Photothermal Materials Characterization
at Higher Temperatures by means of
IR Radiometry
Dissertation
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
der
Fakultät für Physik und Astronomie
an der Ruhr-Universität Bochum
vorgelegt von
Ayman Al Haj Daoud
aus
Maythalon - Nablus / Palästina
Bochum 1999
Mit Genehmigung des Dekanats vom 14.10.1998 wurde die Dissertation in
englischer Sprache verfasst. Eine deutschsprachige Zusammenfassung befindet
sich am Ende der Arbeit.
Dissertation eingereicht am.........................................................................................23.11.1999
Erstgutachter...................................................................................................... Prof. Dr. J. Pelzl
Zweitgutachter......................................................................................... Prof. Dr. N. Marquardt
Tag der Disputation.....................................................................................................10.02.2000
Gedruckt mit der Unterstützung
des Deutschen Akademischen
Austauschdienst-DAAD.
To my parents,
and
To whom I love
Contents
1. Introduction ....................................................................................................................... 1
2. Signal generation process.................................................................................................. 3
2.1 Description of the radiometric signal............................................................................ 3 2.1.1 Basic principles of thermal radiation....................................................................... 3 2.1.2 Photothermal Radiometry........................................................................................ 8 2.1.3 Stationary radiometric signal................................................................................... 9 2.1.4 Modulated radiometric signal................................................................................ 10
2.2 Thermal wave generation and propagation................................................................. 13
3. A generalized model of photothermal radiometry ....................................................... 25
3.1 Equation of energy transfer for absorbing and emitting media .................................. 26 3.2 Radiative transfer ........................................................................................................ 31 3.3 Energy conservation, including conduction and radiation.......................................... 35
3.3.1 Heat diffusion equation for a solid of finite thickness .......................................... 35 3.3.2 Heat diffusion equation for a thermal wave .......................................................... 40
3.4 Derivation of the radiometric signal ........................................................................... 48
4. Experimental Setup ......................................................................................................... 53
4.1 Excitation and detection of thermal waves ................................................................. 53 4.1.1 Excitation of thermal waves .................................................................................. 53 4.1.2 Infrared components and detection ....................................................................... 55
4.2 Electronic equipment and signal processing............................................................... 57 4.3 High temperature cell.................................................................................................. 58 4.4 Calibration of the photothermal experimental setup................................................... 60
5. Experimental results........................................................................................................ 63
5.1 Measurements of thermal waves in reflection ............................................................ 63 5.1.1 Temperature-dependent measurements of silicon samples ................................... 63 5.1.2 Normalization of measurements and quantitative interpretation of room
temperature data .................................................................................................... 72 5.1.3 Normalization of measurements and quantitative interpretation as a function of
temperature............................................................................................................ 81 5.2 Transmission measurements ....................................................................................... 87
5.2.1 Application of thermal wave theory including IR transparency to the transmission measurements................................................................................... 89
6. Application to modern heat insulation materials ....................................................... 101
6.1 Multi-layer Superinsulator Foils ............................................................................... 101 6.1.1 Discussion of results............................................................................................ 107
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures ............................................................................................................. 117
6.2.1 Measurements of reference materials.................................................................. 119 6.2.2 Temperature-dependent measurement on fibre reinforced material ................... 119 6.2.3 Interpretation of the Temperature-dependent measurement on fibre reinforced
materials .............................................................................................................. 127 6.3 Measurements of fibre-reinforced composites with different fibre concentrations.. 145
7. Conclusions and Outlook .............................................................................................. 151
7.1 Introduction............................................................................................................... 151 7.2 Review of the experimental work ............................................................................. 151 7.3 Review of the theoretical work ................................................................................. 151
7.3.1 Derivation of the general heat diffusion equation including radiative transport and transition to the differential equation of the thermal wave........................... 152
7.3.2 Solution of the differential equation of the thermal wave................................... 153 7.3.3 Derivation of the measured radiation signal........................................................ 154
7.4 Experimental results.................................................................................................. 154 7.4.1 Test measurements on Silicon samples at room temperature.............................. 154 7.4.2 Test measurements on silicon samples at higher temperature ............................ 157 7.4.3 Results on multi-layer superinsulation foils........................................................ 157 7.4.4 Results on Carbon-based fibre-reinforced composites........................................ 158
8. Deutschsprachige Zusammenfassung.......................................................................... 161
8.1 Einleitung.................................................................................................................. 161 8.2 Zusammenfassung..................................................................................................... 163
8.2.1 Kurzaufzählung der experimentellen Arbeiten ................................................... 163 8.2.2 Kurzaufzählung zu den theoretischen Arbeiten .................................................. 163
8.2.2.1 Ableitung der allgemeinen Wärmediffusionsgleichung unter Einfluss von Strahlungstransport und Übergang zur Differentialgleichung der thermischen Welle .......................................................................................... 164
8.2.2.2 Lösung der Diffusionsgleichung der thermischen Welle................................ 165 8.2.2.3 Ableitung des gemessenen Strahlungssignals................................................. 166
8.2.3 Experimentelle Ergebnisse .................................................................................. 167 8.2.3.1 Testmessungen an Siliziumproben bei Raumtemperatur................................ 167 8.2.3.2 Testmessungen an Siliziumproben bei höheren Temperaturen ...................... 169 8.2.3.3 Ergebnisse von vielschichtigen Superisolationsfolien.................................... 170 8.2.3.4 Ergebnisse an faserverstärkten Verbundwerkstoffen...................................... 171
Appendix A. Transmission characteristics of different IR materials.............................. 173
Appendix B. The relative spectral sensitivity of the MCT-detector ................................ 175
Appendix C. List of the used components.......................................................................... 176
REFERENCES..................................................................................................................... 177
Acknowledgment .................................................................................................................. 183
Curriculum Vitae ................................................................................................................. 185
1
1. Introduction
Photothermal radiometry, based on intensity-modulated surface heating by a laser
beam in the visible spectral range and on the detection of the thermal wave response in the
near and mid infrared spectral range, has been proposed twenty years ago [Nordal and
Kanstad, 1979] for depth-resolved measurements of the thermal properties of solids.
Meanwhile it has been developed to a measurement technique which is used in industry to
monitor production processes, e.g. by online measurements of coating thicknesses [Petry,
1998]. It has also been shown that this method can successfully be applied to eroded surfaces
[Haj-Daoud, Katscher, Bein, Pelzl, 1998] and real technical layer systems, which are not
ideally smooth and which can be slightly translucent both in the visible and the infrared
spectral range [Bein, Bolte, Dietzel and Haj-Daoud, 1998].
This method, however, has to face problems of interpretation, when the optical
absorption length in the visible spectrum is large, e.g. in comparison to the layer thickness of
coatings. Additional problems arise for photothermal measurements of IR-translucent
materials, since the signal generation process and the theory of thermal waves have not
sufficiently been analyzed so far.
The tasks for my work were to analyze the signal generation process in the case of IR
translucent solids, both experimentally and theoretically, to derive a thermal wave theory
which include slightly IR translucent samples and to run measurement on materials, such as
carbon based fibre reinforced composite and multi-layer foils used for cryogenic insulation
systems, where both conductive and radiative heat transport contribution are expected.
Following this introduction. In chapter 2 a description of the experimental setup is
given. In chapter 3 the basics of the photothermal signal generation process are presented for
IR opaque samples, including the generation and propagation of thermal waves in solids of
finite thickness. In chapter 4, an extension of the usual theory of photothermal radiometry is
derived, which includes slightly IR translucent samples for which both the surface and the
interior of the sample radiate and contribute to the measured signal and for which the internal
heat sinks and heat sources related to the emission and re-absorption of thermal radiation
inside the sample have to be considered. Based on the thermal wave concept, the heat
diffusion equation has been linearized and solved to account for both the conductive and
radiative heat transfer in the solid of finite thickness. Theoretical descriptions are finally
presented for the radiometric signal measured for thermal waves in the reflection
configuration and the transmission configuration. In chapter 5, frequency-dependent test
2 1 Introduction
measurements of “ thermal waves in reflection”, where the thermal wave is excited and
detected at the same surface, are shown and interpreted for silicon samples of the different
thickness, both at room temperature and at higher temperatures. Additionally “thermal waves
in transmission” have been measured for silicon, where the thermal waves are excited at the
front surface and detected at the rear surface of the samples of finite thickness.
Subsequently some technologically relevant materials and systems are measured,
which are applied, e.g. for cryogenic insulation of large-scale applications of
superconductivity in basic nuclear research and nuclear fusion, as heat shields for heat pulse
absorption and heat shock protection in the defense sector. In section 6.1, the method of
“transmitted thermal waves” is applied to multi-layer superinsulation foils at room
temperature. The damping factor for simultaneously conductive and radiative heat transport
has been determined for multi-layer insulation systems consisting of an increasing number of
aluminized mylar and spacer layers. In section 6.2, the method of “thermal waves in
reflection” has been applied on fibre-reinforced composites at different temperatures and on
materials with systematically varied fibre concentrations.
In chapter 7, some conclusions from the present work are summarized and discussed,
and suggestions for future work in this field are given.
3
2. Signal generation process
2.1 Description of the radiometric signal
2.1.1 Basic principles of thermal radiation
The electromagnetic radiation, which is emitted from a body when its temperature is
above absolute zero, is described by its position in the electromagnetic spectrum. Since heated
objects at lower and moderate temperatures radiate energy in the infrared region, this portion
of the electromagnetic spectrum is depicted in greater detail in the lower part of Fig. 2.1
[Hudson, 1969].
Two parameters may be employed in characterizing the radiation, these are the
frequency ν or the wavelength λ , which are related by the [Sparrow, 1978]:
νλ=c (2.1)
Figure 2.1: The electromagnetic spectrum [Hudson, 1969]
The electromagnetic radiation, which is in thermal equilibrium, may be considered as
a “gas” consisting of photons. Because the angular momentum of the photons is integral, it
obeys Bose statistics and the photon gas will have an energy distribution given by the Bose-
Einstein statistics. This photon gas can be considered also as an ideal gas. The occupation
numbers of the quantum states for the photon gas is given by [Landau, 1968; Pointon, 1967]
1]/)exp[(1
−−=
Tkn
Bk µε (2.2)
4 2 Signal generation process
where νε hk = is the energy of the photon with frequency ν , h = 6.625.10 –34 Js the Planck
constant, µ the chemical potential, Bk = 1.381.10-25 J/K the Boltzmann constant, T the
absolute temperature. The thermal equilibrium occurs due to the absorption and emission of
photons by the matter, the number of photons in the matter is not fixed. Therefore as one of
the necessary conditions, that the free energy of the gas photon should be a minimum for a
given temperature T and volume V, we obtain [Landau, 1968; Nolting, 1994]
0,
=
∂∂
=VT
NF
µ (2.3)
The distribution of photons among the various quantum states within energies νε hk = is
therefore given by equ. (2.2) with 0=µ .
1)/exp(1
−=
Tkhn
Bν (2.4)
This is called Planck’s distribution. The number of quantum states in the frequency interval
between ν and νν d+ is 32 /8 cdV ννπ . By multiplying Planck’s distribution with this
quantity, the number of photons in the volume V and inside the frequency interval is given by
νννπ
dTkhc
VdN
B 1)/exp(8 2
3 −= (2.5)
The radiation energy density in this interval of the spectrum is
νννπ
ν dTkhc
hh
VNd
dEB 1)/exp(
8 3
3 −== (2.6)
This formula for the spectral energy distribution is called Planck’s formula. In terms of
wavelength it becomes
λλλ
πd
Tkhchc
dEB 1)/exp(
185 −
= (2.7)
In infrared radiometry we will consider the specific radiation energy ),( TW o λ . This
specific radiation energy can be regards as a Blackbody radiation, which is a special type of
thermal radiation that exists inside an isothermal enclosure and defined as “any body or
material that absorbs completely all incident radiation and that also be the most efficient
radiator” [Hudson, 1969]. We will denote to the specific radiation of Blackbody as
),(0 TW λλ . By taking in consideration the mean velocity of photons [Pointon, 1967] and the
direction of distributions. The energy radiated from semi-infinite space per unit area per unit
time in the given wavelength range may be written as [Hudson, 1969]
2.1 Description of the radiometric signal 5
1)/exp(1
),(2
51
−=
TCC
TW o
λλλ (2.8)
which is a function of only the wavelength and the temperature [Brewester, 1992]. In equ.
(2.8) the constants 1C and 2C are given by
WchC 413.372 21 == π µm4/cm2
388.14/2 == BkchC µm K
The total spectral radiation energy can be obtained by integrating equ. (2.8) over all
wavelengths.
∫∞
==0
4),()( TdTWTW SBoo σλλ (2.9)
where T is the absolute temperature of the surface in Kelvin, 81067.5 −⋅=SBσ W/m2K4 is
Stefan-Boltzmann’s constant. The formula (2.9) is known as Stefan-Boltzmann law, which
states that the total radiation of a blackbody is proportional to the fourth power of the absolute
temperature. Thus relatively small changes in temperature can cause large changes in radiant
emittance [Hudson, 1969]. The spectral radiant energy density of a blackbody at different
temperatures from 300 K to 1500 K is shown in Figure 2.2. The position of the maximum in
the spectral energy depends on temperature. In Figure 2.2 we can see that the spectral radiant
energy at each wavelength increases with temperature and that the peak shifts to shorter
wavelengths with increasing temperature. The position of the maximum can be calculated by
differentiating the Planck function with respect to the wavelength and setting the result equal
to zero [Brewester, 1992], the result is
2898max =Tλ µm K (2.10)
where maxλ refers to the wavelength of maximum spectral radiant emittance, thus the
wavelength at which the maximum spectral radiant emittance occurs varies inversely with the
absolute temperature. This relation called Wien’s displacement law can be used to calculate
the wavelength of maximum blackbody radiation for a given temperature. For example, the
wavelength of maximum spectral radiant energy for a temperature 300 K is 9.6 µm in the far
infrared (compare Figure 2.1).
There are two approximate forms of the spectral radiant energy formula (equ. (2.8)),
which are convenient because of their simplicity and valid when the product Tλ is very
small or alternative, very large. In the limit of small wavelength and / or low temperatures the
6 2 Signal generation process
0 5 10 15 20103
104
105
106
107
108
109
1010
1011
1012
300 K
900 K
500 K
1500 K
W 0 (
λ,T
) /
Wm
-1
λ / µm
Figure 2.2: Spectral radiant emittance of blackbody at various temperatures.
term exp( TC λ/2 ) in the Planck function is much larger than one and the following
expression results:
)/exp(1
),(2
51
TC
CTW o
λλλ = (2.11)
which is called Wien’s limit.
In the case of long wavelength and / or high temperatures the term exp[ TC λ/2 ] can be
expanded in a taylor series. The resulting expression is known as Rayleigh- Jeans limit.
251),(
C
TCTW o λ
λλ = (2.12)
These two limiting expressions for spectral hemisphere blackbody radiant energy are plotted
in Figure 2.3 along with the exact Planck function [Brewester, 1992].
For a real surface, the spectral radiation energy is given by
),(),(),( TWTTW o λλελ = (2.13)
where the spectral emissivity ),( Tλε is a parameter characterizing the radiative properties of
the surface, which is defined as “ the property of a body that describes its ability to emit
radiation as compared with the emission from a blackbody at the same temperature [Siegel,
2.1 Description of the radiometric signal 7
1981]. Thus the emissivity is a function of the type of material and its surface finish and it can
vary with wavelength, direction and the temperature of the material [Hudson, 1969].
Figure 2.3: Spectral hemisphere blackbody radiant energy [Brewester, 1992]
Three types of sources can be distinguished by the way that the spectral emissivity
varies [Hudson, 1969]:
1 - A blackbody, for which 1),( == ελε T
2 - A gray body, for which the spectral directional emissivity is independent of
wavelength and direction, =),( Tλε constant < 1 [Brewester, 1992].
3 - A selective radiator, for which ),( Tλε varies with wavelength and is smaller than 1.
In heat transfer analysis; it is justified to assume that the emissivity of any material at a
given temperature is numerically equal to its absorptance at that temperature. This is known
as Kirchoff’s law [Brewester, 69]. For a gray body it is satisfied that
)()( TT αε = (2.14)
where )(Tα is the absorptivity of a surface at a given temperature. For an opaque material
this relation is can be written
)(1)( TT ρε −= (2.15)
where )(Tρ is the reflectivity of the surface. Following the definitions, equ. (2.15), the
quantity )(Tε can be determined from measurements of the reflectivity [Hudson, 1969].
8 2 Signal generation process
2.1.2 Photothermal Radiometry
Photothermal science encompasses a wide range of techniques and phenomena based
on the conversion of absorbed optical energy into heat. Optical energy is absorbed by the
material and electronic states in atoms or molecules are excited. The excited electronic states
will loose their excitation energy by a series of non-radiative transitions that result in a
general heating of the material [Almond, 1996]. The radiative energy produced by these
transitions is usually in the ultraviolet, visible and infrared portions of the electromagnetic
spectrum. This thermal radiation is also a form of heat transport. Heat transport, defined as
thermal energy transfer from one body to another due to a temperature difference, appears in
two fundamental forms; conduction and radiation. Convection is thermal transport associated
with bulk fluid motion and as such is not only a form of heat transport. The fundamental
mechanism of energy transport in conduction is the direct exchange of kinetic energy between
particles of material. In radiation, the fundamental mechanism of energy transport is by
electromagnetic waves or photons that are emitted and absorbed by the particles of the
material as they undergo state transitions [Brewester, 1992].
There are two technique used in the measurement of radiant energy, photometry and
radiometry. Photometry involves the visual sensation produced by light in the consciousness
of an observer. Thus the methods of photometry are rather psychophysical than physical
[Hudson, 1969]. The methods of radiometry, on the other hand, are more pertinent to the
infrared region and provide broadband measurement.
The photothermal radiometry usually measures the radiation variation, not the total
radiation, since a periodic modulation of the heat source is used to generate a modulated
radiation response. For improved understanding, if we have a body subject to plane harmonic
heating of the form [Almond, 1996] )](cos1[2/0 tI ω+ where 0I is the source intensity, ω
is the angular modulation frequency of the heat source and t is the time, the heating divides
into two parts 2/0I and )][exp(2/0 tiI ω , which produce a dc temperature increase and an
ac temperature modulation Tδ . The resulting temperature modulation is determined by the
specific details of the thermal propagation within the medium (compare chapter 3). The
change in thermal emission produced by modulated heating sample can be derived from the
Stefan Boltzmann law to first order
TTTTW dcSB δσελδ 30 )(4),( += (2.16)
The emitted radiation is directly proportional to the modulated component of the sample
temperature Tδ and to the cube of its local static temperature )( 0 dcTT + [Almond, 1996].
2.1 Description of the radiometric signal 9
Because of the periodic modulation of the heat source it is natural to adopt the principles of
wave physics, and the kind of temperature variations in space and time that are excited in a
body by intensity modulated periodical heating process are denoted as thermal waves [Bein
and Pelzl, 1989].
2.1.3 Stationary radiometric signal
The measured radiometric signal, which is related to the IR radiation emitted by a
solid of stationary temperature T , can be described by [Bein et al., 1995]
λλλελλ dTWTRFCTM ),(),()()()( 0
0∫∞
= (2.17)
where ),( Tλε the spectral emissivity of the sample within the collected solid angle,
),(0 TW λ is Planck’s blackbody radiation, )(λF is the transmittance of the IR optical system
and )(λR is the spectral responsivity of the detector. The constant C may describe the
collected solid angle of the radiant flux, the emitting surface area, the maximum responsivity
maxR of the detector, the amplification factor of the used electronic components etc. In the
case of a gray body in which the emissivity is independent on the wavelength, equ. (2.17) can
be simplified to
λλλλε dTWRFTCTM ),()()()()( 0
0∫∞
= (2.18)
By introducing the quantity
∫
∫∞
∞
=
0
0
0
0
),(
),()()(
)(
λλ
λλλλγ
dTW
dTWRF
T (2.19)
the signal can be transformed into
λλγε dTWTTCTM ∫∞
=0
0 ),()()()( (2.20)
The integral ∫∞
=0
40 ),( TdTW SBσλλ can be solved analytically, and we obtain
4)()()( TTTCTM SBσγε= (2.21)
The quantity )(Tγ depends on the detectable wavelength interval in the infrared,
21 λλλ << , which is limited by the transmittance )(λF of the IR optics system, e.g. filters
10 2 Signal generation process
and lenses, including the window of the high-temperature cell, and by the spectral
responsivity of the detector. According to [Bolte, 1995]. The quantity )(Tγ
4
0 ),()()(
)(
2
1
T
dTWRF
TSBσ
λλλλ
γλ
λ
λ∫
= (2.22)
can thus be considered as a measure of the efficiency of the used IR detection system, to
convert the radiation emitted by a blackbody at constant surface temperature T into a voltage
signal. The characteristics of the detector and the lenses are known [Appendix A and B], thus
the factor )(Tγ can be determined by a numerical integration as shown in Figure 2.4 which is
plotted as a function of temperature.
2.1.4 Modulated radiometric signal
The radiometric signal, which is related to the thermal wave and which can be
considered as a small variation of the stationary radiometric signal )(TM , TfT <<)(δ , can
be deduced by using a taylor expansion to first order with respect to the temperature T [Bein
et al., 1995]:
)()(
)(),( fTTTM
TMTTfM δδ∂
∂+=+ (2.23)
The quantity
)()(
),( fTTTM
TfM δδ∂
∂= (2.24)
describes the measured radiometric signal, and can be approximated as
)(])(
4)([4)(),( 3 fT
TTT
TTTCTfM SB δγ
γσεδ∂
∂+= (2.25)
if the temperature variation of the emissivity is negligible in comparison to the temperature
dependence of Planck’s blackbody radiation. Here the quantity
TTT
TT∂
∂+=′ )(
4)()(
γγγ (2.26)
can be defined, which can be considered as a measure for the efficiency of the used IR
detection system to detect thermal waves [Bolte, 1995]. )(Tγ ′ can be also determined by
numerical integration and is plotted in Figure 2.4 as a function of temperature.
2.1 Description of the radiometric signal 11
Figure 2.4: The dimensionless functions )(Tγ and )(Tγ ′ as a function of temperatures.
Then we obtain the following correlation between the temperature variation )( fTδ
and the variation of the radiation signal
)(4)()(),( 3 fTTTTCTfM SB δσγεδ ′= (2.27)
When both the periodic radiometric signal ),( TfMδ and the stationary signal )(TM
corresponding to the stationary surface temperature are measured, the factor C and the
effective emissivity )(Tε , can be eliminated, and when additionally the stationary
temperature is known, the thermal wave can be determined directly [Bolte, 1995]
∂
∂+
=
TT
TT
TTM
TfMfT
)()(
14
14)(
),()(
γγ
δδ (2.28)
where )(/),( TMTfMδ is known as the thermal contrast. The small variation of the detector
signal which correspond to the thermal waves response, ),( TfMδ , are distinguished from the
background radiation level, )(TM , by filtering the signal with the help of a lock-in amplifier
at the modulation frequency f, of the thermal wave. Nevertheless, the infrared detection is
affected by the limit of incoherent and coherent background fluctuation [Bolte, Gu, and Bein,
1997].
The radiometric signal, ),( TfMδ , contains not only the pure measured thermal waves,
but also the effect of the electronic components in the measurements of the thermal waves,
300 600 900 1200 1500 18000.00
0.05
0.10
0.15
0.20
0.25
0.30
γ'(T)
γ(T)
T / K
12 2 Signal generation process
due to the frequency dependence of the measured thermal wave by measuring system. For the
quantitive interpretation of the measurements and in order to eliminate the frequency response
of the measured system, the measured signal ),( TfMδ of the sample of unknown optical and
thermal properties can be normalized with the help of reference signal obtained for smooth,
homogeneous sample, e.g. glassy carbon. When the surface temperature of the sample and
that of the reference are equal, the combined factor )(TC SB γσ ′ shown in equ. (2.27), which
are related to the characteristics of the IR detection system, are also eliminated by the
normalization procedure, and the material properties of the sample and the reference body can
be compared directly (compare chapter 5 and 6).
2.2 Thermal wave generation and propagation 13
2.2 Thermal wave generation and propagation
In this section the essential feature of heat transfer will be represented, followed by a
discussion of the derived thermal wave and the influence of the optical and thermal properties
on the thermal wave.
Thermal waves, which can be excited in solids by intensity modulated heating, are
governed by the heat diffusion equation [Casslaw and Jaeger, 1984]
),(),,(),(
),(),( txQTtxFt
txTTxcTx
rrr
rr+∇−=
∂∂
ρ (2.29)
The heat flow ),( txFr
in equ. (2.29) is related to the temperature distribution ),( txTr
by
),(),(),( txTTxktxFrrrr
∇−= (2.30)
Here ,ρ c and k are the mass density, specific heat capacity, and thermal conductivity,
respectively, of the solid which in general can vary with the space-coordinates xr
, time t, and
the temperature T.
In this theoretical consideration, we will consider an isotropic homogeneous semi-
infinite medium whose surface is subjected to plane harmonic heating by incident radiation
intensity in the form
)]2cos(1[2
),0( ftI
txI o π+== (2.31)
where f is the modulation frequency of heating. According to Lambert-Beer’s law the
intensity, which is incident on the sample, will partially be absorbed and supply the heating
source in the sample. The heat source distribution is given by
)exp()]cos(1[2
]exp[),0(),(
),( xtI
xtxIdx
txdItxQ o βω
βηββηη −+=−==−=
rr
(2.32)
Here we are primarily interested in the ac component
Re2
),( oItxQ
βη=
r )exp()exp( tix ωβ− (2.33)
and will omit the dc component in the following solution, in which Re stands for “the real part
of”, 1−=i is the imaginary unit, and β is the optical absorption constant of the solid in the
visible light, and η is the photothermal conversion efficiency, defined as the fraction of the
total incident intensity oI transformed into heat. The optical parameters β and η are
functions of the wavelength λ of the incident radiation. The Ar+ laser used in our experiment
has a definite wavelength (514 nm). Therefore β can be considered as a constant parameter.
14 2 Signal generation process
In the case that the diameter of the heating spot on the sample is large in comparison with the
thermal diffusion length and detection area of the detector we can work with a one-
dimensional heat diffusion equation. Thus, we can rewrite the heat diffusion equation by
substituting equ. (2.30) in equ. (2.29) as:
),(]),(
)([),(
)()( txQx
txTTk
xttxT
TcT +∂
∂∂∂
=∂
∂ρ (2.34)
After neglecting the temperature dependency of thermophysical parameters we obtain
ctxQ
x
txTt
txTρ
α),(),(),(
2
2
+∂
∂=
∂∂
(2.35)
where ck ρα /= is the thermal diffusivity of the material.
The heat diffusion equations for a homogeneous solid and the gas region in contact
with the solid are
ss
s
s
sss
ss
c
txQ
x
txT
t
txT
ρα
),(),(),(2
2
+∂
∂=
∂∂
(2.36)
and
2
2 ),(),(
g
ggg
gg
x
txT
t
txT
∂
∂=
∂
∂α (2.37)
The geometry of our problem is shown in Figure 2.5 for an isotropic homogeneous
semi-infinite medium in contact with a gas region.
oI
sW
osT WWW −=
oW
0=gx
0=
=
s
gg
x
lx ∞→sxsxgx
Gas Solid
oI
sW
osT WWW
oW
0gx
0=s
gg
x
lx sxsxgx
Figure 2.5: Schematic of the geometry
2.2 Thermal wave generation and propagation 15
In general, modes of energy transfer across the solid-gas interface include conduction,
convection and radiation. The convection can be neglected as in solids convection is absent
and as the convection in the gas region has no significance for the low temperature changes
associated with thermal waves. To develop the theory of thermal waves, only radiation
coming from the surface and conduction are considered here. Also appropriate boundary
conditions are needed for the analysis of heat conduction / radiation problems. The boundary
conditions specify the thermal condition at the boundary surfaces (gas / solid). At a given
boundary surface, the distribution of temperature can be prescribed, but the heat flux can be
specified. The boundary conditions are:
1. At the surfaces, 0=gx and ∞=sx , shown in Figure 2.5, the temperature
values must be finite.
oss
ogg
TtTx
TtTx
=∞∞=
==
),(:
),0(:0 (2.38)
2. Continuity of the temperature at the (gas / solid) interface.
),0(),(:0, tTtlTxlx sggsgg === (2.39)
To achieve the condition of equ. (2.39) for the temperature continuity at the gas / solid
interface without temperature slip, the required modulated frequency, which used to heat the
sample, can not be too high, which means that slower heating process are only considered,
which give chance for the continuity of temperature at the interface to happen.
3. Continuity of the heat flux at the (gas / solid) interface.
Wx
txTk
x
txTkxlx
sgg xs
ssslx
g
gggsgg +
∂∂
−=∂
∂−==
== 0,
),(),(:0 (2.40)
where the quantity W is the net radiatve heat transfer across the interface, and is given by
44 ),0(),0( oSBosSBsoss TtTWtxWW σεσε −=−== (2.41)
where oW is the heat flux emitted from the surrounding gas at the ambient temperature To,
and ),0( tWs is the radiation emitted from the surface of the solid of emissivity sε .
The total solution for the temperature distribution can be solved by using the ansatz
),()(),( ,,,,,, txTxTtxT sgsgsgsgsgsg δ+= (2.42)
16 2 Signal generation process
which allows to separate the stationary from the time dependent problem. After substituting
the heat source in equ. (2.33), the equations in the regions of solid and gas become
)exp()exp(2
),(),(2
2
tixc
I
x
txT
t
txTss
ss
sos
s
sss
ss ωβρ
βηδα
δ−+
∂
∂=
∂∂
(2.44)
2
2 ),(),(
g
ggg
gg
x
txT
t
txT
∂
∂=
∂
∂ δα
δ (2.45)
where sβ is the optical absorption coefficient of the solid, and sα and gα are the thermal
diffusivity of the solid and gas, respectively.
To solve the homogeneous equ. (2.44) and (2.45) let us assume the periodic
component has a solution of the form
Re),( =txTδ )2exp()( tfixT πδ 2.46)
Omitting the “Re” symbol, substituting equ. (2.46) in equ. (2.45), we obtain [Almond, 96]
0)()(
2
22 =
− xT
i
dx
xTde tfi δ
αωδπ (2.47)
Discarding the exponential time factor, the general solution for the spatial dependence of the
temperature may be written in the form xx eBeAxT σσδ −+=)( 2.48)
where A and B are constants. The quantity απσ /)1( fi+= , which is the solution of the
dispersion relation of thermal waves, is complex and contributes to the phase shift of the
thermal waves.
For the solid, its complex solution can be written as
tixs
xs
xsss eeCeBeAtxT ssssss ωβσσδ ][),( −− ++= (2.49)
where the last term is related to the heat source in the inhomogeneous equation. The constant
sC in equ. (2.49) is determined from the inhomogeneous equ. (2.44) of the solid region,
−
−=2)(12
s
sss
oss
k
IC
βσ
β
η (2.50)
For the gas region in front of the solid and by using this special cell which can be evacuated
to reduce the effects of conduction and convection in the gas region, we can neglect the
convective motion and heat source, a first-order solution is given by [Pelzl and Bein, 1989].
tixg
xggg eeBeAtxT gggg ωσσδ ][),( −+= (2.51)
2.2 Thermal wave generation and propagation 17
The quantities sσ and gσ are
gsgs
fi
,, )1(
απ
σ += (2.52)
for the solid and gas region, respectively.
The incremental radiative emittance Wδ due to a temperature radiation Tδ can be
derived from the Stefan Boltzmann law by substituting equ. (2.41) in equ. (2.40) , considering
only the first order term
),0(),0(),0()0(4)0( 434 tWtWTtTTTW oSBossSBssSBs δσεδσεσε +=−+= 2.53)
The boundary conditions, equ (2.38), (2.39) and (2.40), can be reformulated for the time
dependent solution as
0),(:
),(),(
),0(),(:0
0),0(:0
0
,
=∞∞=
+∂
∂−=
∂
∂−
===
==
==
tTx
Wx
txTk
x
txTk
tTtlTxlx
tTx
ss
xs
ssslx
g
ggg
sggsgg
gg
sgg
δ
δδδ
δδ
δ
(2.54)
The integration constants ggss BABA ,,, can be determined from the boundary conditions eqi.
(2.54) and are calculated as
]1[
)(1
0
GR
GR
CB
A
s
s
s
sss
s
++
++
−=
=
βσ
σβ
(2.55)
where the complex quantity R can be understood as the ratio of the radiative heat loss to the
conductive heat transport of the solid at the interface gas / solid.
ss
sSBs
k
TR
σσε )0(4 3
= (2.56)
The relevant thermophysical parameter in the quantity R is the thermal effusivity
ss cke )( ρ= of the solid, which can be seen when the real amount of R is calculated;
s
sSBs
ckf
TR
)(
)0(22 3
ρπ
σε= (2.57)
The quantity G is defined as
)tanh( ggss
gg
lk
kG
σσ
σ= (2.58)
18 2 Signal generation process
If the thickness of the gas layer is large, )tanh( gg lσ can be approximated by one, we obtain a
real quantity
s
g
ck
ckG
)(
)(
ρ
ρ= (2.59)
which is given by the ratio of the thermal effusivities of the gas and the solid, respectively,
and which can be understood as the ratio of the conductive heat losses in the gas to the
conductive heat transport in the solid at the interface gas/ solid. In general the value of the
quantity G is very small, less than one. For example, for hard foam materials with low
effusivity the value of G is 0.01, and for silicon it is nearly 10-4 (compare Figure 2.6).
The resulting expression for the temperature distribution in the semi-infinite solid of
finite optical absorption constant sβ is
[ ])4/(
2expexpexp
]1[
)(1
122
),( πωβσ
βσβ
σ
βσ
π
ηδ −−
−++
++
−
= tix
s
sxs
s
s
ss
osss
ssss
GR
GR
fe
ItxT
(2.60)
The complex frequency dependent solution at the sample surface, which gives information
about the measurable thermal wave, can be obtained by setting 0=sx in equ. (2.60).
[ ])4/(
2exp
]1[
)(1
122
),0( πω
βσβ
σ
βσ
π
ηδ −
−++
++
−
== ti
s
ss
s
s
ss
osss GR
GR
fe
ItxT (2.61)
The quantity R can be denoted as radiation-to-conduction parameter and depends on the
modulation frequency of heating f and the surface temperature of the sample. Normally, the
value of R is also small in comparison to one. This means that the temperature distribution of
thermal waves is in general independent of the boundary conditions, weather there is assumed
to be purely conductive or purely radiative. According to equ. (2.60), exceptions may arise for
high average sample temperatures, low effusivity values and translucent sample with low sβ -
values. For comparison, two samples (silicon and hard foam) with different effusivities are
shown in Figure 2.6 at 300K and 1000K, which represents the magnitude of R as a function of
frequency. For silicon the absolute value of |R| at 300 K is smaller than the at 1000 k
(compare equ. (2.56), but the absolute value of |R| for hard foam at 300 K is nearly in the
2.2 Thermal wave generation and propagation 19
same order of silicon at 1000 K, which returns to low value of effusivity for hard foam.
Accordingly the values of |R| are increased for the hard foam with increasing temperatures. As
a result, the absolute value of |R| is always smaller than one at room temperature and it
increased at higher temperatures especially at very low frequencies, which shows that the
radiative term is strongly temperature-dependent, it is less important than the conduction term
at room temperature, but equal or more important than that at higher temperatures. From
Figure 2.6 we see that at high frequencies the value of G is more important than the value of
|R|.
For a special case of equ. (2.60), for a semi-infinite opaque solid, we can consider the limit of
a surface heating 0/ ≈ss βσ , for which the thermal energy periodically applied at the surface
is dissipated into the solid by conduction. Additionally we assumed that the parameter G is
also very small. The complex frequency-dependent solution is given by
+
−−−
−
−
++
= 21
1tan
2/12
1
expexp
211
1
2),(
Rx
fti
xf
s
osss
ss
ss
Rfe
ItxT
απ
ω
απ
π
ηδ (2.62)
10-3
10-2
10-1
100
101
102
103
104
105
106
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
G Silicon hard foam
G
|R| Silicon hard foam T=500 K
T=1000 K
T=300 K
T=300 K
Si
α= 88.10-6 m
2/s
e = 15500 Ws1/2
/ m2K
hard foam
α= 0.1*10-6 m
2/s
e = 500 Ws1/2
/ m2K
|R|
f / Hz
Figure 2.6: The radiation-to-conduction parameter |R| as a function of frequency for
different materials and temperatures, in comparison with the parameter G.
20 2 Signal generation process
The complex frequency dependent solution at the sample surface is
+
−− −
++
== 21
1tan
2/12
1
exp
211
1
2),0(
Rti
s
osss
s
Rfe
ItxT
ω
π
ηδ (2.63)
equ. (2.62) depends on the optical and thermal parameters of the sample. For small values of
R , the real part solution can easily be derived from equ. (2.62) and has been given by [Bein
and Pelzl, 1988].
−−
−=
42cosexp
2
1
)(2),(
πα
ππ
απ
πρ
ηδ x
ftfx
f
fck
ItxT
s
os (2.64)
Equ. (2.64) reveals that in a periodically heated solid, space and time dependent temperature
variations are induced by periodic heating, which can be interpreted as thermal waves. In
Figure 2.7 the amplitudes and phases of the thermal waves ),( txTδ in a semi-infinite solid
are shown plotted as function of depth below the surface for different times t . The
exponentially but oscillatory decay of the temperature amplitude and the linear phase
variation with sample depth should be noted [Almond, 1996].
The relevant thermal parameters in equ. (2.64) are the thermal diffusivity α and the thermal
effusivity cke ρ= . The thermal diffusivity is the ratio of the thermal conductivity k and the
thermal capacitance, cρ . The thermal diffusivity gives the ratio at which heat is distributed
in a material and this rate depends not only on the thermal conductivity but also on the rate at
which energy heat can be stored. A large value of α (large k and/ or low cρ ) implies that a
medium is more effective in transferring energy by conduction than it is in storing energy. In
general, metallic solids have higher α values, while nonmetallic solids have lower values of
α [Incropera, 1990]. The combined quantity ck ρ , the thermal effusivity (W s1/2/ K m2), is
the relevant thermophysical parameter, rather than the thermal conductivity, mass density, and
the specific heat capacity separately. The effusivity is the relevant parameter for time-varying
surface heating or cooling processes. In reality, we are familiar with this parameter. If we
touch bodies of equal temperatures but of different affectivities, they do not seem to be
equally hot or cold, instead we feel that one body is “hotter” or “cooler”, According to
21
2211
)()(
)()(
ckck
TckTckTm
ρρ
ρρ
+
+= (2.65)
2.2 Thermal wave generation and propagation 21
where mT is the contact temperature which is a function of the effusivity of the body we touch
[Bein and Pelzl, 1989]. This important property can be understood as the relevant
thermophysical parameter for transient surface heating processes and the heat transition
between layers of different thermal properties, whereas the thermal diffusivity α is the
relevant parameter for time-dependent heat propagation within solid bodies or layers of
constant thermal properties.
According to equ. (2.64), this quantity cke ρ= alone determines the surface
temperature ),0( txT =δ , and it is a measure for the heat energy stored in a solid per degree
of temperature rise after the beginning of a surface heating process [Grigull and Sandner,
1979].
fetxT
1),( ∝δ (2.66)
Figure 2.7: Thermal waves in semi-infinite homogeneous and opaque medium at
different times t3 < t1 < t2
Low values of the thermal effusivity, e.g. in insulator, lead to high oscillation amplitudes of
the surface temperature and it is easy to measure the relatively high surface temperature
oscillations, but the information they can give comes from a region just at the surface. High
values of the effusivity, on the other hand, lead to low surface temperature oscillation as in
good thermal conductor, and the information can come from deeper beneath the surface, but it
is difficult to measure the smaller temperature surface [Bein and Pelzl, 1989]. Thermal
0 1 2 3 4-1.0
-0.5
0.0
0.5
1.0
t2
t1
phas
e
X / a.u.
δ T (
x,t)
/ a.
u.
X / a.u. 0 1 2 3 4
-100
0
100
200
300
400
t3
t2
t1
t3
22 2 Signal generation process
effusivity values are shown along with thermal diffusivity values in Table 2.8. In general, a
high diffusivity material also has a high effusivity but there are exceptions to this rule. The
most important exception is air, which has a high diffusivity because its very low conductivity
is balanced by its equally low density [Almond].
The amplitude damping and phase shift of these waves are directly related to the
effusivity, the thermal diffusivity and the propagation distance x as shown by the thermal
wave solution. Low values of the thermal diffusivity lead to a very rapid attenuation of the
amplitude below the surface, and high thermal diffusivity values contribute to a relatively
deeper penetration of the thermal wave.
Between the periodic heating process according to equ. (2.31) and the thermal
response equ. (2.63) there is a phase lag
4π
απ
φ −−=∆ xf
(2.67)
which increases with the propagation distance x of the thermal wave [Carslaw and
Jaeger, 1969] and varies with the modulation frequency of the heating process. At the surface,
the phase lag is –45° between the heat source and the resulting surface temperature. This
delay corresponds approximately to our experience about the heating of the earth crest by the
sun both in days and during the year. The highest irradiation reaches the earth at 12 hours
noon, while the highest temperature measured during the day is at about 15 hour, which
means that there is a delay of about 3 hours. The phase lag of 45° corresponds with respect to
the number of days in year to 45 days. This means the highest temperature is measured at the
beginning of August [Matthes, 1990].
The thermal diffusion length is defined as fπαµ /= which can be controlled by the
modulation frequency f of heating, a systematic variation of the modulation frequency can be
used for subsurface depth inspections of solid samples, and thus thermal wave techniques are
spatially suited to measure depth-dependent thermal properties. It can be seen that the wave
amplitude is strongly damped; at a distance of
fx παµ /== (2.68)
it decays to e/1 of its initial value. Thus equ. (2.60) can be applied to geometrically
relatively thin sample as long as their thickness is comparable to the thermal wavelength
[Bein and Pelzl, 1989].
2.2 Thermal wave generation and propagation 23
Material Thermal diffusivity
10-6 m2 s-1
Thermal effusivity
Ws1/2 / Km2
Aluminum 98 24000
Copper 116 36900
Silicone 88 15500
V2A-steel 4 7570
Molybdenum 53 20200
Fibre-reinforced material V1 0.45 850
Fibre-reinforced material V4 0.45 780
Hard foam 0.1 500
Nylon 0.06 1440
Graphite 3-130 2800-13000
Sigradur 4.2 2400
Neutral glass 1 0.53 1460
Quartz glass 0.9 1460
Air 18.5 5.8
Table 2.8: Thermophysical parameters of some solids at room temperature [Touloukian,
1973; Incropera, 1990; Simon, 1996].
25
3. A generalized model of photothermal radiometry
In recent years the modulated photothermal radiometry of solids, which consists of the
observation of the modulated infrared emission from a periodically illuminated and heated
sample, has received considerable attention. Its theory was originally developed by [Nordel
and Kanstad, 1979]. In the first theoretical models [Rosencwaig, 1976; Nordal and Kanstad,
1979] the measured radiation was assumed to be sensitive only to the temperature rise of the
sample surface and that the sample is opaque to thermal radiation. Radiation related to IR-
emission from the interior of the sample is absorbed by the sample, and there is no radiation
flux from the interior of the sample in these models. This assumption, however, is not always
correct since some materials are transparent in the infrared and thus subsurface IR radiation
can contribute to the measured signal. Therefore the theory was improved later by [Tom,
O’Hara, and Benin, 1982; Walther and Seidel, 1992 and 1996; Gu, 1993; Sommer, 1994;
Dietzel, Haj-Daoud, Macedo, Pelzl, Bein, 1999]. All these authors demonstrated that the
experimental results are remarkably affected by the infrared radiation inside the sample. This
means that in the case of infrared translucent materials, the signal is not simply proportional
to the modulated surface temperature but it is determined by the superposition of all infrared
radiation fluxes arising from different depths inside of the sample having different phase lags
with respect to the modulated heating beam, and being partially reabsorbed inside the sample.
All these new models relied on the assumption, however, that the heat sources due to the re-
absorption of thermal radiation are not important for the determination of the temperature
distribution of the thermal wave and may thus be neglected, especially at room temperature
and for samples with high material density.
Here in this work, an extension of the PTR theory is presented to include samples
which are slightly translucent to the thermal radiation, so that both the surface and the interior
of the sample radiate and contribute to the measured signal, and in which additionally the
internal heat sources due to the re-absorption of thermal radiation are also considered.
Consequently, the emission and re-absorption of thermal radiation has formally been included
in the heat diffusion equation. Both the radiation fluxes inside the sample and the measured
signal have been calculated from the solution of the radiative transfer equation.
First we will describe and illustrate the basic principles of radiation transfer in the
sample and then we will combine the resulting heat sources / sinks with the heat conduction
equations in order to find the general heat diffusion equation, to calculate the time- and space-
dependent temperature distribution in the sample. By using a suitable linear ansatz for the
26 3 A generalized model of photothermal radiometry
temperature distribution, we can obtain after that the temperature distribution of the thermal
wave. In the following, we will write the complex solution of the thermal wave involving two
terms, namely the usual form used in chapter 2, (equ. 2.49), and the additional solution, which
is related to the internal heat sources due to the re-absorption of thermal radiation. The second
term will be considered as a small perturbation in the temperature distribution of the thermal
waves. Finally we will determine the general complex solution of the temperature
distribution of the thermal wave for slightly IR translucence materials and the radiative heat
flux from which the detected radiometric signal can be derived.
3.1 Equation of energy transfer for absorbing and emitting media
We will consider here a medium that is semitransparent to thermal radiation, as shown
in Figure 3.1, with an absorbing, emitting, and scattering layer of thickness d that is
maintained at a uniform temperature T.
The radiation traveling along a path in the participating medium is attenuated as a
result of absorption and scattering according to Bouguer’s law, and it is enhanced as a result
of emission and incoming scattering along the path, since the radiation emitted in the interior
of a hot, semitransparent body can pass through the medium and finally leave the body
through the boundary surfaces [Oezisik, 1985]. The radiative transfer is usually classified as
being transfer in participating media or transfer between surfaces where opaque surfaces are
considered as participating media in which the radiative emission and absorption are
concentrated in a thin region near the surface of the medium [Brewster, 1992].
Figure 3.1: A schematic of a one-dimensional absorbing-emitting medium
3.1 Equation of energy transfer for absorbing and emitting media 27
The process of absorption, emission, and scattering will be employed to develop a first
order integro-differential equation governing the radiation intensity along a path through a
medium, which is called the equation of transfer. The intensity as described by the equation of
transfer gives the radiation that is locally travelling in a single direction per unit solid angle
and wavelength and is crossing a unit area normal to the traveling direction.
The transfer equation can formally be derived by making an optical energy balance on
a differential element along a single line of sight, as shown in Figure 3.2 [Brewster, 1992].
x∆
T
)(xIλ)( xxI ∆+λ
A∆
∆Ω ∆Ω
x
)(xIλ)( xxIλ
Figure 3.2a: Schematic for the optical energy balance of a differential absorbing and
emitting volume along a single line of sight.
x∆
)( xxI ∆+λ
A∆
∆Ω
∆Ω
),( ΩxIλ
),( Ω′xIλ Ω′∆
x
)( xxIλ),(xIλ
),(xIλ
Figure 3.2b: Schematic for the optical energy balance of a differential absorbing,
emitting volume along a single line of sight with scattering contribution.
28 3 A generalized model of photothermal radiometry
The balance of the optical energy of the differential volume element in x-direction can
be described by
xAxIAxIxAxxI
dxIxA
ATIxAxI
sca
scaB
∆∆Ω∆+∆Ω∆∆+∆Ω∆∆+
=Ω′Ω→Ω′Ω′∆∆Ω∆
+∆Ω∆∆+∆Ω∆ ∫)()()()(
)(),(4
)()()(4
λλλλ
πλλλ
λ
λ
λ
σα
φπ
σε
(3.1)
where the different terms in sequence are the energy in, energy emitted, energy scattered
into Ω -direction, the energy out, energy absorbed, and energy scattered out of Ω -direction,
respectively, and where
)(xI λ is the incident radiant intensity,
)( xxI ∆+λ the intensity leaving the element,
)(TI Bλ the intensity of blackbody radiation at the medium’s temperature T;
A∆ is the projected area of the element normal to the traveling direction ,
∆Ω the solid angle,
x∆ the path length;
)( x∆λε is the emissivity of the element with path length x∆ ,
λσ sca the scattering coefficient, and
)( x∆λα the absorptivity of the element with path length x∆ , respectively.
In order to understand and determine the contribution of scattering into the Ω -direction from
other directions, it is necessary to know the directional distribution of scattered energy. With
reference to Figure 3.2b, the directional distribution of scattered energy is given by the
scattering phase function )( Ω→Ω′φ , which is defined as.
isotropicis scatteringif intoscatteredEnergydirection-intofromscatteredEnergy
lim)( 0 ΩΩΩ′
=Ω→Ω′ →∆xφ (3.2)
For isotropic scattering 1=φ , the energy scattered into the Ω direction from all incoming
Ω′ - directions is given by
∫ Ω′Ω→Ω′Ω′∆∆∆Ω
πλ φσ
π λ
4
)(),(4
dxIAxsca (3.3)
Assuming that the medium is in local thermodynamic equilibrium, T = constant, and that
Kirchhoff’s law is valid [Oezisik, 1995],
])(exp[1)()( xxx ∆−−=∆=∆ λβαε λλ (3.4)
3.1 Equation of energy transfer for absorbing and emitting media 29
where ]exp[ x∆−β is the transmissivity of the participating medium. For a small path length
x∆ the argument of the exponential term in equ. (3.4) is small and the exponential can be
linearized by a Taylor series to give [Brewster, 1992]
........)()()( +∆=∆=∆ xxx λβαε λλ (3.5)
where )(λβ is the absorption coefficient, describing the attenuation of radiation intensity.
)()()(
xIdx
xdIλ
λ λβ−= (3.6)
Based on this equation an alternative definition of the absorption coefficient can be given,
xVV
x ∆∆∆
= →∆ )(lim)( 0 onincidentEnergy
inabsorbedEnergyλβ (3.7)
namely by the fraction of energy absorbed in a small volume element V∆ of length x∆
divided by the energy incident on V∆ .
The scattering coefficient can be written in analogous way as
xVV
xsca ∆∆∆
= →∆ )(lim)( 0 onincidentEnergy
ofoutscatteredEnergyλσ (3.8)
Substituting equ. (3.4) and (3.5) into (3.1), dividing by xA ∆∆Ω∆ , and taking the limit
0→∆x , equ. (3.1) can be written as
∫ Ω′Ω→Ω′Ω′+
−+−=∆
−∆+→∆
πλ
λλλλ
φπ
λσ
λσλβλβλ
4
0
)(),(4
)(
)()()()()()()()(
lim
dxI
xITIxIx
xIxxI
sca
scaBx
(3.9)
The left hand side of equ. (3.9) is the differentiation of )(xI λ with respect to x and the
resulting equation
∫ Ω′Ω→Ω′Ω′+−−=π
λλλλ φ
πλσ
λσλβλβλ
4
)(),(4
)()()()()()()(
)(dxIxIxITI
dx
xdI scascaB (3.10)
is called the equation of radiative transfer for an absorbing, emitting, and scattering medium,
where dxxdI /)(λ represents the increase in the intensity of radiation per unit length along the
direction of propagation and where the right hand terms in sequence are the emission per unit
volume, the absorption per unit volume, the loss by scattering per unit volume and gain by
scattering per unit volume [Oezisik, 1985]. Thus the increase in intensity is the result of a
balance between the increase due to emission and in-scattering and the attenuation due to
absorption and out-scattering by the medium.
The sum of absorption and scattering coefficient gives the total extinction coefficient by the
30 3 A generalized model of photothermal radiometry
medium according to
)()()( λσλβλβ scae += (3.11)
and the transfer equation becomes
∫ Ω′Ω→Ω′Ω′++−=π
λλλ φ
πλσ
λβλβλ
4
)(),(4
)()()()()(
)(dxITIxI
dx
xdI scaBe (3.12)
The albedo for scattering oΩ , defined as the ratio of the scattering coefficient to the extinction
coefficient, is [Siegel, 1981]:
)(
)()(
λβλσ
λe
scao =Ω (3.13)
For scattering alone the albedo is 1)( =Ω λo , while for absorption alone it is 0)( =Ω λo . By
introducing equ. (3.13) into equ. (3.12), we obtain
[ ] ∫ Ω′Ω→Ω′Ω′Ω+Ω−+−=
πλλ
λ φπλ
λλβ λ
4
)(),(4
)()()(1)(
)(
)(1
dxITIxIdx
xdI oBo
e
(3.14)
The last two terms on the right hand side of equ. (3.14) can be combined to give the source
function ),( Ω′ xI λ defined as
[ ] ∫ Ω′Ω→Ω′Ω′Ω+Ω−=Ω′
πλλ φ
πλ
λλ
4
)(),(4
)()()(1),( dxITIxI o
Bo (3.15)
This is the source of intensity along the optical path due to both emission and incoming
scattering. For anisotropic scattering, ),( Ω′ xI λ is a function of Ω and the equation of
transfer then becomes
),()()(
)(1
Ω′=+ xIxIdx
xdI
eλλ
λ
λβ (3.16)
This is an integro-differential equation, since )(xI λ is within the integral of the source
function on the right hand side. For isotropic scattering the phase function φ in equ. (3.15)
becomes equal to unity, and the source function ),( Ω′ xI λ reduces to
[ ] ∫ ΩΩΩ
+Ω−=Ω′
πλλ π
λλ
λ
4
),(4
)()()(1),( dxITIxI o
Bo (3.17)
If scattering is independent of the incidence angle the source function reduces to
[ ] )()()()(1)( xITIxI oBo λλ λλλ
Ω+Ω−=′ (3.18a)
If scattering can be neglected, for 0)( →Ω λo , the source function reduces to
)()( TIxI Bλλ =′ (3.18b)
3.2 Radiative transfer 31
and equ. (3.16) becomes
)()()()()(
xITIdx
xdIB λ
λ λβλβλ
−= (3.19)
The solution of equ. (3.19) for a homogeneous, isothermal medium is obtained by
using an integrating factor. Multiplying through with )exp( xβ gives
xB
xx exTIxIexd
xdIe )()()( ))(()()()(
)( λβλ
λβλλβλ
λβλβ ′=′+′′
(3.20)
and integrating over an element thickness from x=0 to x=d gives
∫ ′′+= ′−−−x
xxB
x xdexTIeIxI0
))(()( ))(()()0()( λβλβλλ λ
λβ (3.21)
where )0(λI is the intensity entering the medium at the boundary surface x=0. Equ. (3.21) is
interpreted physically as the intensity being composed of two terms at point x. The first is the
attenuated incident radiation arriving at x, and the second is the intensity at x resulting from
emission by all thickness elements along the path, reduced by exponential attenuation
between each point of emission x′ and the location x.
3.2 Radiative transfer
Thermal radiation that is absorbed in the interior of the sample will also heat the
sample. If Rq is the total thermal radiation flux in the sample, that can be calculated from the
solution of radiative transfer equation, the power generated by this absorption per unit volume
is given by Rq∇− [Tom, O’Hara and Benin, 1982].
The solution of the radiative transfer equation can be used to obtain the intensity
distribution for a plane layer as shown in Figure 3.3.
The arbitrary two paths S at position x in Fig. 3.3 denote the directions of the spectral
intensities of the thermal radiation, which are represented at the angles θ as shown in Figure
3.3, where θcos/xS = . It will be convenient to adopt a new notation here. The prime
denoting a directional quantity will be replaced by + or -, depending on the directions with
positive or negative θcos , respectively. That means
)(xI +λ corresponds to ≤≤ 10 θ 90° and
)(xI −λ corresponds to 90° ≤≤ 2θ 180°.
where )(xI +λ refers to the radiation incident on the detector on the left hand side of the
sample, and )(xI −λ is the radiation leaving the sample to the right hand side.
32 3 A generalized model of photothermal radiometry
Using these quantities )(xI +λ and )(xI −
λ , the equation of transfer equ. (2.19) becomes
)),(()(),()(),(
cos θλβθλβθ
θλλ
λ xTIxIdx
xdIB=+ +
+
(3.22a)
)),(()(),()(),(
cos θλβθλβθ
θλλ
λ xTIxIdx
xdIB=+ −
−
(3.22b)
A convenient substitution is θµ cos= , then equ. (3.22a) and (3.22b) become
)),(()(),()(),(
µλβµλβµ
µλλ
λ xTIxIdx
xdIB=+ +
+
(3.23a)
)),(()(),()(),(
µλβµλβµ
µλλ
λ xTIxIdx
xdIB=+ −
−
(3.23b)
0=x dx =
xd ′
x′ x
ΩdSr
1θ2θ
),( 1θλ xI + ),( 2θλ xI −
0x dx
xd
x
dS
1
2
),( 1θλ xI + ),( 2θλ xI −
Figure 3.3: Schematic of a solid of finite thickness d with the radiative heat fluxes at
a position x emitted to the right and the left hand side.
By using an integrating factor like in equ. (3.9), equ. (3.24) can be integrated according to the
boundary conditions
),0(),( µµ λλ++ = IxI at x=0 (3.24a)
),(),( µµ λλ dIxI −− = at x=d (3.24b)
3.2 Radiative transfer 33
The solution of the transfer equation is then
∫ ′′+=′−−−
++x xx
B
x
xdexTIeIxI0
)()()(
)),(()(
),0(),( µλβ
µλβ
λλ µµλβ
µµλ
(3.25a)
∫ ′′+=−′−−−
−−d
x
xx
B
xd
xdexTIedIxI)(
)()(
)(
)),(()(
),(),( µλβ
µλβ
λλ µµλβ
µµλ
(3.25b)
The net radiative heat flux )(xqR can be calculated by integrating these intensity
distributions. Considering to the directions ≤≤ 10 θ 90° and 90° ≤≤ 2θ 180°.
)]()([)( xqxqxqR−+ −= λλ (3.26)
where
∫ ∫∞
≥++ Ω+=
00cos
cos)cos,()(θ λλ θθλ dxIdxq (3.27a)
∫ ∫∞
≥−− Ω−=
00cos
cos)cos,()(θ λλ θθλ dxIdxq (3.27b)
We have to make additional approximations, which allow the wavelength and angular
integration in equ. (3.27a) and (3.27b) to be done explicitly. First, we assume according to the
“gray body” approximation of radiative transfer [Siegel, 1981], that the absorption coefficient
)(λβ is independent of wavelength over the relevant wavelength interval of thermal
radiation. Thus βλβ =)( , where β is a constant absorption coefficient characteristic for the
whole spectrum of thermal radiation. After this simplification we obtain
∫ ∫ +++ == 2
0
1
0),(2sin2cos),()(
π
λλλ µµµπθθπθθ dxIdxIxq (3.28a)
∫ ∫ −−− −=−=π
π λλλ µµµπθθπθθ2/
1
0),(2sin2cos),()( dxIdxIxq (3.28b)
and the net flux in the positive x-direction can be written as
[ ]
−= ∫ −+
1
0
),(),(2)( µµµµπ λλ dxIxIxqR (3.29)
The intensities are here substituted by equ. (3.25a) and (3.25b) to yield
′−′−′′+
−−
=
∫ ∫∫ ∫
∫ ∫−′
−′−
−
−−
−−
+
1
0
)(1
0
)(
0
1
0
1
0
)(
)),(()),((
),(),0((
2)(
µβµµβµ
µµµµµµ
πµ
βµ
β
µβ
λµβ
λ
λλdxdexTIdxdexTI
dedIdeI
xqxxd
x
B
xxx
B
xdx
R (3.30)
34 3 A generalized model of photothermal radiometry
The power per unit volume generated by absorption of thermal radiation can be written as the
negative of the divergence of the radiant flux vector )(xqR
−−−
′−′+
′′+
−+
=∇−
∫ ∫
∫ ∫
∫ ∫
∫ ∫
−′−
′−−
−−
−−
+
1
0
1
0
1
0
2)(
1
0
2)(
0
1
0
1
0
)(
)),(()),((
)),((
)),((
),(),0((
2)(
µβµµβµ
µµ
βµ
µµ
βµ
µβµµβµ
π
λλ
λ
λ
µβ
µβ
µβ
λµβ
λ
dxTIdxTI
dxdexTI
dxdexTI
dedIdeI
xq
BB
xxd
x
B
xxx
B
xdx
R (3.31)
For diffuse boundaries, the boundary distributions ),0( µλ+I and ),( µλ −− dI do not depend on
the incidence angle, this means they are independent of µ and can be expressed in terms of
the outgoing diffuse fluxes [Siegel, 1981]
)(1
)()0(1
)0( dqdIqI λλλλ ππ== −+ (3.32)
Typically for photothermal radiometry, the temperature )(xT is always close to the ambient
temperature oT , so that the thermal emission spectrum is close to the blackbody emission
)( oB TIλ
. If the absorbing medium is in radiative equilibrium, which means that the total
energy emitted from volume dV is equal to the total absorbed energy, we can write the
source function as
)(1
))(( 4 xTxTI SBB σεπλ
= (3.33)
After substituting the approximations (3.32) and (3.33) into the previous equ. (3.30) and
(3.31), the net radiative heat flux and the deposited power radiation can be written as
∫
∫
′−′′−
′′−′+−−= −+
d
x
SB
x
SBR
xdxxExT
xdxxExTxdEdqxEqxq
))(()(2
))(()(2))(()(2)()0(2)(
24
0
24
33
βσεβ
βσεβββ
(3.34)
∫∫ ′−′′−′′−′+
−−+=∇− −+
d
xSB
x
SB
SBR
xdxxExTxdxxExT
xTxdEdqxEqxq
))(()(2))(()(2
)(4))(()(2)()0(2)(
142
01
42
433
βσεββσεβ
σεβββββ (3.35)
3.3 Energy conservation, including conduction and radiation 35
where )(xEn denotes the exponential integral [Abramowitz, 1965]
∫−
−=1
0
2 exp)( µµ µ dxEx
nn (3.36)
3.3 Energy conservation, including conduction and radiation
The general energy balance on a volume element in a medium states that the rate of
change of thermal energy stored within the volume is equal to the sum of the net conduction
heat rate into a unit volume, the internal heat sources due to the re-absorption of thermal
radiation and the volumetric rate of thermal energy generation. The internal heat sources can
be written as the negative of the divergence of radiative heat flux )(xqR . The net heat
conduction into a volume element can also be written as the negative of the divergence of
conductive heat flux cF . Thus the energy equation can be written as
QqFtT
c Rc +∇−∇−=∂∂
)(ρ (3.37a)
where TkFc ∇−= .
The energy equation can be utilized by adding Rq− to Tk ∇ to yield [Siegel, 1981].
QqTktT
c R +−∇∇=∂∂
)(ρ (3.37b)
3.3.1 Heat diffusion equation for a solid of finite thickness
To discuss the influence of the variable in the problem of a medium interacting with
radiation, it is convenient to consider a simple geometry. A plane layer of finite thickness is
used in which the temperature and properties of the medium vary only along the x-axis.
To proceed further and enable to advance to analytical solutions for the heat diffusion
equation with appropriate boundary conditions, we shall make additional approximations
which are coherent with our experimental conditions and allow the integrations in equ. (3.34)
and (3.35) to be done explicitly. First )0(+q and )(dq − in equ. (3.34) and (3.35) are the
radiative heat fluxes entering the sample at each surface from the surrounding. In chapter 2,
we have seen that the temperature distribution of thermal waves is in general independent of
the radiative heat fluxes that are emitted by the surroundings. Therefore, we will neglect here
the first two terms of equ. (3.35). The second simplification is that the diameter of the heating
spot on the sample surface is large in comparison with the thermal diffusion length and
36 3 A generalized model of photothermal radiometry
detection area of the detector, so that we can work with a one-dimensional heat diffusion
equation. This means that in first approximations the angle θ can be set to zero. After these
simplifications we can rewrite equ (3.34) and (3.35) in the form
∫∫ ′′−′′= −′−′−−d
x
xxSB
xxx
SBR xdxTxdxTxq )(4
0
)(4 exp)(exp)()( ββ σεβσεβ (3.38)
∫
∫
′′+
′′+−=∇−
−′−
′−−
d
x
xxSB
xxx
SBSBR
xdxT
xdxTxTxq
)(42
0
)(424
exp)(
exp)()(2)(
β
β
σεβ
εσβσβε
(3.39)
)(xqR is the radiative heat flux due to the medium at temperature )(xT and the corresponding
divergence )(xqR∇− characterizes the internal heat sources due to the re-absorption of
thermal radiation.
It is worthy to mention that the exponential integrals in equ. (3.35) and equ. (3.35) can
be solved and that the radiation collected within the solid angle θ can be considered as long
as the temperature distribution in the solid is one-dimensional. To achieve this, the
exponential integrals can be approximated by exponential functions which can then be
integrated analytically [Haj-Daoud, Bein, Dietzel and Pelzl, 2000, to be published]. In reality
the problem is much more complicated, since in the general case the temperature distribution
is two- or three-dimensional, and in this case an analytical solution seems to be impossible.
In the following, we write the heat diffusion equation and the boundary conditions for
a medium with finite thickness and consider only one-dimensional geometry as shown in
Figure 3.4.
By substituting equ. (3.39) into the heat diffusion equation (3.37) , we obtain
∫
∫
′′+
′′+−∂
∂=
∂∂
−′−
′−−
d
x
xxSB
xxx
SBSB
xdtxT
xdtxTxTx
txTk
ttxT
c
)(42
0
)(4242
2
exp),(
exp),()(2),(),(
β
β
σεβ
εσβσβερ
(3.40)
The additional terms in the heat diffusion equ. (3.40) appear when the internal heat sources
due to re-absorption of thermal radiation are considered, especially for slightly IR translucent
materials with a finite IR absorption coefficient β . It is worthwhile to mention that the
radiation transfer plays an important role especially if the medium is heated to high average
sample temperatures or has got a larger porosity allowing heat exchange through the medium
by radiation in addition to conduction heat transfer. The radiative transfer term vanishes for
materials with very high absorption coefficients in the IR-spectrum so that the emitted
3.3 Energy conservation, including conduction and radiation 37
radiation can only come from the surface, and the radiation emitted in the interior of the
medium is completely re-absorbed inside the opaque medium. Mathematically it can be
shown that for large values of β the radiation terms in equ. (3.40) vanish and equ. (3.40)
reduces to the pure heat conduction equ. (2.35). The additional terms in the heat diffusion
equation (3.40) including radiative contributions
oI
)(xqR
01 =gx11 Lxg = 03 =gx 33 Lxg =
),( txF ss),( txF gg
0=sx dxs =
sx1gx 3gx
Gas Solid
oI
)(xqR
Gas
01gx11 Lxg
03gx 33 Lxg
),( txF ss),( txF gg
0sx dxs
sx1gx 3gx
Figure 3.4: Solid of finite thickness, including a schematic representation of the modes
of heat transfers across the gas-solid interfaces, namely conduction, thermal
radiation and illumination in the visible spectrum.
∫∫ ′′−′′− −′−′−−d
x
xxSB
xxx
SBSB xdtxTxdtxTxT )(42
0
)(424 exp),(exp),()(2 ββ σεβεσβσβε (3.41)
The variation of the exponential term in equ. (3.41) has a shorter length scale than the
variation of ),(4 txT , then equ. 3.41 can be simplified
∫∫ ′−′− −′−′−−d
x
xxSB
xxx
SBSB xdtxTxdtxTxT )(42
0
)(424 exp),(exp),()(2 ββ εσβεσβσβε (3.42a)
By intergrating, we obtain
[ ] [ ])(444 exp1),(exp1),()(2 xdSB
xSBSB txTtxTxT −−− −−−− ββ εσβεσβσβε (3.42b)
For large values of the absorption coefficient, the terms in equ. (3.42b) canceled each other.
In equ. (3.40) we omitted the external heat source which has been introduced in the boundary
conditions. To solve the equation (3.40), appropriate boundary conditions are needed, which
are:
38 3 A generalized model of photothermal radiometry
1. At the outer boundaries, 01 =gx and ∞=3gx , the temperature must be finite:
ogg TtTx == ),0(:0 11 (3.43a)
ogg TtlTLx == ),(: 3333 (3.43b)
2. Continuity of the temperature at the gas / solid interfaces:
),0(),(:0 11,11 tTtLTxLx sgsg === (3.43c)
),0(),(:0, 33 tTtdTxdx gsgs === (3.43d)
To fulfill the conditions (3.43c) and (3.43d), the arguments already used for equ. (2.39) in
chapter 2 are applied here, which means the modulation frequencies of heating should not be
too high.
3. Continuity of the heat flux at the gas / solid interfaces:
0,11 ),(),(:011
== =+==sg xssoLxggsg txFQtxFxLx (3.44a)
03, 3),(),(:0 == ===
gs xggdxgsgs txFtxFxdx (3.44b)
where ),( txF are the conduction heat flux. The medium considered here is totally opaque for
the visible light corresponding to an absorption coefficient of ∞→oβ , so that the modulated
laser intensity applied to the surface is dissipated by the conduction into the solid. From
chapter 2, )1(2/ tioo eIQ ωη += and xtxTktxF ∂∂−= /),(),( . Equ. (3.44a) and (3.44b)
become
01
1 ),()1(
2
),(
11==
∂∂
−=++∂
∂−
sgxs
sss
tio
Lxg
ggg x
txTke
I
x
txTk ωη
(3.45a)
dxs
sss
xg
ggg
sg
x
txTk
x
txTk
==∂
∂−=
∂
∂−
),(),(
03
3
3
(3.45b)
In contrast to equ. (2.40), the radiative heat fluxes across the interfaces cannot appear
explicitly in equ. (3.44a) and (3.44b). The reason is that the net radiative heat fluxes across
the interfaces can only be considered once in the calculation, and this has already been done
by introducing the radiative heat fluxes in the heat diffusion equation. To prove this, the heat
diffusion equation (equ. 3.37b)
)),(),((),(
txqtxTkt
txTc R−∇∇=
∂∂
ρ (3.46)
3.3 Energy conservation, including conduction and radiation 39
will be discussed at the interfaces, 0=x and dx = . The energy balance is applied to the
differential volume element of infinitesimal thickness, so that the rate of change of the
thermal energy stored within the volume element, namely tTc ∂∂ /ρ , is very small and can be
neglected. Equ. (3.46) then becomes
0)),(),(( =−∇∇ txqtxTk R (3.47)
By integrating equ. (3.47), we obtain
0),(),( =−∇ txqtxTk R (3.48)
after that substituting equ. (3.38) into equ. (3.48), we finally get
0exp),(exp),(),( )(4
0
)(4 =′′+′′−∂
∂∫∫ −′−′−−d
x
xxSB
xxx
SB xdtxTxdtxTx
txTk ββ σεβσεβ (3.49)
If we consider a relatively opaque solid with respect to thermal radiation, the temperature
variation ),( txT is slower in comparison to the variation of the exponential function and we
can put the term ),( txT in front of the integral, which allows to solve the integral explicitly.
After this simplification, we obtain
0exp),(exp),(),( )(4
0
)(4 =′+′−∂
∂∫∫ −′−′−−d
x
xxSB
xxx
SB xdtxTxdtxTx
txTk ββ βσεβσε (3.50)
and at the interface 0=x , we obtain
0exp),(),(
0
4
0
=′+∂
∂∫ ′−
=
dx
SBx
xdtxTx
txTk ββσε (3.51)
Solving the integral for large values of β , we obtain the first boundary condition
0),(),( 4
0
=′+∂
∂
=
txTx
txTk SB
x
σε (3.52)
and at the interface dx = , we obtain the other boundary condition
0),(),( 4 =′−
∂∂
=
txTx
txTk SB
dx
σε (3.53)
so that the boundary conditions at the interfaces for the continuity of the radiative and
conductive heat fluxes are obtained from the heat diffusion equation.
40 3 A generalized model of photothermal radiometry
3.3.2 Heat diffusion equation for a thermal wave
The solution for the temperature distribution in the gas and solid region can be found
by using the ansatz
),()(),( ,,,,,, txTxTtxT sgsgsgsgsgsg δ+= (3.54)
which allows to separate the stationary temperature distribution from the time dependent
problem. By suppressing the higher order terms ),...,(,),( 32 txTtxT ssss δδ , which in principle
can result from the radiative contributions and by considering only
),()(4)(),( 344 txTxTxTtxT sssssss δ+= (3.55)
we obtain
′′−
′′−
=∂
∂
∫
∫∞
−′−
′−−
s
ss
s
ss
x
sxx
ssSBs
x
sxx
ssSBsssSBs
s
sss
xdxT
xdxTxT
x
xTk
)(42
0
)(424
2
2
exp)(
exp)()(2)(
β
β
σεβ
σεβσβε
(3.56)
for the stationary problem and
′′−
′′−
−∂
∂=
∂∂
∫
∫∞
−′−
′−−
s
ss
sss
xs
xxss
x
sxx
ssss
sss
sss
ssss
xdtxT
xdtxTtxT
xTax
txTk
t
txTc
)(2
0
)(2
2
2
exp),(
exp),(),(2
))((),(),(
β
β
δβ
δβδβδδ
ρ
(3.57)
for the time-dependent first order thermal wave.
In equ. (3.57) the term
3)(4))(( ssSBsss xTxTa σε= (3.58)
contain the information about the stationary temperature distribution in the sample, which in
principle can be determined by solving the integro-differential equation (3.56), In the context
of this work we are not interested in this sufficiently complex mathematical problem, since
the modulated photothermal radiometry used for our measurements normally relies on the
measurements of thermal waves of small amplitudes (µK, mK, up to only a few K) and since
steady state temperature gradient near the surface heated by the modulated laser beam can be
normally neglected.
3.3 Energy conservation, including conduction and radiation 41
The boundary conditions (3.43a)-(3.44b) reformulate for the time dependent thermal wave
are
0:
),0(),(:0,
),0(),(:0
0),0(:0
333
33
11,11
11
==
===
===
==
gg
gsgs
sgsg
gg
TLx
tTtdTxdx
tTtLTxLx
tTx
δ
δδ
δδ
δ
(3.59)
dxs
sss
xg
ggggs
xs
sss
tio
Lxg
gggsg
sg
sg
x
txTk
x
txTkxdx
x
txTke
I
x
txTkxLx
==
==
∂∂
−=∂
∂−==
∂∂
−=+∂
∂−==
),(),(:0,
),(
2
),(:0
03
33
01
1,11
3
11
δδ
δηδ ω
(3.60)
The time dependent solution for the gas region calculated from, compare equ. (2.47) in
chapter2,
2
2 ),(),(
g
ggg
gg
x
txT
t
txT
∂
∂=
∂
∂ δα
δ (3.61)
The ansatz for the complex solution of the thermal wave can be written in a similar
way as in equ. (2.49). Taking into consideration all terms appearing in the inhomogeneous
heat diffusion equation (3.57), under inclusion of the internal heat source )(xqR∇− due to the
re-absorption of thermal radiation, we have to add two further terms in the complex ansatz,
responsible for the radiative transfer through the medium.
tixs
xs
xs
xsss eeEeDeBeAtxT sssssnssns ωββσσδ ][),( +++= −− (3.62)
For the gas region, the ansatz in comparison to equ. (2.51) remains unchanged
tixg
xggg eeBeAtxT gggg ωσσδ ][),( −+= (3.63)
The complex ansatz (3.62) can be divided in two parts
tixs
xs
tixs
xs
ssso
sss
eeEeDeeBeA
txTtxTtxTsssssnssns ωββωσσ
δδδ
][][
),(),(),( )1()(
+++=
+=−− (3.64)
where ),()( txT so
sδ is the classical form for the thermal wave (equ. 2.49), which will be
denoted as unperturbated solution. The second term ),()1( txT ssδ , which is related to the
distribution of the internal heat sources due to the re-absorption of thermal radiation, will be
42 3 A generalized model of photothermal radiometry
regarded as a small perturbation of the temperature distribution of the thermal waves. In the
term of the internal heat sources on the right hand side of equ. (3.57), which describe the
interaction between the radiation field and temperature field, we consider only the ansatz for
not perturbed solution ),()( txT so
sδ of the thermal wave. By introducing this simplification,
the integration on equ. (3.57) can be done analytically.
The integration constants As, Bs, Ds and Es can be determined from the thermal
diffusion equation and the subjected boundary conditions. Substituting the complex solution
equ. (3.56) in equ. (3.47) and evaluating the integration’s, we obtain
[ ]
[ ]0exp
exp))((
exp))((
exp))(())((
exp))(())((
))((2
exp))(())((
))((2
2
)(2
)(2
222
222
222
=
−+
++
−+
−+−
++
+
+−
−−+−+
+−
−−+−
+−−−
−
−
ss
nssnss
ss
sns
sns
x
sssss
nss
ssss
d
nss
ssss
xsssss
nss
ssss
nss
ssss
x
nss
sss
nss
ssssssnsssss
x
nss
sss
nss
ssssssnsssss
kciE
xTaB
xTaA
kciDxTa
BxTa
A
xTaxTaxTakciB
xTaxTaxTakciA
βσβσβ
β
σ
σ
βωρ
σββ
σββ
βωρσβ
βσβ
β
σββ
σββ
βσωρ
σββ
σββ
βσωρ
(3.65)
From the first two terms of equ. (3.65), which correspond to the homogeneous equation, the
complex quantity nsσ can be determined
2222
4212121
−
+
+±
+
+±=
s
s
s
sns
s
s
s
sns
s
s
s
ns RRβσ
βσ
βσ
βσ
βσ
βσ
(3.66)
where the quantity
ss
fi
απ
σ )1( += (3.67)
is the solution of the dispersion relation of the classical thermal wave (chapter 2) and where
the parameter
nss
sSBs
nss
ssns k
xT
k
xTaR
σσε
σ
3)(4))((== (3.68)
is a new radiation-to-conduction parameter, which is different from the former parameter
sssSBss kxTR σσε /)(4 3= existing in chapter 2 only in the new definition of nsσ . The new
3.3 Energy conservation, including conduction and radiation 43
complex solution nsσ , which is a function of two parameters ss βσ / and nsR and which is
affected both by conductive and radiative heat transfer ( nss R,β ) can be rearranged as
22222
421212
−
+
+±
+
+=
s
s
s
sns
s
s
s
sns
s
s
s
ns RRβσ
βσ
βσ
βσ
βσ
βσ
(3.69)
By considering the limit for slightly IR translucent sample, 1)/( <<ss βσ , and by using the
relation for small value, [ ] zz 211 2 +=+ , we can write the terms in the first part under the
square root as
+
+=
+
+
s
sns
s
s
s
sns
s
s RRβσ
βσ
βσ
βσ
42121222
(3.70)
substituting equ.(3.70) into (3.69)
+
−±
+
+=
s
sns
s
s
s
sns
s
s
s
ns RRβσ
βσ
βσ
βσ
βσ
421212222
(3.72)
using these relation also for small value, [ ] )2/(11 2/1 zz −=− , we can write
+
−=
+
−
s
sns
s
s
s
sns
s
s RRβσ
βσ
βσ
βσ
2142122
(3.73)
substituting in equ. (3.72), we obtain
+
−±
+
+=
s
sns
s
s
s
sns
s
s
s
ns RRβσ
βσ
βσ
βσ
βσ
21212222
(3.74)
we can obtain two limit solutions nsσ and from equ. (3.74). By taking the plus sign we get
+≈
s
snssns R
βσ
βσ 21 (3.75)
and by taking the minus sign in equ. (3.74), we get
sns σσ ≈ (3.76)
The value )/( ssnsR βσ of equ. (3.75) in general is small in comparison to unity, and the
resulting solution of sns βσ ≈ is real and cannot describe a thermal wave-type solution
required by the used ansatz (3.62). Thus we can conclude that equ. (3.75) has got no physical
meaning and that the plus sign in equ. (3.69) can be neglected. From equ. (3.76), on the other
hand, we can conclude; that the new complex solution nsσ lies in the neighborhood of the
former complex solution sσ , and that this is a typical thermal wave solution required for the
44 3 A generalized model of photothermal radiometry
case of modulated heating. Consequently the minus sign of equ. (3.69) has to be chosen to
calculate physically realistic numerical solutions of equ. (3.69).
The remaining two terms of equ. (3.65) are fulfilled by setting the coefficients of ss xeβ
and ss xe β− equal to zero. The resulting equations
[ ] 0))((
.))(( 2
22
=−+
−+
+ sssss
nss
ssss
nss
ssss kciD
xTaB
xTaA βωρ
σββ
σββ
(3.77)
[ ] 0exp))((
exp))(( 2)(
2)(
2
=−+
+
+
−
+−−−sssss
nss
ssss
d
nss
ssss kciE
xTaB
xTaA nssnss βωρ
σββ
σββ σβσβ (3.78)
have to solved together with the boundary conditions equ. (3.59) and (3.60). From the
boundary conditions we get
ns
o
ns
ss
ns
ssss k
IGEGDGBGA
ση
σβ
σβ
2)1()1( =
−−
++++−− (3.79)
0)1()1( )()(2 =
+−
−+−++− −+−− d
ns
ss
d
ns
ss
dss
nssnssns eGEeGDeGBGA σβσβσ
σβ
σβ
(3.80)
where the quantity
nss
gg
k
kG
σ
σ= (3.81)
is expected to be small and similarly to chapter 2 will be neglected in the future calculations.
From equ. (3.77), (3.78), and the simplified equations (3.79) and (3.80), the four integration
constants sA , sB , sD and sE for the solid region are calculated as
ns
d
s
nsns
s
ns
s
nsns
d
s
nsns
s
ns
s
nsns
nss
os H
e
R
R
e
R
R
k
IA
nssns )(
322
2
322
21
1
21
1
1
2
σβσ
βσ
βσ
βσ
βσ
βσ
βσ
ση
+−−
−
−
+
+
−
−
−
−
=
(3.82)
3.3 Energy conservation, including conduction and radiation 45
ns
d
s
nsns
s
ns
s
nss
s
nsns
s
ns
s
nsns
nss
os H
e
R
R
R
R
k
IB
nss )(
322322
21
1
21
1
1
2
σβ
βσ
βσ
βσ
βσ
βσ
βσ
ση
+−
−
−
−
−
−
−
+
+
=
(3.83)
where the constant nsH is given by
−
−
+
−
−
−
−
+
−
−
−
−
−
−
−−
−
−
+
+
=
−
+−+−
−
d
s
nsns
s
ns
s
nss
d
s
nsns
s
ns
s
nsns
d
s
nsns
s
ns
ns
d
s
nsns
s
ns
s
nsns
s
nsns
s
ns
s
nsns
ns
s
nssnss
ns
e
R
R
e
R
R
e
R
R
e
R
R
R
R
H
β
σβσβ
σ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
2
2322
2
2
)(2
2322
2
2
)(
322
2
2
322
2
322
21
1
21
1
21
4
21
1
1
21
1
1
(3.84)
nss
nsns
s
ns
d
s
nsns
s
ns
s
nsns
s
ns
s
nsns
s
ns
s
nsns
s
ns
s
nsns
nsss
HR
e
R
R
R
R
Rk
ID
ns
−
−
−
−
−
−−+
−
−
+
++
=
−
322
2
322
2
322
2
0
21
21
1
1
21
1
1
2
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
ση
σ
(3.85)
46 3 A generalized model of photothermal radiometry
nss
nsns
s
ns
s
nsns
s
ns
d
s
nsns
s
nsns
s
ns
d
s
nsnsd
s
nsns
sss
HR
R
eR
R
eR
e
Rk
IE
ss
s
ns
ss
ns
−
−
−
−
+
+
−
−
−
−
=
−+−
+−
322
322
2
2
322
)1(22
)1(
21
21
1
21
1
2
2
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
ση
ββ
βσ
ββ
σ
(3.86)
The general solution of the temperature distribution of the thermal wave in the solid of finite
thickness can be written as
tixs
xs
xs
xsss eeEeDeBeAtxT ssssnssns ωββσσδ ][),( −−− +++= (3.87)
where the integration constants are given by equ. (3.83), (3.84), (3.85) and (3.86).
If we take the limit ∞→sβ for an IR opaque solid, the integration constants (3.83),
(3.84), (3.85) and (3.86) simplify to
[ ][ ] [ ]
−−+−
=
−
−
dss
ds
ss
os
s
s
eRR
eR
k
IA
σ
σ
ση
222
2
11
1
2 (3.88)
[ ][ ] [ ]
−−++
=
− dss
s
ss
os
seRR
Rk
IB
σση
222 11
12
(3.89)
0=sD (3.90)
0=sE (3.91)
where the new conduction to radiation parameter reduce to the former parameter sR (see equ.
(3.68) and (3.76). The integration constants for the additional terms of the thermal wave
solution in equ. (3.62) vanish, which means that the conduction heat transfer is only
considered. The temperature distribution of the thermal wave in this case.
( )tix
d
s
s
xd
s
s
sss
osss ee
eR
R
eR
R
Rk
ItxT ss
s
ss
ωσ
σ
σ
ση
δ −
−
−−
+−
−
+−
+
+=
2
2
)(2
1
11
1
11
11
2),( (3.92)
corresponds to the form already derived by [Gu, 1994].
3.3 Energy conservation, including conduction and radiation 47
Equ. (3.87) has been derived for a solid with finite thickness. For the semi-infinite slightly IR
transparent solid, ∞→d , the temperature distribution of the thermal wave is obtained as
tix
s
nsns
s
ns
s
nsns
x
s
nsns
s
ns
s
ns
s
nsns
nss
oss ee
R
R
e
R
R
k
ItxT sns
ss
nss
ωσ
βσ
β
βσ
βσ
βσ
βσ
βσ
βσ
βσ
ση
δ −
−−
−
−
+
+
−
−
+
+
=
322
1
322
21
1
1
21
1
1
2),( (3.93)
Equ. (3.93) shows that the temperature distribution of the thermal wave depends on several
parameters (compare chapter 5 and 6), that affect the behavior of the thermal wave in solids
according to its thermal and optical properties.
In the limit of an opaque semi-infinite solid we obtain the solution
tix
ss
oss
sek
ItxT ωσ
ση
δ +−=2
),( (3.94)
already derived by [Bein and Pelzl, 1989].
To study the influence of the various parameters that affect the thermal wave, Figure
3.5 shows the thermal wave, equ. (3.93) both the amplitudes and the phases, in a semi-infinite
solid as a function of depth below the surface. The important aspect to be notice that we can
see that at higher value of the effective IR absorption constant sβ , the exponentially but
oscillatory decay of the temperature amplitude and the linear phase variation with sample
depth should be noted, at the intermediate value of sβ , we have transition phase both in the
temperature amplitude and the phase lag. At low value of sβ , the typical behavior for the
thermal wave is vanished and there is no variation in the phase lag, which means that the
material is too transparent in IR spectrum and the thermal wave traveled without variation in
the phase lag. From this figure we can see the influence of radiative transfer by comparing its
with Figure 2.7 in chapter 2.
48 3 A generalized model of photothermal radiometry
0 2 4 6 8 10-2
0
2
4
6
8
10
12
βs1<βs2
<βs3
phas
e / a
.u.
( x / a.u.)
δT (
x) /
a.u.
( x / a.u.)0 2 4 6 8 10
-240
-120
0
120
240
360
βs3
βs2
βs1
∞
βs3
βs2
βs1
βs
Figure 3.5: Temperature distribution of the thermal wave and phase shift in a semi-infinite
solid for different IR absorption constants ∞→<<< 4321 ssss ββββ at a constant
value of the effusivity ckρ .
3.4 Derivation of the radiometric signal
We shall derive the radiometric signal measured by the photoconductive detector
(MCT) placed in front of the sample, as shown in Figure 3.1. The radiative heat flux leaving
the sample towards the direction of the detector is ))(( xTqR (equ. 3.38).
∫∫ ′′−′′= −′−′−−d
x
sxx
sSBss
x
sxx
sSBsssR
s
sss
s
sss xdxTTxdxTTxTq )(4
0
)(4 exp)()(exp)()())(( ββ σεβσεβ (3.95)
To derive the radiometric signal in reflection, where the thermal waves are excited and
detected at the same surface, equ. (3.95) has to be solved for the case 0=sx
∫ ′′−= ′−d
sx
sSBsssR xdxTTxTq ss
0
4 exp)()())(( βσεβ (3.96)
Here, we can omit the negative sign, since only the magnitude of the measured signal is of
interest. In equ. (3.96) the term 4)()( sSBs xTT σε has to be replaced by the black body radiation
))(,(),( 0ss xTWT λλε to account for the effects of the wavelength characteristics of the IR
3.4 Derivation of the radiometric signal 49
optics on the measured signal
∫ ′= −′−d
sxx
ssssR xdxTWTTq sss
0
)(0 exp))(,(),(),( ββλλελ (3.97)
The stationary radiometric signal ),( ss TM λ which can be measured by the detector is
affected by different factors, namely the transmittance of the IR optics system and the spectral
responsivity of the detector, already mentioned in chapter 2 and 4. Following the example of
equ. (2.17), the radiometric signal can be described as
∫ ∫∞
′− ′=d
sx
ssssss xddxTWTRFCTM ss
0 0
0 exp))(,(),()()(),,( λβλλελλλ β (3.98)
In the case of the gray body approximation, in which the emissivity is independent of the
wavelength, equ. (3.98) can be simplified to
∫ ∫∞
′− ′=d
sx
ssssss xddxTWRFTCTM ss
0 0
0 exp))(,()()()(),,( λβλλλελ β (3.99)
By introducing the quantity
4
0
0
0
0
0
0
)(
),()()(
))(,(
))(,()()(
)(sSB
s
s
s xT
dTWRF
dxTW
dxTWRF
Tσ
λλλλ
λλ
λλλλγ
∫
∫
∫∞
∞
∞
== (3.100)
the signal can be transformed into
∫ ′′= ′−d
sx
ssSBssss xdxTTTCTM ss
0
4 exp)()()()( ββσγε (3.101)
which is the stationary radiometric signal that results from the superposition of all infrared
radiation fluxes arising from different depths inside the sample.
The modulated radiometric signal can be described by using the same procedure
mentioned in section 2.1.4, by using equ. (2.24)
)()(
),( fTT
TMTfM s
s
ssss δδ
∂∂
= (3.102)
If the temperature variation of the emissivity is negligible in comparison to the temperature
dependence of Planck’s blackbody radiation, following the example of equ. (2.25), the
modulated radiometric signal can be described as
∫ ′′′= ′−d
ssx
ssssSBssss xdfTxTTTCTfM ss
0
3 )(exp)()()(4),( δβγσεδ β (3.103)
50 3 A generalized model of photothermal radiometry
where the definitions of )( sTγ and )( sTγ ′ are mentioned in chapter 2. If higher average
sample temperatures are considered, where the steady state temperature gradient produced
near the surface by the modulated laser beam can be neglected, equ. (3.103) can be simplified
∫ ′′= ′−d
ssx
sssSBssss xdfTxTTTCTfM ss
0
3 )(exp)()()(4),( δβγσεδ β (3.104)
By introducing the temperature distribution of the thermal wave for the solid of finite
thickness (equ. 3.87) into equ. (3.104), we obtain
⋅⋅+−
⋅+
+
−+
−
−
′=−
+−−−
dEe
D
eB
eA
k
ITxTTCfTM
ss
d
s
s
ns
d
s
s
ns
d
s
nss
ossSBssss
s
nssnss
β
βσ
βσ
ση
γσεδβ
σβσβ
21
1
1
1
1
2)()(4)(),(
2
)()(
3 (3.105)
which is the modulated radiometric signal measured for a solid of finite thickness in
reflection. For a semi-infinite solid, the modulated signal, which can be derived either by
substituting the thermal wave equation (3.93) into equ. (104) or by taking the limit ∞→d of
equ. (3.105), can be described as
ti
s
nsns
s
ns
s
nsns
s
nsns
s
ns
s
ns
s
nsns
s
ns
nss
ossSBssss e
R
R
R
R
k
ITxTTCfTM ω
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
ση
γσεδ
322
322
3
21
1
1
21
121
1
1
2).()(4)(),(
−
−
+
+
−
−
+
++
′= (3.106)
Equ. (3.105) is the radiometric signal, in which we take into account the effects of radiative
heat transport on the temperature distribution of the thermal wave.
If we neglect the effects of radiative heat transport on the temperature distribution of the
thermal wave and substitute the solution obtained by [Gu, 1994], as written in equ. (3.92),
into equ. (3.104), we obtain
d
s
s
s
s
dd
s
s
s
s
d
sss
ossSBssss
s
ss
s
ss
eR
R
ee
R
Re
Rk
ITxTTCfTM
σ
σβσ
σβ
βσ
βσ
ση
γσεδ2
2
)(2
)(
3
1
11
1
11
1
1
1
11
2).()(4)(),(
−
−−−
+−
+−
−
−
−+−
++
−
+′=
(3.107)
3.4 Derivation of the radiometric signal 51
where the former parameters of ss fi απσ /)1( += and sssSBss kxTR σσε /)(4 3= are used
here. In the limit of semi-infinite solid , and considering 1<<sR , we obtain the solution
s
sss
ossSBssss k
ITxTTCfTM
βσσ
ηγσεδ
+′=
1
12
).()(4)(),( 3 (3.108)
already derived by [Dietzel, 1999].
Following the same procedure as for the modulated radiometric signal in reflection,
we can derive the modulated radiometric signal in transmission, where the samples are heated
at the front surface and the thermal response is detected at the rear surface.
The radiative heat flux leaving the sample towards the direction of the detector
(equ. 3.38). is given by
∫∫ ′′−′′= −′−′−−d
x
sxx
sSBss
x
sxx
sSBsssR
s
sss
s
sss xdxTTxdxTTxTq )(4
0
)(4 exp)()(exp)()())(( ββ σεβσεβ (3.109)
To measure the radiometric signal in transmission. The radiometric-measured signal can be
obtained by substituting dxs = in equ. (3.109)
∫ ′′= ′−−d
sxd
sSBsssR xdxTTxTq ss
0
)(4 exp)()())(( βσεβ (3.110)
Following the same procedure and argumentation used in equ. (3.97)-(3.104). The modulated
radiometric signal can be described as
∫ ′′= ′−−d
ssxd
sssSBssss xdfTxTTTCTfM ss
0
)(3 )(exp)()()(4),( δβγσεδ β (3.111)
⋅−
+⋅+
−
−+
+
−
′=−
−
−−−
2
11
2)()(4)(),( 3
dd
sd
ss
s
ns
dd
s
s
ns
dd
s
nss
ossSBssss
ss
s
snssns
eeEedD
eeB
eeA
k
ITxTTCfTM
βββ
βσβσ
β
βσ
βσ
ση
γσεδ (3.112)
which is the modulated radiometric signal for the solid of finite thickness.
53
4. Experimental Setup
In this chapter the experimental setup is described which has been used to excite and
detect the thermal waves. In order to study the thermophysical properties of different solid
materials at high temperature in the range from 300 K to 1000 K, a specially designed high-
temperature cell has been used. The complete measurement system which is schematically
shown in Fig 4.1 consists of: 1- The high temperature vacuum cell to keep the sample at the
desired average temperature and to protect it against oxidation and thermal erosion; 2- The
heating system for the excitation of thermal waves based on a modulated laser beam; 3- The
detection system for the modulated infrared radiation originating from the thermal waves; 4-
Various electronic components (preamplifier, lock-in amplifier, and computer) which are used
to amplify, filter, and register the measured data as function of the modulation frequency of
the thermal waves.
4.1 Excitation and detection of thermal waves
4.1.1 Excitation of thermal waves
The thermal waves have been excited by illuminating the sample surface with the help
of an argon ion laser of a wavelength of λ = 514 nm. An effective beam power of 900 mW
has been applied for the heating spot whose diameter was about 3 mm. This type of laser has
a small IR-contribution in the spectrum [Messing, 1994], which is strongly attenuated by
using two quartz glass prisms, which deflect the laser beam to the sample. Additionally a
spike filter with definite wavelength can be used to prevent direct infrared contributions of the
modulated laser beam to the measured signal. The periodic modulation of the laser beam has
been done by using an acousto-optic modulator. The used acousto-optic modulator relies on
the photoelastic effect [Driscoll, 1978]. The desired periodic reference frequency is supplied
from the lock-in amplifier, which gives a digital signal in the form of square wave voltage.
The voltage is then amplified by an electronic driver, which is used in conjunction with the
acousto-optic modulator. The modulation frequencies can be varied over a frequency range
from 1 mHz to 100 kHz. The modulated laser beam intensity is deflected towards the front
surface of the sample by using two prisms, as shown in Figure 4.1, allowing us to measure “
thermal waves in reflection” (compare chapter 5 and chapter 6), which means the thermal
waves are excited and detected at the same surface. Alternatively the modulated laser beam
54 4 Experimental Setup
can be deflected towards the rear surface of the sample, which enables the measurement of
“thermal waves in transmission” (compare chapter 5). The measurement system setup was
covered also with black cartoon to provide good shielding from laser beam.
Figure 4.1: Schematic of the experimental setup
4.1 Excitation and detection of thermal waves 55
4.1.2 Infrared components and detection
The sample is irradiated periodically by a modulated Ar+ laser beam. The light
absorption at the sample surface causes a periodic temperature variation. This temperature
variation penetrates into the sample producing a temperature field oscillating in time and
space, which is called a thermal wave. The IR optical technique for the detection of the
thermal waves is schematically described in Figure 4.2.
The infrared optics used to detect the IR radiation as illustrated in Figure 4.2 consists
of two BaF2 lenses. The focal length (fL) of the lenses is 14.7 cm, the diameter φ is 10 cm,
and the thickness d is 2.8 cm. These parameters give a maximum solid angle of Ω =0.35
steradians for the collected and focused radiant flux from the sample surface. The location of
the sample has to be in the left focal point of the lens L1. The detection area of the detector
has to be located in the right focal point length of lens L2.
Figure 4.2: Schematic representation of the Infrared detection.
A careful adjustment has to be done both with respect to the focal distances, fL1=fL2,
and the optical axis. According to Fig. A.1 [Appendix A] the BaF2 lenses have good
transmission characteristics in the detected interval of the IR spectrum. The transmittance is
nearly 92% between the wavelengths of 0.5 µm and 12 µm. The investigated samples that will
be used in this work are Silicon; fiber reinforced material and multi-layer superinsulation
foils. These samples can be regard as a gray emitter. According to the spectral emittance for
these samples at temperature T, this lenses are appropriate to measure the IR radiation from
L1
L2
fL1 f
L2
56 4 Experimental Setup
these samples that having an average temperature between 300 K and 1000 K The successful
infrared detection of thermal waves depends on maximizing the infrared radiation collected
by the detector, by making the solid angle collecting the signal large, and by minimizing the
incidence of the excitation source radiation on the detector, see Figure 4.3.The first of these
requirements has been satisfied by the use of the suitable infrared lenses. The problem of
shielding the detector from the excitation source has been overcome by using a germanium
filter in front of the detector, which acts as a cut-on for wavelength above 1 µm and stop the
reflected modulation laser beam from the sample traveling toward the detector by absorption.
Using a designed interference filter that reflects rather absorbs visible light whilst being
highly transmissive to infrared radiation can significantly reduce the problem of spurious
secondary thermal waves produced in the germanium filter. The Ge-filter with 3mm thickness
has nearly a constant transmission of about 60% in the range of wavelength 2 µm and 12 µm
[laser components, 1994].
Figure 4.3: Schematic diagram of the generation and detection of infrared radiation in a photothermal experiment [ Almond, 96]
Detectors play a fundamental role in radiometry. The most important physical
detectors of radiant energy is a semiconductor detectors. A photoconductive Mercury-
Cadmium- Telluride detector (MCT detector) for the radiometric detection of the thermal
response is used to produce an electrical signal in response to radiant energy. The principle of
the detector is to produce an electron-hole pair that lowers the detector resistance by
producing more carriers by the radiant IR-radiation incident on the detector. The change of
the photoconductive resistance produces a change in the voltage drop. This detector allows a
4.2 Electronic equipment and signal processing 57
detectable wavelength interval of 1 µm < λ < 12 µm. The relative spectral sensitivity s(λ) is
described in Fig. B.1 [Appendix B] the detector has to be cooled down by using liquid
nitrogen (77 K) to reduce the background noise [Becherer, 79].
4.2 Electronic equipment and signal processing
The used photoconductive detector converts the modulated radiant flux related to the
thermal wave to an electronic signal. A preamplifier and subsequently a Lock-in amplifier
will then amplify this signal. The radiometric signal arising from the thermal wave response is
a small additional contribution to the total radiant flux emitted by the solid surface. It is
filtered from the high radiation level corresponding to the average sample temperature by
means of a two-phase Lock-in amplifier (Stanford 830 DSP), which has been used to analyze
the measured signal with respect to its amplitude and phase lag relative to the modulated
heating laser beam. The Lock-in amplifier, which is used to detect and measure very small Ac
signals, especially when the small signals are obscured by noise, only detects signals whose
frequencies are very close to the reference frequency.
By optimal electronic adoption of the detector, preamplifier and lock-in amplifier, the
thermal waves signals have been measured nearly almost free of the noise up to 100 kHz as
shown in Figure 4.4.
Therefore Radiometry is based on a number of system components (optical and
electrical apparatus that can be used to describe the transfer of radiant energy from a source to
a detector. There are many physical parameters in the detection system, which are playing am
important role in achieving adequate signal. These parameters are summarized below
[Brewesters, 1992]:
1. The detection solid angle Ω
2. The detection area of the sample As
3. The amplifying of the preamplifier followed the detector Vv
4. The spectral responsivity of the detector )(λR
5. The transmittance of the IR optical system )(λF
The spectral responsivity of the detector is given by
)(.)( max λλ sRR =
where maxR is the given maximum responsivity and )(λs is the relative spectral responsivity.
The factor maxR can be combined together with the other factors described in the geometry (1,
58 4 Experimental Setup
2, and 3) by a constant, which illustrate the characteristic quantity of the experimental setup
(compare chapter2).
10-1
100
101
102
103
104
105
106
10-3
10-2
10-1
100
101
102
(f / Hz)
sigradure neutral glass noise
A
mp
litu
de
/ m
V
Figure 4.4: Photothermal signals, showing noise level with respect to the signals generated in
two examples samples (sigradure and glass- density glass).
4.3 High temperature cell
A specially designed high temperature cell [Huettner, 1991] has been used to keep the
sample at the desired average temperature between 300 and 1000 K. A cross section view of
the cell is shown in Figure 4.5. This cell can be evacuated to reduce oxidation and thermal
erosion at higher temperatures and to reduce the effects of conduction and convection cooling
of the sample in the surrounding air. To keep the sample at the desired average temperature,
an external voltage source is used and connected to the coax-wire winding with a resistance of
50 ohm. The coax-wire is in thermal contact with the sample holder to heat the sample. A
thermocouple is inserted in the coax-wire in order to measure the temperature of the sample
holder directly, which is in consequently has a good contact with the sample. Additionally a
second thermocouple is introduced to measure the surface temperature of the sample, which is
useful in determining the temperatures difference between the surface and the rear of the
sample (compare chapter 5). The modulated laser beam can enter the cell through two
4.3 High temperature cell 59
windows, where a quartz glass is at the left hand side and a BaF2 window at the right hand
side. These two windows allow measuring the thermal waves “in reflection” or “in
transmission”. To reduce thermal background radiation from the cell, special water pipes are
inserted inside the cell, which are cooled by water circulation.
Figure 4.5: Cross-section view of the high temperature cell
1- The sample
2- BaF2 window (thickness d = 2 mm, diameter D = 24 mm)
3- Quartz glass window, (d = 3 mm, D = 50 mm)
4- Coax-wire winding
4b- Sample holder
5- Cooling water
6- Ceramic plate
7- Radiation shield made of copper
8- Sealing ring
9- Vacuum pump
10- Cell support
60 4 Experimental Setup
4.4 Calibration of the photothermal experimental setup
In this section test measurements are presented with samples of well known thermo-
optical properties to insure the quality and reliability of measurements. The test measurements
on glassy carbon (Sigradur) and neutral glass (Schott NG1) are then normalized and
compared with the respective signal generation theory. These measurements are used also for
the calibration of the experimental setup. The main objectives of these tests are to check the
experimental conditions in the frequency interval used, 0,03 Hz ≤≤ f 100 kHz, and to verify
the consistency between experiment and theory of signal generation for materials with
different optical and thermal properties. The typical test is to normalize an opaque
homogeneous sample such as glassy carbon (Sigradur (SIG)) and a transparent sample such as
neutral glass (Scott NG1) as a reference. The normalized phase is given as difference of the
two measured phases 1NGSIGn ϕφϕ −= and under the condition that one sample is slightly
transparent and the other is absolutely opaque,
1. 11 <<SIG
NG
SIG
NG
ββ
αα
(4.1)
we can obtain two simple relationships [Bein, Krueger and Pelzl, 1985]
[ ] 111)cot( NG
NGnf β
πα
ϕ =− (4.2)
[ ] ff NGNG
n 21)cot( 11 +=+ β
πα
ϕ (4.3)
which only depend on the opto-thermal properties of the neutral glass. Thus it is impossible to
obtain any information about the thermal properties of the homogeneous opaque sample. The
results of this phase test is shown in Figure 4.6 and compared with the theoretical solutions
esq. 4.2 and 4.3. In fact very good agreement between experiment and theory has been found,
both for low and high modulation frequencies.
4.4 Calibration of the photothermal experimental setup 61
0 50 100 150 200 250 300-100
0
100
200
300
400
500
600
700
800(f
/ H
z)1/
2 [cot
(ϕn)
+/-
1]
(f / Hz)1/2
Figure 4.6 a+b: Test with photothermal phases measurements for a compact, opaque solid
(glassy carbon, Sigradur) and neutral glass (Schott NG1).
0 10 20 30 40 50-20
0
20
40
60
80
100
(f /
Hz)
1/2 [c
ot(ϕ
n)+
/-1]
(f / Hz)1/2
63
5. Experimental results
Silicon, an IR transparent material within be used for cut-on filters in IR optics, is used
in the following to test the measurement system with respect to its sensitivity for
measurements of IR translucent materials and to test the signal generation theory developed in
chapter 2 and 3. To this finality measurements have been done for samples of different
thickness, at different average sample temperatures (Room temperature up to 250 °C) and in
two measurement configurations:
a) For thermal waves in reflection, where the thermal wave is excited and detected at
the same surface
b) For thermal waves in transmission, where the thermal is excited at the front surface
and detected at the rear surface of the sample
5.1 Measurements of thermal waves in reflection
5.1.1 Temperature-dependent measurements of silicon samples
These measurements have been performed on silicon samples of different thickness
and at different average sample temperatures. All samples had a diameter of 20 mm, and the
sample thicknesses were 2.16, 4.18 and 6.28 mm, respectively. A comparatively large heating
spot diameter of about 8 mm has been used for these measurements with an effective beam
power of 1 Watt, so that the thermal waves can be described by one-dimensional heat
propagation. The measurements have been done as a function of the heating modulation
frequency in the range from 1 Hz up to 20 kHz at various fixed temperatures, namely at, T =
25 °C (room temperature), 100 °C, 150 °C, 200 °C and 250 °C. The measurement system and
the used high-temperature cell, which are schematically shown and described in Figure 4.1
and in chapter 4, allow to measure “ thermal waves in reflection”, where the thermal wave is
excited and detected at the same sample surface.
In Figures 5.1a, 5.1b to 5.3a, 5.3b the photothermal amplitudes and phase lags are
plotted versus the excitation frequency. In Figure 5.1b, it is remarkable to see that the
measured phase lags below about 6 Hz are nearly the same for different temperatures,
whereas considerable changes occur above 8 Hz. This strong temperature dependence of the
phase signals is less pronounced in Figure 5.2b and 5.3b for the thicker samples at
intermediate modulation frequencies, where the changes up to about 100 Hz are
64 5 Experimental results
comparatively moderate and a strong temperature dependent split of the curves is only
measured for higher frequencies. The comparatively stronger variations of both amplitudes
10-1
100
101
102
103
104
120
150
180
210
240
270
Am
plitu
des
/ mV
f / Hz
Pha
ses
/ deg
f / Hz
10-1
100
101
102
103
104
10-3
10-2
10-1
100
101
RT 50 °C 100 °C 150 °C 200 °C 250 °C
Figure 5.1 a+b: Photothermal amplitudes (a) and phase lags (b) measured for a 2 mm thick
silicon sample as a function of frequency at different temperatures.
5.1 Measurements of thermal waves in reflection 65
10-1
100
101
102
103
104
120
150
180
210
240
270
Am
plitu
des
/ mV
f / Hz
Pha
ses
/ deg
f / Hz
10-1
100
101
102
103
104
10-3
10-2
10-1
100
101
RT 100 °C 150 °C 200 °C 250 °C
Figure 5.2 a+b: Photothermal amplitudes (a) and phase lags (b) measured for a 4 mm thick
silicon sample as a function of frequency at different temperatures.
66 5 Experimental results
10-1
100
101
102
103
104
120
150
180
210
240
270
Am
plitu
des
/ mV
f / Hz
Pha
ses
/ deg
f / Hz
10-1
100
101
102
103
104
10-3
10-2
10-1
100
101
RT 100 °C 150 °C 200 °C 250 °C
Figure 5.3 a+b: Photothermal amplitudes (a) and phase lags (b) measured for a 6 mm thick
silicon sample as a function of frequency at different temperatures.
5.1 Measurements of thermal waves in reflection 67
and phase lags of the thinner sample (2 mm silicon) at intermediate frequencies in Figure 5.1a
and 5.1b are probably due to interference effects of the thermal wave based on conductive
heat transport contributions reflected at the rear surface of the sample. For the thicker samples
(Figure 5.2a, 5.2b and 5.3a, 5.3b) the damping of these reflected thermal wave contributions
is stronger leading to fewer changes at intermediate frequencies. The temperature-dependent
split of the curves for higher frequencies, which can be observed for all measured phase lags
independent of the sample thickness (Figure 5.1b, 5.2b and 5.3b) is probably related to the
transparency in the infrared spectrum, which changes with the average sample temperature.
Before any further discussion of the results in detail, we want to check the reliability
of our measurements. For this reason, we present in Figure 5.4a and 5.4b the thermal wave
signals for two silicon samples at room temperature, and the noise signal measured under
equal conditions of focusing of the samples without laser excitation of thermal waves. The
detection of thermal waves for a given measurement setup, as shown in figure 4.1 (chapter 4),
is affected by the total noise of the setup within the measured bandwidth. The total noise is
mainly caused by the noise produced in the detector itself, the noise of the electronic system
following the detector and the noise of the radiation incident on the detector. It can be seen
from Figure 5.5a and 5.5b that the photothermal signals measured at room temperature are
reliable over the whole frequency, although at higher frequencies the signals approach the
noise level.
To control the reliability of our measurements at higher temperatures, we calculated
the standard deviation of the registered signals. Figure 5.5a and 5.5b show examples for the 2
mm thick silicon sample at room temperature and at 150 °C. Similar results have been
measured at 200 °C and 250 °C. It is worth to mention that the choice of the lock-in amplifier
time constant and the number of the measured values at each modulation frequency
(integration numbers) strongly affect the noise limit. At low modulation frequencies (long
periodic time of modulated heating of the sample) we have used rather long time constants
whereas the number of integrations was relatively low. At higher modulation frequencies the
time constant was lower but the integration numbers had been increased, as the signals at
these modulation frequencies are lower and unstable. Table 5.1 shows the values of the
integration time constant and integration number used in the experiment. It can be concluded
from Figure 5.5b that the measurements at higher temperatures, especially at higher
frequencies, are less reliable than the measurements at room temperature. This is probably due
to the reduced thermal contrast at higher temperatures and the difficult condition of focusing
of the heated samples in the closed high-temperature cell [Bolte, Gu and Bein, 1997].
68 5 Experimental results
Frequency Time constant (sec.) Integration numbers
1 Hz< f < 5 Hz 3 16
5 Hz< f < 10 Hz 3 40
10 Hz< f < 1 kHz 1 40
1 kHz< f < 20 kHz 0.3 40
Table 5.1: The values of the time constant and integrations number used by the lock-in
amplifier during the measurements.
As we can see in Figure (5.6a) and (5.6b), which shows the relative signal amplitudes
and the differences phase lags as a function of the cubic temperatures at different fixed
frequencies, at low frequencies (f = 3 Hz and 8 Hz) the relative signal amplitudes increase
with increasing average sample temperature in agreement with the 3)(xT dependence of
according to equ. (3.105) and in chapter 3. At higher frequencies a deviation from the
simple 3)(xT dependence can be observed, which probably is due to profile effect related to
temperature dependent radiation across the sample boundary. According to Figure 6.5b, at
low frequencies there are no variations with the temperature. At higher frequencies
(f = 512 Hz and 3250 Hz) two tendencies are observed; a linear increase with 3)(xT between
room temperature and 150 °C. At higher temperature no clear tendencies found so far.
5.1 Measurements of thermal waves in reflection 69
10-1 100 101 102 103 1040
60
120
180
240
300
360
Am
plitu
des
/ mV
f / Hz
Pha
ses
/ deg
f / Hz
10-1 100 101 102 103 10410-3
10-2
10-1
100
101
Si 2mm Si 6mm noise
Figure 5.4 a+b: Comparison of the photothermal amplitudes (a) and phase lags (b) measured
for two silicon samples with the noise signal measured without thermal wave
excitation.
70 5 Experimental results
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
100
101
Si 2mm at 150 °C standard deviation
Am
plitu
des
/ mV
f / Hz
Am
plitu
des
/ mV
f / Hz
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
100
101
Si 2mm at RT standard deviation
Figure 5.5 a+b: Photothermal amplitudes and standard deviations for the 2 mm thick silicon
sample, measured at room temperature (a) and at 150 °C (b).
5.1 Measurements of thermal waves in reflection 71
0,0 5,0x107
1,0x108
1,5x108
2,0x108
-100
-80
-60
-40
-20
0
20
Rel
ativ
e am
plitu
des
(T / K)3
Rel
ativ
e ph
ase
lags
/ de
g
(T / K)3
0,0 5,0x107
1,0x108
1,5x108
2,0x108
0
2
4
6
8
10 Si2 mm at 3 Hz Si
2 mm at 512 Hz
Si6 mm at 8 Hz Si6 mm at 3250 Hz
Figure 5.6 a+b: Relative signal amplitudes (a) and relative phase lags (b) as a function of the
cubic of the temperature for two silicon samples at different fixed frequencies
where the signals measured at room temperature are taken as reference.
72 5 Experimental results
5.1.2 Normalization of measurements and quantitative interpretation of
room temperature data
For a further quantitative interpretation of the measured photothermal signals, the
measured amplitudes and phases are normalized by using reference signals measured under
equal conditions of focusing and electronic filtering. The advantage of the normalization
procedure is that the frequency response of the measurement system is eliminated, and that
different samples (e.g. samples of different thickness or different temperature, samples of
different optothermal properties can be compared, by using the appropriate theoretical
description of the thermal wave (equ. 3.87) and the appropriate description of the signal
generation process (equ. 3.105).
Starting point of the normalization procedure is the modulated radiometric signal
),()(4)()(),( 3 TfTTTTfCTfM SB δσεγδ ′= (5.1)
which can be represented in the complex form as
)(),(),( ψωδ += tieTfSTfM (5.2)
where ),( TfS is the amplitude of the photothermal signal, while ),( Tfψ is the phase lag
relative to the heating modulation. Analogously the thermal wave response can be represented
as
)(),(),( ϕωδδ += tieTfTTfT (5.3)
where ),( TfTδ is the thermal wave amplitude and ϕ its phase lag relative to the heating
modulation. Satisfying the conditions required for normalization, we obtain for the
normalized signals:
)(
3
3)(
, ),(
),(
)()(
)()(
),(
),(),( RSRS i
RR
SS
RRRR
SSSSi
RR
SSRSn e
TfT
TfT
TTT
TTTe
TfS
TfSTTfM ϕϕψψ
δδ
εγεγ
δ −−
′′
== (5.4)
where the index “S” refers to the sample and “R” to the reference measurement. From equ.
(5.4) we then can obtain the normalized amplitude nS
),(
),(
)()(
)()(
),(
),(),(
3
3
,RR
SS
RRRR
SSSS
RR
SSRSn TfT
TfT
TTT
TTT
TfS
TfSTTfS
δδ
εγεγ
′′
== (5.5)
and the normalized phase
nRSRSn ϕϕϕψψψ =−=−= (5.6)
From the two normalized quantities, we can determine the thermal and optical properties of
the sample, if the thermal and optical properties of the reference sample are well known.
5.1 Measurements of thermal waves in reflection 73
Here the normalization technique is first developed for measurements of samples with
different thickness, but at the same average temperature. This means that the parameters
)(Tγ ′ , )(Tε and T can be eliminated from equ. (5.5) and that only the thermal waves differ.
We then obtain for the normalized signals:
[ ] [ ])()(
2
1)()(
2
121
2121
),,(
),,(
),,(
),,(),,( ddi
S
Sddi
S
Sn
SSSS edTfT
dTfTe
dTfS
dTfSddTfM ϕϕψψ
δδ
δ −− == (5.7)
Another normalization technique can be developed if we compare measurements of
the same sample but at different temperatures. Following the previous process for the
normalized signal, we obtain
[ ] [ ])()(
2
13
222
3111)()(
2
12,1
2121
),(),(
)()()()(
),(),(
),( TTi
S
S
S
STTii
S
Sn
SSSS eTfTTfT
TTTTTT
eTfSTfS
TTfM ϕϕψψ
δδ
εγεγ
δ −−
′′
== (5.8)
The effective emissivity can be eliminated from equ. (5.8) if we assume that the emissivity
only slightly varies with temperature in comparison to the temperature dependence of
Planck’s blackbody radiation. The advantage of this type of normalization lies in fact that the
stationary temperatures T1 and T2 can be measured separately by thermocouples and that the
setup-dependent detection efficiency )(Tγ ′ can be calculated and can be taken into account
according to Figure (2.4) (chapter 2). Consequently, the temperature effect on the thermal
wave can be interpreted directly: The ratio of the thermal wave amplitudes can be determined
from
311
322
2,12
1
)()(
),(),(),(
TTTT
TTfMTfTTfT
nS
S
γγ
δδδ
′′
= (5.9)
and the normalized phases can be interpreted similarly
nSSSSn TTTT ϕϕϕψψψ =−=−= )()()()( 2121 (5.10)
The advantage of the normalized phases lies in the fact that they are not influenced by
the emissivity or the effect of the IR detection system on the temperature-dependent
blackbody radiation, and that they are more reliable for the quantitative interpretation.
The general expression for the measured radiometric signal in the case of “thermal
wave in reflection“ (chapter 3, equ. 3.105) is given by
⋅⋅+−
⋅+
+
−+
−
−
′=−
+−−−
dEe
D
eB
eA
k
ITxTTCfTM
ss
d
s
s
ns
d
s
s
ns
d
s
nss
ossSBssss
s
nssnss
β
βσ
βσ
ση
γσεδβ
σβσβ
21
1
1
1
1
2)()(4)(),(
2
)()(
3 (5.11)
74 5 Experimental results
which we want to use in the following interpretation. Silicon has an extended absorption
length ( 51 ≈−sβ mm), which is in the same order of the samples thickness. Therefore, a model
with finite solid must consider here. According to equ. (5.11) and the integration constants
sA , sB , sD and sE of equ. (5.11), there are five relevant parameters that influence the
amplitude of the photothermal signal, which are nssk σ , ))(( 3sxTa , ssP βα 2/1
1 = ,
ss dP /2/12 α= and ss dP β=3 . If we consider the photothermal phases for the case of fixed
average sample temperature, there are three relevant parameters that influence the phases
signal, namely only ssP βα 2/11 = , ss dP /2/1
2 α= and ss dP β=3 . These three combined
parameters are coupled so that the third parameter can be calculated from the other two
parameters 213 / PPP = . Consequently it seems to be reasonable to start with the quantitative
interpretation of the phases, using to two parameters-space.
The combined quantity ss βα 2/1 is contained the quantity 1/ −ss βµ , which is the ratio
of two characteristic lengths, namely of the thermal diffusion length to the IR absorption
length 1−sβ inside the sample. Owing to the fact that the thermal diffusion length can be
controlled by varying the modulation frequency f, the thermal diffusion length can be adjusted
to the absorption length. If the thermal diffusivity is known, also the thermal diffusion length
is known, which can be compared with the IR absorption length. This means the IR
absorption constant can be determined from the phases measured at higher modulation
frequencies. As the detector is sensitive over an extended IR wavelength interval, sβ is an
effective IR absorption constant, which is characteristic for the IR radiation within the
detectable wavelength interval 2 µm < λ < 12 µm.
Here we will first try to interpret the test measurements performed on silicon by taking
into consideration the theory of IR transparency (chapter 3) and the normalization concept
described in equ (5.7).
To interpret the photothermal signals measured for silicon, we normalize the amplitudes and
phases of the thermal waves measured in reflection for two silicon samples of different
thickness. First we interpret the results measured at room temperature. Figures (5.7a) and
(5.7b) show the normalized phases plotted as a function of the square root of the modulation
frequency and the normalized amplitudes plotted as a function of the inverse square root of
the modulation frequency. The normalization process was done by dividing the thicker
sample by the thinner sample. In the frequency range 1 Hz < f < 100 Hz the normalized
phases show very pronounced relative maxima between 6° < nϕ < 15° depending on the
5.1 Measurements of thermal waves in reflection 75
Figu
re 5
.7 a
+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s m
easu
red
for
two
silic
on s
ampl
es (
6 m
m a
nd 2
mm
thi
ck)
at r
oom
tem
pera
ture
and
com
pare
d w
ith th
eore
tical
app
roxi
mat
ions
.
100
101
102
-30
-20
-100102030
Normalized amplitudes
(f / H
z)-1
/2
Normalized phases / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
(1)
α
1/2 β
= 4.
33 s
-1/2
(2)
α
1/2 β
= 5.
69 s
-1/2
(3)
α
1/2 β
= 9.
85 s
-1/2
76 5 Experimental results
Figu
re5.
8 a+
b: N
orm
aliz
ed p
hase
s (a
) an
d am
plitu
des
(b)
mea
sure
d fo
r tw
o si
licon
sam
ples
(4
mm
and
2 m
m t
hick
) at
roo
m
tem
pera
ture
and
com
pare
d w
ith t
heor
etic
al a
ppro
xim
atio
ns,
whe
re t
he t
herm
al d
iffu
sivi
ty i
s ta
ken
α =
cons
tant
and
the
abso
rptio
n co
nsta
nt β
is v
arie
d.
100
101
102
-30
-20
-100102030
Normalized amplitudes
(f / H
z)-1
/2
Normalized phases / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
α s / m
2 s-1 =
270
*10-6
β s /
m-1 =
300
β s /
m-1 =
230
β s /
m-1 =
180
5.1 Measurements of thermal waves in reflection 77
Figu
re 5
.9 a
+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
sili
con
sam
ples
(4
mm
and
2 m
m t
hick
) at
roo
m
tem
pera
ture
and
co
mpa
red
with
th
eore
tical
ap
prox
imat
ions
, w
here
the
IR
ab
sorp
tion
coe
ffic
ient
β
was
take
n a
s a
cons
tant
and
the
var
iatio
n is
with
the
ther
mal
dif
fusi
vity
α.
100
101
102
-30
-20
-100102030
Normalized amplitudes
(f / H
z)-1
/2
Normalized phases / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
β s / m
-1 =
230
α s /
m2 s
-1 =
150
α s /
m2 s
-1 =
270
α s /
m2 s
-1 =
400
78 5 Experimental results
difference in sample thickness. At lower and higher modulation frequencies, the normalized
phases are nearly constant. Owing to the fact that the signals measured at low and high
frequencies are relatively close to the noise signals as shown in Figure (5.4) and (5.5), the
normalized signals show a random fluctuations at low and mainly at higher frequencies.
According to the normalization procedure (thicker-divided-by-thinner samples) the
normalized amplitudes are smaller than 1 (compare Figure 5.7b). For a quantitative
interpretation of the normalized phases and amplitudes, the model of a solid of finite
thickness that is homogenous, compact and opaque for both the visible light and the infrared
region [Gu, 1993], fails completely. A formally correct numerical approximation of the
amplitudes and phases, with satisfactory agreement between theory and experiment over the
measured frequency interval, can only be obtained on the basis of the model combining
radiative heat transport and conductive heat transport (chapter 3). Figure 5.8 and 5.9 show
selected examples of approximation demonstrate the influence of the variation either in the
thermal diffusivity α or in the IR absorption coefficient β . In Figure 5.8 the value of thermal
diffusivity was hold constant while making the variation of the IR absorption coefficient.
These variations affects the value of the maximum of the normalized phase and produce a
split value of the normalized amplitude only at low frequencies. The normalized theoretical
amplitude can be adjusted to the level of measured normalized amplitude through the
combined quantity [ ]dssss k σεη / , where the quantity ssk σ contains information about the
thermal effusivity of silicon.
In Figure (5.9) the IR absorption coefficient was hold constant and the thermal diffusivity was
varied. As a consequence, the position of the maximum for the normalized phases shifted
towards the right and the left hand side, and the intermediate frequency of the normalized
amplitude only was affected.
As shown in this subchapter only two combined parameters e.g. 1P and 2P can be chosen
independently and determine the approximation of the normalized phases. By using arbitrary
combinations of the two independent parameters, 1P and 2P , for two samples of different
thickness we can produce a variety of theoretical curves, which are in reasonable formal
agreement with the measurements. Figure 5.10 shows the existence regions in gray for
optimal parameters for the two different samples (6 mm and 2mm thick), where the
parameters ss dP /2/12 α= are plotted versus the parameters ssP βα 2/1
1 = , As can be seen
from Figure (5.10) a large variety of possible parameters )(),( 1211 dPdP and
)(),( 2221 dPdP would be able to produce reasonable agreement, the problem, however is to
5.1 Measurements of thermal waves in reflection 79
establish criteria to find physically meaningful parameters. In order to select physically
reliable solutions from the variety of possible numerical approximations in good agreement
with the measurements, we plotted the ratio of the quantities 12
)(/)()(/)( 1323 dd dddPdP ββ=
versus the ratio =)(/)( 1222 dPdP 12
)/(/)/( 2/12/1dd dd αα , where β is the IR absorption
coefficient, α the thermal diffusivity of the silicon sample, and 1d and 2d denote the
thickness of the two silicon samples involved in the normalization procedure. Figure 5.11
shows the resulting existence region (in gray) for the ratios of the parameters 2P and 3P and
the various points of parameters ratios, for which theoretical approximations have been
calculated for the normalized phases and amplitudes. If we require that the material properties
α and β should be independent of the individual thickness of the involved samples, the
geometrical conditions 211222 /)(/)( dddPdP = and 121323 /)(/)( dddPdP = should be
fulfilled, with an identical α -value and β -value for the two samples. From the selected
parameters with good agreement between theory and experiment results, the combined
parameter βα 2/1321 * == PPP is finding to be limited by 69.56.3 2/1 << ss βα . According to
literature data this combined quantity has a value of about 1.70. The higher values found in
our measurements can be due to different reason, due to the samples and due to the
experimental conditions:
1) The oxidized surface layer of the used samples can contribute to an effectively reduced IR
signal measured in front of the heated samples, which has been interpreted as an increased
effective IR absorption coefficient.
2) The limiter heating spot diameter, about 8 mm in comparison to the sample diameter of 20
mm contributes to a three-dimensional temperature distribution in the samples and to
laterals increased heat losses due to conductive heat transport. This effect contributes to
effectively increased value of the thermal diffusivity.
3) Owing to the sample geometry and lateral radiative heat losses from the cylindrical
sample, which have not been suppressed by experimental measurement, e.g. reflection of
the lateral IR radiation, the IR signal measured in front of the samples are lower than
foreseen by the theoretical model. This again contributes to an effectively increased
measure IR absorption coefficient.
4) Finally we have to admit, that the theoretical model for the thermal wave containing
conductive and radiative heat transport is limited due to its construction as a perturbation
solution to only slightly IR translucent samples, This model perhaps is not appropriate for
silicon with its relatively long IR absorption length 56.51 =−IRβ mm.
80 5 Experimental results
Figure 5.10: Existence regions of parameters for reasonable formal agreement between
measurement and theoretical approximation of the normalized phases.
Figure 5.11: Existence region for the ratios of parameters for different samples of different
thickness at room temperature.
0 5 10 15 200
4
8
12
16
20
Si6.28 mm
Si2.16 mm
(3)(2)(1)
P
2 =
α1/
2 /d
[se
c-1]
P1 = α1/2β [sec-1]
0 1 2 3 4 50,0
0,2
0,4
0,6
0,8
1,0
d2/d
1=4.18/6.28
2.16/4.18
2.16/6.28
6.28/2.164.18/2.16d1/d
2=6.28/4.18
Sid1
/ Sid2
Si6.28 mm
/ Si2.16 mm
Si6.28 mm / Si4.18 mm Si
4.18 mm / Si
2.16 mm
P3(
d2)
/ P
3(d
1) =
(β
d) d2
/ (β
d) d1
P2(d
2) / P
2(d
1) = (α
1/2/ d)
d2 / (α
1/2 /d)
d1
5.1 Measurements of thermal waves in reflection 81
5.1.3 Normalization of measurements and quantitative interpretation as
a function of temperature
After studying the results obtained at room temperature, we will study the behavior of
the silicon samples at higher temperatures. Figures 5.12a and 5.12b show the normalized
phases and amplitudes for silicon samples (6 mm and 2 mm thick) at 100 °C, the behavior of
the two normalized signals, phases and amplitudes in principle is similar to that shown at
room temperature. In the range of 1 Hz < f < 100 Hz, the phases show very pronounced
relative maxima, °≈ 12nϕ . At higher frequencies, the normalized phases show smaller
relative minima (compare Figure 5.13a) and the normalized amplitudes show pronounced
maxima with increasing average sample temperature (Figure 5.12b and 5.13b). The random
distribution of the phases is due to the relatively small measured signals, which are close to
the noise signals as shown in Figure 5.4 and 5.5. The theoretical approximation (5.11) is used
to interpret the measurements performed at high temperatures. Figure 5.12 and 5.13 show
measurement and theory for silicon samples (thicker-divided-by-thinner samples) at 100°C
and 200 °C respectively. From the approximation in Figure 5.12 and 5.13 we can see, that an
optimal numerical approximation could only be obtained by allowing distinct values for the
combined parameter for the sample of different thickness. The combined quantity βα 2/1 for
the thinner silicon sample is usually higher. The increase of the combined quantity βα 2/1 for
the thinner sample in comparison to the thicker sample at higher temperatures can be due to
different reasons: firstly, the heat conduction in the thinner samples can improve more due to
charge carrier excitation by the laser beam and surface recombination of the charge carriers.
Secondly the IR radiation is shifted with higher temperatures to shorter wavelength and the
absorption in the oxidation surface layer is relatively more important in the thinner samples,
leading to an effectively increased IR absorption coefficient β .
By comparing the values of the combined parameter βα 2/1 for the thicker sample at higher
temperatures to the values at room temperature, we can observe a small decrease of this
quantity with temperatures, which may be in agreement with literature data, which foresee a
decrease of the thermal diffusivity with temperature [Touloquian and Powell, 1973].
Figure 5.14 shows the existence regions for the ratios of the parameters )(/)( 1323 dPdP and
)(/)( 1222 dPdP for silicon samples of different thickness at the temperatures 100°C, and 200
°C. Common points can be found between the existence regions at higher temperatures and
that at room temperature, which coincide with the ratio of the samples thickness as mentioned
in section 5.1.2. In principle the physically correct solution (Figure 5.13) should be found for
82 5 Experimental results
the ratio values 66.0/)(/)( 121323 == dddPdP and 5.1/)(/)( 211222 == dddPdP , since the
material parameters should be the same for the two samples of different thickness. This is
fulfilled by the solution (1) of Figure 5.13, on the other hand apparently good numerical
approximation (solution (2) of Figure 5.13) are also found at other points in the
)(/)( 1323 dPdP and )(/)( 1222 dPdP space, where the combined quantity βα 2/1 for the thicker
sample is 4 s-1/2 and for the thinner sample is 6 s-1/2. To certify this situation, improved
measurements with higher laser power and increased integrations constant of lock-in
amplifier are necessary to reduce the fluctuations which observed for the normalized signal.
Figure 5.15a and 5.15b show the normalized phases and amplitudes for silicon
samples (4 mm and 2 mm thick) at RT, 200 °C and 250 °C. It is worthy to discuss the
behavior with temperature, because the normalized signals show clear tendencies at higher
temperatures. The normalized phases show in addition to the relative maximum at lower
frequencies, a relative minimum, which gets more pronounced with increasing temperature at
about 120 Hz. The maximum of the normalized phases moves to lower frequencies can be
related to the fact that the thermal diffusivity decreases with temperature (compare Figure
5.9a). The normalized amplitudes show also a pronounced maximum at about 25 Hz.
It is remarkable, that the reliable frequency range which can be used for the
quantitative interpretation of the measured phases in general decreases for all silicon
measurements at higher temperatures. At a first glance, this effect seems to be in contrast to
the background detection limit observed for IR-opaque samples [Bolte, Gu, Bein, 1997a],
which improved with higher temperatures. In the case of the silicon measurements at higher
temperatures, however, the rear surface of the samples is in contact with the coax-wire-heated
sample holder, which has got a higher temperature of about 10 –30 K than the front surface.
This means, the relatively small thermal wave is measured in front of higher fluctuating
temperature background, as Silicon is IR-transparent.
5.1 Measurements of thermal waves in reflection 83
Figu
re 5
.12
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
6 m
m a
nd 2
mm
thic
k) a
t 100
°C
and
com
pare
d w
ith th
eore
tical
app
roxi
mat
ions
.
100
101
102
-30
-20
-100102030
Normalized amplitudes
(f / H
z)-1
/2
Normalized phases / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
theo
retic
al s
olut
ion
(1)
theo
retic
al s
olut
ion
(2)
theo
retic
al s
olut
ion
(3)
84 5 Experimental results
Figu
re 5
.13
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
6 m
m a
nd 4
mm
thic
k) a
t 200
°C
and
com
pare
d w
ith th
eore
tical
app
roxi
mat
ions
.
100
101
102
-30
-20
-100102030
Normalized amplitudes
(f / H
z)-1
/2
Normalized phases / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
the
oret
ical
sol
utio
n (1
) t
heor
etic
al s
olut
ion
(2)
5.1 Measurements of thermal waves in reflection 85
Figure 5.14: Existence region for the ratios of parameters for samples of different thickness
for 100 °C and 200 °C, compared with the existence region at room temperature
(light gray).
0 1 2 3 4 50,0
0,2
0,4
0,6
0,8
1,0
(2)(1)
(3)(2)
(1)
Si6 mm
/ Si4 mm
at 200 °C Si
6 mm/ Si
2 mm at 100 °C
P3(
d 2) /
P3(
d 1) =
(β
d)d
2 /
(β d
) d1
P2(d
2) / P
2(d
1) = (α
1/2/ d)
d2 / (α
1/2 /d)
d1
86 5 Experimental results
Figu
re 5
.15
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
4 m
m a
nd 2
mm
thic
k) a
t
di
ffer
ent t
empe
ratu
res,
in c
ompa
riso
n w
ith th
eore
tical
app
roxi
mat
ions
.
100
101
102
-30
-20
-100102030
Normalized amplitudes
(f / H
z)-1
/2
Normalized phases / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
RT
200
°C
250
°C
5.2 Transmission measurements 87
5.2 Transmission measurements
Subsequently, the theory of thermal waves including IR transparency will be applied
to the analysis of frequency-dependent thermal wave measurements of silicon samples using
the transmission configuration, where the samples are heated at the front surface and the
thermal response is detected at the rear surface. The front surface of the samples has been
sprayed with graphite in order to maximize the absorbed laser beam power with an increased
photothermal conversion efficiency η . A comparatively large heating spot diameter of about
15 mm has been used for these measurements with an effective beam power of 1 Watt to
approach the condition of one-dimensional heat propagation in the samples of 20 mm
diameter. Additionally, the lateral side of the samples has been wrapped with an Aluminum
foil, in order to reduce lateral radiative heat losses. The path of the laser beam was laterally
covered with black cartoon to provide good shielding of the IR lenses, IR filter and detector
from the laser beam and to avoid any stray light contributions on the detector. The
measurements have been carried out at room temperature using the range of heating
modulation frequencies from 0.03 Hz to 100 Hz. In order to obtain quantitative information
on the optothermal properties of the samples, the signals measured for the samples of
different thickness are normalized against each other.
Figures 5.16a and 5.16b show the photothermal transmission signals, amplitudes and
phase lags respectively, which have been measured at the rear surface of the different silicon
samples (2 mm, 4 mm and 6 mm thick) as a function of frequency. As can be seen from
Figure 5.16a, at low frequencies the signal amplitudes decrease with the increasing thickness
of the sample, while at higher frequencies, relatively high and well-defined signals are
observed, which are nearly independent of the sample thickness. The measured signals for all
samples are above the noise limit of the equipment over the whole measured frequency range
[0.03 Hz, 100Hz] and far away from the background noise at the higher frequencies. In part
the observed frequency dependence of the measured raw signals (below about 1 Hz) is due to
the frequency characteristics of the used electronic equipment, which can be eliminated by
normalization of the signals with the help of a measured reference signal (Compare Figure
5.17a and 5.17b). The phase lags shown in Figure 5.16b show nearly the same behavior for
all samples, with exceptions only at the intermediate frequencies, 0.2 Hz < f < 10 Hz, where
different curvatures are observed.
88 5 Experimental results
10-2 10-1 100 101 102180
240
300
360
420
Am
plitu
des
/ mV
f / Hz
Pha
ses
/ deg
f / Hz
10-2
10-1
100
101
102
10-3
10-2
10-1
100
101
noise measurement
Si 2 mm Si 4 mm Si 6 mm
Figure 5.16 a+b: Photothermal amplitudes (a) and phase lags (b) measured as a function of
frequency for silicon samples of different thickness in transmission.
5.2 Transmission measurements 89
By comparing Figure 5.16a with Figures 5.1a, 5.2a and 5.3a, the decrease of the signal
amplitude in the case of the transmission signals is much weaker with increasing frequency
than in the case of the reflection signals. The relatively high and well-defined signal
amplitudes and the relatively small changes of the phases at the higher frequencies in Figure
5.16a and 5.16b, respectively, can be due to one reason: the high IR transparency of silicon
which leads to the phenomena, that the IR response measured in the transmission
configuration contains information about the modulated heating process from all portions
inside the sample that the information from the region just beneath the front surface has got a
relatively higher weight due to the higher time-averaged temperature distribution )(xT close
to the heated front surface, and that the direct radiative contribution from the rear surface are
comparatively unimportant.
5.2.1 Application of thermal wave theory including IR transparency to
the transmission measurements
The general expression for the measured radiometric signal in the case of “thermal
wave in transmission“ (chapter 3, equ. 3.112) is given by
⋅−
+⋅+
−
−+
+
−
′=−
−
−−−
2
11
2)()(4)(),( 3
dd
sd
ss
s
ns
dd
s
s
ns
dd
s
nss
ossSBssss
ss
s
snssns
eeEedD
eeB
eeA
k
ITxTTCfTM
βββ
βσβσ
β
βσ
βσ
ση
γσεδ (5.12)
in which we want to use in the following interpretation.
For the further interpretation, the signal measured for the thicker sample will be
divided by the signal measured for the thinner sample. Similarly to the procedure used in
section 5.1.2, the normalized signals are then described by
[ ] [ ])()(
2
1)()(
2
121
2121
),,(
),,(
),,(
),,(),,( ddi
S
Sddi
S
Sn
SSSS edTfT
dTfTe
dTfS
dTfSddTfM ϕϕψψ
δδ
δ −− == (5.13)
where the parameters )( sTγ ′ , )( ss Tε and )( sxT have been assumed to be equal for the two
samples and have thus been eliminated. According to equ. (5.13), the normalized amplitudes
nS can then be written as
),,(
),,(
),,(
),,(),,,(
2
1
2
121 dTfT
dTfT
dTfS
dTfSddTfS
S
S
S
Sn δ
δ== (5.14)
90 5 Experimental results
and the normalized phase as
[ ] [ ] nSSSSn dddd ϕϕϕψψψ =−=−= )()()()( 2121 (5.15)
Figures 5.17a and 5.17b show the normalized phases and amplitudes, which both are
plotted as a function of the square root of the modulation frequency. At low modulation
frequencies, at about 1 Hz, the phases show very pronounced relative maxima, 10° < nϕ
<23°. According to the normalization procedure (thicker-divided-by-thinner sample) the
normalized amplitudes are smaller than 1 (Figure 5.17b) and show very pronounced relative
minima below 1 Hz.
If we try to approximate the normalized measured signals according to equ. (5.12),
(5.14) and (5.15), where the stationary temperature distribution is assumed to be constant
along the path of the radiation inside the samples, =)( sxT constant, the normalized phases
and amplitudes in Figure 5.18a and 5.18b e.g. for the 6 mm and 4 mm thick silicon samples,
deviate completely from the theoretical solutions.
While the measured normalized phases (Figure 5.18a) show a relative maximum at about 1
Hz, the theoretical phase approximations show relative minimum, which depending on the
value of the parameter βα 2/11 =P move from lower to higher frequencies. The measured
normalized amplitudes show a pronounced minimum below 1 Hz, whereas the theoretical
amplitudes approximations show not very pronounced minima, moving with increasing
parameters 1P from the lower to the higher frequencies.
Since the experimental data and the theoretical approximations show such a systematic
discrepancy, we have to admit that the basic assumptions used for the derivation of the
thermal wave equation (equ. 3.58) and for the calculation of the thermal wave solution
(chapter 3, equ. 3.87), namely that the thermal wave solution is a small linear temperature
variation which is independent of the time-averaged local temperature distribution )(xT
inside the sample and which can be calculated by assuming a constant temperature
distribution for the whole sample =)(xT const, is not correct, at least when transmitted
thermal waves are measured in a sample in which due to a good IR transparency the thermal
wave contribution from hotter and colder region are measure simultaneously. Therefore, in
the following, the depth-dependent time-averaged background temperature )(xT will be
considered in the detection process of the radiative flux, whereas the effect of the temperature
distribution )(xT on the thermal wave )( fTδ continues to be neglected, due to its expected
smaller effect.
5.2 Transmission measurements 91
Figu
re 5
.17
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
diff
eren
t sili
con
sam
ples
(6
mm
, 4 m
m a
nd 2
mm
th
ick)
at r
oom
tem
pera
ture
as
a fu
nctio
n of
mod
ulat
ion
freq
uenc
y an
d pl
otte
d ve
rsus
the
squa
re r
oot o
f th
e
mod
ulat
ion
freq
uenc
y.
02
46
810
-30
-20
-100102030
ln (normalized amplitudes)
(f /
Hz)
1/2
Normalized phases / deg
(f /
Hz)
1/2
02
46
810
-2-1012
Si 6
mm /
Si 2
mm
Si 4
mm /
Si 2
mm
Si 6
mm /
Si 4
mm
92 5 Experimental results
Figu
re 5
.18
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
6 m
m a
nd 4
mm
thic
k) a
t roo
m
te
mpe
ratu
re a
nd c
ompa
red
with
theo
retic
al a
ppro
xim
atio
ns, b
y as
sum
ing
cons
tant
tim
e av
erag
ed
te
mpe
ratu
re d
istr
ibut
ion
T (x
) =
con
st in
side
the
sam
ples
.
02
46
810
-30
-20
-100102030
ln(normalized amplitudes)
(f /
Hz)
1/2
Normalized phases / deg
(f /
Hz)
1/2
02
46
810
-2-1012
αs /
m2 s-1
=88*
10-6
β
s / m
-1 =
100
=8
8*10
-6
=
180
=2
00*1
0-6
= 2
00
=6
00*1
0-6
= 1
80
5.2 Transmission measurements 93
Strating from equ. (3.111), the radiometric signal can be written as
∫ −−
∂
∂+=
d
ssxd
sss
SBsss dxfxTexTTTxT
TTCfTM ss
0
)(3 ),()()(
4
)()()(4),( δβ
γγσεδ β
(5.16)
where temperature profiles of the following form will be investigated
n
ssss d
xTTxT
−∆+= 1)( (5.17)
with Ts the time-and space-averaged temperature of the sample, sT∆ the temperature
difference between the front and rear surface of the sample, sx the depth coordinate of the
sample, d the sample thickness and n an integer number 1,2,3,…etc. Figure 5.18 shows the
stationary temperature distributions in the sample with the variation of the exponent n. As the
actual temperature distribution inside the sample cannot be measured, various temperature
profiles with different values n will be tested for the interpretation of the experimental results.
After inserting the temperature profile into equ. (5.16) the first order term in x will only be
considered and higher order terms in x are neglected.
0,0 0,2 0,4 0,6 0,8 1,0300
302
304
306
308
310
n=10
7
5
3
2
n=1
T(x
s / d
) / K
xs / d
Figure 5.19: Stationary temperature distribution in the sample as a function of the profile
parameter n.
94 5 Experimental results
The radiometric signal can then be written for the example of the n = 2 temperature profile as
),(),(),( fTMfTMfTM bs
ass δδδ −= (5.18)
where ),( fTM asδ is the radiometric signal obtained from the uniform stationary temperature
distribution, and ),( fTM bsδ is the additional term due to the temperature profile n = 2:
( ) ( )
∫−−⋅
∆+
∂∂∆+
+=
d
ssxd
s
sssss
sSBsssas
dxfxTe
TTT
TTTTTfTM
ss
0
)(
3
),(
)(
4)()(4),(
δβ
γγσεδ
β (5.19)
( ) ( )
∫ −−⋅
∆∆+
∂
∂∆++=
d
sssxd
s
sss
sssSBsss
bs
dxxfxTe
d
TTT
TTTT
TTfTM
ss
0
)(
2
),(
)(4
8)(6)(4),(
δβ
γγσεδ
β
(5.20)
The quantity )(Tγ is defined according to equ. (2.22) in chapter 2. The thermal wave for a
solid of finite thickness (3.87) can be written as
tixs
xs
xs
xsss eeEeDeBeAtxT ssssnssns ωββσσδ ][),( −−− +++= (5.21)
where the integration constants are given by (3.82), (3.83), (3.85) and (3.86). By substituting
equ. (5.21) into the integrals (5.19) and (5.20), we obtain
−+⋅+
−
−+
+
−
′∆+=−
−
−−−
2
11
2).()(4)(),( 3
dd
sd
ss
s
ns
dd
s
s
ns
dd
s
nss
ossssSBsss
as
ss
s
snssns
eeEedD
eeB
eeA
k
ITTTTCfTM
βββ
βσβσ
β
βσ
βσ
ση
γσεδ (5.22)
and
( ) ( )
( ) ( )
( ) ( )
+
−++
−+
−
−−
+
++
+
−+
∆⋅
∂
∂∆++∆+=
−−
−−
−
de
ed
Eed
D
de
ed
B
de
ed
A
Tk
I
TTTT
TTTTCfTM
s
dd
s
sds
s
nss
dd
nss
s
ns
s
nss
dd
nss
s
ns
s
ss
o
sssssSBss
bs
s
ss
s
ns
s
ns
βββ
σβσββ
σ
σβσββ
σ
ση
γγσεδ
βββ
βσ
βσ
421
122
11
1
11
1
2
)(43
4)(4)(),( 2
(5.23)
5.2 Transmission measurements 95
From the above equation (5.22) and (5.23) we see that the measured IR signal
additionally depends on the temperature difference sT∆ between the front and the rear surface
of the sample.
Figure 5.20a and 5.20b show the normalized phases and amplitude respectively for the
6mm and 4 mm thick silicon samples in comparison with theoretical approximations based on
the parameters 2=n , =∆ mmT6 78 K, =∆ mmT4 72 K for the depth-dependent temperature
distributions and 69.1)(6
2/1 =mmSiβα s-1/2 and 94.1)(
4
2/1 =mmSiβα s-1/2 for the sample
material parameters 1P for the continuous curve. The broken curve is based on the parameters
2=n , =∆ mmT6 65 K, =∆ mmT4 50 K and =mmSi6
)( 2/1 βα 69.1)(4
2/1 =mmSiβα s-1/2, and the
broken-pointed curve is based on the parameters 2=n , =∆ mmT6 75 K, =∆ mmT4 72 K and
69.1)(6
2/1 =mmSiβα s-1/2 and 94.1)(
4
2/1 =mmSiβα s-1/2. The better agreement between
measured data and theoretical approximation is obtained for the continuous curve. The
relatively large steady-state temperature differences used for the approximation are justified in
a graphite spraying of the front surface, which had to be done to reduce the optical reflectance
of the samples and to increases the absorbed fraction of the incident laser beam, lead to a front
surface layer of reduced thermal effusivity and to an increased IR absorption coefficient in a
thin layer close to the front surface. The differences in the 1P values of the sample of different
thickness may be related to the thin layer carbon particles diffused into silicon substrate and
this effect may affect the measured signal more when the total thickness of the sample is
smaller.
Figure 5.21a and 5.21b show theoretical approximations based on different forms
( =n 2, 3 and 5) of the depth-dependent temperature distribution (Figure 5.19). The
continuous curve in Figure 5.21 represents the theoretical solution based on =n 2, =∆ mmT6
78 K and =∆ mmT4 72 K, the pointed broken curve is the solution based on =n 3, =∆ mmT6 52
K and =∆ mmT4 49 K, and the broken curve has been calculated with the parameters =n 5,
=∆ mmT6 31 K and =∆ mmT4 29.5 K, where the values of the quantity 1P have been taken the
same for all curves, for which 69.1)(6
2/1 =mmSiβα s-1/2 and 94.1)(
4
2/1 =mmSiβα s-1/2. From
the theoretical results in Figure 5.21, we can conclude that the reasonable approximations
totally depend on the information about the time averaged temperature distribution )(xT in
the sample, and that at higher values of n, the temperature difference T∆ can become smaller.
This means, steep temperature gradient lead to more realistic approximations.
96 5 Experimental results
Figures 5.22a, 5.23a, 5.22b and 5.23b show the normalized phases and amplitudes
measured for the 6 mm and the 2 mm thick silicon sample, respectively for the 4 mm and the
2 mm thick silicon samples, in comparison with theoretical approximations. The theoretical
approximations in Figure 5.22 based on the parameters 2=n , =∆ mmT6 70 K,
69.1)(6
2/1 =mmSiβα s-1/2 and 5.2)(
2
2/1 =mmSiβα s-1/2 for all curves and make the variations by
the depth-dependent temperature distributions for the 2 mm sample, where for the broken-
pointed curve =∆ mmT2 25 K, for the continuous curve =∆ mmT2 30 K and for the broken curve
=∆ mmT2 35 K. According to the Figure 5.23, the theoretical approximation based on
parameters 2=n , =∆ mmT4 62 K, 94.1)(46
2/1 =mmSiβα s-1/2 and 5.2)(
2
2/1 =mmSiβα s-1/2 for
all curves and the variations are done by the depth-dependent temperature distributions for the
2 mm sample, where for the broken-pointed curve =∆ mmT2 20 K, for the continuous curve
=∆ mmT2 25 K and for the broken curve =∆ mmT2 30 K. The better agreement between
measured data and theoretical approximations in Figure 5.22 and 5.23 are obtained for the
continuous curve. The higher value of 1P for the 2 mm thick contributes to the same reason
mentioned above. From the results of this chapter we can conclude that for IR translucent materials,
especially in the transmission measurements, the local distribution of the stationary
temperature inside the sample has to be taken into account for a reliable quantitative
interpretation with respect to the thermo-optical parameters. Experimentally, this can be
achieved by localized temperature measurements by means of thermocouples, at least at the
front and the rear surface of the samples. Furthermore, we have to conclude that the
decoupling of the diffusion equation of the thermal wave (3.57) from the diffusion equation of
the time-averaged temperature distribution (3.56) may not be allowed for IR-translucent
materials, when higher temperature differences and larger IR absorption length 1−β occur.
In general, the values obtained in the transmission measurements of thermal waves
5.269.1 2/1 << ss βα are below the values obtained in the reflection measurements
69.56.3 2/1 << ss βα and very close to the expected literature data 269.1 2/1 << ss βα . This is
due to the fact that the radiation losses in the transmission measurements have been reduced
by wrapping the samples with a reflecting Al-foils and a larging the heating spot so that effect
of three dimensional heat propagation are reduced.
5.2 Transmission measurements 97
Figu
re 5
.20
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
6 m
m a
nd 4
mm
thic
k) a
t roo
m
te
mpe
ratu
re a
nd c
ompa
red
with
the
oret
ical
app
roxi
mat
ions
bas
ed o
n th
e d
epth
-dep
ende
nt t
empe
ratu
re
d
istr
ibut
ion
(5.1
7).
02
46
810
-30
-20
-100102030
ln(normalized amplitudes)(f
/ H
z)1/
2
Normalized phases / deg
(f /
Hz)
1/2
02
46
810
-2-1012
98 5 Experimental results
Figu
re 5
.21
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
6 m
m a
nd 4
mm
thic
k) a
t roo
m
te
mpe
ratu
re a
nd c
ompa
red
with
the
oret
ical
app
roxi
mat
ions
bas
ed o
n th
e d
epth
-dep
ende
nt t
empe
ratu
re
d
istr
ibut
ion
(5.
17)
with
dif
fere
nt e
xpon
ents
n.
02
46
810
-30
-20
-100102030
ln(normalized amplitudes)(f
/ H
z)1/
2
Normalized phases / deg
(f /
Hz)
1/2
02
46
810
-2,0
-1,5
-1,0
-0,50,0
0,5
1,0
1,5
2,0
n=2
n=3
n=5
5.2 Transmission measurements 99
Figu
re 5
.22
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
6 m
m a
nd 2
mm
thic
k) a
t roo
m
te
mpe
ratu
re a
nd c
ompa
red
with
the
oret
ical
ap
prox
imat
ions
bas
ed o
n th
e d
epth
-dep
ende
nt t
empe
ratu
re
dis
trib
utio
n (5
.17)
.
02
46
810
-30
-20
-100102030
ln (normalized amplitudes)
(f /
Hz)
1/2
Normalized phases / deg
(f /
Hz)
1/2
02
46
810
-2-1012
100 5 Experimental results
Figu
re 5
.23
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) m
easu
red
for
two
silic
on s
ampl
es (
4 m
m a
nd 2
mm
thic
k) a
t roo
m
te
mpe
ratu
re
02
46
810
-30
-20
-100102030
ln (normalized amplitudes)(f
/ Hz)
1/2
Normalized phases / deg
(f /
Hz)
1/2
02
46
810
-2-1012
101
6. Application to modern heat insulation materials
In this chapter heat insulation materials and systems are analyzed, in which – owing to
voids, an effective low mass density and relatively low thermal conductivity – the radiative
heat transport can also play a certain role. Apart from foam materials [Doermann and
Sacadura, 1995], textiles and ceramics [Mehling, Kuhn, Valentin and Fricke, 1995; Ebert and
Fricke, 1998] such effects also are relevant in multi-layer superinsulation foils consisting of
highly reflective aluminized mylar foils, which serve to isolate the superconducting magnetic
coils, e.g. of particle accelerators, such as the Large Hadron Collider CERN, or of Tokamaks,
such as Tore-Supra. In such large scale applications of superconductivity, the energy costs for
cooling are very high and require optimization of the insulation system.
Radiative heat transport is also very important in carbon-based fibre-reinforced
composites used as heat shields in the defense sector. During the absorption of intense heat
pulses of very short duration these materials can be heated to higher temperatures, where the
radiative heat transport increases considerably and where simultaneously the materials
degradate.
Carbon based fibre-reinforced composites, which combine the properties of two or
more materials and in which the weakness of the matrix material is compensated by the
strength of the fibres, have found wide spread application, mainly due to their good
mechanical behavior, combining excellent elastic properties, crack resistance with a low
specific weight and good thermal shock resistance. The achieved thermophysical properties
depend on the size, type, concentration and orientation of the fibres embedded in the base-
matrix [Jastrzebaki, 1977; Clauser, 1975]. These composite materials also offer mechanical
constructive solutions, which cannot be realized by other materials. Further advantages are
excellent biocompability, low thermal expansion coefficient and form stability at high
temperature, which allow the most diverse applications, e.g. as implants in surgery or as
leading edge heat shields of aircraft or in the reentry of spacecraft.
6.1 Multi-layer Superinsulator Foils
In this section, the results of photothermal measurements of the effective thermal
transport properties, respectively of the shielding properties of multi-layer superinsulation
foils are presented. The examples shown here refer to the multi-layer system Lydall DAM
supplied by CERN, Geneva. The analysed multi-layer system Lydall DAM consists of
external metallized mylar foils of 25 µm thickness both at the front and rear of the multi-layer
102 6 Application to modern heat insulation materials
system, in the interior various sequences of spacer (Cryotherm 234) and metalized foils
follow each other. The internal metalized foils have got a thickness of 6 µm (Figure 6.1). The
measurements have been run on an external foil separately and on multi-layer systems
consisting of the external foil at the front surface and up to seven sequences of spacer and
internal metalized foils. The measurements have been done using a comparatively large
heating spot diameter of about 15 mm with an effective beam power of 1 Watt, so that the
thermal waves can be described by one-dimensional heat propagation and lateral heat losses
along the aluminized foils can be neglected. Similarly to the measurements an silicon in the
transmission configuration of thermal waves, the measurement system namely the path of the
laser beam and the line of sight between sample and detector have separately been covered
with black cartoon to provide good shielding of laser stray light contributions. The
measurements presented here have been carried out at room temperature and normal pressure
using a range of the heating modulation frequency from 0.03 Hz to 20 Hz.
Figure 6.1: Schematic of the sample support and the multi-layer superinsulator foils.
A special sample holder is used to fix the multi-layers of the Aluminized foils. The
sample holder consists of a quartz glass at the front surface and a black plastic film, which
serve to absorb the incident laser light. In the black film, a thermal wave is produced which is
transmitted across the multi-layer samples by conduction and radiation. The aluminized foils
are inserted between the heated black film and a second thin glass plate with a hole in its
center to allow the transmitted radiative flux to be measured by the IR detector. The distance
6.1 Multi-layer Superinsulator Foils 103
between the heated black film and the second glass can be changed by screws, to increase or
reduce the voids space between the various foils. The modulated laser beam travels through
the thin quartz glass plate, which is totally transparent in the visible spectral range, to be
absorbed by the black film. As a result, thermal waves are generated in the heated sample
support. The IR-transmission signal for the heated sample support is measured separately
before inserting the aluminized foils (Figure 6.2) and will be used as a reference. Thus, the
heated sample support acts as modulated heat emitting a modulated heat flux. This modulated
heat flux is attenuated by the different multi-layers of the aluminized foils. Figure 6.2 shows
Figure 6.2: Photothermal transmission signals measured for the heated sample support (O),
signal attenuation measured for various multi-layer systems consisting of external
foil and an increasing number of internal foils (Lydall DAM):
external foil (25 µm)
×× external foil + spacer layer(243) + internal layer (6 µm)
++ external foil + 3 x (spacer layer + internal layer)
∆∆ external foil + 5 x (spacer layer + internal layer)
◊◊ external foil + 7 x (spacer layer + internal layer) + external foil
∗ ∗ noise signal of the equipment without laser heating
10-2
10-1
100
101
102
10-3
10-2
10-1
100
101
102
Am
plitu
de /
mV
(f / Hz)
104 6 Application to modern heat insulation materials
the IR transmission signals, which have been measured at the rear surface of the samples and
which are specially suited to measure the insulation properties, both with respect to radiation
and heat conduction. With the increasing number of insulating layers the IR transmission
signal decrease. The measured signals for different Aluminum foils are shown in Figure 6.2.
The signal denoted by (O ) is measured for the black film sample support separately and is
generally used as reference signal. The signals ( ) measured for the external 25 µm thick foil
are above the noise limit (∗∗) of the equipment over the whole measured frequency range [0.03
Hz, 20Hz], whereas the signals of the other samples containing an increasing number of
spacer and internal layers (××, ++, ∆∆), respectively an additional external foil (25 µm) at the
rear surface (◊◊) reach the noise limit already at about 10 Hz, 4 Hz, 2 Hz, and 1 Hz. By
comparing the IR-transmission signals of these aluminized foils with the silicon samples
measured in transmission (Figure 5.16a), we see that the measured signals for all silicon
samples are above the noise limit of the equipment over the whole measured frequency range
[0.03 Hz, 100Hz] and far away from the background noise, whereas only the very first signals
of the multi-layer foils with a small number of foils and at very low frequencies are above the
noise limit. In part the observed frequency dependence of the measured raw signals (below
about 1 Hz) is due to the frequency characteristics of the used electronic equipment, which
can be eliminated by normalization of the signals with the help of a measured reference
signal, e.g. normalization of the signals ( , ×, +, ∆, ◊) measured for the various insulation
layers by the signal (O) measured for the heated sample support (Comp. Figure 6.4 and 6.5).
In order to get higher signals free of noise for larger modulation frequencies, or when
composite samples containing a larger number of insulation layers are measured or when
measurements are run under reduced air pressure, the used heating power of the laser beam
has to be increased. It is worth to mention that the required values of the Lock-in amplifier
integration time constant and the number of the measured values at each modulation
frequency (integration numbers) depend strongly on the modulation frequency. The range of
frequency used in these measurement are relatively low, from 0.03 Hz to 20 Hz. At very low
frequencies (high periodic time of the modulation heating exposed on the sample) we have to
use a high integration time constant whereas the integration numbers can be relatively low,
because the signal is quite stable once the Lock-in has enough time to register the value of the
measured signal at that modulation frequency. At higher modulation frequency the integration
time constant can be lower but the integration numbers have to be increased, as the signals at
higher modulation frequencies are lower and unstable. Since the signal reaches the noise limit
depending on the number of layers (Figure 6.2), the integration time constant should be higher
6.1 Multi-layer Superinsulator Foils 105
to permit the lock-in amplifier to register more stable signals. Table 6.1 shows the values of
the integration time constant and integration number used in the experiment.
Frequency Time constant (sec.) Integration numbers
f = 0.03 Hz 30 8
0.03 Hz< f < 0.1 Hz 10 8
0.1 Hz< f < 0.5 Hz 10 16
0.5 Hz< f < 1 Hz 10 40
1 Hz< f < 10 Hz 3 40
10 Hz< f < 20 Hz 1 40
Table 6.1: The values of the time constant and integrations number used by the lock-in
amplifier during the measurements.
Figure 6.3 shows the phase values of the transmission signals for the various samples,
which give information about the retardation of the thermal response at the rear surface with
respect to the modulated heating process at the front surface. This means that the positive
phase values in Figure 2, which are due to the chosen representation, have got no direct
physical meaning and that only normalized relative values, e.g. the differences between the
signals measured for the heated sample support (O) and the various insulation layers
( , ×, +, ∆, ◊) give information on the retardation and attenuation of the heat transport
across the increasing number of insulation layers. As expected, the phase of the noise signal
(*) is completely random and the noise limits observed for the measured phases of the various
samples ( , ×, +, ∆, ◊) compare with the noise limits of the amplitudes (Figure 6.2).
106 6 Application to modern heat insulation materials
Figure 6.3: Photothermal phases measured for the heated sample support (O) and phase
shifts measured for various multi-layer systems consisting of external foil and
an increasing number of internal foils (Lydall DAM):
external foil (25 µm)
×× external foil + spacer layer(243) + internal layer (6 µm)
++ external foil + 3 x (spacer layer + internal layer)
∆∆ external foil + 5 x (spacer layer + internal layer)
◊◊ external foil + 7 x (spacer layer + internal layer) + external foil
∗∗ phase of the noise signals
10-2
10-1
100
101
102
-60
0
60
120
180
240
300
360
420
480
Pha
se /
deg
( f / Hz)
6.1 Multi-layer Superinsulator Foils 107
6.1.1 Discussion of results
For the quantitative interpretation, the measured amplitude and phase signals have also
been normalized with the help of reference signals. This allows eliminating the influences of
the measurement system on the measured signals, e.g. of the heating power used, the
frequency characteristics of the electronic equipment, etc (compare chapter 5).
In this work the measurements are performed for multi-layer systems consisting of the
external foil at the front surface and up to seven sequences of spacer and internal metalized
foils, and the measured signals will be compared with the signals of the heated sample
support. Therefore we will use an appropriate normalization process by dividing the signals of
the samples with increasing number foils (mls) by the heated support sample (hss). By
applying the same process as used in section 5.1.2, we obtain for the normalized signals:
[ ]
[ ])()(3
3
)()(
),,(
),,(
)()()(
)()()(
),,(
),,(),,,(
hssmlsi
R
S
RR
SS
hssmlsii
R
Sn
RS
RS
ehssTfT
mlsTfT
xTTT
xTTT
ehssTfS
mlsTfShssmlsTfM
ϕϕ
ψψ
δδ
εγεγ
δ
−
−
′′
=
=
(6.1)
where the parameters, )(Tγ ′ , )(Tsε and 3)(xT can be eliminated as the heated sample
support exists in all measurements. From equ. (6.1), the normalized amplitude nS can be
written as
),,(
),,(
),,(
),,(),,,(
hssTfT
mlsTfT
hssTfS
mlsTfShssmlsTfS
S
S
S
Sn δ
δ==
(6.2)
and the normalized phase
[ ] [ ] nSSSSn hssmlshssmls ϕϕϕψψψ =−=−= )()()()(
(6.3)
Further ahead in this section, we will compare the multi-layer systems with each other, the
normalized amplitudes and phases can be written analogically to equ. (6.2) and (6.3)
))(,,(
))(,,(
))(,,(
))(,,())(,)(,,(
2
1
2
1
21
nS
nS
nR
nSnnn mlsTfT
mlsTfT
mlsTfS
mlsTfSmlsmlsTfS
δ
δ== (6.4)
[ ] [ ] nnSnSnSnSn mlsmlsmlsmls ϕϕϕψψψ =−=−= ))(())(())(())((2121
(6.5)
where 1n and 2n are the number of multi-layers.
Figure 6.4 shows the measured signals (Figure 6.2) in a special normalized form,
which is useful for the quantitative interpretation of transmission signals [Bein, Gibkes,
Mensing, Pelzl, 1994]. First, the signals of the samples containing Lydall DAM insulation
layers ( , ××, ++, ∆∆, ◊◊) are normalized by the signal measured for the heated sample support
108 6 Application to modern heat insulation materials
alone (O). Then the normalized signals are plotted in logarithmic form versus the square root
of the modulation frequency
From Figure 6.4, it is noteworthy that there is a different slope of the normalized amplitudes
at higher frequencies, namely of the signal of the external foil alone in contrast to the
signals (××, ++, ∆∆, ◊◊) obtained for the samples including one or more spacer and internal layers.
0 1 2 3 4 5-7
-6
-5
-4
-3
-2
-1
ln(n
orm
aliz
ed a
mpl
itude
)
( f / Hz )1/2
Figure 6.4: Normalized amplitudes of the samples with increasing number of insulation
layers, where the heated support sample is taken as a reference.
external foil (25 µm)
×× external foil + spacer layer (243) + internal layer (6 µm)
++ external foil + 3 x (spacer layer + internal layer)
∆∆ external foil + 5 x (spacer layer + internal layer)
◊◊ external foil + 7 x (spacer layer + internal layer) + external foil
At first glance one can already justify that the curves (××, ++, ∆∆, ◊◊) coincide with the results
interpreted according to the model of a solid of finite thickness which is homogenous,
compact and opaque for both the visible light and the infrared region [Gu, 1993], and in
which only conductive heat transfer is considered. The slop of the curve of the sample ( ) at
6.1 Multi-layer Superinsulator Foils 109
higher frequencies may be analog to what has been observed for the transmission signals of
silicon (Figure 5.16a), where the signals measured for higher frequencies continue to be far
away from the noise limit can only be possible, if the heat transport across the sample
contains radiative heat transport contributions.
Subsequently, we will first analyze the measured curves according to the model where
radiative heat transport inside the sample can be neglected, the model of a solid of finite
thickness, which is homogenous, compact and opaque for both the visible light and the
infrared region. The thermal wave at the rear surface of the solid can be written according to
equ. (3.92) in chapter 3 as
)(22
),( tidd
ss
osss e
eekI
tdTssss
ωσσσ
ηδ
−
= − (6.6)
The photothermal transmission signal can then be written as
),()()(4),( 3 tdTTTTCfTM sSBs δγσεδ ′= (6.7)
according to equ. (6.4) and (6.5) the normalized amplitudes and phases can be written in the
form [Bein, Gibkes, Mensig and Pelzl, 1994];
2/1
)2exp()2cos(2)2exp(
)2exp()2cos(2)2exp(
),(
),(
−+−
−+−
=
=
sss
rrr
sr
rs
r
sn
fff
fff
e
e
fTM
fTMS
τπτπτπ
τπτπτπηη
δδ
(6.8)
)cot()tanh()cot()(tan1
)cot()tanh()cot()(tan
)(tan)(tan1
)(tan)(tan)(tan
rrss
rrss
rs
rsrs
ffff
ffff
τπτπτπτπ
τπτπτπτπ
+
−=
ΦΦ+Φ−Φ
=Φ−Φ
(6.9)
The quantity τ in equ. (6.8) and (6.9) represents the thermal diffusion time and the indexes s
and r refer to the sample and the reference measurement, respectively. The thermal diffusion
time of reference and sample, respectively, are given by
sr
srsr
d
,
,,
ατ
2= (6.10)
In Figure 6.5 the normalized amplitudes (samples containing Lydall DAM insulation layers
divided by the heated sample support) are presented in a form, which in the extrapolation to
110 6 Application to modern heat insulation materials
very low frequencies gives information on the damping of the steady state heat transport
based on radiation and conduction by the various layers. Thus the steady state heat transport is
reduced by the external 25 µm thick foil ( ) to a value of only 5%, whereas the composite
insulations with the spacer and internal layers (××, ++, ∆∆) and the additional external foil at the
rear surface (◊◊) reduce the steady state heat transport to about 3.7%, 3.2%, 2.6% and 2.5%,
respectively.
10-2 10-1 100 101 1020,00
0,01
0,02
0,03
0,04
0,05
0,06
Nor
mal
ized
am
plitu
de
( f / Hz ) Figure 6.5: Normalized amplitudes of the samples with increasing number of insulation
layers, in comparison with theoretical solutions.
external foil (25 µm)
×× external foil + spacer layer (243) + internal layer (6 µm)
++ external foil + 3 x (spacer layer + internal layer)
∆∆ external foil + 5 x (spacer layer + internal layer)
◊◊ external foil + 7 x (spacer layer + internal layer) + external foil
On the other hand, the decrease of the normalized amplitudes at the higher frequencies, which
can be approximated by
6.1 Multi-layer Superinsulator Foils 111
( ) 2121 //lnln)(ln fe
e
S
SS sr
sr
rs
r
sn ττπ
ηη
−+
≈
= (6.11)
gives the possibility to derive the effective thermal diffusion time and according to equ. (6.10)
the effective thermal diffusivity from the normalized amplitudes.
Equ. 6.11 is a straight line in logarithmic representation, in agreement with the measurements
shown in Figure 6.6, which can be used to determine the thermal effusivity by direct linear
extrapolation from high and intermediate frequencies to the values at f 1/2 = 0.
For a reliable measurements, it is thus necessary to have enough measured data points
available at the intermediate and higher frequencies where the quantity ln(Sn) has to be
approximated by a linear function of f ½. This additional condition is restrictive for the
0 1 2 3 4 5-7
-6
-5
-4
-3
-2
-1
ln(n
orm
aliz
ed a
mpl
itude
)
( f / Hz )1/2
Figure 6.6: Normalized amplitudes of the samples with increasing number of insulation
layers, in comparison with theoretical solutions.
external foil (25 µm)
×× external foil + spacer layer (243) + internal layer (6 µm)
++ external foil + 3 x (spacer layer + internal layer)
∆∆ external foil + 5 x (spacer layer + internal layer)
◊◊ external foil + 7 x (spacer layer + internal layer) + external foil
112 6 Application to modern heat insulation materials
measured system, since the measurements at high frequencies are limited by the noise limit of
the experimental setup. In order to obtain a sufficiently large frequency interval with reliable
measurement data, the beam power of the laser used must be high enough to obtain detectable
temperature amplitudes of the thermal waves at the rear surface of the samples.
The different theoretical approximations with increasing negative slopes in Figure 6.6
refer to effective thermal diffusivities decreasing according to the ratio 0.51: 0.44: 0.19: 0.14:
0.07.
As long as the measured signals are above the noise limits (e.g. f = 1 Hz for the signals
◊, or f = 4 Hz for the signals +), a principal agreement between the theoretical approximations
according to equ. (6.11) and the measured amplitudes can be observed in Figure 6.6,
especially for the curves (××, ++, ∆∆, ◊◊) which reveals that the measurements can be interpreted
based on a purely diffusive model of heat transport, independent of the fact that both
conductive and radiative heat transport contributions may be present. The radiative heat
transport can probably only be identified and measured separately under reduced air pressure
and with appropriate spacer layers, maintaining larger void spaces between the consecutive
internal layers.
For the relative changes of the thermal diffusivity, a more reliable interpretation can be
obtained when the normalized phase signals, equ. (6.9), are considered as shown in figure 6.7,
the interpretation of which depends on a smaller number of parameters, namely only on the
thermal diffusion time of reference and sample as shown in equ. (6.9). The normalized phases
in Fig. 6.7 are approximated with a good agreement of the theoretical solutions according to
equ. (6.9). The different theoretical approximations with increasing negative slopes in Figure
6.7 give information on the effective thermal diffusivity, which decreases with the number of
insulation layers according to the ratio: 0.49: 0.41: 0.19: 0.13: 0.08. The data obtained here
for the phases are more reliable than the values obtained for the amplitudes (Figure 6.6).
From the agreement between measured phases and theory, we can here also conclude, that
radiative heat transport contributions can not be identified under normal air pressure
conditions and without larger voids between the various sublayers, maintained by the
appropriate spacer layers.
As can be seen in Figures 6.6 and 6.7, the signal ( ), measured separately for the aluminized
25 µm thick of the foil, deviate at higher frequencies from what is expected for conductive
heat transport and may be correlated with the additional radiative heat transport. To describe
the complete signal, the radiometric transmission signal (3.112) derived in chapter 3
6.1 Multi-layer Superinsulator Foils 113
⋅−
+⋅+−
−+
+
−′=−
−−−−
2112)()(4)(),( 3
dd
sd
ss
s
ns
dd
s
s
ns
dd
snss
ossSBssss
ss
s
snssns eeEedD
eeB
eeA
kI
TxTTCfTMββ
ββσβσ
β
βσ
βσσ
ηγσεδ
(6.12)
will be used, and the normalized signal for the external foil (ef) can be written as
[ ])()(
),,(
),,(
),,(
),,(),,,( hssefii
R
S
R
Sn
RSehssTfS
efTfS
hssTfM
efTfMhssefTfM ψψ
δδ
δ −== (6.13)
0 1 2 3 4 5-360
-300
-240
-180
-120
-60
0
Nor
mal
ized
pha
se /
deg
( f / Hz )1/2
Figure 6.7: Normalized phases of the samples with increasing number of insulation
layers, in comparison with theoretical solutions
external foil (25 µm)
×× external foil + spacer layer (243) + internal layer (6 µm)
+ + external foil + 3 x (spacer layer + internal layer)
∆∆ external foil + 5 x (spacer layer + internal layer)
◊◊ external foil + 7 x (spacer layer + internal layer) + external foil
114 6 Application to modern heat insulation materials
0 1 2 3 4 5-7
-6
-5
-4
-3
-2
-1
∞βs
βs3
βs2
βs1
βs1<βs2
<βs3
ln(n
orm
aliz
ed a
mpl
itude
)
( f / Hz )1/2
Figure 6.8: Normalized amplitude of the 25-µm external foils, in comparison with
theoretical solutions considering radiative heat transport contributions.
Figure 6.8 shows example of theoretical approximation based on equ. (6.13) and
(6.12), in comparison the normalized amplitude for the external foil, which show good
agreement between theory and experiment at a intermediate value of the absorption
coefficient βs2 for the external foil.
To demonstrate the effect of additional spacer and internal layers on the heat
insulation problem, normalized signals of samples with different numbers of spacer layer and
internal layer, normalized against each other are shown in Figure 6.9 and 6.10, in comparison
with theoretical approximations according to equ. (6.4) and (6.5). From the extrapolation at
very low frequencies, the damping factor for steady state radiative and conductive heat
transport is about 90% for the system (external foil + 3(space layer + internal layer)) in
comparison to the system (external foil + (space layer + internal layer)), about 80% for the
system (external foil + 5(space layer + internal layer) and 72% for the complete multi-layer
system Lydall DAM in comparison to the system consisting of (external foil + (space layer +
internal layer))
6.1 Multi-layer Superinsulator Foils 115
Figure 6.9: Normalized amplitudes for different multi-layer insulations consisting of external
foil, and different numbers of spacer layer and internal layer (Lydall DAM):
+ ef + 3 x (sl + il) / ef + sl + il
∆ ef + 5 x (sl + il) / ef + sl + il
◊ ef + 7 x (sl + il) + ef / ef + sl + il
The signals increasing again at higher frequencies in Figure (6.9) or Figure (6.19) are due to
the noise contributions.
The different theoretical approximations with increasing negative slopes in Figure
(6.9) and (6.10) give information on the effective thermal diffusivity, which decreases with
the number of insulation layers according to the ratio: 0.43: 0.31: 0.17.
According to these measurements, which have been run under the conditions of
ambient temperature and pressure, the heat transport across the multi-layer superinsulation
foil system lydall DAM is mainly diffusive. This is probably due to the air between the
various layers and due to the fact that the various insulation layers are in narrow contact with
each other. In order to identify radiative heat transport under cryo conditions and to separate it
from the diffusive heat transport, multi-layer system with larger voids between the various
insulation layers should be measured.
0 1 2 3-4
-3
-2
-1
0
1
ln(n
orm
aliz
ed a
mpl
itude
)
( f / Hz )1/2
116 6 Application to modern heat insulation materials
Figure 6.10: Normalized photothermal phases for different multi-layer insulations consisting
of external foil, and different numbers of spacer layer and internal layer (Lydall
DAM):
+ ef + 3 x (sl + il) / ef + sl + il
∆ ef + 5 x (sl + il) / ef + sl + il
◊ ef + 7 x (sl + il) + ef / ef + sl + il
To improve these measurements from the point of view of Photothermal Radiometry
(PTR), a higher heating power of the laser beam should be used and the lock-in integration
time constant and the number of integrations should probably also to be increased.
0 1 2 3-240
-180
-120
-60
0
60
120
Nor
mal
ized
pha
se /
deg
( f / Hz )1/2
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 117
6.2 IR transparency and radiative heat transfer in fibre-reinforced
materials at higher temperatures
In recent years, carbon based fibre-reinforced composite materials have found special
interest due to their wide spread applications, e. g. in communication systems, medical
instruments, computers, and in militaries. This is mainly due to their good conduct either in
mechanical, electrical or thermal properties. Such material have been studied at room
temperature earlier [Bolte, 1995], when in depth-dependent measurements of the thermal
properties of carbon and glass fibre reinforced materials based on the radiometric detection of
thermal waves, the layer structure was analyzed. Figure 6.11 shows the phase lags as
measured for various composite materials at room temperature, which in general consist of
thin woven cover layers, two-dimensional surface-parallel carbon fibres and carbon matrix
materials. They differ by different types of carbon fibres, different carbon matrix materials
and by the fibres of the cover layer (polyester, glass or carbon). Another part of the samples is
sandwich structures consisting of cover layers of protective varnish, surface-parallel glass or
carbon fibre structures and foamed bulk material. Among the variety of measured samples
four families of distinct frequency-dependent photothermal profiles can be distinguished from
the measurement IM400, T800, V1, V2, V4, R, and V5. To understand the behavior
of these families, Figure 6.12 shows the normalized phases of the samples R, T800 and V1,
which are plotted versus the square root of the modulation frequency. The interpretation of
these plots [Bolte, 1995] showed that one could approach the behavior of the normalized
phases on the bases of different layer solution, with well distinct thermal and optical
parameters for the different layers [Bein et al., 1995].
For these fibre-reinforced materials, which are candidate material for heat pulse absorption
and heat shields in the defence sector [Simon, 1996] the thermal properties at higher
temperatures are mainly of interest. Thus the composite materials R, V1 and V4, which are
representative of each family, have been selected and in systematic measurements based on
IR detection of thermal waves, frequency-dependent measurements at higher temperatures
have been done.
118 6 Application to modern heat insulation materials
RV5
- V2
- V4
- IM400
- T800
- V1
x
+
Figure 6.11: Photothermal phases, measured for a variety of fibre reinforced materials as
function of the modulation frequency [Bolte, 1995].
R
V1
T800
Figure 6.12: Normalized measured phases of the samples V1 and R, plotted versus the square
root of the modulation frequency.
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 119
6.2.1 Measurements of reference materials
Measurements of the fibre-reinforced composites are presented here together with
measurements of reference samples, which are necessary for the physical interpretation of the
results. Figure 6.13a and 6.13b show the photothermal amplitude and phases as a function of
frequency in the range from 0.5 Hz up to 5 kHz at room temperature for the materials. In part
the observed frequency dependence of the measured raw signals (below about 1 Hz) is due to
the frequency characteristics of the used electronic equipment, which can be eliminated by
normalization of the signals with the help of a measured reference signal.
For the materials Sigradur, neutral glass and V2A steel, the frequency-dependent of the
raw signal amplitudes and phases in Figure 6.13a and 6.13b show smooth decreasing curves,
which correspond to homogeneous compact samples. The fibre-reinforced material V4 shows
a very characteristic deviation from the relatively smooth curves of the reference samples.
Thus one can immediately conclude that V4 has a depth-dependent thermal structure, which is
due to the mechanical structure of the fibre-reinforced material. The material V4 consists of a
glass fibre woven cover layer, two-dimensional surface-parallel carbon fibers and epoxy
matrix materials [Bolte, 1995].
6.2.2 Temperature-dependent measurement on fibre reinforced material
In the present work frequency-dependent measurements of fibre-reinforced composite
materials are done at higher temperatures. A comparatively large heating spot diameter has
been used for these measurements with an effective beam power of 1 Watt, so that a model of
one-dimensional heat propagation can describe the thermal waves. The measurements have
been done as a function of the heating modulation frequency in the range from 0.03 Hz up to
10 kHz at various fixed temperatures. The high-temperature cell (Figure 2.5) was used to keep
the samples at higher temperatures in order to avoid thermal erosion in the presence of
oxygen. Due to the temperature stability limit of the samples, the actual measurements have
been done at constant sample temperature between room temperature and 250 °C. The
measurement system, schematically shown in Figure 2.1, allows to measure “ thermal waves
in reflection”, where the thermal wave is excited and detected at the same surface.
The measurements have been done for the following samples.
a) Fibre-reinforced material V1, with the thickness of 6,03 mm
b) Fibre-reinforced material V4, with the thickness of 6,99 mm
c) Fibre-reinforced material R, with the thickness of 4,15 mm
120 6 Application to modern heat insulation materials
Figure 6.13 a+b: Photothermal amplitudes (a) and phases (b) measured as a function of
frequency at room temperature for a fibre-reinforced material and different
reference materials (glass carbon, neutral glass and V2A steel).
10-1 100 101 102 103 104120
180
240
300
360
420
Am
plitu
des
/ mV
f / Hz
Pha
ses
/ deg
f / Hz
10-1 100 101 102 103 10410
-1
100
101
102
103
Sigradur Neutral glass V
2A steel
Fibre reinforced material V4
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 121
Figure 6.14 a+b: Photothermal amplitudes (a) and phase lags (b) with respect to the heating
modulation measured at different sample temperatures for the sample of
homogeneous V2A steel.
10-1
100
101
102
103
104
105
150
200
250
300
350
Pha
se /
deg
f / Hz
10-1
100
101
102
103
104
105
10-2
10-1
100
101
102
RT 100°C 250°C
Am
plitu
de /
mV
f / Hz
122 6 Application to modern heat insulation materials
Figures 6.14 to 6.18 show the measured raw signals, the photothermal amplitudes and
the phase lags with respect to the heating modulation registered for the carbon-based fibre
reinforced heat shield materials (V1, V4, R), glassy carbon (Sigradur) and V2A steel as
function of the modulation frequency at the average sample temperatures of 20 °C, 100 °C ,
200 °C and 250 °C. We can see that in all cases the signal amplitudes increase with increasing
temperature and that this is in line with the T3 dependence of equ. (3.105). The decay of the
amplitude at very low frequencies is due to the frequency characteristics of the electronic
components of the measurement system. All materials except V2A steel show a temperature-
dependent split of the measured phase lags at higher frequencies. For the sample of
homogenous glassy carbon, splitting starts at about 10 Hz reaching the maximum difference
at about 3 kHz. For the fibre-reinforced composites (V1, V4, R), splitting starts at a frequency
of about 100 Hz and reaches its maximum difference at 5 kHz. In order to interpret these
measurements, the normalization procedure described in section 5.1.2 will be used, which
allows eliminating the frequency-dependence of the measurements system.
By comparing Figure 6.14b with Figure 6.15b, 6.16b, 6.17b and 6.18b, we can see the
difference of the raw phase lags for different average sample temperatures at higher
frequencies. It is evident that V2A steel is totally opaque for the visible and infrared
spectrum. As will be shown, the temperature dependent splitting of the measured phase lags at
higher frequencies can be interpreted by temperature dependent changes of a characteristic
combined thermo-optical material parameter describing the effects of IR transparency for
these materials.
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 123
Figure 6.15 a+b: Photothermal amplitudes (a) and phase lags (b) with respect to the heating
modulation measured at different sample temperatures for the sample of
homogeneous glassy carbon (Sigradur).
10-2
10-1
100
101
102
103
104
-150
-100
-50
0
50
100
150
Pha
se /
deg
f / Hz
10-2
10-1
100
101
102
103
104
10-2
10-1
100
101
102
RT 100°C 200°C
Am
plitu
de /
mV
f / Hz
124 6 Application to modern heat insulation materials
10-2
10-1
100
101
102
103
104
-150
-100
-50
0
50
100
150
Pha
se /
deg
f / Hz
10-2
10-1
100
101
102
103
104
10-2
10-1
100
101
102
RT 100°C 200°C
Am
plitu
de /
mV
f / Hz
Figure 6.16: a+b: Photothermal amplitudes (a) and phase lags (b) with respect to the heating
modulation measured at different sample temperatures for the carbon-based
fibre reinforced heat shield material V1.
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 125
10-2
10-1
100
101
102
103
104
-150
-100
-50
0
50
100
150
Pha
se /
deg
f / Hz
10-2
10-1
100
101
102
103
104
10-2
10-1
100
101
102
RT 100°C 200°C
Am
plitu
de /
mV
f / Hz
Figure 6.17: a+b: Photothermal amplitudes (a) and phase lags (b) with respect to the heating
modulation measured at different sample temperatures for the carbon-based
fibre reinforced heat shield material V4.
126 6 Application to modern heat insulation materials
10-2
10-1
100
101
102
103
104
-100
-50
0
50
100
150
200
Pha
se /
deg
f / Hz
10-2
10-1
100
101
102
103
104
10-2
10-1
100
101
102
RT 100°C 200°C
Am
plitu
de /
mV
f / Hz
Figure 6.18 a+b: Photothermal amplitudes (a) and phase lags (b) with respect to the heating
modulation measured at different sample temperatures for the carbon-based
fibre reinforced heat shield material R.
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 127
6.2.3 Interpretation of the Temperature-dependent measurement on
fibre reinforced materials
For glassy carbon and the fibre reinforced materials we can work with a semi-infinite
model, in contrast to silicon (section 5.1.3) with its rather extended IR absorption length
( 51 ≈−sβ mm). Thus, for the quantitative interpretation of the signals we can use the theory
developed in equ. (3.106), chapter 3.
ti
s
nsns
s
ns
s
nsns
s
nsns
s
ns
s
ns
s
nsns
s
ns
nss
osssSBssss e
R
R
R
R
k
ITxTTCfTM ω
βσ
βσ
βσ
βσ
βσ
βσ
βσ
βσ
ση
γσεδ
322
322
3
21
1
1
21
121
1
1
2).()(4)(),(
−
−
+
+
−
−
+
++
′=
(6.14)
Equ. 6.14 is the radiometric signal for a semi-infinite solid, taking into consideration the
radiative and conductive heat transport. Apart from the photothermal efficiency sη , there are
three relevant parameters that influence the amplitude of the photothermal signal, which are
nssk σ , ))(( 3sxTa and ssP βα 2/1
1 = , and only one relevant parameter that influences the phase
of the photothermal signal, which is ssP βα 2/11 = . The parameter nsR , which is the ratio of
the radiative heat loss to the conductive heat transport of the solid at the interface gas / solid,
can be taken from equ. (3.68)
nss
ssns k
xTaR
σ))0(( =
= (6.15)
where nsσ is given according to equ. (3.66) by
22222
421212
−
+
+−
+
+=
s
os
s
osns
s
os
s
osns
s
os
s
ns RRβσ
βσ
βσ
βσ
βσ
βσ
(6.16)
The investigated materials in this chapter are only slightly transparent in the IR spectrum,
which allows to use the former approximation 1/ <sos βσ with the result osns σσ = where
sos
fi
απ
σ )1( += (6.17)
128 6 Application to modern heat insulation materials
The real amount of nsR can then be calculated to be about
s
sSBsns
ckf
TR
)(
)0(22 3
ρπ
σε= (6.18)
where )0(sT is the time-averaged temperature of the sample surface, which has been measured
by inserting thermocouples near the heating spot on the front surface of the sample. Another
thermocouple has been inserted in the coax-wire-heated-sample support to measure the
temperature of the sample holder rsT , which is in good thermal contact with the rear surface
of the sample. Table 6.1 shows the measured front and rear surface temperatures of the
Sigradur sample and of the fibre-reinforced sample V1 during the photothermal
measurements and the corresponding values of the real amount of nsR for different
frequencies. From Table 6.1, one can see that the values of nsR are small in comparison to
one and that in first order equ. (6.14) can be simplified to
ti
s
nsnss
ossSBssss e
k
ITxTTCfTM ω
βσσ
ηγσεδ
+′=
1
12
).()(4)(),( 3 (6.19)
From equ. (6.19), the real solution can be derived
+
−−
+
+
′=1
1arctan
4cos
112).()(4)(),(
2
3
µβπ
ω
µµβ
βσ
ηγσεδ
s
s
s
nss
ossSBssss t
k
ITxTTCfTM
(6.20)
For an opaque sample in the infrared spectrum, likeV2A steel ∞→sβ , the measured signal
can be written as
ti
nss
ossSBssss e
k
ITxTTCfTM ω
ση
γσεδ2
).()(4)(),( 3 ′= (6.21)
By comparing the signals measured for an IR-opaque solid with those of a slightly IR
translucent solid, we can get for the normalized amplitude
s
s
ttransparen
opaquen S
SS
β
µµβ
112
+
+
== (6.22)
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 129
|Rns| / 10-6
f / Hz
Trs / °C
Tfs/ °C
100 1000 4000
25 37 0,19 0.06 0,03
100 82 2 0,66 0,33
Sigradur
268 216 38 12 6
24 53,31 1,6 0,52 0,26
99,4 96 9,5 3 1,5
V1
268,6 218 110 35 18
Table 6.1: Stationary sample temperatures of glassy carbon and fibre-reinforced composite
during measurements and corresponding values nsR . 100
and for the normalized phases
+
=−=1
1arctan
µβϕϕϕ
sttransparenopaquen (6.23)
where the thermal diffusion length is fs
πα
µ = . Equ. (6.22) and (6.23) can be rewritten as
( )2
221
ssssttransparen
opaquen
ff
S
SS
βα
πβα
π++== (6.24)
and
+=−=
fs
ttransparenopaquenπβα
ϕϕϕ/1
1arctan (6.25)
We see from equ. (6.24) and (6.25) that the effect of IR transparency depends on the
combined quantity ss βα , which has already been discussed in section 5.1.3.
The normalization process follows the procedures already described in section 5.1.2.
First the different samples measured at the same temperature can be normalized against each
other
130 6 Application to modern heat insulation materials
[ ]
[ ]))(())((
))(())((
))(,,(
))(,,(
)(
)(
))(,,(
))(,,())(,)(,,(
RSi
sR
sS
RR
SS
RSi
sR
sSssn
sSsS
sSsS
eRTfT
STfT
T
T
eRTfS
STfSRSTfM
βϕβϕ
βψβψ
βδβδ
εε
ββ
ββδ
−
−
=
=
(6.26)
and secondly the same sample measured at different temperatures can be normalized
[ ] [ ])()(
2
13
222
3111)()(
2
12,1
2121
),(),(
)()()()()()(
),(),(
),( TTi
S
S
S
STTii
S
Sn
SSSS eTfTTfT
xTTTxTTT
eTfSTfS
TTfM ϕϕψψ
δδ
εγεγ
δ −−
′′
== (6.27)
where the measurements at room temperature are always taken as a reference. In this case the
information obtained from the normalized phases is independent of the combined quantities
)(Tγ ′ , )(Tε and 3)(xT .
Figure 6.19a and 6.19b show theoretical calculation for normalized phases and
amplitudes at different values of the IR absorption coefficient for a slightly transparent
sample, where the signal for the totally opaque solid is used as reference signal. The thermal
properties of glassy carbon (Sigradure) are used for the calculations. From equ (6.23) and
(6.24), one can see that different IR absorption coefficient can produce large measurable
effect which can be found if the thermal diffusion length µ and the absorption length in the
infrared 1−sβ are comparable. This means, mainly for higher modulation frequencies the
effects become visible.
Figure 6.20a and 6.20b show the normalized phase and amplitude for Sigradur with V2A steel
as IR-opaque reference, in comparison to the theoretical approximation. From the obtained IR
absorption length =−1sβ 3.33 µm one can conclude that glassy carbon can be considered to be
nearly opaque for IR radiation at 20 °C. Here it has to be mentioned that the value =sβ 3⋅105
m-1 is characteristic for the detected IR wavelength interval, 2 µ m – 12 µ m.
In Figure 6.21a and 6.21b, the normalized phases and amplitudes are shown for the Sigradur
sample at higher temperatures, for which the measurements at room temperature are used as
reference signal. The normalized amplitudes shown in Figure 6.21b reach constant values at
low frequencies, and these values vary according to the dependence on the stationary
temperature and the thermal effusivities of the samples measured at different temperatures.
The major factor playing an important role for the normalized signals [Sigradur (RT) /
Sigradur ( 100 °C)] and [Sigradur (RT) / Sigradur ( 200 °C)] at low frequencies is the ratio
33 / TRT TT of the temperatures. If we compare the phases measured for glassy carbon at higher
average sample temperature with those measured at room temperature, we see that the
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 131
Figu
re 6
.19
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) fo
r sl
ight
ly I
R tr
ansp
aren
t sam
ples
, whe
re a
n IR
opa
que
sam
ple
is
ta
ken
as r
efer
ence
.
100
101
102
-50510152025303540
Normalized Phase / deg
(f / H
z) 1
/2
10-2
10-1
100
012345678910
βs /
m-1 =
2*1
04
βs /
m-1 =
5*1
04
βs /
m-1 =
2*1
05
βs /
m-1 =
5*1
05
Normalized amplitude(f
/ Hz)
-1/2
132 6 Application to modern heat insulation materials
Figu
re 6
.20
a+b:
Nor
mal
ized
pha
se (
a) a
nd a
mpl
itude
(b)
for
gla
ssy
car
bon
(Sig
radu
r) w
ith V
2A s
teel
as
ref
eren
ce, i
n
co
mpa
riso
n to
a th
eore
tical
app
roxi
mat
ion,
whe
re a
val
ue o
f β
= 3
⋅105 m
-1 is
obt
aine
d fo
r Si
grad
ur a
t roo
m
te
mpe
ratu
re.
100
101
102
-30
-15015304560
Normalized phase / deg
(f /
Hz)
1/2
10-2
10-1
100
0,00
0,05
0,10
0,15
Normalized Amplitude
( f /
Hz)
-1/2
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 133
normalized phases increase continuously above about 16 Hz (Figure 6.21a). A similar effect
becomes visible in the signal amplitudes. This effect can be interpreted according to the
model of IR transparency increasing with temperature according to equ. (6.23) and (6.24).
This means, the combined quantity ss βα plays the important role in these variations.
Figure 6.22a and 6.22b show the normalized phase and amplitude for [Sigradur (RT) /
Sigradur (100 °C)], in comparison with theoretical curves, where we use the value of the IR
absorption coefficient at room temperature =sβ 3⋅105 m-1, while varying the value of the
absorption coefficient at 100 °C. From Figure 6.22a and 6.22b, we have good agreement
between theory and experiment, if the value of the IR absorption coefficient at higher
temperatures is comparably below the value at room temperature. According to literature data
the thermal diffusivity at these temperatures is approximately constant [HTW GmbH, 1999].
Thus we can conclude, the decreasing value of the combined parameter ss βα at higher
temperatures is mainly due to the absorption coefficient alone.. Figure 6.23a and 6.23b show
the normalized phases and amplitudes for the normalization [Sigradur (RT) / Sigradur (100
°C)] and [Sigradur (RT) / Sigradur (200 °C)], in comparison to the theoretical
approximations. By using the value of the absorption coefficient obtained for Sigradur at
room temperature as reference, we can obtain information on the absorption coefficient at
higher temperatures
The following values of the effective absorption coefficient have been obtained, by
holding the thermal diffusivity constant 610*2.4 −=sα m2 / s:
Sample temperature / °C /Sigradurβ m-1
24 °C (RT) 3,0*105
100 5,1*104
200 2,6*104
Table 6.2: Effective absorption coefficient in the infrared spectrum for glassy carbon
(Sigradur) at different temperatures.
Thus we can conclude that the absorption coefficient decreases with increasing
temperature, which means that the sample become transparent to thermal radiation with
increasing temperature. This may be due to two reasons: (i) the material parameter sβ
decreases with temperature and (ii) the IR radiation is shifted with higher sample
134 6 Application to modern heat insulation materials
temperatures towards the detectable wavelength interval, which is limited to 2 µm < λ < 13
µm by the used IR lenses and detector.
The deviation between theory and experiment which still exist especially at higher
frequencies can be due to the simplification that a wavelength-independent IR absorption
coefficient is considered in the theory. This question can be resolved in additional
measurements by using IR filters with definite smaller wavelength intervals.
Figure 6.24 shows the normalized amplitude for V2A steel against the carbon fibre-
reinforced material V1 at room temperature, where we have two regions one at low
frequencies and the other at high frequencies, which are related to the 2-layer structure of the
material V1. The modulation frequency corresponds to the thermal diffusion length, which
increases from low values at the very surface to higher values with increasing penetration
depth, 2/1−∝ fx , to reach only deeper below the surface the characteristic value of the bulk
material. Therefore, low frequencies give information about the bulk matrix, which consists
from polyamide, and high frequencies give information on the characteristic properties of the
cover layer, which consists of a thin woven cover layer of two dimensional surface-parallel
carbon fibres. From Figure 6.24, the values of the absorption coefficients obtained from the
theoretical approximation are 4103.2 ⋅=sβ m-1 for the polyamide matrix at low frequencies
and 61024.1 ⋅=sβ m-1 for the carbon fibre layer at high frequencies. It is known that these
carbon fibre-reinforced materials are nearly IR opaque at 20 °C [Simon, 1996]. In Figure 6.25a and 6.25b the measurements at room temperature are used as reference for the
measurements at higher temperatures. The normalized phases in figure 6.25a show nearly no
changes up to about 200 Hz and strongly increase above 400 Hz, especially at the higher
average sample temperature. Similar effects become visible in the signal amplitudes at higher
frequencies, whereas the differences between the various measurements at low frequencies is
mainly due to the 3)(xT dependence of the signal (equ. 6.14). After determining the value of
the absorption coefficient from Figure 6.24, it is possible to interpret the measured
temperature-dependent changes quantitatively. Figure 6.26a and 6.26b show the normalized
phase and amplitude for [V1 (RT) / V1 (100 °C)], in comparison with theoretical curves,
where we use the value of the IR absorption coefficient obtained for the cover layer at room
temperature =sβ 1.24⋅106 m-1 (Figure 6.24), while varying the value of the IR absorption
coefficient at 100 °C. The deviations between theory and experiment in Figure 6.26a and
6.26b are probably due to layer structure of the sample V1 that is not considered in the
approximation.
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 135
Figu
re 6
.21
a+b:
Nor
mal
ized
pha
ses
(a)
and
am
plitu
des
(b)
for
gla
ssy
car
bon
(Sig
radu
r), w
here
the
nea
rly
IR
-opa
que
Si
grad
ur s
ampl
e at
20
°C is
use
d as
ref
eren
ce f
or n
orm
aliz
atio
n.
100
101
102
-30
-20
-100102030405060
Normalized phase / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
RT /
100°
C R
T /
200°
C R
T / 1
00°C
RT
/ 200
°C
Normalized amplitude
(f / H
z) -1
/2
136 6 Application to modern heat insulation materials
Figu
re 6
.22
a+b:
Com
pari
son
of th
e no
rmal
ized
pha
ses
(a)
and
ampl
itude
s (b
) w
ith th
eore
tical
app
roxi
mat
ions
for
gla
ssy
carb
on
w
here
the
near
ly I
R-o
paqu
e Si
grad
ur s
ampl
e w
ith β
s =
3. 105 m
-1 a
t 20
°C is
use
d as
ref
eren
ce.
100
101
102
-30
-20
-100102030405060
Normalized phase / deg
(f /
Hz)
1/2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
RT
/ 100
°C
βs /
m-1 =
3*1
04
βs /
m-1 =
5*1
04
βs /
m-1 =
8*1
04
Normalized amplitude(f
/ Hz)
-1/
2
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 137
Figu
re 6
.23
a+b
: Nor
mal
ized
pha
ses
(a)
and
am
plitu
des
(b)
for
gla
ssy
car
bon
(Si
grad
ur),
whe
re
the
nea
rly
IR
-opa
que
S
igra
dur
sam
ple
at 2
0 °C
is u
sed
as
ref
eren
ce f
or n
orm
aliz
atio
n, i
n c
ompa
riso
n w
ith t
he t
heor
etic
al
a
ppro
xim
atio
ns, β
s (10
0 °C
) =
5.1. 10
4 m-1
, βs (
200
°C)
= 2.
6. 104 m
-1.
100
101
102
-30
-20
-100102030405060
100
°C /
RT
200
°C /
RT
Normalized phase / deg
(f/ H
z)1/
2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
100
°C /
RT
200
°C /
RT
Normalized amplitude(f
/ Hz)
-1/2
138 6 Application to modern heat insulation materials
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
Nor
mal
ized
am
plitu
de
(f / Hz)-1/2
Figure 6.24: Normalized amplitude for the fibre-reinforced material V1 with V2A steel as
reference, in comparison with theoretical approximations, different IR
absorption coefficient are assumed for the surface layer and the bulk material.
Figure 6.27a and 6.27b show the normalized phases and amplitudes for the
normalization’s [V1 (RT) / V1 (100 °C)] and [V1 (RT) / V1 (200 °C)], which are compared
with theoretical approximations based on eq’s. 6.23 and 6.24. The combined parameter
ss βα decreases significantly with temperature.
From thermal wave measurements in transmission, we know that the thermal
diffusivity of these composites only weakly decreases with temperature,
1.1)100(/)( =°CRT ss αα and 25.1)200(/)( =°CRT ss αα [Simon, 1996]. Thus we can
conclude, that the decreasing value of the combined parameter ss βα at higher temperatures
is mostly due to a decreasing effective absorption coefficient, which means that the material
V1 becomes more transparent to IR radiation with increasing temperature. This may be due
also to two reasons: (i) the material parameter sβ decreases with temperature and (ii) the IR
radiation is shifted with higher sample temperatures into the detectable wavelength interval
(Figure 2.2).
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 139
By considering the variation of the thermal diffusivity with temperature in the
combined quantity ss βα, the following values of the effective absorption coefficient, have
been obtained, here the value of the thermal diffusivity at room temperature is
7105.4 −⋅=sα m2 / s.
Sample temperature / °C /1Vβ m-1
24 °C (RT) 1,24*106
100 3,1*105
200 2,2*105
Table 6.3 Effective IR absorption coefficients for the carbon based fibre-reinforced
composite V1 at different temperatures
Additionally other fibre-reinforced materials have been investigated, V4 which
consists of a carbon fibre HTA7 with an epoxy matrix, and the material R, which consists of
cover layers of protective varnish and surface-parallel carbon fibre structures and foam
material in the bulk. Figure 6.28a and Figure 6.28b show the normalized phases and
amplitudes for the normalization [V4 (RT) / V4 (100 °C)] and [V4 (RT) / V4 (200 °C)]. The
normalized phases in figure 6.28a show nearly no changes up to about 100 Hz and strongly
increase above 100 Hz, especially at the higher average sample temperature. Similar effects
become visible in the signal amplitudes. The behavior shown by the material V4 corresponds
to the effects observed for the glassy carbon and V1. Figure 6.29a and Figure 6.29b show the
normalized phases and amplitudes for the normalization [R (RT) / R (100 °C)] and [R (RT) /
R (200 °C)]. The normalized phases in figure 6.29a show nearly no changes up to about 150
Hz and strongly increase above 150 Hz, especially at the higher average sample temperature.
The normalized amplitudes in contrast have another behavior: at high frequencies the
normalized amplitudes drop considerably in comparison to other carbon materials. This
astonishing effect perhaps can be explained by the evolution of this experiment. Here, first the
measurements had been done at room temperature then at 200 °C and finally at 100 °C.
During this process the sample R eroded thermally at about 160 °C. Owing to the fact that the
char layer of the eroded surface which consists of small particle of soot is discontinuous, its
effusivity value decreases considerably in comparison to the value of the continuous varnish
layer. This effect probably contributes to the smaller value of the normalized amplitudes at
the very high frequencies, as shown in Figure 6.29b, whereas in the phase the effect of IR
transparency at high frequencies may even be increased (Figure 6.29a).
140 6 Application to modern heat insulation materials
Figu
re 6
.25
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) fo
r th
e ca
rbon
-bas
ed f
ibre
rei
nfor
ced
com
posi
te V
1,
whe
re t
he
si
gnal
s m
easu
red
at 2
0 °C
are
use
d as
ref
eren
ce f
or n
orm
aliz
atio
n.
100
101
102
-30
-20
-100102030405060
RT/
100
RT/
200Normalized phase / deg
(f / H
z)1/
2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
RT/
100
RT/
200
Normalized amplitude
(f / H
z)-1
/2
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 141
Figu
re 6
.26
a+b:
Com
pari
son
of
the
nor
mal
ized
pha
ses
(a)
and
am
plitu
des
(b)
with
the
oret
ical
app
roxi
mat
ions
for
the
f
ibre
-rei
nfor
ced
com
posi
te V
1, w
here
the
near
ly I
R-o
paqu
e sa
mpl
e at
20
°C, β
s = 1
.24. 10
6 m-1
, is
used
as
r
efer
ence
.
100
101
102
-30
-20
-100102030405060
Normalized phase / deg
(f / H
z)1/
2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
β s(
100
°C) /
m-1 =
1.8
6*10
5
β s(
100
°C) /
m-1 =
3.1
0*10
5
β s(
100
°C) /
m-1 =
4.6
5*10
5
Normalized amplitude
(f / H
z)-1
/2
142 6 Application to modern heat insulation materials
Figu
re 6
.27:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (a
) fo
r th
e ca
rbon
-bas
ed f
ibre
-rei
nfor
ced
com
posi
te V
1, w
here
the
sig
nals
mea
sure
d at
20
°C a
re u
sed
as r
efer
ence
sig
nals
for
nor
mal
izat
ion,
in c
ompa
riso
n to
the
theo
retic
al p
prox
imat
ions
,
βs (
100
°C)
= 2.
95. 10
5 m-1
, βs (
200
°C)
= 2.
0. 105 m
-1.
100
101
102
-30
-20
-100102030405060
RT/
100°
C R
T/20
0°C
Normalized phase / deg
(f / H
z)1/
2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
RT/
100°
C R
T/20
0°C
Normalized amplitude
(f / H
z)-1
/2
6.2 IR transparency and radiative heat transfer in fibre-reinforced materials at higher temperatures 143
Figu
re 6
.28
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) fo
r th
e ca
rbon
-bas
ed f
ibre
rei
nfor
ced
com
posi
te V
4, w
here
the
si
gnal
s m
easu
red
at 2
0 °C
are
use
d as
ref
eren
ce f
or n
orm
aliz
atio
n.
100
101
102
-30
-20
-100102030405060
RT/
100
RT/
200
Normalized phase / deg
(f / H
z)1/
2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
RT/
100
RT/
200
Normalized amplitude
(f / H
z)-1
/2
144 6 Application to modern heat insulation materials
Figu
re 6
.29
a+b:
Nor
mal
ized
pha
ses
(a)
and
ampl
itude
s (b
) fo
r th
e ca
rbon
-bas
ed f
ibre
rei
nfor
ced
com
posi
te R
, whe
re th
e si
gnal
s
m
easu
red
at 2
0 °C
are
use
d as
ref
eren
ce f
or n
orm
aliz
atio
n.
100
101
102
-30
-20
-100102030405060
RT/1
00 R
T/2
00Normalized phase / deg
(f / H
z)1/
2
10-2
10-1
100
0,0
0,4
0,8
1,2
1,6
2,0
RT/1
00 R
T/2
00
Normalized amplitude(f
/ Hz)
-1/2
6.3 Measurements of fibre-reinforced composites with different fibre concentrations 145
6.3 Measurements of fibre-reinforced composites with different
fibre concentrations
Here, the measurements are presented for fibre-reinforced composites, which consist
of polycarbonate matrix material with a systematic variation of the carbon fibre content,
namely with 5%, 10% and 20% fibre concentrations. Figure 6.30a and 6-30b show the
photothermal amplitudes and the phase lags as function of frequency in the range from 1 Hz
up to 10 kHz at room temperature. The photothermal amplitudes of the samples with 5% and
10% fibre concentration show at low frequencies nearly the same magnitudes, while the
amplitudes of the sample with 20% fibre concentration have got a clearly lower magnitude at
low frequencies. This can be explained by the fact that a sample with higher fibre content will
have a higher effusivity value and simultaneously reach a smaller temperature and
temperature gradient due to laser heating during the measurements. At higher frequencies the
amplitude of the sample with 20% fibre content has a smaller negative slope and thus behave
like the signals of a more opaque sample, whereas the signals of the sample with smaller fibre
content show a larger negative slopes, similar to samples, which are more translucent in the
IR spectrum. To analyze the effect of fibre concentration at ambient temperature, the signals
obtained for the different concentration have been normalized by using the signal measured
for glassy carbon (Sigradur) at ambient temperature, which can be considered in good
approximation as opaque reference sample both in the visible and the infrared spectrum. The
frequency dependent changes of the measured normalized phases and amplitudes shown in
figure 6.31a and 6.31b can be explained by a concentration-dependent increase of the
combined quantity ss βα . In this case two explanations seem to be reasonable: (i) the
thermal diffusivity and heat conduction improve with the increasing fibre concentration, and
(ii) the effective IR absorption coefficient increases with the fibre concentration as the larger
number of fibres per volume represent a larger number of scattering centers for the IR
radiation inside the sample.
In first approximation we try to estimate the relative variation of the thermal
diffusivity with the fibre content from Figure 6.30a. According to (6.21) the radiometric
signal Mδ can be written as
[ ]4/3
)(22).()(4)(),( πω
π
ηγσεδ −′= tio
ssSBssss enef
ITxTTCfTM 6.30
where the effusivity may depend on the fibre concentration )(nee = . We shall consider here
only the low frequencies, for which we can neglect the radiative heat transport (chapter 3) in
146 6 Application to modern heat insulation materials
comparison with the conductive heat transport. Figure 6.30 shows the normalized amplitudes
for the normalization [carbon fiber (10%) / carbon fiber (5%)] and [carbon fiber (20%) /
carbon fiber (5%)] plotted as a function of frequency. From the normalized values at low
frequencies
)(
)(
)(
)(
1
2
2
1
ne
ne
nS
nSSn == (6.31)
we can estimate that effusivity at a fibre content of 10 % is about %)5(024.1%)10( ee ⋅= and
at 20 % fibre concentration it is %)5(3.1%)20( ee ⋅= . The thermal diffusivity α is related
with the thermal effusivity by [ ]22 )(/)()( ncnen ρα = , so that for the ratios of the thermal
diffusivities of different fibre content follows
2
1
2
2
2
1
2
1
)(
)(
)(
)(
)(
)(
⋅
=
nc
nc
ne
ne
n
n
ρρ
αα
(6.32)
If we assume that it is mainly the thermal conductivity which changes due to the increased
fibre concentration [Haj Daoud, Bein and Pelzl, 1996], the ratios of the thermal diffusivity
can be estimated to vary according to 05.1%)5(/%)10( =αα and 7.1%)5(/%)20( =αα .
Considering these variations of the thermal diffusivity in the combined quantity ss βα , we
can calculate the values of the effective IR absorption coefficient, where the thermal
diffusivity 710*5.2 −=sα m2 s-1 for the 5% fiber content sample is known from literature:
Fibre
concentration
Measured value
ss βα / s-1/2
Estimated value
sα / m2 s-1
/sβ m-1
5% 21 0,25.10-6 4,2.104
10% 55 0,26.10-6 1,05.105
20% 175 0,33.10-6 2,1.105
Table 6.4 Effective absorption coefficients in the infrared for the fibre-reinforced
composites with different fibre concentration.
6.3 Measurements of fibre-reinforced composites with different fibre concentrations 147
Figure 6.30: Normalized signal amplitudes of fibre reinforced material with different fibre
concentrations, where the fibre concentration 5 % is taken as reference.
Nor
mal
ized
am
plitu
de
(f / Hz)100 101 102
0
1
2 fibre concentration 10% / fibre concentration 5% fibre concentration 20% / fibre concentration 5%
148 6 Application to modern heat insulation materials
10-1
100
101
102
103
104
105
240
270
300
330
360
Am
plitu
de /
mV
f / Hz
Pha
se la
g / d
eg
f / Hz
10-1
100
101
102
103
104
105
10-1
100
101
102
103
Polycarbonate matrix+ 5% carbon fibre Polycarbonate matrix+ 10% carbon fibre Polycarbonate matrix+ 20% carbon fibre
Figure 6.31 a+b: Photothermal amplitudes (a) and phase lags (b) measured for fibre
reinforced materials with different fibre concentrations.
6.3 Measurements of fibre-reinforced composites with different fibre concentrations 149
100 101 102-10
0
10
20
30
40
Nor
mal
ized
am
plitu
des
(f / Hz)-1/2
Nor
mal
ized
pha
ses
/ deg
(f / Hz)1/2
10-2 10-1 1000,0
0,4
0,8
1,2
1,6 fibre concentration 5% fibre concentration 10% fibre concentration 20%
Figure 6.32 a+b: Normalized signal amplitudes (a) and phases (b) of fibre reinforced
material with different fibre concentrations, in comparison with
theoretical approximations, where Sigradure is taken as reference.
151
7. Conclusions and Outlook
7.1 Introduction
The applicability of the photothermal IR radiometry to infrared translucent materials
has been analyzed here, both from the experimental and the theoretical point of view. This
technique is usually based on the assumption that only radiation from the surface contributes
to the measured signal [Nordal and Kanstad, 1979]. In solids, however, which are transparent
in the infrared spectrum, respectively slightly translucent, e.g. silicon, carbon-based fibre-
reinforced composites, and materials with larger voids, the subsurface radiation can contribute
to the measured signal, and the temperature distribution inside the sample can additionally be
affected by the internal heat sinks and sources related to thermal radiation.
7.2 Review of the experimental work
In this work, frequency-dependent photothermal measurements, based on modulated
surface heating by means of an argon ion laser and detection of the thermal wave response by
means of a MCT detector have been applied to solids which are transparent, respectively
slightly translucent in the near and mid infrared spectrum. In detail, measurements have been
run on silicon test samples, carbon based fibre-reinforced composites and multi-layer
superinsulation foils consisting of aluminized mylar foils. Part of the measurements has been
done at room temperature and part of the measurements has been done at higher temperatures,
up to 250°C by using a specially designed High-temperature cell. Depending on the thickness
and the transmittance of the samples in the infrared spectrum, two types of geometrical
configurations have been used in the measurements: (i) “thermal waves in transmission”
which are excited at the front surface and detected at the rear surface of the sample have been
measured for the silicon samples and the multi-layer superinsulation foils, and (ii) “thermal
waves in reflection” which are excited and detected at the same sample surface have been
measured both for silicon test samples and for the carbon based fibre-reinforced composites.
7.3 Review of the theoretical work
In order to interpret the experimental data, an extension of the usual theory of the
photothermal radiometry (PTR) has been derived which includes samples, which are slightly
152 7 Conclusions and Outlook
translucent to the thermal radiation, so that both the surface and the interior of the sample
radiate and contribute to the measured signal, and in which additionally the internal heat
sources and sinks due to the re-absorption and emission of thermal radiation affect the
temperature distribution of the thermal wave.
7.3.1 Derivation of the general heat diffusion equation including
radiative transport and transition to the differential equation of
the thermal wave
In order to derive the heat diffusion equation for thermal waves in IR-translucent
media, first the general energy balance of a volume element in an absorbing and emitting
medium which is semi-transparent to thermal radiation has been studied. This balance states
that the rate of change of thermal energy stored within the volume element is equal to the sum
of the net heat conduction rate into the volume, the internal heat sources due to the re-
absorption of thermal radiation and the heat sinks due to emission of thermal radiation, and
the energy generation related to an external source.
The net internal heat sources/ sinks, on the other hand, have been derived from the
radiative heat flux as the negative of the divergence. The radiative heat flux is defined as the
integral of the intensity distribution of the thermal radiation. Therefore, first the radiative heat
transport equations have been solved to give the intensity distribution of the radiation. This
distribution can be interpreted physically as the intensity emitted by all volume elements
along the path of the radiation, reduced by exponential attenuation between the position of
emission and the position, where the intensity has to be determined. In this derivation two
assumptions have been used: (i) the internal radiation may correspond to the gray body
approximation and (ii) the time-averaged temperature may be close to the ambient
temperature. The gray body approximation states that the local emissivity is wavelength- and
direction-independent. The assumption about the time-averaged sample is typical for the
modulated photothermal radiometry, which requires only small temperature differences.
The resulting general heat diffusion equation is a nonlinear integro-differential
equation, which cannot be solved analytically. In order to tackle the problem of the modulated
photothermal radiometry, the concept of thermal waves has then been introduced, which
considers the thermal wave as a small temperature oscillation superposed to the general
temperature distribution. Based on this concept and by considering only the linear thermal
wave term, the stationary temperature distribution and the time dependent problem of the
thermal wave have been separated. Owing to the radiative contributions, the resulting
7.3 Review of the theoretical work 153
differential equation of the thermal wave is coupled to the differential equation of the
stationary temperature distribution by the cubic power of the stationary temperature.
In contrast to the previous work in this field [Tom, O’Hara, and Benin, 1982; Sommer,
1994; Paolini and Walther, 1997; Dietzel, Haj Daoud, Macedo, Pelzl, and Bein, 1999], the
differential equation of the thermal wave thus considers the internal heat sources/ sinks
related to the re-absorption and emission of thermal radiation inside the solid.
7.3.2 Solution of the differential equation of the thermal wave
This differential thermal diffusion equation of the thermal wave has been solved for
one-dimensional temperature distribution, as the diameter of the heating spot on the sample is
large in comparison to the thermal diffusion length and to the detection area of the detector.
Another restriction used for the solution is that the stationary averaged temperature
distribution existing in the differential equation of the thermal wave is constant. This
assumption can be fulfilled, especially if higher average sample temperatures are considered,
where the steady state temperature gradient produced near the surface by the modulated laser
beam can be neglected.
The external heat source due to modulated laser heating has been introduced into the
thermal wave solution through the boundary conditions, by restricting the solution to totally
opaque solids in the visible spectrum.
The time- and space- dependent temperature distribution of the thermal wave has been
calculated by using a suitable linear ansatz, which contains four terms, namely the usual two
terms appropriate for an IR-opaque solid with conductive heat transport alone, and two new
terms to account for the radiative heat transfer inside the sample. For the derivation of the
thermal wave, a perturbation method has been used by inserting only the usual two terms of
the unperturbed thermal wave into the internal heat sources and sinks related to the thermal
radiation. Thus the solution should apply for thermal waves, whose temperature distribution
does not deviate too much from the usual thermal wave of IR opaque solids.
The amplitude of the thermal wave for the IR translucent solid of finite thickness
depends on five combined thermo-optical parameters, namely the quantities η/e, 3)(xT ,
α1/2β, α1/2/d, and β d. Here, the quantity η/e, which is the absorptance in the visible spectrum
divided by the thermal effusivity, determines the amplitude of the temperature oscillation; the
cubic of the time-averaged sample temperature 3)(xT determines the order of magnitude of
the radiative transport; α1/2β is the product of the square root of the thermal diffusivity with
154 7 Conclusions and Outlook
the absorption coefficient in the infrared spectrum, α1/2/d is the square root of the thermal
diffusion time of the sample, and the quantity β d determines the attenuation of the thermal
radiation. If the time-averaged temperature of the sample is known from an additional
measurement, the phase of the thermal wave depends only on three parameters, namely the
quantities α1/2β, α1/2/d, and β d, which are coupled, so that there are only two independent
parameters.
Once the general solution for thermal waves in an IR-translucent solid of finite
thickness had been found, several special cases have been derived, namely for the semi-
infinite IR-translucent solid, which applies to rather thick samples, and the limit of an
increased IR-absorption coefficient (ß → ∞) has been considered, giving the usual thermal
wave solution for the IR opaque solid.
7.3.3 Derivation of the measured radiation signal
Consequently, the photothermal radiometric signal measured by the photoconductive
detector (MCT) has been derived starting from the radiative heat flux calculated from the
solution of the radiative transfer equation and from the thermal wave solution. In the
quantitative interpretation of the radiometric signal, the technical factors affecting the signal,
namely the transmittance of the IR optic system and the spectral responsivity of the detector
have been taken into account. Depending on the position of the detector with respect to the
sample, the radiometric signals have been determined both for the reflection and the
transmission configuration of thermal waves.
7.4 Experimental results
7.4.1 Test measurements on Silicon samples at room temperature
The measurements performed on Silicon samples of different thickness at room
temperature and higher temperatures have been compared with the theoretical model, both in
the reflection and the transmission configuration.
In the reflection configuration, as already mentioned above, the measured phases
depend only on two independent parameters, and for the calculation of the numerical
approximation the two parameters P1 = α1/2β and ss dP /2/12 α= have been varied in a first
step. For the normalized phases obtained from the measurements of two samples of different
thickness ds1 and ds2 a larger variety of possible parameter combinations P1 (ds1), P2 (ds1)
7.4 Experimental results 155
and P1 (ds2), P2 (ds2) have been found which formally give reasonable agreement between
the measured data and the theoretical curves, forming two extended existence regions for
possible solutions in the two-dimensional parameter space P1, P2. In order to distinguish
physically realistic from only formally correct solutions in the parameter space P1, P2, the
corresponding ratio =)(/)( 1323 dPdP 12
)(/)( dd dd ββ has been plotted versus the ratio
=)(/)( 1222 dPdP 12
)/(/)/( 2/12/1dd dd αα . By requiring that the material properties α and β
should be independent of the individual thickness of the involved samples, which means that
the geometrical conditions 211222 /)(/)( dddPdP = and )(/)( 1323 dPdP 12 / dd= should be
fulfilled, the physically relevant parameters have then been selected.
From the resulting optimal parameters P2 and P3, fulfilling the geometrical conditions
and with good agreement between theoretical curves and experimental data, the combined
parameter P1 = P2⋅P3 = α1/2β has been calculated. The resulting values, 3.6 s-1/2 < α1/2β < 5.7
s-1/2, obtained from the normalized phases measured at room temperature for the different
samples of 2 mm, 4 mm and 6 mm thickness, are higher than the literature data by a factor of
2 to 3. These differences can be due to different reasons, namely due to the properties of the
measured samples and due to the conditions of the experiment:
(1) The oxidized surface layer of the used silicon samples probably contribute to an
effectively reduced IR signal measured in front of the heated samples. In the
theoretical interpretation, the reduced IR signal has been interpreted as an increased
effective IR absorption coefficient.
(2) The limited heating spot diameter, about 8 mm in comparison to the sample diameter
of 20 mm contributes to a three-dimensional temperature distribution in the samples
and to increase lateral heat transport due to conduction. This effect contributes to an
apparently increased value of the thermal diffusivity.
(3) In addition, owing to the sample geometry and the lateral radiative heat losses from
the heated spot and the cylindrical sample, which are not suppressed by any
experimental condition, e.g. by reflection of the lateral IR radiation, the IR signal
measured in front of the samples is lower than foreseen by the theoretical model. This
again contributes to an effectively increased measured IR absorption coefficient.
(4) Finally we have to admit, that the theoretical model for the thermal wave containing
conductive and radiative heat transport is limited due to its construction as a
perturbation solution to only slightly IR translucent samples. Thus the model may
156 7 Conclusions and Outlook
perhaps not be appropriate for silicon with its relatively large IR absorption length of
56.51 =−IRβ mm.
The results of the transmission measurements, namely the normalized phases obtained
from silicon samples of different thickness, showed principal deviations from the
corresponding theoretical solutions which could not be removed by systematic variations of
the parameters P1 and P2, as done for the reflection measurements of thermal waves.
Instead, a stationary temperature gradient in the sample with an increased stationary
temperature at the laser-heated front surface of the sample and a lower stationary temperature
at the rear surface which is only heated indirectly by conductive and radiative heat transport,
had to be assumed. The IR response measured in the transmission configuration contains
information about the modulated heating process from all positions inside the sample. The
information from the hotter region just beneath the front surface, however, has got a relatively
higher weight due to the cubic power of the higher stationary temperature 30)( =xT close to
the heated front surface. Consequently, the radiative contributions close to the rear surface are
comparatively less important. To account for this temperature difference, in the signal
generation process a depth dependent stationary temperature distribution has been introduced
while the thermal wave solution was assumed to be unchanged. To this finality various
temperature profiles with different temperature gradients have been tested, and it has been
shown, that the normalized phases show good agreement for steeper temperature gradients,
nearly independent of the chosen finite temperature difference. For the quantitative
interpretation of the phases realistic low values of the combined thermo-optical parameter P1
were obtained between 1.7 s-1/2 <α1/2β < 2.5 s-1/2 which are close to literature data. This may
be due to the fact that the radiation losses in the transmission measurements have been
reduced by wrapping the samples laterally with reflecting Al foils and that a specially large
heating spot (15 mm) was used, so that three-dimensional conductive heat propagation and
radiative heat losses have been reduced.
Additionally one has to admit that the concept of the linear thermal wave and the
derivation of the thermal wave solution under the assumption of a constant time-averaged
temperature distribution are doubtful in IR transparent solids. For such solids, the
experimental conditions have to be well controlled to coincide with the assumptions used for
the theoretical derivations.
7.4 Experimental results 157
7.4.2 Test measurements on silicon samples at higher temperature
The results obtained at higher temperatures from the reflection measurements of the
rather IR-transparent silicon samples in principal agree with the theoretical solutions
calculated according to the systematic determination of the existence regions for the
parameters P1 and P2 and the selection of the physically relevant parameters by assuming
sample-independent material parameters α and β. The combined value P1 = α1/2β increases,
however, with the temperature. This effect may be realistic, since with at higher average
sample temperature, the distribution of the thermal radiation shifts to shorter wave lengths,
which may selectively be absorbed in the samples.
The reliable frequency range which can be used for the quantitative interpretation of
the measured phases in general decreases for all silicon measurements at higher temperatures.
At a first glance, this effect seems to be in contrast to the background detection limit observed
for IR-opaque samples, which improved with higher temperatures. In the case of the silicon
measurements at higher temperatures, however, the rear surface of the samples is in contact
with the coax-wire-heated sample holder, which has got a higher temperature of about 10 –30
K than the front surface. This means, the relatively small thermal wave is measured in front of
higher fluctuating temperature background, as Silicon is IR-transparent. To avoid this
problem in the experiment, a higher integration time constant and a higher integration number
should be used in the filtering process of the lock-in amplifier. A different sample holder
heating the samples laterally would also improve the measurement conditions, as the
temperature gradient in the samples would be decrease.
7.4.3 Results on multi-layer superinsulation foils
Measurements of thermal waves in transmission have been run on the multi-layer
superinsulation foils under the conditions of ambient temperature and pressure. The
superinsulation foils consist of aluminized mylar foils, an external mylar foil of 25 µm
thickness both at the front and rear of the multi-layer system and of a sequence spacer layers
(Cryotherm 234) and aluminized mylar foils of 6 µm thickness.
From the measurements, one can conclude that the heat transport across the composite
insulation layer system can be described quantitatively by using the method of “transmitted
thermal waves“. Additionally one can conclude that the heat transport across the composite
insulation layer system is purely diffusive, at least as more than one single layer is considered.
158 7 Conclusions and Outlook
This is probably due to the air between the various layers and due to the fact that the various
insulation layers are in narrow contact with each other.
From the results the relative damping factors with respect to radiative and conductive
heat flux across the foils and the relative decrease of the thermal diffusivity, which are the
relevant thermal parameters to characterize thermal insulation foils, have been determined for
various examples of multi-layer systems, composed of the layers of the Lydall DAM
superinsulation foil. For the measurement performed on the 25 µm thick external foil alone, it
was possible to identify the radiative heat transport existing in addition to the conductive heat
transport. In order to identify the radiative heat transport for more than one layer and under
cryo conditions the superinsulation foils should probably have larger voids between the
various layers.
As the measured signals are close to the limit of noise, especially at higher
frequencies, an improvement for the measurements requires a higher heating power of the
laser and improved filtering conditions for the lock-in amplifier.
7.4.4 Results on Carbon-based fibre-reinforced composites
For the results obtained for the fibre-reinforced composite materials, a good agreement
between the measurements at higher temperatures and the theoretical approximations based
on the model including IR transparency has in general been obtained. As the fibre-reinforced
samples are relatively thick, both thermally and with respect to their IR absorption length, the
measured amplitudes and phases can be approximated by using the model for the semi-
infinite solid. The advantage of this model is, that the phases depend only on one combined
thermo-optical parameter, name the quantity ssP βα 211
/= . Consequently the approximation
process of the measured curves and the determination of the relevant parameter is much
easier.
In general it has been observed, that the parameter ssP βα 211
/= decreases with
increasing temperature. For some of the samples, the thermal diffusivity has been measured
separately and it has been found that the major effect on the combined quantity ss βα 2/1 is
related to IR transparency increasing with the average sample temperature.
Stronger deviations between measurement and theoretical approximation only have
been observed at higher modulation frequencies and at lower effective IR absorption
coefficients. This can be explained by the layer structure of the composite materials which
consist of a thin woven cover layer and the bulk material below. The approximation surely
7.4 Experimental results 159
could be improved with a two-layer model allowing IR transparency and considering the
scattering effects related to the fibre structure.
The results obtained for the composite material with a systematic variation of the fibre
content have been explained by using a concentration-dependent increase of the combined
parameter ss βα 2/1 . According to the approximations of the normalized measured signals at
low modulation frequencies; the relative thermal effusivty increases with the increasing fibre
content. By considering this effect also for the thermal diffusivities, the IR absorption
coefficients also increase with the fibre content. This latter result also seems to be realistic, as
the increased numbers of fibres per volume represent more scattering centers for the internal
IR radiation.
161
8. Deutschsprachige Zusammenfassung
8.1 Einleitung
Photothermische Infrarotradiometrie, die auf Anregung thermischer Wellen durch
intensitätsmodulierte sichtbare Laserstrahlung und Detektion im nahen und mittleren
infraroten Spektralbereich basiert, wurde vor ca. zwanzig Jahren als tiefenaufgelöste
Messmethode thermischer Eigenschaften vorgeschlagen [Nordal and Kanstad, 1979]. Seit
dieser Zeit hat sich die photothermische Infrarotradiometrie bis zu einer Messmethode
entwickelt, die zur Überwachung industrieller Produktionsprozesse eingesetzt werden kann
[Petry, 1998]. Es konnte auch gezeigt werden, dass diese Methode erfolgreich eingesetzt
werden kann z.B. zur Analyse erodierter Oberflächen [Haj-Daoud, Katscher, Bein, Pelzl,
1999] und technischer Schichtsysteme, deren Oberflächen nicht ideal glatt sind und die im
sichtbaren und optischen Spektralbereich leicht transparent sein können [Bein, Bolte, Dietzel
and Haj-Daoud, 1998].
Die Interpretation der Messungen mit dieser Methode bereitet jedoch verschiedene
Probleme, falls die optische Absorptionslänge im sichtbaren sehr groß ist (z. B. im Vergleich
zu den Schichtdicken). Zusätzliche Probleme entstehen bei Anwendung der photothermischen
Infrarot-Radiometrie auf IR-transparente Materialien, da für diese Fälle der
Signalentstehungsprozess und die Theorie thermischer Wellen bis jetzt noch nicht
ausreichend erforscht sind.
Die Aufgabe meiner Arbeit bestand darin, den Signalentstehungsprozess bei IR-
transparenten Medien experimentell und theoretisch zu analysieren, die Theorie thermischer
Wellen für IR-transparente Stoffe abzuleiten und Messungen an Materialien durchzuführen,
bei denen sowohl Wärmetransport durch Leitung als auch durch Strahlung erwartet werden
können. Solche Proben waren z. B.: faserverstärkte Verbundwerkstoffe, auf Kohlenstoffbasis
und kälteisolierende technische Mehrschichtsysteme aus Aluminiumbeschichteten
Kunststofffolien.
Im Anschluss an diese Einleitung wird in Kapitel 2 eine Beschreibung des
experimentellen Aufbaus präsentiert. In Kapitel 3 werden die Grundlagen des
Signalentstehungsprozesses der photothermischen Infrarot-Radiometrie für IR-opake Körper
dargestellt. Darüber hinaus wird an dieser Stelle die Theorie der Anregung und Ausbreitung
thermischer Wellen in Festkörpern endlicher Dicke diskutiert.
162 8 Deutschsprachige Zusammenfassung
In Kapitel 4 wird diese Theorie der üblichen photothermischen Infrarot-Radiometrie
für Proben erweitert, die im Infraroten leicht transparent sind. Dies führt dazu, dass
Abstrahlung sowohl von der Oberfläche als auch aus dem Inneren zum Signal beiträgt und die
mit Strahlungsemission und Reabsorption verbundenen Wärmequellen und Senken innerhalb
der Probe berücksichtigt werden müssen. Basierend auf dem Konzept der thermischen Wellen
wurde für diesen Fall die Wärmediffusionsgleichung liniearisiert und gelöst, wobei sowohl
dem Wärmetransport durch Strahlung als auch dem durch Leitung Rechnung getragen wurde.
Abschließend werden in diesem Kapitel die theoretischen Beschreibungen des
radiometrischen Signals in Reflexions- und Transmissionskonfiguration gegeben.
Im Kapitel 5 werden frequenzabhängige Testmessungen in Reflexionskonfiguration
präsentiert, bei denen die thermische Welle an der gleichen Oberfläche angeregt und
detektiert wird. Messungen an Silizium-Proben verschiedener Dicke wurden bei
Raumtemperatur und höheren Temperaturen gemessen und interpretiert. Messungen
thermischer Wellen in Transmission, bei denen die Detektionsseite der Anregungsseite
gegenüberliegt wurden ebenfalls gemessen und werden hier zusätzlich dargestellt.
Anschließend werden in den folgenden Kapiteln Untersuchungen vorgestellt an
verschiedenen technologisch relevanten Materialien, die z.B. großflächig zur
Wärmeisolierung von supraleitenden Komponenten in der Kernforschung eingesetzt werden
oder die in der Verteidigungstechnik als Schutzschilde gegen sog. Hitze-Blitze bzw. Wärme-
Schockwellen eingesetzt werden. Im Kapitel 6.1 wird die Methode der transmittierten
thermischen Wellen bei Raumtemperatur auf vielschichtige Superisolationsfolien angewandt,
die aus aluminiumbeschichteten Mylar-Folien und Spacermaterial bestehen. Der
Abschwächungsfaktor für gleichzeitigen Leitungs- und Strahlungstransport konnte für
Isolationsfolien verschiedener Dicke bestimmt werden. In Kapitel 6.2 wurde die Methode der
thermischen Wellen in Reflexion zur Untersuchung faserverstärkter Verbundmaterialien
benutzt. Dabei wurden systematische Messungen in Abhängigkeit der Probentemperatur und
der Faserkonzentration durchgeführt.
Im Kapitel 7 wird dann abschließend eine Zusammenfassung der vorliegenden Arbeit
gegeben, bei der die wesentlichen Schlußfolgerungen noch einmal diskutiert werden und ein
Ausblick auf weiterführende Ansätze gegeben wird.
8.2 Zusammenfassung 163
8.2 Zusammenfassung
In der vorliegenden Arbeit wurde sowohl vom theoretischen als auch vom
experimentellem Standpunkt aus der Anwendbarkeit der photothermischen
Infrarotradiometrie auf infrarot transparente Materialien untersucht. Üblicherweise basiert die
verwendete Messtechnik auf der Annahme, dass lediglich Strahlung direkt von der Oberfläche
zum Messsignal beiträgt [Nordal and Kanstad, 1979]. In Festkörpern jedoch, die transparent
oder durchlässig im Infraroten sind (z. b. Silizium, Faser-verstärkte Verbundwerkstoffe und
Materialien mit größeren Leerräumen), können auch Strahlungsbeiträge von unterhalb der
Oberfläche zum Messsignal beitragen. Darüber hinaus kann die Temperaturverteilung im
Inneren der Probe durch strahlungsbedingte Wärmequellen oder –senken beeinflußt werden.
8.2.1 Kurzaufzählung der experimentellen Arbeiten
Im Rahmen dieser Arbeit wurden frequenzabhängige photothermische Messungen
durchgeführt, bei denen mit Hilfe eines Ar+-Lasers die Probe moduliert geheizt wurde,
während die Detektion der thermischen Wellen mit Hilfe eines sog. MCT Detektors erfolgte.
Solche Messungen wurden auf verschiedene Festkörper angewandt, die transparent bzw.
durchscheinend im nahen bis mittlerem Infrarot waren. Konkret wurden folgende Proben
untersucht: Silizium, Faser-verstärkte Verbundmaterialien und mehrschichtige Super-
isolationsfolien. Diese Isolationsfolien bestanden aus einer Vielzahl einzelner aluminium-
beschichteter Mylar-Folien.
Die Experimente wurden im Temperaturbereich zwischen Raumtemperatur und ca.
250° abhängig von Probendicke, Transparenz und Wärmetransporteigenschaften in zwei
geometrischen Konfigurationen durchgeführt:
(1) „Thermischer Wellen in Transmission“, die an der Vorderseite angeregt
wurden und an der Rückseite detektiert wurden, wurden für die Silizium-Proben und die
Superisolationsfolien gemessen.
(2) „Thermische Wellen in Reflektion“, die an der Probenvorderseite angeregt und
detektiert werden, dienten zu Testmessungen an den Silizium-Proben und wurden zur
Untersuchungen an den Verbundwerkstoffen eingesetzt.
8.2.2 Kurzaufzählung zu den theoretischen Arbeiten
Zur Interpretation der experimentell gewonnenen Daten wurde die übliche Theorie zur
Beschreibung der photothermischen Infrarotradiometrie (PTR) dahingehend erweitert, dass
164 8 Deutschsprachige Zusammenfassung
auch Festkörper mit leichter Transparenz für thermische Strahlung beschrieben werden
können. Diese erweiterte Theorie berücksichtigt dabei, dass zusätzlich zur Abstrahlung von
der Oberfläche auch Strahlung aus tieferliegenden Schichten zum Signal beiträgt und dass
aufgrund von Absorption und Re-Emission thermischer Strahlung zusätzliche Wärmequellen
und Senken entstehen, die die Temperaturverteilung der thermischen Welle beeinflussen
können.
8.2.2.1 Ableitung der allgemeinen Wärmediffusionsgleichung unter Einfluss von Strahlungstransport und Übergang zur Differentialgleichung der thermischen Welle
Um die Wärmediffusionsgleichung für thermische Wellen für den Fall IR-
transparenter Stoffe abzuleiten, wurde zuerst die allgemeine Energiebilanz eines
Volumenelementes in einem emittierendem und absorbierendem Medium, das semi-
transparent für thermische Strahlung ist, aufgestellt. Diese Energiebilanz besagt, dass die
Änderung der thermischen Energie in dem Volumenelement gleich der Summe der
Wärmeleitung in das Volumenelement, den inneren Wärmequellen durch Absorption, den
Wärmesenken durch Emission thermischer Strahlung und den Wärmequellen durch äußere
Anregung ist.
Die Bilanz der inneren Wärmequellen bzw. Wärmesenken wurde als negative
Divergenz des Energiestromes durch Strahlung bestimmt werden. Dieser Energiestrom durch
Strahlung ist definiert als Integral über die Intensitätsverteilung der thermischen Strahlung.
Daher mußten zuerst die Gleichungen zur Beschreibung des Strahlungstransportes gelöst
werden, um die Intensitätsverteilung der Strahlung zu bestimmen. Diese Verteilung kann
physikalisch interpretiert werden als im ganzen Körper emittierte Intensität, die exponentiell
zwischen der Position der Abstrahlung und der Position, an der die Intensität bestimmt wird,
abgeschwächt wird. Bei der Ableitung wurden zwei vereinfachende Annahmen benutzt:
(i) Die interne Strahlung soll als Graukörperstrahlung beschrieben werden können.
(ii) Die mittlere Probentemperatur soll näherungsweise der Umgebungstemperatur
entsprechen.
Die Annahme eines Graukörpers besagt dabei, dass das lokale Emissionsvermögen
wellenlängen- und richtungsunabhängig ist. Die Annahme (ii) ist typisch für die
photothermische Radiometrie, da bei dieser Methode nur mit sehr kleinen
Temperaturoszillationen gearbeitet werden kann.
Die resultierende Wärmeleitungsgleichung ist eine nichtlineare Integro-Differential-
Gleichung, die im allgemeinen analytisch nicht mehr gelöst werden kann. Um dieses Problem
8.2 Zusammenfassung 165
für die modulierte photothermische Infrarotradiometrie zu lösen, wurde das Konzept der
thermischen Wellen eingeführt, das auf kleinen Temperaturoszillationen, der zeitlich
gemittelten Temperaturverteilung überlagert, beruht. Basierend auf diesem Konzept und bei
ausschließlicher Berücksichtigung linearer thermischer Wellen kann die stationäre
Temperaturverteilung vom zeitabhängigen Problem der thermischen Wellen separiert
werden. Wegen der Strahlungsbeiträge ist die resultierende Differentialgleichung der
thermischen Welle mit der Differentialgleichung der stationären Temperaturverteilung über
die dritte Potenz der stationären Temperaturverteilung gekoppelt.
Im Gegensatz zu früheren Arbeiten auf diesem Gebiet [Tom, O’Hara, and Benin,
1982; Sommer, 1994; Paolini and Walther, 1997; Dietzel, Haj Daoud, Macedo, Pelzl, and
Bein, 1999] werden in der Diffusionsgleichung für die thermischen Wellen jetzt innere
Wärmequellen bzw. Wärmesenken aufgrund von Absorption und Emission thermischer
Strahlung berücksichtigt.
8.2.2.2 Lösung der Diffusionsgleichung der thermischen Welle
Diese differentielle Wärmediffusionsgleichung wurde nun für eine eindimensionale
Temperaturverteilung gelöst. Diese Vereinfachung ist möglich, da der Heizfleckdurchmesser
auf der Probe im Vergleich zur thermischen Diffusionslänge und der aktiven Detektionsfläche
sehr groß ist. Eine weitere Einschränkung für die Lösung ist, dass die in der
Differentialgleichung vorkommende mittlere stationäre Temperaturverteilung konstant ist.
Diese Annahme ist besonders im Falle hoher Probentemperaturen erfüllt, wenn der durch die
Lasereinstrahlung hervorgerufene, oberflächennahe Temperaturgradient vernachlässigt
werden kann.
Die externe Heizquelle durch die intensitätsmodulierte Laserstrahlung wurde durch
eine spezielle Randbedingung berücksichtigt, wobei allerdings die dadurch erhaltenen
Lösungen auf optisch vollkommen opake Körper beschränkt sind.
Die orts- und zeitaufgelöste Temperaturverteilung der thermischen Wellen wurde
durch einen geeigneten linearen Ansatz berechnet, der im wesentlichen aus vier Termen
besteht. Dies sind zum einen die zwei üblichen Terme, die in einem IR-opaken Körper den
Leitungstransport beschreiben, und darüber hinaus zwei neue Terme, die den
Strahlungstransport innerhalb der Probe berücksichtigen. Für die Ableitung der thermischen
Welle wurde eine Störungstheorie benutzt, bei der nur die üblichen zwei Terme der
herkömmlichen thermischen Welle zur Berechnung der Beiträge der strahlungsbedingten
Wärmequellen und Wärmesenken benutzt wurden. Daher sollte die Lösung nur solche
166 8 Deutschsprachige Zusammenfassung
Probleme beschreiben können, bei der die Temperaturverteilung nicht zu stark von der
üblichen thermischen Welle IR-opaker Festkörper abweicht.
Die Amplitude der thermischen Welle für IR-transparente Körper endlicher Dicke
hängt von fünf kombinierten thermo-optischen Materialparameter ab. Diese Parameter sind
e/η , 3)(xT , βα 2/1 , d/2/1α , und dβ . Die Größe e/η , die der Quotient aus
Absorptionsvermögen im sichtbaren und dem Wärmeeindringkoeffizienten ist, bestimmt die
Amplitude der Temperaturoszillationen; die dritte Potenz der mittleren stationären
Probentemperatur 3)(xT bestimmt die Größenordnung des Strahlungstransportes; βα 2/1 ist
das Produkt aus der Wurzel des Temperaturleitwertes und des Absorptionskoeffiezienten im
Infraroten, d/2/1α ist die Wurzel der thermische Diffusionszeit und die Größe dβ bestimmt
die Abschwächung der thermischen Strahlung. Ist die mittlere Probentemperatur aus einer
zusätzlichen Messung bekannt, so hängt die Phasenlage der thermischen Welle nur von drei
Parametern ab: nämlich den miteinander gekoppelten Parametern βα 2/1 , d/2/1α , und
dβ .Dies bedeutet, dass im Endeffekt nur zwei unabhängige Parameter zu berücksichtigen
sind, um die gemessenen Verläufe der Phasen anzunähern.
Nachdem die allgemeine Lösung für thermische Wellen in IR-transparenten Körpern
gefunden war, konnten verschiedene Spezialfälle abgeleitet werden. Zur Beschreibung sehr
dicker Proben wurde der Spezialfall des halbunendlichen Festkörpers abgeleitet, während der
Grenzfall β → ∞ wieder die übliche thermische Welle eines IR-opaken Festkörpers lieferte.
8.2.2.3 Ableitung des gemessenen Strahlungssignals
Ausgehend von der Strahlungstransportgleichung und unter Berücksichtigung der
Lösung der thermischen Welle, konnte das Strahlungssignal berechnet werden, das durch die
auf den photoleitenden MCT-Detektor fallende Strahlung hervorgerufen wird. In der
quantitativen Interpretation des radiometrischen Signals wurden dabei verschiedene
technische Kenngrößen berücksichtigt, die das Signal beeinflussen (z. B. das
Transmissionsverhalten der IR-Optik, spektrale Empfindlichkeit des Detektors). Abhängig
von der Position des Detektors relativ zur Probe, konnten die radiometrischen Signale für die
Transmissions- und Reflexionskonfigurationen thermischer Wellen berechnet werden, die
dann zur Interpretation der Messungen benutzt wurden.
8.2 Zusammenfassung 167
8.2.3 Experimentelle Ergebnisse
8.2.3.1 Testmessungen an Siliziumproben bei Raumtemperatur
Die Messungen an Siliziumproben verschiedener Dicke, die bei Raumtemperatur
sowohl in der Transmissions- als auch der Reflexionskonfiguration vorgenommen wurden,
wurden mit dem theoretischen Modell verglichen.
Wie schon erwähnt, hängen in der Reflexionskonfiguration die gemessenen Phasen
nur von zwei unabhängigen Parametern ab. Zur Berechnung der theoretischen Lösungen
wurden daher die zwei Parameter βα 2/11 sP = und ss dP /2/1
2 α= in einem ersten Schritt
variiert. Für die normierten Phasen, die aus Messungen an zwei Proben unterschiedlicher
Dicke 1sd und 2sd gewonnen wurden, wurde eine größere Vielfalt an möglichen
Parameterkombinationen P1 (ds1), P2 (ds1) and P1 (ds2), P2 (ds2) gefunden, die formal eine
vernünftige Übereinstimmung zwischen den experimentellen Daten und den theoretischen
Kurven liefern. Es zeigt sich , dass die möglichen Lösungen zwei ausgedehnte Gebiete im
zweidimensionalen Parameterraum 1P und 2P bilden. Um aus diesen Gebieten formal
möglicher Lösungen die physikalisch realistischen Lösungen zu ermitteln, wurde für die an
der Signalnormierung beteiligten Proben unterschiedlicher Dicke das Verhältnis
=)(/)( 1323 dPdP 12
)(/)( dd dd ββ gegen das Verhältnis =)(/)( 1222 dPdP
12)/(/)/( 2/12/1
dd dd αα aufgetragen. Unter der Annahme dass die Materialparameter α und β
unabhängig von der jeweiligen Dicke der Probe sein sollten, was bedeutet dass die
geometrischen Bedingungen 211222 /)(/)( dddPdP = und )(/)( 1323 dPdP 12 / dd= erfüllt sein
sollten, konnten dann die physikalisch sinnvollen Lösungen identifiziert werden.
Aus den resultierenden optimalen Parametern 2P und 3P , die sowohl die
geometrischen Bedingungen erfüllen als auch gute Übereinstimmung zwischen Messung und
Theorie liefern, wurde der kombinierte Parameter βα 2/1321 =⋅= PPP berechnet. Die
resultierenden Werte 3.6 s-1/2 < α1/2β < 5.7, die man aus den normierten Phasen bei
Raumtemperatur für die verschiedenen Probendicken von 2 mm, 4 mm und 6 mm bestimmen
kann, sind um einen Faktor von ca. 2-3 größer als die Literaturwerte. Diese Unterschiede
können verschiedene Ursachen haben, die entweder in den Proben selbst zu suchen sind oder
die Versuchsbedingungen betreffen:
(1) Die Oxidschicht an der Probenoberfläche der benutzten Proben trägt vermutlich zu
einem effektiv reduzierten IR-Signal an der Vorderseite der geheizten Probe bei. In
168 8 Deutschsprachige Zusammenfassung
der theoretischen Interpretation würde dann dieses reduzierte IR-Signal als erhöhter
effektiver IR-Absorptionskoeffizient interpretiert.
(2) Der auf ca. 8mm begrenzte Heizfleckdurchmesser führt bei einem Probendurchmesser
von ca. 20mm zu einer dreidimensionalen Temperaturverteilung in den Proben, die
dazu führt, dass sich der laterale Wärmetransport durch Leitung erhöht, dieser Effekt
führt zu einem scheinbar erhöhten Wert des Temperaturleitwertes.
(3) Zusätzlich ist wegen der Probengeometrie und der strahlungsbedingten lateralen
Wärmeverluste, die nicht durch experimentelle Bedingungen unterdrückt werden
können (z. B. durch Reflexion der lateralen Strahlung) das gemessene IR-Signal
niedriger als theoretisch erwartet, was wiederum als größerer effektiver IR
Absorptionskoeffizient interpretiert wird.
(4) Als letztes muß an dieser Stelle noch erwähnt werden, dass das theoretische Modell
für thermische Wellen bei gleichzeitigem Wärmetransport durch Leitung und
Strahlung aufgrund der theoretischen Annahmen (Störungstheorie) nur für leicht IR-
transparente Proben anwendbar ist. Daher sollte die Anwendbarkeit auf Silizium mit
einer relativ großen Absorptionslänge für IR-Strahlung ( 1−IRβ = 5.56 mm) nur
eingeschränkt gegeben sein.
Die Ergebnisse der Transmissionsmessungen an Silizium, besonders die normierten
Phasen von Siliziumproben unterschiedlicher Dicke, zeigten prinzipielle Abweichungen von
den entsprechenden theoretischen Lösungen, die nicht durch eine systematische Variation der
Parameter 1P und 2P angenähert werden konnten, wie dies bei den Reflexionsmessungen der
Fall war.
Statt dessen wurde ein Temperaturgradient in der Probe angenommen, bei dem eine
erhöhte stationäre Probentemperatur an der lasergeheizten Vorderseite einer niedrigeren
Rückseitentemperatur gegenübersteht. Die niedrigere Rückseitentemperatur erklärt sich dabei
durch die nur indirekte Heizung durch Leitungs- bzw. Strahlungstransport. Das in der
Transmissionskonfiguration gemessene IR-Signal enthält Informationen über den modulierten
Heizprozess von jeder Position innerhalb der Probe. Die Information aus dem wärmeren
Probengebiet direkt unterhalb der geheizten Oberfläche hat jedoch eine relativ gesehen höhere
Gewichtung, da die zum Signal beitragende Infrarotabstrahlung mit der dritten Potenz der
stationären Teperaturverteilung 30)( =xT skaliert. Dem zufolge tragen Strahlungsbeiträge
von Position dicht an der Probenrückseite relativ weniger zum Strahlungssignal bei. Um diese
Temperaturunterschiede zu berücksichtigen, wurde in der Signalentstehungstheorie eine
tiefenabhängige stationäre Temperaturverteilung eingeführt während die thermische Welle als
8.2 Zusammenfassung 169
unverändert betrachtet wurde. Bis zur hier vorliegenden endgültigen Form wurden
verschiedene Temperaturverteilungen mit verschiedenen Temperaturgradienten getestet, und
es zeigte sich, dass die normierten Phasen gut angenähert werde können, wenn relativ große
Temperaturgradienten benutzt werden, wobei die gewählten absoluten
Temperaturunterschiede nahezu keinen Einfluß haben. Bei der quantitativen Interpretation der
gemessenen Phasen ergaben sich relativ niedrige Werte des kombinierten thermo-optischen
Materialparameters 1P zwischen 1.7 s-1/2 <α1/2β < 2.5 s-1/2, die nahe an den Literaturwerten
liegen. Dies kann daran liegen, dass die Strahlungsverluste bei den Transmissionsmessungen
durch ein seitliches Umwickeln der Proben mit reflektierender Aluminiumfolie reduziert
werden konnten und ein besonders großer Heizfleckdurchmesser von ca. 15mm benutzt
wurde, so daß dreidimensionaler Wärmetransport ebenfalls reduziert werden konnte.
Allerdings muß an dieser Stelle auch zugegeben werden, dass das Konzept linearer
thermischer Wellen und die Ableitung dieser thermischen Wellen unter der Annahme einer
zeitlich gemittelten und konstanten Temperaturverteilung in IR-transparenten Festkörpern nur
eingeschränkt gültig sein kaum. Für solche Festkörper müssen die experimentellen
Bedingungen sorgfältig kontrolliert werden, um mit den Annahmen der theoretischen
Ableitungen übereinzustimmen
8.2.3.2 Testmessungen an Siliziumproben bei höheren Temperaturen
Die Resultate, die bei höheren Probentemperaturen an den relativ IR-transparenten
Siliziumproben in der Reflexionskonfiguration erzielt wurden, stimmen prinzipiell gut mit
den theoretischen Lösungen überein, bei denen nach einer systematischen Bestimmung der
Existenzgebiete der Parameter 1P und 2P die physikalisch relevanten Lösungen ermittelt
wurden, wobei wieder von probenunabhängigen Materialparametern α und β ausgegangen
wurde. Der kombinierte Wert βα 2/11 =P steigt dabei mit der Temperatur. Dieser Effekt kann
realistisch sein, da sich bei höherer mittlerer Probentemperatur die Verteilung der thermischen
Strahlung zu kürzeren Wellenlängen verschiebt, die in den Proben besonders stark absorbiert
werden könnten.
Der Frequenzbereich, der zur quantitativen Interpretation der gemessenen Phasen
herangezogen werden kann, verringert sich jedoch für alle Siliziummessungen mit steigender
Temperatur. Auf den ersten Blick scheint ein solcher Effekt im Widerspruch zu
Detektionsgrenzen zu stehen, die man bei Infrarot-opaken Körpern beobachtet, und die sich
mit steigender Temperatur zu höheren Grenzfrequenzen verschieben. Im Fall von Messungen
170 8 Deutschsprachige Zusammenfassung
an Siliziumproben jedoch muß berücksichtigt werden, dass die Probenrückseite in Kontakt
mit dem Koax-Draht-geheizten Probenhalter steht und dass dort eine um ca. 10-30 K im
Vergleich zur Probenvorderseite erhöhte stationäre Temperatur herrscht. Dies bedeutet, dass
die geringen Temperaturoszillationen der thermischen Welle aufgrund der IR-transparenz des
Siliziums vor einem relativ höheren, flukturienden Strahlungshintergrund gemessen werden
müssen. Um dieses experimentelle Problem zu kompensieren, sollten für den
Filterungsprozess des Lock-In-Verstärkers höhere Zeitkonstanten und eine größere
Integrationszeit verwendet werden. Ein modifizierter Probenhalter, bei dem die Heizleistung
an den Seiten eingebracht wird, würde die experimentellen Bedingungen ebenfalls verbessern,
da insbesondere der Temperaturgradient in der Probe reduziert würde.
8.2.3.3 Ergebnisse von vielschichtigen Superisolationsfolien
An den mehrschichtigen Superisolationsfolien Lydall DAM wurden bei
Raumtemperatur und Umgebungdruck Messungen thermischer Wellen in der
Transmissionskonfiguration durchgeführt. Die untersuchten Superisolationsfolien bestehen
aus aluminisierten Mylar-Folien und sog. Spacer-Schichten, wobei eine 25 µm dicke Mylar-
Folie auf der Proben Vorder- und Rückseite angebracht sind, während sich dazwischen
Sequenzen aus Spacer-Schichten (Cryotherm 234) und 6µm dicker Mylar-Folien befinden.
Aus den Messungen wird ersichtlich, dass der Wärmetransport durch das isolierende
Schichtsystem quantitativ durch thermische Wellen in Transmission beschrieben werden
kann,. Darüber hinaus zeigte sich, dass der Wärmetransport durch das Schichtsystem rein
diffusiv ist, sobald mehr als nur eine einzelne Folie betrachtet werden. Dies ist vermutlich auf
die Luft zwischen den Schichten und den engen Kontakt der Schichten zueinander
zurückzuführen.
Aus den Ergebnissen konnten unter Berücksichtigung des Wärmetransportes durch
Leitung und Strahlung die relativen Dämpfungsfaktoren für verschiedene Mehrschicht-
systeme und der relative Abfall des Temperaturleitwerts bestimmt werden. Diese beiden
Parameter sind die relevanten thermischen Parameter zu Charakterisierung der
Superisolationsfoilen.
Die Messung, die an der einzelnen 25 µm dicken äußeren Folie durchgeführt wurde,
ermöglichtes, den neben der Wärmeleitung ebenfalls vorhandenen Wärmetransport durch
Strahlung zu identifizieren. Um Strahlungstransport ebenfalls bei Mehrschichtsystemen unter
Tieftemperaturbedingungen zu identifizieren, müssten vermutlich die Abstände zwischen den
einzelnen Folien größer sein.
8.2 Zusammenfassung 171
Da die gemessenen Signale sehr nah am Rauschen sind, insbesondere bei hohen
Anregungsfrequenzen, müssten zur Verbesserung der Messungen höhere Heizleistungen
benutzt und zugleich die Lock-In Filterung verbessert werden.
8.2.3.4 Ergebnisse an faserverstärkten Verbundwerkstoffen
Für die Hochtemperatur Messungen an den faserverstärkten Verbundwerkstoffen
konnte prinzipiell eine gute Übereinstimmung mit den theoretischen Approximationen erzielt
werden, wobei das Modell benutzt wurde, das eine leichte Infrarottransparenz berücksichtigte.
Da die faserverstärkten Verbundwerkstoffe sowohl thermisch als auch hinsichtlich ihrer
Absorptionslänge im Infraroten relativ dick waren, konnten theoretische Näherungen der
Amplituden und Phasen auf der Basis eines Modells für einen halbunendlichen Festkörper
vorgenommen werden. Der Vorteil eines solchen Modells ist, dass die theoretischen Phasen
nur von einem kombinierten thermo-optischen Materialparameter ssP βα 211
/= abhängen.
Dies vereinfacht natürlich den Approximationsprozess und demzufolge auch die Bestimmung
der relevanten physikalischen Parameter.
Prinzipiell konnte ein Abfallen des Parameters ssP βα 211
/= mit steigender
Temperatur beobachtet werden. Da für einige Proben der Temperaturleitwert sα separat
gemessen wurde, ergab es sich, dass der größere Effekt in den Änderungen von ss βα 2/1 auf
eine mit der mittleren Probentemperatur ansteigende effektive Infrarottransparenz
zurückzuführen ist.
Größere Abweichungen zwischen der Theorie und den Messergebnissen konnten nur bei
höheren Modulationsfrequenzen und niedrigeren effektiven Absorptionskoeffizienten im
Infraroten beobachtet werden. Diese Abweichungen können durch die Schichtstruktur der
faserverstärkten Verbundwerkstoffe, die aus einer gewebten Deckschicht und darunter
liegendem Bulkmaterial bestehen, erklärt werden. Die theoretischen Näherungen könnten
jedoch sicherlich durch ein Zwei-Schicht-Modell, das IR-Transparenz und Streuung an den
Fasern berücksichtigt, verbessert werden.
Die Messungen, die an faserverstärkten Verbundwerkstoffen mit systematischer
Änderung der Faserkonzentration vorgenommen wurden, können durch einen
konzentrationsabhängigen Anstieg des kombinierten thermo-optischen Parameters ss βα 2/1
mit steigender Faserkonzentration erklärt werden. Aus den Näherungen für die normierten
Signalamplituden bei niedrigen Frequenzen wird deutlich, dass die relativen thermischen
Wärmeeindringkoeffizienten mit steigender Faserkonzentration ebenfalls ansteigen. Nimmt
172 8 Deutschsprachige Zusammenfassung
man einen solchen Effekt ebenfalls für den Temperaturleitwert an, so bedeutet dies ein
ansteigen des IR-Absorptionskoeffizienten mit der Faserkonzentration. Dieses Ergebnis
erscheint realistisch, da die steigende Faserkonzentration mit einer Erhöhung der
Konzentration von Streuzentren für interne thermische Strahlung einhergeht.
173
Appendix A. Transmission characteristics of different IR
materials
The transmittance of melt grown MgF2 2.1 mm thick (A), CaF2 1.2 mm thick (B), SrF2
1.5 mm thick(C) and BaF2 1.5 mm thick (D)
Figure A.1: Spectral transmission properties of different IR materials [Savage, 1985].
Figure A.2: Transmission characteristic of LiF and CaF2 at different thickness
[Bergmann and Schaefer, 1993].
174
Figure A.3: Spectral transmission properties for different IR materials
(thickness 2 mm) [Hudson, 1969]
Figure A.4: Spectral transmission properties for a silicon sample (thickness 3 mm)
[Laser components, 1994].
Figure A. 5: Spectral transmission properties for the Germanium filter used in this
work (thickness 3 mm) [Laser components, 1994].
175
Appendix B. The relative spectral sensitivity of the MCT-detector
Figure B.1: Relative spectral sensivity of the detector used in this work.
176
Appendix C. List of the used components
Equipment
Model
Manufacture
Ar+-Laser
Modulator
Modulator driver
Lock-in amplifier
IR detector
Preamplifier
Infrared lenses
IR filter
Computer
High Temperature Cell
Series 2000 (2020)
LM080
MD-080D
SRS 830 DSP
J15D12-M204-S02M-60
BaF2
Ge
PC486
Spectra Physics
Laser Components
Laser Components
Stanford-Research-System
Judson-Infrared
Electronic Workshop (RUB)
Dr.Karl Korth oHG
Laser Components
Epson
Made by AG
Festkoerperspektroskopie
group and RUB workshop
177
REFERENCES
Abramowitz and Stegun, 1965 M. Abramowitz, I. A. Stegun, Handbook of
mathematical functions, Dover, New York, 1965, chap.
5.
Almond and Patel, 1996 D. Almond, P. Patel. Photothermal science and
techniques, Chapman and Hall, 1996
Becherer and Grum, 1979 R. J. Becherer, F. Grum, Optical radiation
measurements, Volume1, Radiometry, Academic press,
INC, 1979
Bein, Krueger and Pelzl, 1986 B. K. Bein, S. Krueger, J. Pelzl. Photoacoustic
measurements of effective thermal properties of porouse
limiter graphite. Can. J. Phys. 64 (1986), 1208-1216
Bein and Pelzl, 1989 B. K. Bein, J. Pelzl, Analysis of Surfaces Exposed to
Plasmas by Nondestructive Photoacoustic and
Photothermal Techniques" In: PLASMA
DIAGNOSTICS, Vol. 2, Surface Analysis and
Interactions, (Eds. O. Auciello, D.L. Flamm), Academic
Press, 1989, 211-326.
Bein et al, 1994 B.K. Bein, J. Gibkes, A. Mensing, J. Pelzl, Measurement
of the Thermophysical Properties of Porous Materials at
High Temperatures, High Temp.-High Pressures 26
(1994), 299-307.
Bein et al, 1995 B. K. Bein, J. Bolte, M. Chirtoc, C. Gruss, J. Pelzl, K.
Simon, Characterization of carbon and glass fibre
reinforced Material by Photothermal Methods, Advances
in Thermal inslulation (Proc. Of the Eurotherm Seminar
N° 44), 1995, 191-198
Bein et al, 1998 B. K. Bein, J. Bolte, D. Dietzel, A. Haj-Daoud,
Characterisierung technischer Schichtsysteme mittels IR-
Radiometrie thermischer Wellen, tm. Technische
Messen, 387-395, Oldenburg, 65 (1998), 387-395
178
Bein et al, 1999 B. K. Bein, D. Dietzel, A. Haj-Daoud, G. Kalus, F.
Macedo, J. Pelzl, Photothermal characterization of
amorphous thin film, Surface & Coatings Technology,
1999, in print
Bergmann and Schaefer, 1993 Bergmann, Schaefer, Lehrbuch der Experimentalphysik,
Bd. 3 Optik. Walter de Gruyter, 1993
Bolte, 1995 J. Bolte, Untersuchung der thermischen Eigenschaften
von faserverstärkten Verbundwerkstoffen mittels
frequenz-abhängiger photothermischer Radiometrie.
Diplomarbeit, Ruhr-Universität Bochum, 1995
Bolte, Gu and Bein, 1997a J. Bolte, J. H. Gu, B. K. Bein, Background fluctuation
limit of infrared detection of thermal waves at high
temperature. High Temp.- High Pressure 29 (1997), 567-
580
Bolte, Bein and Pelzl, 1997b J. Bolte, B.K. Bein, J. Pelzl, IR Detection of Thermal
Waves − Effect of Imaging Conditions on the
Background Fluctuation Limit, In: Quantitative InfraRed
Thermography QIRT’96, (Eds. D. Balageas, G. Busse a.
C.M. Carlomagno), Eurotherm Series 50, Edizioni ETS,
Pisa, (1997), 9-14.
Brewester, 1992 M. Quinn Brewster, Thermal radiative transfer and
properties, John wiley and Sons, Inc, 1992
Carslaw and Jaeger, 1959 H. S. Carslaw, J. C. Jaeger, Conduction of heat in solids.
Oxford, Clarendon press, 1959
Clauser, 1975 H. R. Clauser, Industrial and engineering materials, Mc
Graw-Hill Inc., Tokyo, 1975
Dietzel, 1997 D. Dietzel, Untersuchung der Temperaturabhängigkeit
der photothermischen Infrarotradiometrie an
faserverstärkten Verbundwerkstoffen, Diplomarbeit,
Ruhr-Universität Bochum, 1997
Dietzel et al, 1999 D. Dietzel, A. Haj-Daoud, F. Macedo, J. Bolte, J.
Pelzl, B. K. Bein, IR transparence and radiative heat
transport of fibre-reinforced materials at higher
temperatures, Photoacoustic & Photothermal Phenomena
179
(eds. F. Scudieri a. M. Bertolotti), AIP Conf. Proc. 463,
1999, 321-323
.Doermann and Sacadura, 1995 D. Dooermann, J. F. Sacadura, Prediction of thermal
performance of open cell foam insulations. Advance in
thermal insulation, Proceeding of the Eurotherm Seminar
N° 44, Espinho-Purtugal, 1995, 25-43.
Driscoll, 1978 W. G. Driscoll, Handbook of optics, New York, Mc
Graw-Hill, 1978
Ebert and Fricke, 1998 H. P. Ebert, J. Fricke, High Temp. -High Pressures 30
(1998), 655-669
Fotherring and Glastech, 1994 U.Fotherringgham, F. Lentes, Glastech. Ber. Glass Sci.
Technol. 67 (1994) No.12
Grigull and Sdanders, 1988 G. Grigul, H. Sandner. Waermeleitung. Springer Verlag,
Berlin, 1988.
Gu, 1993 J. H. Gu, Photothermische Radiometrie für Festkörper
bei Hohen Temperaturen, Dissertation, Ruhr-Universität
Bochum, 1993
Gusev, Mandelis and Bleiss, 1993 V. Gusev, A. Mandelis, R. Bleiss, Theory of second
harmonic thermal wave generation: 1D geometry, Int. J.
Thermophysics, 1993
Haj Daoud, Bein and Pelzl,1996 A. Haj-Daoud, B. K. Bein, J. Pelzl, Thermophysical
properties of mixed systems- Textiles, packaging
material, Living Tissues, contribution to the meeting of
the EU-Network (BRITE Euram), Control of migration
profiles and structural evolution in thin and non-compact
materials by photothermal methods, Reims, December
1996
Haj Daoud et al, 1998 A. Haj-Daoud, U. Katscher, B. K. Bein, J. Pelzl, H.
Bach, W. Oswald, Technical damage analysis of a
mechanical seal based on thermal waves and correlated
with EDX and SEM, Photoacoustic & Photothermal
Phenomena (eds. F. Scudieri a. M. Bertolotti), AIP Conf.
Proc. 463, 1999, 383-385
180
Haj Daoud etal, 2000 A. Haj-Daoud, B. K. Bein, D. Dietzel J. Pelzl, Three-
dimensional radiative heat transports in a one-
dimensional temperature fields, 2000, to be published
Hess, 1989 P. Hess, Photoacoustic, Photothermal and
Photochemical Processes at surface and in thin film,
Volume 47, Springer-Verlag, Berlin, 1989.
HTW GmbH, 1999 Hoch-Temperatur-Werkstoffe GmbH, 1999
Hudson, 1969 R. D. Hudson. Infrared system engineering, Wiley-
Interscience, New York, 1969
Huettner, 1989 R. Huettner, Aufbau eines Experiments zur temperatur
abhängigen photothermischen Infrarotradiometrie.
Diplomarbeit, Ruhr-Universität Bochum, 1989
Huettner et al, 1991 R. Huettner, B. K. Bein, J. H. Gu, J. Pelzl, D. L.
Balageas, A. A. Deom, IR Radiometric detection of
small-amplitude thermal waves between 500 K and 1000
K, In: Proc. 7 th Intern. Topical Meeting Photoacoustic
and Photothermal Phenomena, Doorwerth, NL, 1991, 26-
30, (Ed. D. Bicanic).
Incropera and DeWiTT, 1990 F. P. Incropera, D. P. DeWiTT, Fundamentals of heat
and mass transfer, John Wiley and Sons, 1990
Jastrzebaki, 1977 Z.D. Jastrzebaki, The nature and properties of
engineering materials. Wiley, New York, 1977
Kalus et al, 1997 G. Kalus, B. K. Bein, J. Pelzl, H. Bosse, A.
Linnenbruegger, Characterization of tribological
protective films and wear by IR radimetry of thermal
waves, QIRT 96-Eurotherm Series 50-Edizioi ETS, Pisa,
1997, 203-208.
Kohlrausch 1985 F. Kohlrausch, Praktische Physik. B. G. Teubner
stuttgart, 1985
Landau and Lifshitz, 1968 L. D. Landau, E. M. Lifshitz, Statistical physics,
Pergamon press Ltd, 1968
Laser comp, 1994 laser components, Optische filter, 1994
Matthes, 1990 G. Matthes, Lehrbuch der Hydrogeologie, Band 2.
Borntraeger-Verlag, Berlin, 1990.
181
Mehling et al,1995 H. Mehling, J. Kuhn, M. Valentin, J. Fricke,
Measurements of IR optical properties and emitted
radiation of semitransparent ceramics. Advance in
thermal insulation, Proceeding of the Eurotherm Seminar
N° 44, Espinho-Purtugal, 1995, 85-99.
Mensing, 1994 A. Mensing, IR-Radiometrie thermischer Wellen an
plasmagespritzten NiCoCrAIY-Beschichtungen.
Diplomarbeit, Ruhr-Universität Bochum , 1994
Nolting, 1994 W. Nolting, Grundkurse theoretische Physik 6.
Zimmermann Neufang, 1994
Nordal and Kanstad, 1979 E. Nordal, S. O. Kanstad, Photothermal radiometry,
Phys. Scripta 20 (1979) , 659
Oezisik, 1985 M. N. Oezisik, Heat transfer A basic approach,
McGraw-Hill Book Company, New York, 1985.
Paoloni and Walther,1997 S. Paoloni, H. G. Walther, Photothermal radiometry of
infrared translucent materials, J. Appl. Phys. 82 (1997),
101-106
Petry, 1998 H. Petry, Online-Messungen von Lackschichtdicken mit
thermischen Wellen, tm. Technische Messen, Oldenburg,
Nov. 1998
Pointon, 1967 A. J. Pointon, An Introduction to statistical physics for
students, Longmans, Green and Co Ltd, 1967
Rosenwaig and Gersho, 1976 A. Rosencwaig, A. Gersho, Theory of the photoacoustic
effects with solids, J. Appl. Phys. 47, 64 (1976).
Sanderson, 1955 J. A. Sanderson, Emission, Transmission, and Detection
of the infrared, Guidance, Princeton, N. J.: Van
Nostrand, 1955
Santos and Miranda, 1981 R. Santos, L. Miranda, Theory of the photothermal
radiometry with solids, J. Appl. Phys. 52 (1981), 4194-
4198
Savage, 1985 J.A. Savage, Infrared optical materials and their
antireflection coatings. Bristol: Hilger, 1985
182
Seidel and Walther, 1996 U. Seidel, H. G. walther, The influence of the spectral
emissivity on the signal phase at photothermal
radiometry, Rev.Sci, Instrum. 67 (1996), 3658-3663
Siegel and Howell 1981 R. Siegel, J. R. Howell, Thermal radiation Heat
transfer, McGraw-Hill Book Company, New York, 1981
Simon,1996 K. Simon, Private Mitteilung, 1996
Sommer, 1994 T. Sommer, Untersuchungen zur Temperatur-
abhaengigkeit und Ortsaufloesung der Photothermischen
Radiometrie an technischem Graphit. Diplomarbeit,
Ruhr-Universität Bochum, 1994
Sparrow and Cess, 1978 E. M. Sparrow, R. D. Cess, Radiation heat transfer,
McGraw-Hill Book company, 1978
SRS, 1992 Stanford Research Systems. SR830 Basics, 1992
Storm, 1951 M. L. Storm, Heat conduction in Simple Metals, J.
Applied Physics. 22(1951), 940-951
Tom, O’Hara and Benin, 1982 R. D. Tom, E. P. O’Hara, D. Benin, A generalized model
of photothermal radiometry, J. Appl. Phys. 53 (1982),
5392-5399
Touloukian et al, 1973 Y. S. Touloukian, R. W. Powell, C. Y. Ho, M. C.
Nicolaou. Thermalphysical properties of matter, 10, IFI /
Plenum, New York, 1973.
Touloukian et al, 1973 Y. S. Touloukian, R. W. Powell, C. Y. Ho, M. C.
Nicolaou. Thermalphysical properties of matter, 1-6, IFI
/ Plenum, New York, 1973.
Walther et al, 1992 H. G. Walther, U. Seidel, W. Karpen, G. Busse,
Application of modulated photothermal radiometry to
infrared transparent samples, Rev.Sci. instrum. 63
(1992), 5479
183
Acknowledgment
With high eternity, I would like to express my thanks and convey my sincere gratitude
to my adviser Prof Josef Pelzl, for his guidance, encouragement, advise, help and support in
every aspect of this work, and for providing the facilities that were necessary for the
realization of this thesis.
I would like also to express my sincere gratitude and appreciation to Prof. Bruno. K.
Bein, the leader of the Photoacoustic and Photothermal group, where I conducted my
research. With his scientific attitude and his high research spirit, he has been able to steer my
work always in the right direction. He has been an inexhaustible source of inspiration and a
tireless continuous stimulating discussion partner. I am saying nothing but the truth when
admitting that his guidance, his open door policy and his openness have contributed
significantly to the good working atmosphere in Bochum, and that his constructive critics in
the course of the work helped me to come ahead. To him and to Prof. Pelzl I say: words will
never express my gratitude towards you.
Furthermore, I extend my sincere thanks and appreciation to Dipl.-Phys. D. Dietzel,
who was introduced to the photothermal technique by me and then he became a source of
ideas. I thank him for all his support and for his good friendship. Parts of the presented work
have been done with his help.
I would also to thank and convey my special regards to our experient Dipl.-Phys. J.
Gibkes for his stimulating and fruitful discussions that led to an improved understanding in
this field both experimentally and the theoretically and for good advices in the measuring
programs and the calculation program. Also special thanks and appreciation to the friendly
character of Dr. R. Hüttner for his good ideas for improving the conditions of the experiment
and his good argumentation in explanations.
For the excellent technical assistance, special thanks and appreciation to Mr. D.
Krüger for his useful help concerning all technical problems.
I would like to address my special thanks to my colleagues, with whom I shared the
office, Dr. J. Bolte and Dipl.-Phys. G. Kalus. Both of them introduced me into the
experimental work. Sincere gratitude to Dr. J. Bolte, for his valuable advices and for
introducing me into all the little secrets in the field the Photothermal Radiometry. Dr. Bolte
had started to study the effects of radiation transport in addition to the conduction transport
and threw light on the road in the beginning of my work.
184
I would also like to express my deep thanks and convey my sincere gratitude to the
friendly Dipl.-Phys. D. Spoddig and Dipl.-Phys. F. Niebisch for their endless help in the
technical computer problems and numerical simulation during my stay at Ruhr-Universität-
Bochum and for the good atmosphere in our office, especially to Frank Niebisch, who spent a
lot of time supplying his expertise helping me in all stages of this work.
Special thanks also to the electronic workshop, especially Mr. Niesler for helping in
the construction of the special preamplifier used in the work, and to Mr. Stauche for
preparing the silicon samples.
Thanks to Dipl.-Phys. K. Simon, WIS Munster, who made the fibre-reinforced heat
shield materials available , and to Dr. A. Poncet, CERN-Geneva, who made the
superinsulation foils available, also to Dr. F. Machado de Macedo, University of Braga,
Portugal, for part of the fibre-reinforced samples. Special thanks for Prof. A. Zihlif,
University of Jordan, Amman-Jordan, for stimulation discussions with respect to the
properties, both electrical and thermal of carbon-based fibre-reinforced composite materials.
Last but definitely not least I want to thank the whole group in Bochum for the
pleasant working atmosphere and uncountable discussions during this work, especially Mrs.
V. Kubiak, Dr. C. Gruss, who introduced me into the Flying-spot technique, Dipl.-Phys. U.
Katscher, for supplying technical samples with eroded surface, Dr. J. Pflaum, for helping in
preparing GMR-seminar, Dipl.-Phys. V. John, Dipl.-Phys. H. Althaus, Dr. I. Delgadillo-
Holtfort, Dr. R. Meckenstock, Dipl.-Phys. M. Kaack, Dipl.-Phys. D. Kurowski.
I would like to express my gratitude to the Deutscher Akademischer Austauschdienst
(DAAD) for the financial support during my stay at the Ruhr-Universität-Bochum and in
Goettingen during my stay at the Goethe Institute.
Special thanks to my Uncle Zuhdi Daoud and Maria in München for their
encouragement and moral support. I am very grateful to my brother Dr. Eng. A. Batta and
his family during my staying in Bochum for helping me from A to Z and sharing my efforts
and pleasure. Special thanks to all my friends in Bochum, especially Dipl.-Phys. I. Alawneh
for helping in the correcting of the Dissertation, Dipl.-Phys. S. Saleem and Dr. M. Shabat
for their general helps.
I wish to express my deep gratitude and sincere thanks to my parents in Palestine for
their endless moral support, encouragement and love throughout my life. Last but not least, to
them and to all members of my family I say: words will never express my gratitude towards
you.
185
Curriculum Vitae
Name: Ayman Nazmi Ahmad Al Haj Daoud.
Date and Place of Birth: 05/09/1965, Nablus, West Bank, Palestine.
Nationality: Palestinian.
Home address: District Court, P. O. Box 246, Nablus-West Bank, Palestinian Authority, Via
Israel
Marital Status: Single.
E-mail: [email protected], [email protected]
Tel: Germany / 0049 234 32 27964, Nablus / 00972 9 2380337
Academic Background:
1. School Education:
Elementary Education 1971-1976: 1 st - 6 th elementary calss.
Preparatory Education 1977-1979: 1 st - 3 rd preparatory calss.
Secondary Education 1980-1983: 1 st 6 th secondary calss.
Obtained the General Secondary Education Certificate Examination
(Scientific Specialization) in 1983. Marks obtained 80.6%
2. University Education
Bachelor of Science in Physics (1987); B. Sc. In Physics, with grade average
80.1% (Degree: very Good) from An-Najah National University, Nablus-West
Bank.
3. Higher Education:
Master of science in Physics (1992), Major: Condensed Matter Physics, with
grade average 84.1% (Degree: excellent) from University of Jordan, Amman,
Jordan.
Title of Master Thesis: Electrical Properties of Conductive Material “Nickel
Coated Carbon Fibre-Nylon 66” at Microwave
frequency.
Publication: “OPTIMUM MEASURING CONDITIONS OF SHIELDING
EFFECTIVENESS FOR CONDUCTIVE POLYMER
COMPOSITE”, (a) A. N. Haj-Daoud, (a) A. M. Zihlif, (b) M. K.
Abdul-Aziz, Materials Letters 15 (1992) 104-107, North
Holland. (a) Physics Department (b) Electrical Engineering
Department
186
4. Teaching Experience
1989-1992 Teaching assistance at the University of Jordan.
1992-1995 Teaching as a physics teacher at high schools.
Since 1995 I received a scholarship from Deutscher Akademischer
Austauschdienst (DAAD) as a Ph. D. student at the Institute of
Experimentalphysik III of the Ruhr-Universität-Bochum.
Title of Ph. D. Thesis: Photothermal Materials Characterization at Higher Temperatures by
Means of IR Radiometry.
Supervision by: Prof. Dr. J. Pelzl
Publications:
(1) B. K. Bein, J. Bolte, D. Dietzel, A. Haj-Daoud, Characterisierung technischer
Schichtsysteme mittels IR-Radiometrie thermischer Wellen, tm. Technische Messen,
387-395, Oldenburg, (65) Nov. 1998
(2) B.K. Bein, J. Bolte, A. Haj-Daoud, V. John, F. Niebisch, Background fluctuation limit
of IR detection of thermal waves – Basic principls and application to photothermal
characterization of biological materials and living tisues,Photoacoustic &
Photothermal Phenomena (eds. F. Scudieri a. M. Bertolotti), AIP Conf. Proc. 463, 93-
95, 1999.
(3) B. K. Bein, D. Dietzel, A. Haj-Daoud, G. Kalus, F. Macedo, J. Pelzl, Photothermal
characterization of amorphous thin film, Surface & Coatings Technology, 1999, in
(4) D. Dietzel, A. Haj-Daoud, F. Macedo, J. Bolte, J. Pelzl, B. K. Bein, IR
transparence and radiative heat transport of fibre-reinforced materials at higher
temperatures, Photoacoustic & Photothermal Phenomena (eds. F. Scudieri a. M.
Bertolotti), AIP Conf. Proc. 463, 321-323, 1999.
(5) D. Dietzel, A. Haj Daoud, F. Macedo, J. Pelzl, B.K. Bein, IR Transparency and
Radiative Heat Transport of Carbon-based Fibre-reinforced Composites at High
Temperatures, submitted for publ. High Temp. – High Pressures.
(6) D. Dietzel, A. Haj-Daoud, F. Macedo, K.Simon, B.K. Bein, J. Pelzl, Thermophysical
Properties of Fibre-reinforced Composites at Higher Temperatures, Verhandl. DPG
5/1998, S.1052, 62. Physikertagung, Regensburg.
187
(7) D. Dietzel, A. Haj Daoud, F. Macedo, J. Pelzl, B.K. Bein, Systematic study of second
harmonic thermal wave detection based on IR radiometry, Photoacoustic & Photo-
thermal Phenomena (eds. F. Scudieri a. M. Bertolotti), AIP Conf. Proc. 463, 50-52,
1999.
(8) A. Haj-Daoud, B. K. Bein, J. Pelzl, Thermophysical properties of mixed systems-
Textiles, packaging material, Living Tissues, contribution to the meeting of the EU-
Network (BRITE Euram), Control of migration profiles and structural evolution in
thin and non-compact materials by photothermal methods, Reims, December 1996.
(9) A. Haj-Daoud, U. Katscher, B. K. Bein, J. Pelzl, H. Bach, W. Oswald, Technical
damage analysis of a mechanical seal based on thermal waves and correlated with
EDX and SEM, Photoacoustic & Photothermal Phenomena (eds. F. Scudieri a. M.
Bertolotti), AIP Conf. Proc. 463, 383-385, 1999.
(10) A. Haj-Daoud, B. K. Bein, D. Dietzel J. Pelzl, Three-dimensional radiative heat
transports in a one-dimensional temperature fields, 2000, to be published.
(11) C. Gruss, R. Hüttner, A. Haj-Daoud, B.K. Bein, J. Pelzl, The oscillating spot: A new
active IR radiometry technique for thermal characterization of solids, Photoacoustic &
Photothermal Phenomena (eds. F. Scudieri a. M. Bertolotti), AIP Conf. Proc. 463,
114-116, 1999.
(12) F. Macedo, J. Ferreira, F. Vaz, L. Rebouta, A. Haj Daoud, D. Dietzel, B.K. Bein,
Photothermal characterization of sputtered thin films and substrate treatment,
Photoacoustic & Photothermal Phenomena (eds. F. Scudieri a. M. Bertolotti), AIP
Conf. Proc. 463, 536-538, 1999.
(13) J. Bolte, A. Haj-Daoud, B.K. Bein, High-temperature background fluctuation limit of
IR detection of thermal waves, Gordon Research conference 1997, Photoacoustic and
Photo thermal Phenomena, Sept 1997.