical
energy on the front face as well as the rear face
emissivity in the range of sensitivity of the IR detector, can strongly aﬀect the measurement. As a
matter of fact, depending on these sample surface
features, the heating can result not suﬃcient to
guarantee a high enough temperature rise: the
typical case is represented by metallic samples with
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*
Corresponding author. Tel.: +39-22-125-8274.
E-mail address: [email protected] (F. Cernuschi).
polished surfaces (in this case the temperature
measurement by IR methods is diﬃcult). On the
contrary, translucent or porous samples (i.e. air
plasma spray ceramic coatings) do not guarantee
that the absorption of optical energy takes place
only on the front surface: this can cause ﬂash by of
the detector during the experiment. In order to
eliminate these problems it is common practice
darkening both sample surfaces by thin ﬁlms of
graphite or by other materials deposited using
diﬀerent techniques.
But in principle, the presence of these two
darkening layers could modify also signiﬁcantly
the measured thermal diﬀusivity in respect to the
true thermal diﬀusivity value of the sample. An
estimation of the diﬀerence between true and apparent thermal diﬀusivity can be performed both
theoretically and experimentally. In this work
some modelling of the thermal diﬀusivity for a
three layers system is presented. Moreover from
the experimental point of view, the comparison of
thermal diﬀusivity values obtained performing laser ﬂash measurements on some reference samples
1350-4495/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V.
PII: S 1 3 5 0 - 4 4 9 5 ( 0 2 ) 0 0 1 3 1 - 7
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F. Cernuschi et al. / Infrared Physics & Technology xxx (2002) xxx–xxx
using Eqs. (1)–(4),
a ¼ K=qC results:
(darkened by graphite layers but following diﬀerent procedures) is given.
the
thermal
diﬀusivity
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a¼h
In order to estimate the inﬂuence of the thin
graphite layers deposited onto the two surfaces of
the sample, a simple model for the thermal conductivity of a multilayer system can be used. In
particular, taking into account that the layers are
perpendicular to the direction of heat conduction,
the Voigt–Reuss model for in serie materials (see
Fig. 1) can adopted [4]. In this case the total
thermal conductivity K is:
K¼
2dl
ðl0 þ 2dlÞ
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CpCL f qCL
Cp0 ð1 f Þq0
þ
qCL f þ ð1 f Þq0 qCL f þ ð1 f Þq0
ð3Þ
ð4Þ
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Cp ¼
l2 ð2dl þ l0 Þ2
¼
a
a
"
#
2
ð2dlÞ2
e0 þ e2CL
l20
¼
þ
þ 2dll0
aCL
K0 KCL
a0
ð2Þ
Moreover, observing that the density q and the
speciﬁc heat Cp of this two phases system can be
expressed as follows:
q ¼ f qCL þ ð1 f Þq0
For comparison purposes it could be useful to
consider the ratio l2 =a:
Fig. 1. A sketch of the sample blackened by two thin graphite
layers.
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ð6Þ
Thus the percent diﬀerence between l2 =a and l20 =a0
can be expressed as:
2 ðl2 =aÞ ðl20 =a0 Þ
2dl
a0
¼
l20 =a0
l0
aCL
2dl
qCL CpCL
K0
þ
þ
l0
q0 Cp0
KCL
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ð7Þ
Since in laser ﬂash experiments the thickness of
sample is usually taken thick enough to have the
ratio l20 =a0 > 330 times the heating duration s (in
order to be allowed to consider the laser pulse as a
Dirac pulse [5,6]) and in our speciﬁc case
s ¼ 800 ls, in a ﬁrst approximation
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ it is possible to
substitute in Eq. (7) l0 ﬃ 0:26a0 :
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ðl2 =aÞ ðl20 =a0 Þ
2dl2
2dl
ﬃ
þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
l0 =a0
0:26saCL
0:26
qCL CpCL
e0
þ
ð8Þ
e0
KCL
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
where e0 ¼ q0 C0 k0 is the thermal eﬀusivity of the
sample. Obviously, this equation is less precise
than Eq. (7) but a rough estimation of the error
can be obtained without any detailed information
on the sample thickness.
In order to semi-quantitatively estimate the effect of the graphite layer on the thermal diﬀusivity
measurement, information about the thermal
properties of this blackening layer is required.
After some preliminary trials, two diﬀerent coatings have been selected. The ﬁrst one––obtained
painting a slurry of graphite and acetone––consists, in the case of a metallic polished substrate, of
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where f is the volumetric percentage of the two
thin layers and KCL and K0 are the thermal conductivity of the graphite layer and of the sample
respectively. For layers with uniform thickness dl,
the volumetric percentage f can be expressed in
terms of the thickness of both the sample (l0 ) and
the layers:
f ¼
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ð1Þ
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ðf =KCL Þ þ ðð1 f Þ=K0 Þ
ð5Þ
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2. Modelling
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ð2dl þ l0 Þ
2 2 i
e þeCL
l2
ð2dlÞ2
þ 2dll0 K00 KCL
þ a00
aCL
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islands of graphite uniformly distributed on the
sample surface covering about 18% of the overall
area of the surface as shown by Figs. 2 and 3. On
the contrary for a substrate with diﬀerent roughness and wettability (for example POCO graphite)
it consists of a dense and uniform layer (see Figs. 4
and 5). The typical macroscopic thickness of this
‘‘layer’’ is of about 3 and 6 lm in the former and in
the latter cases respectively. Similarly to the second
case of the painted coating, the other layer, deposited by spraying graphite, results dense and
quite uniform but with a thickness of about 24 lm
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Fig. 4. SEM image of the dense graphite coating painted onto
the POCO graphite AXM-5Q sample.
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Fig. 2. SEM image of the columnar graphite coating deposited
onto the AISI304 stainless steel sample.
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Fig. 3. SEM close-up view of the columnar graphite coating
deposited onto the AISI304 stainless steel sample.
Fig. 5. SEM close-up view of the dense graphite coating
painted onto the POCO graphite AXM-5Q sample.
(see Figs. 6 and 7). To roughly estimate the thermal properties of these graphite layers, and their
eﬀect on the measured thermal diﬀusivity of the
sample, some further modelling is required. Speciﬁcally, in the case of the columnar layer, if the
sample surface is very reﬂective, a possible model
for the thermal conductivity is an in-parallel two
phase system as shown in Fig. 8 where the lack of
graphite (i.e air or vacuum) could be schematised
as a perfect insulator (i.e. Klack ﬃ 0) and the analytical equation is [4]:
KCL ¼ fKGraphite þ ð1 f ÞKlack
ð9Þ
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KCL ﬃ fKGraphite
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The opposite situation corresponds to a sample
surface with a very high absorption coeﬃcient; in
this case it is still possible to apply the in-parallel
two phase system (i.e. Eq. (9)), but the lack of
graphite can be represented as a perfect conductor
(i.e. Klack KGraphite ). In these two cases, Eq. (9)
can be simpliﬁed as follows:
and
KCL ﬃ ð1 f ÞKlack
ð10Þ
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Thermal conductivity of graphite and the lack of
coating could be ﬁxed as 24 W/mK [10] and 1000
W/mK respectively; by using these two values for
the thermal conductivity of the graphite layer, the
Eq. (8) can be to graphically represented as a
function of the thermal eﬀusivity e0 as shown in
Fig. 9. In the case of the graphite dense layers,
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Fig. 6. SEM image of the dense graphite coating sprayed onto
the AISI304 stainless steel sample.
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Fig. 7. SEM close-up view of the dense graphite coating
sprayed onto the AISI304 stainless steel sample.
Fig. 8. A scketch of the sample coated with a columnar
graphite coating. The white columns represent the lack of
coating material.
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Fig. 9. Eq. (8) as a function of e0 for (a) the columnar coating
painted onto a reﬂecting substrate, (b) the columnar coating
painted onto an absorbing substrate, (c) the painted dense
coating and (d) the sprayed dense coating. Computations have
been performed considering a thickness of 3, 6 and 24 lm for
columnar, dense painted and dense sprayed coatings respectively. The density was estimated by applying Eq. (3) with the
theoretical density of graphite equal to 2.25 g/cm2 . The speciﬁc
heat of graphite was ﬁxed equal to 690 J/Kg °C. The thermal
conductivity of the coating was estimated for the three cases by
using Eqs. (10) and (11) respectively and the values were (a) 4.5
W/mK, (b) 1000 W/mK, (c) and (d) 11 W/mK. For AISI304
and POCO AXM-5Q the thermal eﬀusivity e0 was equal to 8049
J/m2 Ks1=2 and to 10249 J/m2 Ks1=2 respectively.
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where the lowercases 1, 2 refer to open and closed
porosity percentages respectively (i.e. the total
porosity f ¼ p1 þ p2 ). The functions U and W describe the eﬀect of open and closed porosity; they
X
are both equal to ð1 pÞ , where the value of the
exponent X depends mainly from the shape of
porosity and its orientation in respect the heat
diﬀusion as explained in details elsewhere [7–9]. In
particular, for open porosity X ¼ 1:66, while for
lamella shaped close porosity oriented with the
thickness parallel to the direction of the heat diffusion (typically produced by the spraying process)
X could be reasonably chosen ¼8. Fixing, as a
tentative porosity value, f ¼ 0:25 with a 20–5%
distribution between open and closed porosity, it is
possible, similarly to the previous case, to graphically represent Eq. (8) as a function of the thermal
eﬀusivity e0 as shown in Fig. 9 as well.
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3. Experimental
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For the experimental activity, two diﬀerent reference materials have been selected: the POCO
Graphite AXM-5Q and the AISI304 stainless steel
(Fig. 10). The ﬁrst material is highly absorbing
while the stainless steel, when its surfaces are ﬂat
and polished, is highly reﬂective. This means that
Graphite does not require any darkening of the
surfaces before the laser ﬂash measurements but
AISI304 does. This also means that for the ﬁrst
material it is possible to directly compare values
referring to the same sample in the coated and in
the uncoated conditions; on the contrary, in the
case of AISI304 it is possible to compare the experimental results obtained by coating the sample
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thermal conductivity could be estimated considering a three-phases model where the matrix consists of graphite, the other two phases consist of
open and lamella shaped closed porosity. In this
case the equation for the thermal conductivity is
[7]:
KGraphite
p1
KCL ¼
U
Wðp2 Þ
2
ð1 p2 Þ
p2
þW
ð11Þ
Uðp1 Þ
ð1 p1 Þ
Fig. 10. Eq. (7) as a function of k0 for (a) AISI304 painted, (b)
AISI304 sprayed, (c) POCO painted, (d) POCO sprayed.
Computations have been performed considering the same
thermal properties and thickness of coatings as reported within
Fig. 9. The thickness l0 of the two samples is the one reported
within the text. For AISI304 and POCO, literature thermal
properties have been considered [10].
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using the two selected procedures with the values
reported in the literature. Measurements have been
carried out on 10 mm diameter samples 2.026 mm
(AISI304) and 5.019 mm (POCO) thick respectively. Table 1 summarises the experimental values
of the apparent thermal diﬀusivity. The accuracy
on the numbers refers to the statistical uncertainty
estimated by repeating many times (typically 10)
the measurement on the same sample for each
surface treatment. Measurements have been carried out at 63 and 152 °C for AISI304 and POCO
samples respectively. The experimental results
conﬁrm qualitatively that the sprayed coating affect more signiﬁcantly the thermal diﬀusivity value
if compared to the painted coating. In order to
compare the theoretical predictions and the experimental results, within the left side of Eqs. (7)
and (8), the literature ( or the reference) value and
the two experimental values can be used as a0 and
a respectively; l0 has been ﬁxed equal to the
aforementioned values for the two samples.
Moreover it is possible to obtain l by adding to l0
2dl equal to 6, 12 and to 48 lm for painted and
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Acknowledgements
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This work has been developed within the frame
of ‘‘Ricerca di Sistema’’ D.L. MICA 26/01/2000.
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[1] W.P. Parker, R.J. Jenkins, C.P. Butter, G.L. Gutter, G.L.
Abbott, J. Appl. Phys. 32 (1961) 1679.
[2] ASTM C714-72 standard test method for thermal diﬀusivity of carbon and graphite by a thermal pulse method,
ASTM, 1972.
[3] BS7134: Section 4.2: 1990; Method for the determination
of thermal diﬀusivity by the laser ﬂash (or heat pulse)
method. British Standards Institution, 1990.
[4] A.D. Brailsford, K.G. Major, Brit. J. Appl. Phys. 15 (1964)
313.
[5] R.E. Taylor, K.D. Maglic, in: K.D. Maglic et al. (Eds.),
Compendium of Thermophysical Propert of Measurement
Methods, Survey of Measurement Techniques, vol. 2,
Plenum Press, New York, 1984.
[6] R.E. Taylor, K.D. Maglic, in: K.D. Maglic et al. (Eds.),
Compendium of Thermophysical Propert Measurement
Methods, Recommended Measurement Techniques and
Practices, vol. 2, Plenum Press, New York, 1992.
[7] P. Scardi, M. Leoni, F. Cernuschi, A. Figari, J. Am.
Ceram. Soc. 84 (4) (2001) 827.
[8] B. Schulz, High Temp.-High Press. 13 (1981) 649.
[9] D.A.G. Bruggeman, Ann. Physik 24 (1935) 636.
[10] Web site www.matls.com.
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sprayed coatings respectively. Table 2 summarises
the results of this computation and reports also the
values furnished by the theoretical simulations
using Eqs. (7) and (8) respectively. Obviously, the
data obtained by using Eq. (7) and the experimental values show a better agreement if compared to the data furnished by using Eq. (8).
Moreover both experimental data and theoretical predictions show that darkening by painting
aﬀects thermal diﬀusivity measurements by laser
ﬂash in a less signiﬁcant way if compared with
spraying. In any case the relative error induced by
painting resulted below 2% for a wide range of
solids. Moreover it would be possible to roughly
compensate the eﬀect of darkening on the thermal
diﬀusivity measurements carried out laser ﬂash, by
a suitable inversion procedure based on the theoretical modelling.
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