Photothermal Materials Characterization at Higher ...

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


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,


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].


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].


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


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.


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


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.


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


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)


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)


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)


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


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.


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)


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


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].


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.


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


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.


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)


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


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


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:


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)


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.


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.


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


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


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


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)


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)


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].


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.


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)


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


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,


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


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




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




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].


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.


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.


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).


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.


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.


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)


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


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


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


(f / H

z)-1

/2


(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


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


(f / H

z)-1

/2


(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


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


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.


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


5.1.3 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


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.


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


(f / H

z)-1

/2


(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)


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


(f / H

z)-1

/2


(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)


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


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


(f / H

z)-1

/2


(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.


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.


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)


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.


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


(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


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


(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


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.


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)


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.


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.


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


(f /

Hz)

1/2

02

46

810

-2-1012


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


(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


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


(f /

Hz)

1/2

02

46

810

-2-1012


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


(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)


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


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).


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 + spacer layer(243) + internal layer (6 µm)




∗∗ 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)


6.1.1 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


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 + spacer layer (243) + internal layer (6 µm)




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


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


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.






On the other hand, the decrease of the normalized amplitudes at the higher frequencies, which

can be approximated by


( ) 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.







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


⋅−

+⋅+−

−+

+

−′=−

−−−−

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 + 3 x (spacer layer + internal layer)




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))


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


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.


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.


6.2.1 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


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


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


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.



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


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.


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


fibre reinforced heat shield material V4.


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



fibre reinforced heat shield material R.


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)


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)


|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


[ ]

[ ]))(())((

))(())((

))(,,(

))(,,(

)(

)(

))(,,(

))(,,())(,)(,,(

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


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


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


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


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.


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


(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


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


(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


/ Hz)

-1/

2


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


(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


/ Hz)

-1/2


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).


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).


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


(f / H

z)-1

/2


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


(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


(f / H

z)-1

/2


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


(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


(f / H

z)-1

/2


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


(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


(f / H

z)-1

/2


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


/ 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


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.


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%


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.


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


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


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.


7.4.2 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.


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


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


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


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


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.


8.2.3 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


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


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


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.


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


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

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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

print

(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.

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