Dissertation. Zur Erlangung des Grades eines Doktors der Naturwissenschaften in der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum

Transcription

1 Invetigation of Thermal Tranport in Layered Sytem and Micro-tructured Semiconductor Device by Photothermal Technique and Finite Element Simulation Diertation Zur Erlangung de Grade eine Doktor der Naturwienchaften in der Fakultät für Phyik und Atronomie der Ruhr-Univerität Bochum vorgelegt von Jean Lazare Nzodoum Foting au Jaunde Kamerun Bochum 004

2 Mit der Genehmigung de Dekanat vom wurde die Diertation in englicher Sprache verfat Eingereicht am: Diputation am: Gutachter: Prof. Dr. J. Pelzl. Gutachter: Prof. Dr. A. Wieck ii

3 To my uncle Takam André Pombo Gaton who paed away repectively in 003 and in 004. May their oul ret in peace iii

4 Content Content... iv Abreviation and Nomenclature...viii 1 Introduction Motivation Objective of the Work Thei Overview... Fundamental and Goal of the Photothermal Technique Introduction The concept of thermal wave Relevant phyical parameter Review of the main detection method Concluion Surface and Suburface Effect of Thermal Tranport in Layered Sytem Introduction Photothermal IR radiometric ignal Decription of the meauring ytem Decription of the invetigated ample Theoretical model of thermal wave for layered ytem Interpretation of the modulated IR radiometric ignal Dicuion of phyical effect in the invetigated ample Surface effect Optical effect: coating emi-tranparency Topological effect Suburface effect: lateral thermal tranport Theoretical development Comparion of 1-D and 3-D thermal wave propagation Effect of heating pot radiu and uburface thermal propertie Interpretation of ignal phae baed on 3D thermal tranport Determination of the propertie of lateral heat tranport Dicuion of reult Concluion Determination of Thermal Tranport Propertie of Two-layer Structure uing the Concept of the Phae Extremum Motivation The concept of the Phae Extremum iv

5 4..1 Phyical ignificance of the obervable phae extrema Theoretical background Application to experimental meaurement Methodology and dicuion Determination of the thermophyical propertie Efficient localization of the phae extremum Application to meaurement at high temperature Dicuion on the reliability of the Extremum Method Solution from any meaured point of the calibrated phae Comparion of reult General reult for on-line interpretation in indutrial application Graphic of thermal reflection coefficient Graphic of ratio of the effuivitie Graphic of thermal diffuion time Example of on-line interpretation Interpretation of modulated IR ignal obcured by the background fluctuation Theory of the tranformation of the invere normalized phae Application of the functional tranform to meaurement: The problem of convergence Phae interpretation obcured by the background fluctuation Concluion Detection of Local Inhomogeneitie of Thermal Tranport and Localization of Heat ource in Micro-caled Sytem baed on Spot Diplacement Motivation Review of the main reult of 3-D thermal wave propagation Diplacement between heating and detection pot Theoretical background Simulation of controlled diplacement between the two pot Localization of heat ource Comparion of experimental meaurement and theoretical reult Scaling of thermal localization of hot pot Comparion with meaurement baed on Thermoreflectance Concluion Efficient Simulation of Thermal Wave Problem with the ANSYS-aided Finite Element Technique Introduction Finite element concept Theoretical foundation Fundamental tep Finite element modeling Pre-proceing Computation Pot-proceing v

6 6.4 Efficient imulation of thermal wave problem Convergence of the numerical olution Non-uniform mehing and meh refinement Control of convergence with the reference phae hift Methodology of imulation Finite element control of theoretical reult Simulation of thermo-elatic ignal Theoretical bai Methodology Example of application Concluion Finite Element Invetigation of Heating Procee in Micro-tructured Semiconductor Device Motivation Feature of two experimental technique in micro-thermal analyi Scanning Thermal Microcopy SThM Scanning Thermal Expanion Microcopy SThEM FE imulation of thermal and thermo-elatic ignal AC heating and DC heating The ac-electrically heated conducting line Phyical ytem and model Thermal ocillation and thermal expanion Frequency-dependent thermal expanion Invetigation of thermal and thermo-elatic ignal in HEMT-device Power diipation in tranitor Baic tructure of tranitor Origin and location of the power diipation Influence of the ytem Drain Gate Source on the ignal Structure exempted from the ytem D-G-S Structure with D-S Structure with D-G-S Comparion of experimental meaurement and FE reult Influence of the modulation frequency Contribution of the tip to the ignal Poition of the problem Modeling of the tip tructure Si-tip in mechanical contact with a GaA-tructure Temperature ocillation along the tip axi Derivation of the thermal expanion of the tip Thermal ocillation along the ample urface Influence of the material of the tip Influence of the heated ubtrate Influence of a fluid film between tip and ample Concluion vi

7 8 Concluion and Future Work Concluion Review of the experimental work Review of the numerical-analytical work Review of the numerical work uing finite element Future work Extenion of the Extremum Method Calibration procedure for determining the temperature of hot pot D finite element imulation Appendix A: Characteritic of the invetigated ample Appendix B: Material propertie ued for FE imulation Bibliography Publication and Conference contribution Curriculum vitae Acknowledgement vii

8 Abreviation and Nomenclature Abreviation 1-D One dimenional -D Two dimenional 3-D Three dimenional D Drain CLTE Coefficient of Linear Thermal Expanion FEM Finite Element Method FET Field Effect Tranitor G Gate GaA-ample Sample of Galium arenide material HEMT High Electron Mobility Tranitor HMA High Effuivity Metallic Alloy HSS High Speed Steel InSb-ample Sample of Indium antimonide material MCT-detector Mercury Cadnium Telluride HgCdTe detector S Source Si-ample Sample of ilicon material Si-tip Tip of ilicon material SJEM Scanning Joule Expanion Microcopy SthEM Scanning Thermal Expanion Microcopy SThM Scanning Thermal Microcopy Pt-tip Tip of platinium material W-tip Tip of tungten material viii

9 Nomenclature Symbol Unity Name T Time τ Thermal diffuion time of the coating ε β σ SB β β e η k c ρ k α = ρ c e = R ij P αβ k ρ c = α 1/ β 1 m W m 1 m 1 K W m K J kg K kg 3 m m W m 1 1/ 1/ K K -4 Emiivity Coefficient of optical aborption Stefan-Boltzmann contant Coefficient of optical aborption Coefficient of linear thermal expanion Photothermal converion efficiency Thermal conductivity Specific heat capacity Ma denitiy Thermal diffuivity Thermal effuivity Thermal reflection coefficient between layer i and j Combined thermo-optical parameter f Hz Modulation frequency of heating ω = π f Hz Angular frequency α µ th = m Thermal diffuion length π f W I o m Incident beam power ix

10 r H r D d HD T r, r t r T, f, t m m m K Radiu of the heating pot Radiu of the detection pot Diplacement ditance between the heating and the detection pot Abolute temperature δ K Modulated temperature ν u r, r t m Poion-number Diplacement vector δ d z m Vertical thermal expanion r W Q r 3 m Heat generation rate x

11 1. Introduction 1.1 Motivation The generic term layered ytem refer to ample coniting of a baic material or ubtrate on which can be identified one or more layer of differing optical and thermophyical propertie. Generally, thee layered material ytem are built up to fulfill ome pecific purpoe, e.g. to erve a thermal barrier or alternatively a thermal conductor, and o a true knowledge of the individual characteritic of the participating component i beneficial to find out the bet tructuring. Thu, the performance of the entire tructure depend upon the integrity of the individual component a well a the interface between layer. Thi i why thermal characterization of uch ytem i of great importance ince it allow to point out the governing thermal tranport propertie and to detect the eventual tructuring defect. However, one of the major problem often reide in the contamination and even in the deterioration of the ample under invetigation by inappropriate experimental method. Thee problem are overcome or avoided by uing the photothermal technique which are more uited for performing thermal invetigation ince they are non-detructive and do not require any ample preparation. On the other hand, thee layered ytem are exploited for the deign and manufacturing of micro-tructured device, e.g. micromechanical and microelectronic device. A for the microelectronic device, they operate very fat and thu produce high power denitie, implying necearily new challenge in the field of thermal management. The ituation i even wore a the dimenion of thee device are getting dratically maller and maller ince in thi cae the phenomena of tranport and diipation become more complex. For example, it ha been etablihed, that the cloe proximity of interface and the extremely mall volume of heat diipation trongly modifie thermal tranport, thu aggravating problem of thermal management [Cahill et al., 003]. Main obeion in invetigating both in the theoretical and in the experimental point of view thee o tiny tructure reide either in the determination of the local thermal conductivity/diffuivity [Milcent et al., 1995; Langer et al., 1996; Hartmman et al., 1997; Ruiz et al., 1998; Milcent et al., 1998; Hui et al, 1999; Gervaie et al., 000] or in the localization of hot pot [Bolte et al., 1998; Bolte, 1999] or even in the calibration of the abolute temperature [Schaub, 001], by uing different photothermal method. The main experimental approache in thermal microcopy are reported in [Price et al., 000; Pelzl et al., 001]. 1

12 1. Objective of the work However, the veritable difficulty in dealing with thee high power device i the identification and detection of the location of high temperature hot pot a well a the management of the diipated heat. Another major difficulty come from the fact that material which are brought into contact with each other to make up a unique device have different depoition temperature and thermal expanion coefficient [Prorok and Epinoa, 00]. Such a material compoition with o different characteritic can unavoidably lead to damage including for example cracking and de-lamination. There i therefore an abolute neceity to examine and tudy the mechanim which can help to enhance the thermal performance of thee extremely thin device. In principle, two particular experimental technique, namely the Scanning Thermal Microcopy SThM and the Scanning Thermal Expanion Microcopy SThEM, are aimed at invetigating and controlling the thermal movement in thee kind of device but due to the limitation impoed to experimental meaurement by the hotile dimenion of the tructure, numerical imulation motly baed on the Finite Element Method FEM contitute a very important tool to predict the thermal behaviour of device and then dicu about their performance via faithful model. Thi option ha at leat the advantage of limiting the high cot related to the production of layout and of avoiding hazardou indutrial tet. 1. Objective of the work The preent reearch work had three main objective: The firt tak wa to invetigate everal layered tructure by mean of photothermal technique and to find out concrete olution for a more rapid and efficient quantitative interpretation of the meaured ignal. Another important tak wa to propoe an alternative photothermal method which can help to get more reliable information on the lateral tranport propertie and allow the identification of heat ource in micro-caled ytem. The third objective of thi contribution wa to invetigate by mean of Scanning Thermal Expanion Microcopy and finite element imulation, the hot pot in ome elected micro-tructured emiconductor device following their excitation by a modulated heat ource. 1.3 Thei overview Thi work, which can be globally ubdivided into two part including the photothermal characterization of layered ytem from macrocopic to microcopic cale and the finite element invetigation of thermal and thermo-elatic ignal in micro-tructured device, i organized in the following way: Chapter recall the baic concept of thermal wave and review the different experimental configuration which are deigned for the generation and the detection of photothermal ignal in olid.

13 1.3 Thei overview Chapter 3 dicue the phyical effect of thermal tranport in variou layered tructure with the thickne of the thin film or coating ranging from about 0.8 to 3 µm. Thee phyical effect are revealed by the phae and amplitude of the modulated IR radiometric ignal meaured at the urface of the ample. The urface effect occurring in the range of high modulation frequencie, namely at low penetration depth, are found to be due either to the optical characteritic of the coating or to the topology of the ample urface. The uburface effect, identified in the limit of very low modulation frequencie, are found to be related to lateral thermal tranport in the ubtrate material of high thermal diffuivity and effuivity. In thi framework, a 3-D theory of thermal wave propagation in two-layer ytem i developed and correlated to the meaured ignal, which then help to dicriminate the main thermophyical property favouring the lateral heat propagation in the invetigated ample. Chapter 4 introduce a new evaluation method baed on the relative extremum of the calibrated meaured ignal phae and applie it to the determination of thermal and phyical propertie of two-layer tructure via two combined thermophyical quantitie. The propoed method, which i efficient, rapid and more accurate, allow to generate general reult for online interpretation in indutrial application and alo provide the poibility to interpret the meaured ignal that may be obcured by the background fluctuation. Chapter 5 decribe a new photothermal method, baed on controlled diplacement ditance between the heating and the detection pot, which help to get more reliable information on the lateral thermal tranport propertie and on the localization of heat ource in micro-caled tructure. Although the principle of diplacement ditance between excitation and detection pot i not new, the originality of thi method reide in the fact that it allow to tudy the meauring condition on the ignal phae and the relevant range of modulation frequencie. It i hown, that the relative extrema obervable on the calibrated phae and which increae with increaing diplacement ditance between the two pot can be exploited to localize the heat ource with good preciion. In order to find out at what ditance uch a localization of heat ource by mean of meaurement in the neighbourhood i till poible, the caling of thermal localization of hot pot from macrocopic to microcopic i performed by adapting the ample to more realitic condition of thermal microcopy and by decreaing gradually the value of the experimental parameter, namely the radii of the heating and the detection pot a well a the diplacement ditance between the center of the two pot. A the imulation of thermal wave problem i omewhat difficult, chapter 6 i enrolled to how by mean of tip and example how uch type of problem hould be efficiently handled with the Finite Element Technique. In thi framework, a new approach baed on the phae hift of the thermal wave at the urface of a homogenou and opaque reference material Φ ref = -45, labelled a reference phae hift, i propoed to help judging the convergence of the numerical olution. In order to verify the reliability of the developed finite element cheme, ome ignificant theoretical reult preented in chapter 3 and in chapter 5 are compared with the reult of finite element imulation. 3

14 1.3 Thei overview Chapter 7 deal with the invetigation of heating procee in micro-tructured device by uing ANSYS-aided finite element imulation. Here, the thermal ocillation in thin device and the thermo-elatic diplacement of their urface are imulated and compared with experimental reult obtained with Scanning Thermal Expanion Microcopy. The contribution of the tip on hot pot on the device to the ignal i alo invetigated. In thi framework, the influence of the material of the tip on one hand, and the influence of the material of the heated ubtrate on the other hand, a well a the incidence of a fluid bridge linking the tip and the ample are methodically analyed and dicued. Chapter 8 conclude thi work and provide inight on future challenge. 4

15 . Fundamental and Goal of the Photothermal Technique.1 Introduction The photothermal technique ue the concept of thermal wave and for emiconductor material additionally the concept of plama wave, generated in the ample by an intenitymodulated laer beam. Thermal wave, which are produced by intenity-modulated localized heating of a olid and which can be decribed by the heat diffuion equation, are temperature ocillation which vary a a function of pace and time. Due to the diffuive propagation, the thermal wave amplitude i damped and the phae lag between the thermal wave and the modulated excitation increae with the propagation ditance. In thi chapter, the baic concept of thermal wave are preented, followed by a review of the different experimental configuration that are deigned for generating and detecting thee thermal wave.. The concept of thermal wave Three fundamental mechanim for tranferring heat from one region to another are worth to be mentioned: Radiation i the mechanim in which heat i tranferred directly by electromagnetic wave. The radiation doe not require a heat tranfer medium, and can occur in vacuum. Convection refer to the tranfer of heat by the flow of a hot or cold fluid. Conduction i characterized by the tranfer of heat from one place to another in a material by a random movement and colliion of carrier atom and molecule in a ga, electron and phonon in a olid, but without the collective flow of the material itelf. Thi third mechanim i the underlying heat tranfer proce in thermal wave. Thu, from the tatement of energy conervation applied to a non-deformable olid, in the abence of internal heat generation, the claical form of the heat diffuion equation i given by: T ρ c + div q = 0.1 t where, q = k gradt, according to the Fourier law of heat conduction which uppoe an iotropic olid [André De Vriendt, 198]. The negative ign in the Fourier law indicate that the heat flow from hot to cold area. By auming a homogenou material, equ..1 become 5

16 . The concept of thermal wave 1 T r, t T r, t = α t. with α = k / ρ c. k i the thermal conductivity of the material, c and ρ repreent the pecific heat capacity and the ma denity, repectively. α i the thermal diffuivity. Thi latet quantity meaure the ability of the material to aborb heat on a tranient bai. Thermal wave are pecial cla of olution to equ.. and are uually claified into two group: periodic thermal wave and puled thermal wave. i For the puled thermal wave, a very hort pule i applied uniformly over the ample urface. The propagation of uch a pule i decribed by a different one dimenional olution to equ... It ha the form: T x, t x Io 1 4αt = e 1/.3 ρc 4παt In equ..3, I o i the incident laer intenity. At the ample urface, the temperature decay monotonically like the reciprocal of the quare root of time t. Beneath the urface, the ituation i rather different. There, the temperature pule tart at zero and increae more or le exponentially in time, reache it maximum, and then decay approximately a the reciprocal of the quare root of the time [Favro and Han, 1998]. ii Periodic thermal wave are generated by applying a periodic heat ource to the olid urface. Such periodic ource have a non-zero average value, and hence contain a dc component a well a an ac component. High enitive detection technique are capable to reject the dc component, in uch a way that the periodic thermal wave are decribed a if they had no average value. If a periodic ource ha been uniformly applied over the front urface of a emi-infinite olid, then under conideration of the appropriate boundary condition [Bein and Pelzl, 1989], the reulting thermal wave can be decribed by a one-dimenional olution to equ.., η 1 π π π δ, = Io f f T x t exp x coπft x. 4 ρck πf α α 4 In equ..4, f i the modulation frequency of heating. Thu, a thermal wave can be defined a the repone of a medium to a periodic or puled heat ource. Only the periodic thermal wave are matter of concern in thi work. A can be een from the olution of the thermal wave.4, the amplitude i coniderably damped a the penetration depth increae. Thi penetration depth, x, i limited to a ditance approximately equal to the thermal diffuion length x µ = α / πf. The thermal diffuion length, which depend on the modulation frequency of heating, how that thermal depth profiling tudie can be achieved by varying the modulation frequency. Thi 6 th

17 .3 Relevant phyical parameter quantity indicate the depth at which a thermal wave technique can be effective. It i therefore regarded a the characteritic length cale of meaurement [Almond and Patel, 1996]. On the other hand, one can ee in equ..4, that the phae lag relative to the heating modulation which ha the value 45 at the olid urface varie with the propagation ditance..3 Relevant phyical parameter It can be oberved in equ..4, that both the amplitude and the phae of the thermal wave depend on thermophyical parameter. Thi mean, frequency-dependent meaurement thermal depth profiling of the amplitude and the phae give information on the effective propertie of the invetigated olid and can provide additional inquirie on poible defect affecting the thermal tranport in the ample. The acceible parameter are the thermal diffuivity α = k / ρ c and effuivity e ρ c k =, where k, c, and ρ, are the thermal conductivity, the pecific heat capacity and the ma denity, repectively. One can ee in equ..4, that mall value of the thermal effuivity lead to high urface temperature ocillation and that large value of the thermal diffuivity lead to a rapid attenuation of the amplitude below the urface. Thi certifie that the thermal diffuivity i the relevant thermophyical parameter which govern the heat propagation inide homogeneou olid wherea the thermal effuivity i the relevant parameter for tranient urface heating procee [Bein et al., 199]..4 Review of the main detection method The photothermal technique developed for thermal characterization of material are baed on the ame principle: generation of a thermal wave through periodic heating of the ample urface by a modulated laer beam, followed by detection of the ignal induced by the local temperature increae. A review of the main experimental method i preented below:.4.1 Photothermal Deflection Spectrocopy The photothermal deflection technique or optical beam deflection or Mirage Effect wa introduced in 1980 by Bocara and co-worker [Bocara et al., 1980]. The method relie on periodic heating of the ample urface by a modulated light beam or pump beam, e.g. an Ar ion laer of wavelength = 514 nm. The heat diffuion in both the ample and the urrounding medium e.g., air produce a temporarily varying gradient in the refractive index which can be detected by the deflection of the probe beam, e.g. a He-Ne laer of wavelength 7

18 .4 Review of the main detection method = 638 nm. Analyi of the deflected beam provide information on the thermal and optical propertie of the ample. The pump and the probe beam can be poitioned either in a parallel or in a perpendicular configuration [Salazar et al., 1993; Salazar et al., 1996]. Thi technique ha been widely ued for the meaurement of the thermal conductivity/diffuivity of compoite [Inglehart et al., 1985; Macedo and Ferreira, 1999] or of CVD coating [Fournier, 001] but i alo propoed for the meaurement of the optical and electronic propertie of emiconductor material [Forget, 1993]. An important tudy preenting the effect of the nonlinear variation of the refractive index with higher air temperature in front of the ample ha been performed [Gru, Bein and Pelzl, 1999]..4. Thermoreflectance The Thermoreflectance technique exploit the local change in the ample optical reflectivity, which change i induced by the modulated temperature generated by the heating laer; and additionally for emiconductor, by the plama wave. Thi experimental method conit of meauring the intenity variation of a probe beam reflected at the ample urface. The relative change of the ample reflectivity due to the modulated temperature i calculated by δr R dr = 1 δt = CTδT.5 R dt The temperature coefficient C T, depend on the probe beam wavelength and on the ample material. For emiconductor material, a term indicating the contribution of plama wave i included: δr R = 1 R R δt T + 1 R R δn = CTδT n + C δn n.6 The Thermoreflectance i uited for the meaurement of thermal and electronic parameter of emiconductor material [Fournier, 199; Kiepert et al.; 1999, Dietzel, 001] but ome other extenion of the method have been deigned for monitoring the ample temperature [Gru et al., 1997; Schaub, 001]. The thermoreflectance microcopy can deliver thermal image with high patial reolution by meauring the variation of the reflection coefficient with temperature [Teier et al., 003]. 8

19 .4 Review of the main detection method I ϕ Thermally modulated Reflexion Thermoelatic Effect h ν n Φ IR-Radiometry h ν n Mirage-Effect Φ p Microphon Τ PZT Photoacoutic Effect Photopyroelectric Effect Figure.1: Scheme of the different configuration for the detection of thermal wave through intenity-modulated laer radiation [Pelzl and Bein, 1990]. 9

20 .4 Review of the main detection method.4.3 Photothermal Radiometry In thi experimental technique, an IR-radiation detector monitor the variation of the infrared radiation emitted from the heated urface of a ample. In the abence of a thermal perturbation thermal wave in the ample, there i however a tationary radiometric ignal which i formally proportional to the fourth power of the local tatic temperature, according to the Stefan-Boltzmann law [Smith et al., 1968; Hudon, 1969]. The modulated radiometric ignal due to the variation of the infrared radiation i therefore proportional to the emiivity, to the cube of mean urface temperature, and to the thermal wave which i very mall in comparion with the time-averaged urface temperature [Bein et al., 1989]. Thi technique i very enitive at high temperature, but le at ambient temperature [Forget, 1993]. During experimental meaurement, the MCT detector which ha a wavelength enitivity from to 1 µm i cooled to liquid nitrogen temperature to avoid the generation of a photocurrent due to the aborption of thermal radiation emitted by the detector itelf at ambient temperature [Almond and Patel, 1996]. The IR radiometry technique ha been ued for the invetigation of fibre-reinforced compoite [Bolte, 1995; Dietzel, 1997; Haj-Daoud, 000] and ha alo been explored for the temperature meaurement of wire [Borca-taciuc and Chen, 1997]..4.4 Thermoelatic detection Here, a region of the ample urface which i affected by thermal wave experience a thermo-elatic deformation, which can be materialized by the reflection of a probe beam directed along the ample urface. The ignal i meaured by uing a poition enitive diode. However, the meaured ignal only give indirect information on the temperature ditribution and therefore one mut refer to theoretical model baed on thermal wave to interpret the meaured data [Varei, 1998; Bolte, 1999]..4.5 Photoacoutic pectrocopy The theoretical fundament of the photoacoutic ignal ha been developed by Roencwaig and co-worker [Roencwaig and Gerho, 1976]. Since then, everal extenion of the theory have been propoed [Ferneliu, 1980; Bennett and Patty, 198; Pelzl, Klein, Nordhau, 198; Bein and Pelzl, 1983] and a wide range of experimental work uing thi technique have been or are till performed [Krüger et al., 1987; Gibke, 199; Malinki, 00; Gibke et al., 004]. The Photoacoutic pectrocopy work according to the following cheme: A ample i irradiated by a modulated light beam, which i then aborbed by the material and converted into heat. The heat that diffue to the ample urface and into the urrounding ga of the photoacoutic cell lead to a thermal expanion of the ga. The 10

21 .5 Concluion reulting preure ocillation are detected a ound by a microphone and the photoacoutic ignal i meaured uing a lock-in amplifier..4.6 Photopyroelectric method Thi detection method conit of meauring the temperature increae of a ample excited by a modulated laer beam, by placing a pyroelectric tranducer enor in thermal contact with the ample [Chirtoc and Mihailecu, 1989]. If the enor i placed at the rear face of the ample where excitation take place, then the configuration i aid invere front. Otherwie, the configuration i aid tandard back. Theoretical development in the one-dimenional approximation for the thermal wave propagation have earlier hown that the pyroelectric ignal depend on the optical, thermal and geometrical parameter of the olid/pyroelectric ytem [Mandeli and Zver, 1985]..4.7 Near field technique The detection method reviewed above are labelled a far-field technique ince no contact with the urface of the invetigated ample i required. Near-field technique have been developed to enable an efficient evaluation of the urface of material. In thee technique, a thermal probe i canned over the ample urface to extract the required information from the ample. More detail concerning two of the near-field technique, which are particulary uited to invetigate tructure in the micro- and ubmicrocale range, are given in chapter 7..5 Concluion In general non-detructive technique are modern tool uited for the invetigation of ample ince they don t necearily need olid contact and can be applied to ample without any pecial preparation. The amplitude and phae of the ignal meaured uing thee technique provide information on the olid characteritic and can additionally give inquirie on poible defect in the invetigated material. In the following chapter, we exploit thi contactle property of the photothermal IR radiometric technique to characterize variou layered olid tructure. 11

22 3. Surface and Suburface Effect of Thermal Tranport in Layered Sytem 3.1 Introduction In indutrial application, everal device are built up from the aembly of many layer of different thermal and phyical characteritic. Since thee device are deigned to fulfill ome pecific requirement, e.g. to erve a thermal barrier for the prevention of damage in electronic circuit or alternatively a good conductor for the enhancement of the conducting capabilitie of an entire tructure, the choice of the contituting layer might not be hazardou. Thi i why one of the reaon of caring about thee layered ytem reide in the knowledge of the thermal tranport and phyical propertie of the different component and eventually in the aement of the poible defect which can obcure the performance of the whole ytem. Thu, the thermal characterization of uch ytem provide a large quantity of information which can help the invetigator to indicate or recommend the more appropriate tructuring to come out with the bet device. Among the available technique deigned for thermal invetigation of material, the photothermal technique are more uited ince they are non-detructive and do not require any particular ample preparation. In thi chapter, variou layered ytem coniting of thin film or coating depoited on ubtrate are invetigated with the help of the IR Radiometric technique. The average total thickne of each of thee material i about 3 mm while the thickne of the thin film range from about 0.8 to 5 µm. The phae and amplitude of the modulated IR ignal meaured at the urface of thee ample are correlated with theoretical prediction in order to determine the unknown thermal and phyical propertie. Some phyical effect revealed by the meaured ignal are analyed and dicued, which effect can be claified into two group: Surface effect at higher modulation frequencie or at maller penetration depth, and uburface effect at lower modulation frequencie or at larger penetration depth. In particular, the comprehenion of the phyical effect in the limit of very low modulation frequencie i made poible with the aid of theoretical develepment baed on 3-D theory of thermal wave propagation in layered olid. 1

23 3. Photothermal radiometric ignal 3. Photothermal radiometric ignal Since the IR radiometric technique ha been mainly ued in the frame of thi work for everal experimental invetigation, we preent the theoretical concept and conideration which help to analye and interpret the meaured IR ignal Stationary and modulated radiometric ignal According to Bein et co-worker [Bein et al., 1995], the meaured tationary radiometric ignal relative to the IR radiation emitted by a olid of tationary temperature T, conidered a a gray body, can be decribed by 0 o M T = Cε T F R W, T d 3.1 In equ.3.1, ε T i the pectral emiivity of the ample within the collected olid angle, W o, T the pectral Planck blackbody radiation, F the tranmittance of the IR optical ytem and R the pectral reponivity of the detector. C i a contant which depend on the characteritic of the IR detector collected olid angle of the radiant flux, maximum reponivity and the electronic ytem. By introducing the quantity o F R W, T d 0 γ T = 3. 0 o W, T d equ.3.1 can be rewritten in a implified form 0 o M T = Cε T γ T W, T d 3.3 The integral in the above equation can be analytically retrieved and equ.3.3 i reduced a follow: 4 = Cε T γ T σ SB 3.4 M T T 4 In equ.3.4, σ SB i the Stefan-Boltzmann contant and σ SBT the Planck blackbody radiation. The quantity γ T depend on the detectable wavelength interval in the infrared, 13

24 3. Photothermal radiometric ignal < <, which i limited by the tranmittance F of the IR optical ytem. The quantity 1 γ T can be conidered a a meaure of the efficiency of the ued IR detection ytem, to convert the radiation emitted by a blackbody at a contant urface temperature into a voltage ignal [Bolte et al., 1995]. Local thermal perturbation due to heating of the olid urface by a modulated laer ource Ar + laer induce a mall variation of the tationary ignal, formally given by 3.4. Thee perturbation or thermal wave produce a modulated radiometric ignal which can be formally interpreted by uing the firt order Taylor expanion with repect to the temperature, M f, T M T + δt, f, t M T = δm T, f, t = δt f, t 3.5 T By combining equ.3.4 and equ.3.5, a correlation between the thermal wave generated at the ample urface δ f, t and the reulting modulated IR radiometric ignal i obtained: T * 3 δ M T, f, t = Cε T γ T T 4σ δt x 0, f, t 3.6 SB = In fact, tarting from the expreion of the tationary radiometric ignal given by equ.3.4, equ.3.6 reult a a Taylor expanion limited to the firt order of M T + δt, f, t with repect to the temperature, under the aumption that the temperature ocillation at the ample urface δ f, t i very mall in comparion with the time-averaged urface T temperature, T, and that the temperature variation of the emiivity i negligible in comparion with the temperature dependence of the Planck black-body radiation. It can be ee in equ.3.6 that the modulated IR radiometric ignal i proportional to the thermal wave at the ample urface. The quantity T T γ * T = γ T + γ T can be conidered a a meaure of the efficiency of the ued IR detection ytem to detect the thermal wave [Bolte et al., 1995]. The complex modulated urface temperature, δ f, t, mentioned in equ.3.6 can be written in term of amplitude and phae, T δt x i πft+ Φ = 0, f,t = A f e 3.8 In thermal wave application, the amplitude, δ T A f, and the phae lag relative to the modulation heating, Φ f, are the reult of phyical interet ince they contain the complete information on the thermal tranport and phyical propertie of the invetigated ample. However, a one can ee in equ.3.6, the modulated IR radiometric ignal i alo affected by the effect of electronic component due to the frequency dependence of the meaured thermal 14 =

25 3. Photothermal radiometric ignal wave by the meauring ytem. At thi tage, any quantitative interpretation of the meaured ignal can only be poible through a ignal calibration. 3.. Signal calibration In order to eliminate the influence of the electronic component on the meaured ignal and for quantitative interpretation, the meaured ignal are calibrated with the ignal meaured at the urface of a mooth and homogenou material, conidered a reference and whoe thermal, optical and geometrical characteritic are well known. Two typical example of material that are often ued a reference are the Glay Carbon Sigradur, which i highly opaque, and the Neutral Gla NG, which i rather tranparent. The opaque reference material i adopted in thi work for ignal calibration. Thu, the modulated IR ignal meaured at the urface of the reference material can be written a follow: * 3 δ M T, f,t = C f ε T γ T T 4σ δt x 0, f,t 3.9 r r SB r r = If meaurement are performed on the ample and after on the reference material under the ame experimental condition, then the combined factor C f γ * T related to the detection ytem can be eliminated and the invere normalized ignal reulting from the calibration of the ignal meaured on the ample with the ignal meaured on the reference material i expreed a 1 δ M T, f r ε A f S r r n f = = exp{ i[ Φ r f Φ f ]} δ M T, f ε A f 3.10 In equ.3.10, A r f and A f repreent the amplitude of the thermal wave generated at the top urface of the reference material and of the ample, repectively. The invere normalized phae i defined a the difference between the meaured phae of the reference material and the meaured phae of the unknown ample. Φ f = Φ f Φ f 3.11 n r Thu, through ignal calibration the unknown thermal, geometrical and optical characteritic of the ample are retrieved from the available propertie of the conidered reference material. Before interpreting and dicuing the ignal meaured on variou olid ample, we firt introduce the main component of the experimental et-up. 15

26 3. Photothermal radiometric ignal 3.3 Decription of the meauring ytem The meauring ytem comprie three main component: i An argon ion laer beam /nm = 514 which i modulated by mean of an acoutooptical modulator. The laer intenity i about 1W and the modulation frequencie can be varied between 0.03 Hz and 100 khz, allowing meaurement of thin film of le than 1 µm up to layer of 3-4 mm. ii For the radiometric detection of the thermal repone, a photoconductive mercury cadmium telluride HgCdTe detector of urface area of about mm and an IR optical ytem coniting of two Barium difluorid BaF lene and a filter are ued, which allow a detectable wavelength interval of 1 / µm 1. iii The electronic ytem conit of a pre-amplifier and a two-phae lock-in amplifier. By optimal electronic adaptation of detector and pre-amplifier, the thermal wave ignal can be meaured free of noie over a frequency interval of 0.1 Hz-100 khz. The lock-in amplifier erve to filter the mall thermal repone from the relatively high IR radiation correponding to the average urface temperature, T, and analye the IR ignal with repect to it amplitude and phae lag relative to the heating modulation. The computer i aimed at coordinating the et-up and proceing all ueful data. Laer Modulator Detector High-Temp. Cell IR-Optic Sample Lock-In Amplitude Phae Preamplifier Computer Figure 3.1 Scheme of the experimental et-up for the meaurement of IR radiometric ignal 16

27 3.4 Decription of the invetigated ample 3.4 Decription of the invetigated ample The ample invetigated in the frame of thi work, identifiable by their individual code ee Appendix A, were globally ubdivided into two group in regard with with their ubtrate material. The two ubtrate material conited of high peed teel HSS and high effuivity metallic alloy HMA. The coating depoited on top of thee ubtrate wa generally a diamond-like carbon DLC and the difference between the ample in regard with their urface layer wa pointed out by the different variant of the DLC. For example, the ample codified C6 and F5 conited of the ame ubtrate material on which ha been depoited two variant of the ame material DLC. Inverely, the ample C4 and C6 were different only by their ubtrate material. Geometrically, the invetigated ample had either a parallelepipedic form with a lateral ize of about 10 mm or a cylindrical form with a diameter of about 10 mm. The thickne of the coating ranged from 1 to 5 µm for a mean thickne of the ubtrate of about 3 mm. More detail about all invetigated ample are provided in Appendix A. 3.5 Theoretical model of thermal wave for layered ytem The generic term layered or layer ytem refer to ample coniting of a baic material or ubtrate on which can be identified one or more layer of differing optical and thermophyical propertie. Figure 3. a + b how the chematic repreentation of a two-layer ytem and of a three-layer ytem, repectively. In each cae, the top of the ample i urrounded by a thick layer of ga air wherea the bottom material i conidered a emiinfinite, o that the temperature fluctuation at the end ide i negligible. Several theoretical formulation of the thermal wave in layered tructure have been propoed [Egee et al., 1986; Tilgner et al., 1986]. y y Subtrate Subtrate z z a b Figure 3. a + b:from left to right Scheme of a two-layer ytem a and of a three-layer ytem b. The top of the ample i excited by a modulated laer beam to generate thermal wave. 17

28 3.5 Theoretical model of thermal wave for layered ytem But thee formulation contain quantitie which are uele becaue normally negligible and thu one cannot have a rapid view of the main factor which govern the propagation of thermal wave in the olid. The theoretical model for thermal wave we propoe are free of uch uele quantitie. Here, the thermal wave are imply expreed a a function of combined parameter. Without further detail, the expreion of thermal wave at the top urface of the ample are preented below for a two- and a three-layer model. A one can oberve in equ.3.6 or in equ.3.9, the ignal amplitude i proportional to the amplitude of the thermal wave while the ignal phae i the phae of the thermal wave. For purpoe of quantitative interpretation, the reult of theoretical approximation phae and amplitude are correlated with the normalized meaured data to extract the unknown propertie of the ample invetigated. Thu, tarting from the hypothei that the laer pot i very large at the ample urface, large enough to neglect any lateral heat diffuion in the material, the expreion of the complex urface temperature for the two-layer and for the three-layer model ha been analytically derived a a function of the modulation frequency and the relevant combined thermophyical quantitie. For a highly opaque and emi-infinite threelayer model of ample Fig. 3.b, whoe upper urface i ubjected to plane harmonic radiation intenity in the form [ 1 + co ] I x = 0, t = I o / πft 3.1 the modulated complex urface temperature i given by π iπft 4 ηioe δt f, t = e πf [ 1+ i πfτ ] [ 1+ i πfτ i ] 1+ i πfτ + πfτ i 1+ Rie + RiRibe + Ribe [ 1+ i πfτ ] [ 1+ i πfτ i ] 1+ i πfτ + πfτ i 1 R e + R R e R e i i ib ib [ ] [ ] 3.13 In equ.3.13, the indexe, i, and b tay for the urface layer, the intermediate layer and the ubtrate, repectively. The two combined quantitie R i and R ib are the thermal reflection coefficient between the coating and the intermediate layer, and between the intermediate layer and the ubtrate, repectively. Thee coefficient are given by gi 1 gib 1 R i = Rib = 3.14 g + 1 g + 1 i ib The two combined parameter defined in

29 3.5 Theoretical model of thermal wave for layered ytem g e / e i = and g ib = ei / eb 3.15 i are repectively the ratio of effuivitie urface layer to intermediate layer and the ratio of effuivitie intermediate layer to ubtrate. The other combined parameter ued in 3.13 τ = d / α and τ i / i i = d α 3.16 are the thermal diffuion time of the thermal wave in the urface layer and the thermal diffuion time of the thermal wave in the intermediate layer, repectively. Thee characteritic time are defined by the thermal diffuivity α and the thickne d of the correponding layer. In the abence of an intermediate layer between the urface layer and the ubtrate, d i = 0, andτ = 0, and the complex urface temperature valid for a two-layer ytem can be deduced i from expreion 3.13 by replacing index i with index or b: δt f, t π iπft 4 ηioe 1+ Rb = 3.17 e πf 1 R b [ 1+ i πfτ ] e [ 1+ i πfτ ] e The thermal reflection coefficient in 3.17 i defined by g b 1 R b = 3.18 g + 1 b In 3.18, g b repreent the ratio of effuivitie coating to ubtrate. The complex urface temperature obtained for the three-layer and for the two-layer model of ample, and given repectively by 3.13 and 3.17 uppoe a highly opaque ample for which the optical penetration depth µ = 1/ β i very mall and therefore negligible. However, if the thin film opt or coating depoited on the aborbing ubtrate i emi-tranparent, e.g. in the viible pectrum, the term exp β d which did not appear for example in the expreion of the thermal wave given by 3.17, due to complete aborption of the incident radiation at the urface β d >> 1, cannot longer be omitted ince in thi cae the optical thickne or optical aborption length, β d, become finite. Under conideration of coating emitranparency, the modulated complex urface temperature can be expreed by equation In thi equation, another combined quantity, α β, appear a a function of the thermal diffuivity and the optical aborption coefficient. Expreion 3.13, 3.17 and 3.19 how, that thermal wave generated at the urface or a region limited by the optical aborption length, propagate into the ample and interact with feature of different thermal propertie to produce interference effect which are detectable at the urface. That mean, the meaured ignal are coniderably influenced by the thermophyical propertie of the ample. 19

30 3.5 Theoretical model of thermal wave for layered ytem δt f, t = e π iπft 4 ηioe 1 + i πf πf 1 α β [ 1+ i πfτ ] 1+ R + be 1 i πf [ ] 1+ i πfτ 1 Rbe α β 1 + i πf + gb 1+ i πfτ α β e e + gb Rbe 1+ i β d πfτ 3.19 According to 3.10, the invere normalized ignal reulting from the calibration of the ignal meaured at the urface of an opaque two-layer ytem with the ignal meaured at the urface of the opaque reference material R b = 0 i given by S 1 n f ε η e ε η e b [ 1+ i πfτ ] e [ 1+ i πfτ ] e r r b = 3.0 r 1+ R 1 R In the limit cae of higher modulation frequencie, correponding to a mall diffuion length of the thermal wave and therefore to a mall penetration depth, according to x µ th = α / πf, S 1 n f ε r ηr e = ε η e r 3.1 and in the limit cae of lower modulation frequencie, correponding to larger penetration depth of the thermal wave, S 1 n f ε r ηr e 0 = ε η e b r 3. That mean, at maller penetration depth the invere normalized ignal S 1 f 1/ n i proportional to the thermal effuivity, e, of the coating and at larger penetration depth it i proportional to the thermal effuivity, e b, of the ubtrate. In equ.3.1 and 3., e r tay for the thermal effuivity of the homogeneou reference material. By comparing the invere ignal at the two limit cae, the optical propertie are eliminated and the ratio of thermal effuivitie coating to ubtrate can be directly determined: 0

31 3.6 Interpretation of the meaured modulated IR ignal S S 1 n 1 n f f e = 0 e b 3.3 In the following ection, the amplitude and phae of the modulated IR ignal meaured at the urface of everal tructured ample are analyed, interpreted and dicued. 3.6 Interpretation of the meaured modulated IR ignal Generally, two type of thermal wave meaurement, giving different information on the material propertie can be done: In the tranmiion configuration of thermal wave, modulated heating and detection of the thermal wave repone take place on the oppoite urface while in the reflection configuration modulated excitation and detection take place at S n -1 Φ n / deg f / Hz 1/ f / Hz -1/ Figure 3.3 a + b: from left to right Invere normalized amplitude a and normalized phae b, calculated a a function of the modulation frequency for 1-D thermal wave propagation in a twolayer ytem. Variation of the ratio of the effuivitie of urface layer to uburface material are compared with the data meaured for a coated ample C6. the ame urface. The tranmiion configuration i more appropriate for ample of finite thickne while the reflection configuration i adopted for the invetigation of ample regarded a emi-infinite. It hould be boted that the terminologie finite ample and emiinfinite ample are in connection with the thermal diffuion length that mean the characteritic length at which a thermal wave meaurement i effective. 1

32 3.6 Interpretation of the meaured modulated IR ignal To be meaured, the ample invetigated in thi work were placed in the reflection configuration ince their average thickne wa ufficient large. R b g b e b / W 1/ m - K -1 e / W 1/ m - K -1 τ / d / m α / m Table 3.1: Thermal tranport propertie of the ample C6 obtained through correlation between experimental meaurement and theoretical prediction. Figure 3.3a and Figure 3.3b how repectively the frequency-dependent amplitude and the frequency-dependent phae of the modulated IR ignal meaured at the urface of a ample C6, a DLC on HSS, in comparion with theoretical approximation full line baed on 1-D thermal wave propagation in olid. The thermal tranport and phyical propertie obtained via the combined thermophyical parameter τ and g b R b according to 3.14, 3.15 and 3.16 are reported in Table 3.1. Additionally, the ratio of the effuivitie coating to ubtrate i varied ytematically, with the value e /e b = {9.5, 1.00, 0.41, 0., 0.13} of the theoretical curve from the top to the bottom. A one can oberve in Figure 3.3, the further theoretical curve indicate the amplitude and phae of the ignal that can be meaured at the urface of a ample where, either the thin film or coating of better thermal propertie g b > 1 ha been depoited on the ubtrate, or alternatively the coating ha been depoited on a ubtrate of better thermal propertie g b < 1. The ratio g b = 1 correpond to the cae of a homogenou ample. However, a one can clearly oberve in Figure 3.3a, there i a mall deviation in the range of very low modulation frequencie between the meaured amplitude and the theoretical approximation. Figure 3.4 how the amplitude and phae of the IR ignal meaured at the urface of another ample F3, a DLC on HMA. One can ee, that the thermal diffuion time τ = d /α, which give information on the thermal thickne, or indirectly on the geometrical thickne d of the urface layer, can be determined by approximating the normalized meaured phae. Once the thermal diffuivity α of the coating i known, the layer thickne can be retrieved by non-detructive and non-contact thermal wave meaurement; or inverely if the layer thickne i known, then the expected quantity i the thermal diffuivity. Good approximation between the theoretical curve and the meaured data i oberved in Figure 3.3 and Figure 3.4 at the intermediate modulation frequencie, e.g. in Figure 3.4b for the interval 15 < f /Hz 1/ < 150. In the low and high frequency limit, however, the theoretical phae and amplitude value baed on 1-D theory of heat tranport in the two-layer ytem, deviate coniderably from the experimental reult, a one can ee alo in Figure 3.4b.

33 3.6 Interpretation of the meaured modulated IR ignal 40 τ =4.µ 5 0 τ =37.9µ S n Φ n / deg f / Hz -1/ f / Hz 1/ Figure 3.4 a + b: from left to right Invere normalized amplitude and normalized phae, calculated for 1-D thermal wave propagation in a two-layer ytem. Variation of the thermal diffuion time of the urface layer are compared with the data meaured for a coating on a metal alloy of higher thermal effuivity F3. Φ n / deg f / Hz 1/ Φ n / deg f / Hz 1/ Figure 3.5 a + b: from left to right Invere normalized phae of the modulated radiometric ignal for two meaured coated ample a-a3 and b-f3 in comparion with a theoretical approximation auming an opaque ytem and baed on 1-D theory of thermal wave propagation. In order to purue the dicuion about the deviation between theory and experiment at low and at high modulation frequencie, the meaured phae of Figure 3.4b are once more repreented in Figure 3.5b. Additionally, Figure 3.5a repreent the calibrated phae of the ample A3 another variant of DLC on HSS. While in Figure 3.5a, there i a mall deviation of the meaured phae, in the limit of very low modulation frequencie, with repect to the 3

34 3.7 Dicuion of phyical effect in the invetigated ample theoretical approximation baed on 1- D thermal wave propagation, a rather coniderable deviation at that frequency range can be oberved in Figure 3.5b between the meaured phae and the theoretical reult. At high modulation frequencie, two type of deviation of the meaured phae with repect to the theoretical approximation are viible. Wherea in Figure 3.5a the deviation of the meaured phae ymbol are directed above the theoretical curve olid line in the frequency interval, 50 < f /Hz 1/ < 350, the deviation of the calibrated meaured phae of Figure 3.5b i directed below the theoretical approximation, however in a reduced frequency interval, 100 < f /Hz 1/ < 350. Another important remark i, that in the range of high modulation frequencie the meaured phae of Figure 3.5a deviate from the limit Φ n f = 0 and increae continuouly to poitive value while the meaured phae of Figure 3.5b how rather a relative maximum below the limit Φ n f = 0 and then decreae with increaing modulation frequencie. All thee deviation of the experimental phae and amplitude with repect the theoretical approximation ugget the exitence of phyical effect revealed by the modulated IR ignal of the correponding invetigated ample. In regard with the above obervation, thee phyical effect can be claified in two group: Surface effect which are localized at high modulation frequencie or at maller penetration depth and uburface effect which are localized at low modulation frequencie or at larger penetration depth. 3.7 Dicuion of phyical effect in the invetigated ample Surface effect The deviation oberved between the calibrated meaured phae and the theoretical approximation baed on 1-D thermal wave propagation in a highly opaque two-layer ytem can be at firt ight attributed to the emi-tranparency of the urface layer. However, a the exitence of a relative phae minimum in the range of intermediate frequencie indicate the tranition coating to ubtrate in the frame of a two-layer model of ample, the appearance of a relative phae maximum in the range of high modulation frequencie may alo indicate the tranition between a thin layer at the very urface of the ample and the coating. For thi, the urface effect can be ubdivided into two group: optical effect and topological effect Optical effect: emi-tranparency of the coating The optical effect are eentially in connection with the emi-tranparency of the coating. In fact, the theoretical approximation ued above for the interpretation of the amplitude and phae of the meaured IR ignal aume a highly opaque ample, where the heat ource 4

35 3.7 Dicuion of phyical effect in the invetigated ample coincide with the urface of the coating due to a very mall optical aborption length or optical penetration depth. In thi cae, there i no contribution to heat generation caued by the aborption of light in the ubtrate. If the thin film depoited on the highly opaque ubtrate i emi-tranparent e.g. in the viible pectrum, then a thermal wave generation at the ubtrate urface may be poible due to a coniderable optical aborption length induced by the finite value of the optical aborption coefficient. Reinterpretation of the meaured phae of Figure 3.5a i hown in Figure 3.6. A can be een, the meaured phae agree well with the theoretical approximation taking into account the emi-tranparency in the viible pectrum of the thin film depoited on the highly opaque ubtrate. A aid before, the level of thi emi-tranparency i controlled by the magnitude of the optical aborption coefficient. A complete characterization of the ample A3 i done by correlating the theoretical approximation with the calibrated meaured phae and the extracted propertie are reported in Table Φ n / deg f / Hz 1/ Figure 3.6: Invere normalized phae of the modulated IR ignal for a DLC-coated teel ample A3: Semi-tranparency of the thin film in the viible pectrum. The dahed curve below the meaured phae at the frequencie 50 < f /Hz ½ < 350 repreent the theoretical approximation baed on 1-D thermal wave propagation auming a highly opaque ample. 5

36 3.7 Dicuion of phyical effect in the invetigated ample e b g b e d / m τ / α / m -1 α 1/ β β / m Table 3.: Characteric of the ample A3, taking into account the emi-tranparency of the thin film layer depoited on the highly opaque ubtrate. In Figure 3.6, the calibrated meaured phae take the value Zero at f /Hz 1/ 15 and then increae continuouly with increaing modulation frequencie. The upper limit of thee meaured phae depend on the magnitude of the combined thermo-optical quantity, P αβ = α 1/ β, therefore on the optical aborption coefficient of the coating. A example, for P αβ = , the upper limit of the calibrated meaured phae would be about Φ n = 18 upper theoretical curve in Figure 3.6. It i ueful to mention, that the coating i aumed to be highly opaque in the IR pectrum β >> 1. Otherwie a finite value of the optical IRd aborption in that pectrum ha to be conidered to account for the radiative contribution both from the urface and the uburface of the material. To indicate uch a double emitranparency of the layer in both the viible and the IR pectrum, the modulated IR ignal are expreed a follow: δ M 3 ' f, T = 4Cε T γ D σ SB T ßIR exp βirx' δ T x', f dx Topological effect While in Figure 3.5a or in Figure 3.6, the calibrated meaured phae affected by the coating emi-tranparency ymbol take the value Zero and increae continuouly with the modulation frequency toward poitive value, the calibrated meaured phae repreented in Figure 3.5b preent rather a relative maximum in the range of high modulation frequencie, namely at about f /Hz 1/ = 5, and then decreae with increaing frequency. Referring to the relative phae minimum that can alo be oberved in Figure 3.5b at the modulation frequency of f /Hz 1/ 50 and which indicate the tranition between the uburface material and the coating, the appearance of a relative maximum in the range of very high modulation frequencie or at very low penetration depth, can be explained by the exitence of an additional layer of reduced thermal tranport propertie at the very urface of the coating. Reaement of the meaured phae of Figure 3.5b, taking into account the exitence of uch a layer at the very urface of the coating i preented in Figure 3.7a. In addition, other calibrated meaured phae for the ample F5 DLC on HSS are inerted for comparion. One can oberve, that at high modulation frequencie the calibrated meaured phae of the two ample poe the ame urface characteritic ame thin layer at the very urface of 6

37 3.7 Dicuion of phyical effect in the invetigated ample Φ n / deg Φ n / deg f / Hz 1/ f / Hz 1/ Figure 3.7a: Invere normalized phae of two ample F5-, F3-* howing the behavior of a three-layer ytem. The meaured data are compared with theoretical approximation baed on 1-D thermal wave propagation. Figure 3.7b: Evolution of the relative phae maximum in the range of high modulation frequencie with the ratio of effuivitie thin film-to-coating, g i = , 0.375, e e i e bhss e bhma α / m -1 α i / m -1 d / m d i / m Table 3.3: Sample characteritic howing the exitence of a thin layer at the very urface of the coating. the coating and ame coating but differ with each other by their uburface material. The reult of the quantitative interpretation are conigned in Table 3.3. A one can ee in thi table, the thickne of the additional layer which ha been depoited on the coating i extremely mall, d /m = The difference between the ubtrate of the two ample i pointed out by the value of thermal effuivitie, that i 7530 W 1/ m - K -1 for the HSSuburface material, and W 1/ m - K -1 for the HMA-uburface material. According to the calibrated theoretical phae repreented in Figure 3.7b, the higher the ratio g i of effuivitie between the thin layer at the very urface and the coating, the higher the value of the relative phae maximum. However, all relative extrema that can occur on the meaured phae do not automatically ignify the tranition between two conecutive 7

38 3.7 Dicuion of phyical effect in the invetigated ample layer. In the next ection we how, that the relative extremum oberved in the limit of very low modulation frequencie i rather induced by thermal tranport effect Suburface effect: Lateral thermal tranport Since in the limit of very low modulation frequencie, correponding to very large penetration depth, the 1-D theory of thermal wave propagation ha been found inufficient to explain the reaon of deviation between layer model and the data meaured for the amplitude and phae in reality, a 3-D theory of thermal wave propagation in two-layer ytem ha been developed, which take into account the finite ize of the heating pot. Firt, the main feature of thi theory are preented: Theoretical development Conidering temperature-independent thermal propertie and the uperpoition principle, T r, θ,z,t = T r, θ,z + δt r, θ,z,t, for the olution of the heat diffuion equation, the diffuion equation for thermal wave propagating in a olid medium can be written in cylindrical coordinate a δt j r, θ, z j, t 1 δt j r, θ, z j, t = α j r t r r r + δt j r, θ, z j, t + r θ δtj r, θ, z j, t δq j r, θ, z j, t + z j ρc j 3.5 In equ.3.5, the index j refer to the ga layer g, the urface layer and the ubtrate b. The upper urface of the ample, defined by z = 0, i heated by a Gauian haped laer beam of radiu r H, which can be modulated at the frequency f. r H i the radiu at which the Gauian ditribution ha dropped to 1/e² of it peak value. Applying the uperpoition principle, the acpart of the volumetric heat ource in the urface layer i given by Ioη r δq r, z, t = exp β exp z exp i ft β π 3.6 πr r H H In equ. 3.6, I 0 i the incident beam power, η the photothermal converion efficiency, and β the optical aborption contant of the coating. Due to the cylindrical ymmetry introduced by the laer geometry, the ac temperature i invariant with repect to the angular coordinate. Neglecting the effect of heat loe to the urrounding gae, which i jutified for ample urface with effuivity value much larger than the effuivity of the ga atmophere [Bein and Pelzl, 1989], the boundary condition accompanying the above equation 3.5 are defined by: 8

39 3.7 Dicuion of phyical effect in the invetigated ample 9 a the negligible temperature at outer boundarie, 0 0,, = = t z r T g g δ 3.7a 0,, = = t l z r T b b b δ 3.7b b the temperature continuity at the interface ga/coating and coating/ubtrate, 0,,,, t z r T t l z r T g g g = = = δ δ 3.8a 0,,,, t z r T t l z r T b b = = = δ δ 3.8b c and the heat flow continuity at the interface ga/coating and coating/ubtrate. 0,t r,z q,t l r,z q g g g = = = δ δ 3.9a 0,t r,z q,t l r,z q b b = = = δ δ 3.9b By uing the Hankel tranformation and ubequently it invere [Abramowitz and Stegung, 1965; Öziik, 1997], defined repectively by: = 0 rdr r J r F F o 3.30a = 0 d r J F r F o 3.30b the Laplacian operator in cylindrical coordinate, taking into account the cylindrical ymmetry, i tranformed according to 3.30a a follow: + = z z r r r r Then, applying the Hankel tranformation to equ.3.5 yield for each medium j = g,, b the following ordinary differential equation b g j e k Q r I t z T dz t z T d j j z j j H o j j j j j j j j,,,,,, = = β π β η δ δ 3.3 for which the general olution i given by ft i z j z j z j j HT j e e C e B e A t z T j j j j j j π β δ,, + + = 3.33

40 3.7 Dicuion of phyical effect in the invetigated ample 30 Since the heat ource i preent only in the urface layer, Q g and b Q are identically equal to zero and therefore uch a term C g or C b doe not exit. In equation 3.3 and 3.33, j j f, σ + = and b,, j / 1 g f i j j = + = α π σ 3.34 The quantitie k j and j σ are the thermal conductivity and the thermal wave vector of the indexed layer, repectively. For quantitative interpretation of the meaured ignal, only the theoretical expreion of the thermal wave at the ample urface i of interet and thu the ueful integration contant are A, B and C. By inerting equ.3.33 into equ.3.3 for the index j =, one obtain: /, 1 8 exp /, 1, H o H o f r k I f Q k r I f C β β π η β β π η = = 3.35 with 8 exp 4 exp 0 H H o H r r rdr r J r r Q = =, according to 3.30a. By applying the Hankel tranformation to the boundary condition given by 3.7, 3.8 and 3.9, and then dividing the tranformed equ. 3.9a by the tranformed equ.3.9b, the three integration contant are interrelated in the following equation: =, 1,,,, 1, 1,, f C f G f f G f A f G f G f B β 3.36 with coth coth, 1 g g g g g g g g g l e e l k k f G σ σ α α σ σ + + = + + = In equ.3.36, G can be further implified by auming an expanded ga layer air in front of the ample, that mean 1 l coth air air, and by remarking that for 0, G,f e g /e,, G,f k g /k. i an integration variable in the Hankel pace. In general the thermal effuivity and conductivity of the ga layer air are very mall in comparion with the ame propertie for mot material [Bein and Pelzl, 1989]. Following thi, parameter G can be neglected and therefore equ reduce to

41 3.7 Dicuion of phyical effect in the invetigated ample 31 /, 1 8 exp,,, H o f r f k I f A f B β π η Taking into account the above approximation on the parameter G, the two following combination of the Hankel tranformed boundary condition, namely [3.8a + 3.9b] and [3.8a 3.9b], give repectively the expreion of the integral contant f, A b and f, B b which are then reintroduced in the Hankel tranformed boundary condition 3.7b to yield [ ] [ ] b b H b o d f g f g f f r d f R f k I f B exp 1,, /, 1 /, 1 8 exp exp, 1,, β β β π η 3.38 which according to 3.37 lead to the determination of another integration coefficient [ ] [ ] b b H b o d f g f g f d f f r d f R f k I f A exp 1,, /, exp, R /, 1 8 exp exp, 1,, b β β β π η 3.39 Since the model i conidered to be highly opaque, the optical aborption coefficient i infinite that mean, β and therefore the heat ource in the coating coincide with the ample urface. In thi cae, the optical penetration depth 1/β i very mall and the optical thickne β d i very large, o that the magnitude of the integration coefficient C, f given by 3.35 become extremely mall and can be neglected. A conequence, the integration contant given by equ.3.38 and equ.3.39 take repectively the reduced form [ ] exp, 1, 8 exp, b H o d f R f k r I f B π η 3.40a [ ] exp, 1, 8 exp, R, b b H o d f R f k r f I f A π η 3.40b

42 3.7 Dicuion of phyical effect in the invetigated ample 3 According to equ.3.30a, and with repoect to equation 3.40a and 3.40b the Hankel tranformed urface temperature i given by: ft i b b H o HT e d f R d f R f k r I t z T π π η δ exp, 1 exp, 1, 8 exp 0,, + = = 3.41 and according to equ.3.30b, the ac complex urface temperature can be calculated a + = = 0 exp, 1 exp, 1, 8 exp 0,, ft i o b b H o e d r J d f R d f R f r k I t z r T π π η δ 3.4 In equation 3.38, 3.39, 3.40, 3.41 and 3.4, d indicate the coating thickne. The quantity 1, 1,, + = f g f g f R b b b 3.43 formally repreent the thermal reflection coefficient at the interface urface layer ubtrate, with the combined parameter, f g b given by b b k k f, g σ σ + + = 3.44 Baed on the expreion of the thermal wave at the ample urface 3.4, the modulated IR radiometric ignal meaured over the detection pot area of radiu r D can be calculated a: = 4,, 3 * T T T f C t T f M SB ε σ γ δ ft i r o b b H o e d r J d f R d f R f r rdr k I D π π η 0 0 exp, 1 exp, 1, 8 exp For remind, the factor C f decribe the frequency characteritic of the detection device, and the factor * T γ decribe the efficiency of thermal wave detection [Bolte, 1995]. T i the time-averaged urface temperature; SB σ and T ε are repectively the Stefan-Boltzmann contant and the emiivity of the coating. The factor 4 3 * T T T f C σ SB ε γ i a real

43 3.7 Dicuion of phyical effect in the invetigated ample quantity, which can be eliminated by an appropriate calibration procedure. A a real quantity it cannot affect the phae of the meaured thermal wave. r H r D e β Surface layer Subtrate α αb eb Figure 3.8: Scheme of the -layer ample and of the experimental condition baed on concentric heating and detection pot Comparion of one- and three dimenional thermal wave propagation Here approximation baed on 1-D and 3-D thermal wave propagation are compared with each other and with the data, meaured for a hard coating of diamond-like carbon of.8 µm thickne on two ubtrate material of different thermal propertie high peed teel HSS and a high effuivity metallic alloy HMA. In Figure 3.9a one can ee, that difference between 3-D and 1-D thermal wave propagation become remarkable only in the limit of the low heating modulation frequencie, correponding to large penetration depth, that mean in the uburface material. While according to 1-D theory broken line the normalized phae clearly tend to the value Zero at the very low frequencie, the 3-D heat propagation continuou line induce coniderable deviation of the normalized phae in that frequency range. The imilarity of the two theoretical curve at the intermediate and high modulation frequencie ugget that the lateral heat tranport only play a role in the low frequency range, f /Hz 1/ < 0, correponding to large penetration depth in the uburface material. A imilar behaviour can be deduced from the meaured phae Fig. 3.9b: 33

44 3.7 Dicuion of phyical effect in the invetigated ample i A already een above, in the limit of high modulation frequencie, f /Hz 1/ > 150, correponding to very mall penetration depth, the normalized phae of the two ample indicate the preence of an additional thin layer of reduced effuivity at the very urface of the coating, which i common to the two ample x, Φ n / deg Φ n / deg f / Hz 1/ f / Hz 1/ Figure 3.9a: Normalized phae according to 1-D broken line and 3-D thermal wave propagation full line a function of the quare root of the modulation frequency. The radii of heating mm and detection pot 0.8 mm ued for 3-D thermal wave calculation agree with the meauring condition of Fig. 3.8 Figure 3.9b: Normalized phae meaured for a hard coating DLC depoited under equal condition on two ubtrate of different thermal propertie F5-HSS, F3-HMA. The phae meaured for the coated ample have been calibrated uing the ignal of a mooth homogeneou ample of glay carbon. ii Alo, the difference at intermediate frequencie, 5 < f /Hz 1/ < 100, are due to different ratio of the effuivitie coating-to-ubtrate, g b = e /e b, and due to an eventually different thermal diffuion time τ = d /α of the coating of the two ample [Bennet and Patty, 198; Bein et al., 1989]. iii In the difference at the very low frequencie, f /Hz 1/ < 15, the 3-D character of thermal wave propagation become obviou, with a tronger lateral heat propagation in the ubtrate of comparatively larger thermal diffuivity and effuivity Effect of heating pot radiu and uburface thermal propertie In general, 1-D thermal wave propagation can be applied for the interpretation of modulated photothermal meaurement, when the heating pot i large in comparion to the thermal diffuion length, r H >> µ th = [α /π f ] 1/. Thi can be verified in Fig. 3.10a and in 34

45 3.7 Dicuion of phyical effect in the invetigated ample Fig. 3.10b for theoretical approximation calculated for increaing heating pot radii. A can be een, the reult baed on 3-D thermal wave propagation and calculated for a heating pot radiu of r H = 3 mm perfectly agree with the approximation baed on 1-D heat tranport. The effect of lateral heat propagation increae with increaing value of the thermal diffuivity α b of the ubtrate material, a can be een in Fig. 3.11a+b. For a low thermal diffuivity value of α b = m / the theoretical approximation i cloe to the cae of 1-D Φ n / deg r H =3mm 8 r D =0.8mm Φ n / deg r H =3mm 8 r D =0.8mm f / Hz 1/ f / Hz 1/ Figure 3.10 a + b: from left to right Range of higher a and lower modulation frequencie b of the normalized phae meaured for a DLC coating on a metallic alloy C4-, in comparion with theoretical olution baed on 3-D thermal wave propagation with increaing heating pot radii full line. The detection pot radiu r D of the three theoretical olution i contant and agree with the meauring condition Φ n / deg α b = m / Φ n / deg α b = m / f / Hz 1/ f / Hz 1/ Figure 3.11a+b: from left to right Range of intermediate a and lower modulation frequencie b of the normalized phae meaured for a DLC coating on a metallic alloy C4-, in comparion with theoretical olution baed on 3-D thermal wave propagation at increaing value of the thermal diffuivity α b of the ubtrate material. The theoretical approximation rely on contant heating and detection pot radii r H = mm, r D = 0.8 mm, in agreement with the meauring condition. 35

46 3.7 Dicuion of phyical effect in the invetigated ample thermal wave propagation, for an intermediate value of α b = 10-6 m / the theoretical approximation i in good agreement with the meaured data, and for a larger value of the thermal diffuivity, e.g. α b = m /, the the effect of 3-D thermal wave propagation become even more ignificant, and the relative phae maximum in the low frequency range, which i characteritic for 3-D thermal wave propagation, hift to higher modulation frequencie, e.g. to f /Hz 1/ 8 Fig. 3.11b. A hown in Figure 3.10a+b and Figure 3.11a+b, the deviation from 1-D thermal wave propagation increae with decreaing heating pot radii and with increaing thermal diffuivitie α b of the uburface material and, a hown in Figure 3.11b, the normalized phae are rather enitive to variation of the thermal diffuivity α b of the uburface material Φ n / deg e b =600 W 1/ m - K f / Hz 1/ Figure 3.1: Normalized phae calculated for 3-D thermal wave propagation full line at different effuivity value e b of the uburface material, in comparion with data meaured for a DLC coating on a metallic alloy of high effuivity C4-. In Figure 3.1 the normalized phae, meaured for another type of hard coating on a high effuivity metallic alloy, are compared with theoretical olution calculated for 3-D thermal wave propagation full line at different effuivitie e b of the uburface material. The variation of the effuivity e b lead to variation of the relative phae minimum at intermediate modulation frequencie Fig. 3.1, where according to Fig. 3.9a the difference between 1-D and 3-D thermal wave propagation are negligible. From the point of view of thermal propertie, the coating of thi ample i homogeneou and can perfectly be approximated by a -layer olution, even in the high frequency limit Fig The reult of Figure 3.1 confirm that the effuivity i the relevant thermal parameter both for tranient and harmonic urface heating procee and for the heat tranition between the different layer of 36

47 3.8 Interpretation of ignal phae baed on 3D thermal tranport a olid, wherea the reult of Figure 3.11 confirm that the thermal diffuivity i the relevant parameter governing time-dependent heat propagation inide a olid and inide a layer of contant thermal diffuivity. By combining meaurement of the normalized phae at intermediate frequencie, where the relative phae minimum give information on the ratio of the effuivitie coating-toubtrate, with phae meaurement at the very low frequencie, where information on the thermal diffuivity of the uburface material i obtained from the relative phae maximum, local inhomogeneitie of the thermal propertie of the coating and of the ubtrate can be identified: i In the cae of meaured contant thermal diffuivity α b of the ubtrate, one can aume that meaured local variation of the ratio of the effuivitie coating-to-ubtrate refer to local variation of the coating. ii If the locally meaured thermal diffuion time of the coating i contant, one can aume on the other hand, that meaured local variation of the ratio of the effuivitie coating-to-ubtrate refer to local variation of the ubtrate. Auming locally invariant thermal propertie of the coating, a complete thermal characterization of the uburface material can then be obtained, with eparate information on the thermal conductivity, k b = e b α b 1/ 3.46 and the volume heat capacity C b = ρ b c b = e b /α b 1/ Interpretation of ignal phae baed on 3D thermal tranport Figure 3.13a and Figure 3.13b how the invere normalized phae meaured for two couple of ample, C4, C6 and X4, X7, repectively. The meaured phae indicate that for each couple of ample, the ame coating ha been depoited on ubtrate of different thermal propertie. Such a remark ha already been done omewhere above. That the two ample poe the ame urface characteritic provide the poibility to obtain information about the thermal propertie of the ubtrate material and in the meantime information about the layer growth and ubtrate preparation. In order to how how uch an information can be extracted, we refer to the theoretical expreion of the modulated IR ignal baed on 1-D thermal wave propagation and etablih the ratio of ignal Sample A to Sample B equ

48 3.8 Interpretation of ignal phae baed on 3D thermal tranport S A / B f 1/ M = M A B ε T, f = T, f ε A B * T { γ T T * T { γ T T 3 3 } } A A η e η e A A B B 1+ R 1 R 1+ R 1 R b A b A b B b B exp[ 1 + i d exp[ 1 + i d exp[ 1 + i d exp[ 1 + i d A A B B / µ / µ / µ / µ A A B B ] ] ] ] 3.48 In the limit cae of mall penetration depth, that mean at higher modulation frequencie, x µ = α / π f 0, the ratio of ignal take the form f S A / B f ε T { γ T T * 3 A A A e 1 / M AT, f A = = 3.49 f,x 0 M BT, f * 3 η f,x 0 B ε BT { γ T T } B e B } η Since the urface characteritic cancel each other; that mean, ε A = ε B, and η A = η B for the optical propertie, and e A = e B for the thermal effuivitie, the only difference of the ratio S A/B f -1/ f, x 0 from the value 1 can be attributed to the poible temperature dependence of the detection proce. S A / B f 1 / f,x 0 M = M A B T, f T, f f,x 0 = * { γ T T * { γ T T 3 3 } } A B 3.50 Under thi conideration, the ample B with lower effuivity of the ubtrate mu then have a relative high mean temperature, o that according to 3.50 a value cloe to 1 can be expected. 5 5 Φ n / deg C6 C4 3D theory Φ n / deg X7 X4 3D theory f / Hz 1/ f / Hz 1/ Figure 3.13 a + b :from left to right Invere normalized phae meaured for two et of ample, C4 & C6 a, and X4 & X7 b, in comparion with theoretical approximation baed on 3-D heat propagation. 38

49 3.8 Interpretation of ignal phae baed on 3D thermal tranport Determination of the propertie of lateral thermal tranport By comparing the theoretical approximation with the experimental reult in the entire meaured domain, Figure 3.14b and Figure 3.15b how a lope in the frequency range 0. < f /Hz -1/ < 1.0. The oberved lope i attributed to 3-D propagation effect which are very important in the ample with higher effuivity of the ubtrate. From thee normalized amplitude Fig. 3.14b and Fig. 3.15b, the ratio of effuivitie of the two ubtrate High Speed Steel-HSS and High Effuivity Metal alloy-hma i retrieved: C4/C6 e b HSS / e b HM = a X4/X7 e b HSS / e b HM = b S A/B C4/C6 S A/B C4/C f / Hz -1/ f / Hz -1/ Figure a + b from left to right Normalized amplitude C4/C6 at maller penetration depth a, and in the total meaurable domain b, in comparion with a theoretical approximation according to a two-layer model taking into account the 3D heat propagation S A/B X4/X7 S A/B X4/X f / Hz -1/ f / Hz -1/ Figure a + b from left to right Normalized amplitude X4/X7 at maller penetration depth a, and in the total meaurable domain b, in comparion with a theoretical approximation according to a two-layer model taking into account the 3D heat propagation. 39

50 3.8 Interpretation of ignal phae baed on 3D thermal tranport From theoretical approximation of the meaured phae in the mean frequency domain; we obtain according to Figure 3.13a and to Figure 3.13b, the following value for the ratio of effuivitie of the two ubtrate: C4/C6 e b HSS / e b HM = a X4/X7 e b HSS / e b HM = b From approximation of the normalized phae at very low modulation frequencie, Fig.3.13a and Fig. 3.13b, the ratio of thermal diffuivitie of the two ubtrate i obtained: C4/C6 α b HSS / α b HM = a X4/X7 α b HSS / α b HM = b According to the reult 3.51a, 3.5a and 3.53a, the ratio of thermal conductivitie of the ubtrate for the ample C4 and C6 give: k k b HSS b HM e = e b HSS b HM α α b HSS b HM = a According to the reult 3.51b, 3.5b and 3.53b, the ratio of thermal conductivitie of the ubtrate for the ample X4 and X7 give: k k b HSS b HM e = e b HSS b HM α α b HSS b HM = b On the other hand, the theoretical ratio i given by: k k b HSS b HM 3 = 80 W/K m = W/K m Dicuion of reult The ratio of thermal conductivitie of the ubtrate for the ample C4 & C6 3.54a, and for the ample X4 & X7 3.54b i relatively high with repect to the theoretical ratio given by equ By comparing the ratio 3.54a with the theoretical value on one hand, and the ratio 3.54b with the theoretical ratio on the other hand, the relative incertitude i given repectively by 11.3% and 0%. Thee relative dicrepancie between derived and 40

51 3.9 Concluion theoretical ratio might be attributed to the nature of the common coating depoited on the ubtrate of ample C4 and C6, and on the ubtrate of ample X4 and X7, which can play a major role in the modification of the thermal propertie of the ubtrate. In the meantime, thee difference between experimental and theoretical reult can alo be explained by poible change in the propertie of the ubtrate thermal and mechanical during their tailoring or during coating depoition. 3.9 Concluion In thi chapter, the modulated IR radiometric ignal meaured at the urface of variou layered ytem have revealed three type of effect of thermal tranport that have been claified into two major categorie: The urface effect oberved in the range of high modulation frequencie have been found to be in connection with either the optical characteritic emi-tranparency in the viible pectrum of the urface layer or the topology preence of a thin layer of reduced thermal tranport propertie at the very urface of the coating. In the cae of coating emi-tranparency, it ha been oberved, that the invere normalized phae deviate from the limit Zero and increae continuouly toward poitive value whoe maximum depend on the magnitude of the optical aborption coefficient, β. A for the topological effect, the exitence of a very thin layer at the very urface of the coating i manifeted on the calibrated meaured phae by the appearance of a relative maximum in the range of high modulation frequencie. The uburface effect, which are alo materialized by the exitence of a relative maximum on the calibrated meaured phae in the limit of very low modulation frequencie, have been found to be induced by lateral thermal tranport in the uburface material. Obervation have been made, that thee lateral heat loe become more important with increaing value of the thermal diffuivity of the ubtrate material and that the variation of the effuivity of the uburface material produce change mainly at the intermediate frequencie, where the tranition between the coating and the ubtrate i identified on the meaured phae by a relative minimum. Thee reult have confirmed that the thermal diffuivity i the relevant thermophyical parameter which govern timedependent heat propagation inide olid and that the thermal effuivity i the relevant thermophyical parameter for both tranient and harmonic urface heating procee. Thu, the appearance of relative extrema on the normalized meaured phae can provide a large quantity of information about the ample characteritic and o the location where thee extrema occur on the frequency-dependent phae may help to extract the unknown thermal tranport and phyical propertie of the invetigated ample. Starting from thee important obervation, a new evaluation method ha been developed, which i baed on the relative extremum of the calibrated meaured phae lag between the periodical modulated excitation of the thermal wave and the detected thermal repone. In the next chapter, we 41

52 3.9 Concluion introduce the new concept of the Phae Extremum of thermal wave and apply the method, labelled a Extremum Method, to the analyi, interpretation and dicuion of the phae of the modulated IR ignal meaured at the urface of opaque two-layer tructure. 4

53 4. Determination of Thermal Tranport Propertie of Two-layer Structure uing the Concept of the Phae Extremum 4.1 Motivation We have een in the previou chapter that the preence of everal phyical effect thermal, topological and optical on the modulated IR ignal meaured at the urface of variou layered ytem, make difficult the quantitative interpretation of the meaured data. It ha been particularly noted, that in addition to the relative extremum which occur on the calibrated meaured phae in the range of intermediate frequencie and which indicate the tranition coating-to-ubtrate, the appearance for certain type of ample of further relative phae extrema repectively in the range of low and of high modulation frequencie, reveal repectively the exitence of thermal effect 3D thermal wave propagation and topological effect thin film on top of the coating. Thu, thee relative phae extrema are the poible location from which complete or at leat eential information about the invetigated ample can be obtained. In thi chapter, we introduce a new concept baed on the relative minimum or maximum of the modulated IR ignal phae meaured at the urface of two-layer ytem. We demontrate, that the meaurable phae extremum and the modulation frequency at which the relative extremum occur, which are the key parameter governing the method, lead to determination of the relevant thermal tranport and phyical propertie of the invetigated ample via two combined thermophyical quantitie, namely the thermal reflection coefficient and the thermal diffuion time. In ection 4., the main feature of the Extremum Method are preented. In ection 4.3, the method i applied to example of real two-layer ytem: hard coating the thermal effuivity of which i maller than the effuivity of the ubtrate coniting of tool teel and a metal alloy ample NiTi hape memory alloy, which due to urface polihing and local demixing exhibit a two-layer tructure with the effuivity of the urface layer above the effuivity of the bulk material. In ection 4.4, dicuion about other alternative approache are performed, which confirm the reliability of the Extremum Method. Then general reult generated for on-line interpretation in indutrial application are preented in ection 4.5. In ection 4.6, a functional tranform i developed, which allow to obtain information on the thermal tranport propertie from any other value of the phae lag and the correponding heating modulation frequency in the neighbourhood of the relative minimum or maximum of the calibrated phae. For thi, the functional tranform method i 43

54 4. The concept of the Phae Extremum applied to example of real coating which are perfectly decribed by the two-layer model or which coniderably deviate from the model of an opaque two-layer ytem due to the coating tranparency or due to the preence of an additional layer of reduced thermal tranport propertie at the very urface of the coating. Furthermore the functional tranform i applied to example of meaurement, which at higher frequencie at the poition of the relative minimum of the calibrated phae may be obcured by the background fluctuation limit of modulated IR radiometry or by the cell reonance effect in photoacoutic detection. The lat ection dicue the application potential of both the Extremum Method and the functional tranform method. 4. The concept of the Phae Extremum 4..1 Phyical ignificance of the obervable phae extrema Figure 4.1 how the phae of the meaured modulated IR ignal for three coated ample, calibrated with the meaured phae of a homogenou and opaque ample ued a reference, here the glay carbon. One can immediately remark, that each of thee invere normalized phae lag poee at leat one relative extremum, which according to Figure 4.1a can be approximately identified for the ample A3 and C6 at the point [45 Hz, ] and [75 Hz, -4.3 ], repectively and for F3 at about {[7 Hz, ]; [50 Hz, -0.3 ]; [8 Hz, -1.7 ]}. The relative extremum a maximum exhibited by the invere normalized phae of the polihed NiTi ample x i identifiable in the neighbourhood of [4 Hz, 3.44 ]. The tangent of the meaured phae decribed in Figure 4.1a are reported in Figure 4.1b. In Fig. 4.1 the meaured phae of the ample A3 and C6 and the polihed NiTi ample x are characteritic of a typical two-layer ytem, howing in addition that the ample A3 i emi-tranparent by it coating while C6 can be aumed opaque. Each of thee meaured phae preent a relative minimum in the range of intermediate frequencie, which indicate the tranition coating-to-ubtrate. In comparion with the meaured phae of the ample A3 and C6, the three relative extrema oberved on the meaured phae of ample F3 have different ignification: The firt relative maximum recorded in the range of very low modulation frequencie i attributed to 3-D heat propagation in the uburface material of high thermal diffuivity. The relative minimum occuring in the range of intermediate frequencie attet of the tranition between the uburface material and the coating while the econd relative maximum which occur in the range of high modulation frequencie prove the exitence of a thin layer at the very urface of the coating. The phae extremum which i 44

55 4. The concept of the Phae Extremum of concern in thi tudy relie on the tranition between the coating and the ubtrate in the framework of a highly opaque two-layer tructure Φ n / deg f / Hz 1/ tan Φ n f / Hz 1/ Figure 4.1 a + b: from top to bottom Invere normalized phae a and the correponding tangent b of the meaured modulated IR ignal for different layered ytem: Identification and characterization of the phae extrema. 45

56 4. The concept of the Phae Extremum 4.. Theoretical background A we have een in chapter 3 ection 3.5, equ. 3.17, the expreion of the thermal wave at the urface of a two-layer ytem coniting of a highly opaque β -1 0 coating of thickne d on an aborbing emi-infinite ubtrate β -1 b 0, can be written a: η 1+ R exp[ 1 + i πfτ ] I o b δt x = 0, f, t = exp[iπft π / 4] 4.1 e πf 1 R exp[ 1 + i πfτ ] b Here, η i the photothermal converion efficiency which determine the fraction of the total incident light intenity I o tranformed into heat, f the heating modulation frequency, t the time, and e the effuivity of the coating. The two combined thermophyical parameter ued in equ.4.1 and which have already been decribed in Chapter 3 are given by Rb 1 e / eb 1+ e / eb = 4. for the thermal reflection coefficient of the thermal wave and τ = α 4.3 d / for the thermal diffuion time of the coating. α i the thermal diffuivity of the coating. The amplitude of the thermal wave, which can be calculated from equ.4.1, i given by δt f 4 η 1 exp 4 co4 exp 8 I Rb πfτ πfτ + Rb πfτ o = 4.4 e πf 1 Rb exp πfτ co πfτ + Rb exp 4 πfτ and the phae hift relative to the heating modulation i given by 1+ Rb exp πfτ in πfτ Rb exp 4 πfτ tanφ f = Rb exp πfτ in πfτ Rb exp 4 πfτ A can be een in equ.4.5 the phae hift of the thermal wave relative to the heating modulation only depend on the two combined thermophyical parameter, R b and τ. Since the meaured ignal, both the amplitude and the phae are affected by the frequency characteritic of the meaurement device, a calibration with the help of a homogeneou reference ample, decribed by the two-layer ytem with equal effuivity value for coating and ubtrate i neceary compare Chap.3. According to equ.4., the 46

57 4. The concept of the Phae Extremum thermal reflection coefficient i then R b = 0 and according to equ.4.5 the phae hift of the reference ample i decribed by tanφ r f = In invere normalization, the expreion for the calibrated phae i given by tanφr tanφ tanφn f = tan[ Φr f Φ f ] = 1+ tanφr tanφ = Rb exp πfτ in πfτ 1 [ Rb exp πfτ ] 4.7 The exitence of a relative extremum, a relative minimum or maximum, in the repreentation of the invere normalized phae lag tanφ n veru f /Hz 1/ a hown in Figure. 4.1b implie, that the minimum or maximum hould fulfill the condition, tan Φ 4 exp [1 exp 4 ]co n f Rb πτ πfτ R b π fτ πfτ = = 0 f [1 R exp 4 ] b πfτ [1 + R b exp 4 πfτ ] in πfτ 4.8 which can be written a tan πτ f extr 1 Rb exp 4 πτ fextr = R exp 4 πτ f b extr at the frequency f extr of the relative minimum or maximum. According to equ. 4.7 the value of the relative minimum or maximum Φ n f extr = Φ n extr i then decribed by tanφ f n extr Rb exp π fextr τ in π fextr τ = tan Φn extr = [ R exp π f τ ] b extr The two equation 4.9 and 4.10 depend on two meaurable quantitie, namely the value tan Φ n extr and the frequency f extr of the extremum. On the other hand, the two equation depend on two combined thermal parameter: the thermal reflection coefficient R b 4. and the thermal diffuion time τ 4.3 of the urface layer of different thermal propertie. Invere olution of equ.4.9 and 4.10, which decribe the two combined thermophyical parameter R b and τ a function of the two meaurable quantitie tan Φ n extr and f extr can eaily be obtained in analytical form. To thi aim, equ.4.9 i reolved for [ R b 1 tan πτ fextr exp πτ fextr ] = tan πτ f extr 47

58 4. The concept of the Phae Extremum By inerting equ.4.11 in the quare of equ.4.10 [tan Φ n extr ] 4[ Rb exp π fextr τ ] [in π fextr τ ] = 4.1 {1 [ R exp π f τ ] } b extr the thermal reflection coefficient can be eliminated, and the reulting equation co 4 πτ = [tan Φ ] 4.1b f extr n extr can be reolved for the thermal diffuion time, given a a function of the two meaurable quantitie tan Φ and f extr : n extr 1 τ {arcco[tan ]} = Φ n extr π fextr Alternative form of thi olution are, e.g. 1 1 tanφ n extr τ = arctan 4.13a S 4π fextr 1+ tanφ n extr 1 arcco [1 tan ]/ = + Φn extr 4π fextr τ 4.13b 1 1 τ arctan 1 = 4.13c 16π f 4 extr tanφ n extr Once the thermal diffuion time ha been calculated according to equ.4.13, the quare of the thermal reflection coefficient can be determined from equ.4.1 R b 1 tan πτ fextr = exp4 πτ fextr tan πτ f extr A can be een from equ.4.13, the combined quantity τ f extr i only a function of the value of the relative phae extremum Φ n extr, and thu the thermal reflection coeffient R b only depend on one of the meaurable parameter, namely the value of the phae extremum Φ n extr : Rb 1 tan{0.5 arcco[tan Φn extr ]} = exp{arcco[tanφ ]} n extr tan{0.5 arcco[tan Φ ]} n extr 48

59 4.3 Application to experimental meaurement If the effuivity of the urface layer i maller than the effuivity of the ubtrate, e < e b, e.g. in the cae of a hard coating on tool teel or on a metal alloy of high effuivity, and a relative minimum i found for the calibrated meaured phae Cf. Figure 4.1, the olution of equ.4.15 i given by Rb = 1 tan{0.5 arcco[tan Φn extr ]} exp{0.5 arcco[tanφ ]} n extr 1+ tan{0.5 arcco[tan Φn extr ]} 4.15a In the cae of the polihed urface layer on the hape memory alloy NiTi, where the effuivity of the urface layer i larger than the effuivity of the bulk of the material, e > e b, o that a maximum i oberved for the calibrated meaured phae, the olution for the thermal reflection coefficient i given by Rb = + 1 tan{0.5 arcco[tan Φn extr ]} exp{0.5 arcco[tanφ ]} n extr 1+ tan{0.5 arcco[tan Φn extr ]} 4.15b Thu, the knowledge of the two meaurable quantitie tan Φ n extr and f extr allow to get acce to the two combined thermophyical quantitie [equ.4.13 and equ.4.15], which according to equ.4. and equ.4.3 lead to the extraction of the thermal tranport propertie of the invetigated ample. In ection 4.3, the developed theory i applied to experimental meaurement preented in Figure Application to experimental meaurement Methodology and dicuion In Figure 4.a theoretical olution according to equ.4.7 are compared with the phae meaured a a function of the heating modulation frequency for everal hard coating on tool teel, and on a metallic alloy of high effuivity. Additionally the phae meaured for the polihed urface of a NiTi hape memory alloy ample x are hown. In fact, the frequency and the value of the phae extremum are obtained from the calibrated meaured phae repreented in Figure 4.1b. Then the combined thermophyical parameter are determined a decribed above and introduced in equ.4.7 to generate the theoretical olution. The reult are reported in Table 4.1. The reaon of deviation between theory and experiment have already been tudied and preented in chapter 3. It can be oberved in Figure 4.b, that the relative maximum of the invere normalized phae for the polihed NiTi 49

60 4.3 Application to experimental meaurement 10 0 Φ n / deg f / Hz 1/ tan Φ n f / Hz 1/ Figure 4. a+ b: from top to bottom Theoretical olution for one-dimenional thermal wave propagation in a two-layer model, in comparion with the invere normalized data meaured for real two layer ytem: hard coating,, on two different ubtrate material and a polihed NiTi hape memory alloy ample x. 50

61 4.3 Application to experimental meaurement Sample Symbol tan Φ n extr f extr / Hz τ / µ R b g b NiTi x A F C Table 4.1: Frequency and value of the relative phae extremum allowing to get acce to the thermal diffuion time and the thermal reflection coefficient for different ample. ample x, which i characteritic for a firt layer with an effuivity above that of the bulk material, i found at rather low frequency, f max 5 Hz, while the relative minima of the coated ample are found at coniderably higher frequencie, e.g. at f min.5 khz. Thi i due to the fact, that the coating conidered here are relatively thin with low thermal diffuion time τ = d /α while the firt layer of the NiTi ample affected by polihing and de-mixing of the alloy i comparatively large Determination of the thermo-phyical propertie Once the combined parameter τ and g b R b have been determined, the thermophyical propertie of the invetigated ample can be eaily retrieved according to equ.4. and equ.4.3. For example, from the reult mentioned in Table 4.1, and knowing the value of ubtrate effuivity for the ample C6, e bs = 7530 W 1/ m - K -1 and it coating thickne, d C6 =.8 µm; the thermal diffuivity and the thermal effuivity of the coating are determined according to α C6 = d C6 /τ and e C6 = g b e bc6, repectively. In Figure 4.3, the meaured phae of the ample C4 and C6 are repreented and compared with the correponding theoretical approximation. A already een in chapter 3, the meaured phae indicate that the two ample have imilar urface characteritic. Correlation between experimental meaurement and theoretical approximation baed on the Extremum Method allow to remark, that the common coating of the two ample i lightly tranparent in the range of modulation frequencie f min /Hz 1/ > 15, the calibrated meaured phae are lightly above the theoretical curve which aume a highly opaque ytem. The data calculated for the ample C6 are available in Table 4.1. For the ample C4, the relative minimum given by tanφ n extr = i localized at the frequency f min 4.7 khz and the ratio of effuivitie coating-to-ubtrate i given by g b = Since the two ample differ with each other only by their uburface material, that mean e C6 = e C4, the unknown effuivity of the uburface material for the ample C4 can be directly extracted according to e bc4 = g bc6 /g bc4.e bc6. In the meantime, the knowledge of the thermal diffuion time of the coating for thi ample C4 lead to the determination of the related thermal diffuivity. 51

62 4.3 Application to experimental meaurement Table 4. ummarize the main characteritic given and extracted of the two invetigated ample. In thi table, d T repreent the total thickne of the ample tan Φ n f / Hz 1/ Figure 4.3: Invere normalized phae of the IR ignal for two ample hard coating on teel-c6, and hard coating on high metal alloy-c4 preenting the ame urface characteritic. The experimental meaurement are correlated with theoretical approximation baed on the Extremum Method. ample τ / µ α / m / d / µm g b e /W 1/ m - K 1 e b /W 1/ m - K 1 d T / mm C C Table 4.: Thermal and phyical propertie of two ample preenting the ame urface characteritic Efficient localization of the phae extremum The value and the frequency of the relative extremum, repectively tan Φ n extr and ƒ extr, are the key parameter governing the Extremum Method. Thi i why a poor etimation of thee important parameter can lead to coniderable error in the interpretation of the meaured phae. For illutration we reconider the meaured phae of Figure 4.3: It appear at firt ight, that the relative minimum of the two calibrated meaured phae can be found at 5

63 4.3 Application to experimental meaurement the ame modulation frequency f extr = khz, correponding to thermal diffuion time of value τc4 = 5.7µ and τc6 = 7.µ, repectively. Theoretical curve dahed line baed on thee value of the thermal diffuion time are alo repreented in Figure 4.4. A one can ee, e.g. for the ample with larger ratio of effuivitie, the theoretical curve dahed line i hifted to the left hand ide with repect to the meaured phae while for the ample with maller ratio of effuivitie the theoretical curve dahed line i coniderably hifted to the right hand ide. Such a ituation can lead to a confuion concerning the actual level of the coating emi-tranparency and alo to a poor appreciation of the ample characteritic, namely the coating thickne or it thermal diffuivity tan Φ n C4 C6 poor etimation of the minimum Theory f / Hz 1/ Figure 4.4: Invere normalized phae of the modulated IR ignal. Theoretical approximation baed on the Extremum Method. Comparion of reult: Efficient localization full line and poor etimation of the phae minimum dahed line. In order to avoid uch error, a four-point baed interpolation involving the ignificant meaured data of the invere normalized phae at the location of the relative extremum, ha to be performed to find out the appropriate and exact value of the frequency and the phae extremum. Thu, by comparing the thermal diffuion time of Table 4. for the ample C4 and C6, with the data obtained through a poor aement of the coordinate of the relative extremum, namely τc4 = 5.7µ and τc6 = 7.µ, the relative error made on the thermal diffuion time i etimated a τ / = 9.5 % and τ / = 1.5 %. 4 6 τ C 53 τ C

64 4.3 Application to experimental meaurement Application to meaurement at high temperature In Figure 4.5, the calibrated meaured phae of a layered tructure are repreented a a function of the modulation frequency for different meauring temperature, and compared with theoretical approximation baed on the Extremum Method. From the obervation, the following remark can be made: i At room temperature and at 118 C, the meaured phae a & b exhibit the behaviour of a two-layer tructure with a emi-tranparent coating. According to Fig. 5 a + b, the calibrated meaured phae take the value Zero at about f /Hz 1/ = 100 and then increae continuouly with increaing modulation frequencie comp. Chap. 3. If the behaviour of the meaured phae lag i imilar from one temperature to another, one can however ee that the obtained value for the thermal reflection coefficient and the thermal diffuion time vary from room temperature to 118 C. That mean, the level of coating emi-tranparency i modified a the temperature change. Thi aertion i proven by the poition of the minimum a & b. ii At 48 C, the calibrated meaured phae alo take the value Zero at the frequency of about f /Hz 1/ = 100. However, intead of increaing continuouly, thee phae reach a maximum at about f /Hz 1/ = 175 before decreaing with increaing modulation frequencie. Such a behaviour of the calibrated meaured phae in the limit high modulation frequencie ha already been een and decribed in chapter 3. It indicate the preence of a very thin layer at the very urface of the coating. With increaing temperature, namely T = 418 C and T = 518 C, the meaured phae maximum decreae while the modulation frequency at which the extremum occur hift from f /Hz 1/ = 175 to f /Hz 1/ = 150. At T = 518 C particularly, the relative maximum i recorded below the value Zero. Thi variation of the relative phae maximum alo mean a variation of the ratio of effuivitie thin layer on top of the coating to coating. In the meantime, the phae minimum ocillate around the value Φ n = -15 C a temperature change, thu proving that the ratio of effuivitie coating-to-ubtrate alo varie with the temperature. 54

65 4.3 Application to experimental meaurement ϕ n / deg -5 ϕ n / deg R b = τ = 31µ -15 R b = τ = 3.µ f / Hz 1/ f / Hz 1/ a b ϕ n / deg -5 ϕ n / deg R b = τ = 33µ -15 R b = τ = 3.6µ f / Hz 1/ f / Hz 1/ c d 10 ϕ n / deg Figure 4.5: Calibrated phae of a layered ample A5 meaured at different temperature. a T = Room temperature RT b T = 118 C c T = 48 C d T = 418 C e T = 518 C -15 R b = τ = 31.µ f / Hz 1/ e 55

66 4.3 Application to experimental meaurement Correlation thermal reflection coefficient temperature In Figure 4.5, the ratio of effuivitie coating-to-ubtrate i repreented a a function of temperature. A one can remark, there i a random dependence of thi ratio on temperature but two ub-domain can be ditinguihed in which a regular ditribution of the ratio with repect to the temperature i effective, namely the interval RT room tempe. to T = 48 C and the interval T = 48 C to T = 518 C. Thi can be explained by the nature of the two phyical effect that are revealed by the meaured phae, namely the emi-tranparency of the coating and the emergence of a very thin layer at the very urface of the coating a the temperature increae g b Temperature / o C Figure 4.5: Variation of the ratio effuivitie coating-to-ubtrate a a function of temperature Correlation thermal diffuion temperature Figure 4.6 how the variation of the thermal diffuion time of the coating a a function of temperature. Here alo, there i a random dependence of τ on the temperature. Since the thermal diffuion time i a function of the thermal diffuivity and the layer thickne, one can conclude that the variation of thi combined quantity i in connection with the change in the topology with increaing temperature, the ample tranit from a two-layer to a three-layer model and with the change of the thermal propertie of the ample a the temperature increae. 56

67 4.4 Dicuion on the reliability of the Extremum Method τ / µ Temperature / o C Figure 4.6: Variation of the thermal diffuion time of the coating a a function of temperature 4.4 Dicuion on the reliability of the Extremum Method The aim of thi ection i to check out whether there i another alternative approach better than the Extremum Method, which can lead to more reliable information about the ample characteritic from other point of the meaured phae. Thi implie the poibility to get acce to the thermal wave reflection coefficient and the thermal diffuion time from any meaured point of the calibrated phae Solution from any meaured point of the calibrated phae Starting from equ.4.7 derived above, two olution giving the thermal reflection coefficient a a function of the thermal diffuion time, the calibrated meaured phae and the modulation frequency can be obtained: R b in πfτ in πfτ πfτ = + 1 exp 4.16a tan[ Φ n f ] tan[ Φ n f ] R in πfτ in πfτ πfτ b = exp 4.16b tan[ Φ n f ] tan[ Φ n f ] 57

68 4.4 Dicuion on the reliability of the Extremum Method The function 4.16a decribe the negative reflection coefficient which indicate the depoition of a coating on ubtrate of better thermal propertie wherea the function 4.16b decribe the poitive reflection coefficient which certifie the depoition of a coating of better thermal tranport propertie on a ubtrate. To implify our invetigation, we only conider the negative thermal reflection coefficient. The objective of thee invetigation i to find out whether from two random point of the meaured phae, it i poible to obtain value of the thermal reflection coefficient and of the thermal diffuion time which are more reliable than the reult provided by the Extremum Method, thi later method being eentially baed on one fundamental point of the meaured phae Study of the function R b = R b τ In Figure 4.7b, two meaured point are conidered at the left hand ide of the phae extremum and the repreentative curve R b = R b τ generated by each of the meaured point i hown in Figure 4.7a. A one can clearly ee, there i a very mall interection angle between the two curve. R b x x x x x10-5 τ Φ n / deg f / Hz 1/ a b Figure 4.7 a + b: Determination of the combined parameter from two meaured point at the left hand ide of the phae extremum no obervable interection point. A econd tet, two other meaured point are conidered at the left hand ide of the phae extremum Figure 4.8b and the correponding repreentative curve provide an interection point I Figure 4.8a. The coordinate of thi point are given by Iτ = 6.6 µ and R b = For comparion, the reult obtained from the Extremum Method are τ = 6.4 µ and R b = At firt ight, the two reult eem to be cloe but one can however remark in Figure 4.8b, that the theoretical curve generated by the two meaured point i omewhat hifted to the left hand ide with repect the meaured phae. 58

69 4.4 Dicuion on the reliability of the Extremum Method R b I Φ n / deg x x x x x10-5 τ f / Hz 1/ a b Figure 4.8 a + b: Determination of the combined parameter from two meaured point at the right hand ide of the phae extremum. Interection point Iτ=6.6 µ, Rb= Reult from the Extremum Method: τ=6.4 µ, Rb= Now, by conidering two other meaured point located repectively at the left hand ide and at the right hand ide of the phae extremum Figure 4.9b, the interection point I of the repreentative curve Figure 4.9a i identified at τ = 6.4 µ, R b = Depite the fact that the value of the thermal diffuion time i quite in agreement with that obtained from the Extremum Method τ = 6.4 µ, R b = , there i a diagreement about the value of the thermal reflection coefficient. The diagreement between the theoretical olution generated by taking into τ = 6.4 µ, R b = and the experimental meaurement can be clearly een in Figure 4.9b, namely in the frequency range 5 < f / Hz 1/ < R b x x x x x10-5 τ I Φ n / deg f / Hz 1/ a b Figure 4.9 a + b: Determination of the combined parameter from two meaured point at the left and right hand ide of the phae extremum. Interection point Iτ = 6.4 µ, R b = Reult from the Extremum Method: τ = 6.4 µ, R b =

70 4.4 Dicuion on the reliability of the Extremum Method R b x x x x x10-5 τ I Φ n / deg f / Hz 1/ a b Figure 4.10 a + b: Determination of the combined parameter from two other meaured point at the left and right hand ide of the phae extremum. Interection point Iτ = 6.3 µ, R b = Reult from the Extremum Method: τ = 6.4 µ, R b = For the latet invetigation two other meaured point are conidered; one point at the left hand ide, very far from the location of the relative minimum, in the range of low modulation frequencie and the other point in the range of high modulation frequencie Figure 4.10b. The interection of the two repreentative curve i given by Iτ = 6.3 µ and R b = Here alo, the thermal diffuion time can be conidered to be in agreement with the value obtained from the Extremum Method. However, the obtained value for the thermal reflection coefficient i rather far away from the expected value. Thi diagreement can be explained by the fact that the two conidered meaured point are located in two frequency domain where the modulated IR ignal are ubjected to phyical effect thermal and optical, namely 3-D thermal wave propagation in the limit of very low modulation frequencie and emi-tranparency effect at high modulation frequencie Compare Chap Comparion of reult A mall interection angle a hown in Figure 4.7a already indicate that an alternative method for the determination of the combined thermophyical quantitie, baed on any point of the meaured phae cannot be reliable. Occaionally, acceptable olution uing uch an approach can be obtained Figure 4.8b. However, according to Figure 4.9b and Figure 4.10b, the large diagreement between the meaurement and the generated theoretical curve in the range of intermediate frequencie doe not favour thi alternative method which conider any point of the meaured phae. It i therefore jutified to conclude, that determining the two combined quantitie R b and τ from any point of the meaured phae different from the relative phae extremum i a random procedure which rather lead to contradictory reult. 60

71 4.5 General reult for on-line interpretation in indutrial application Only the Extremum Method provide the more reliable information about the thermo-phyical propertie of the invetigated ample. Following thi comparion, we demontrate in the next ection how experimental meaurement can be directly handled by uing the propoed evaluation method. 4.5 General reult for on-line interpretation in indutrial application There i an increaing need for on-line interpretation. The indutrial invetigator or the experimentalit would like to make a rapid report on the unknown invetigated ample with repect to their thermal and phyical parameter. For thi, a variety of quetion are often of concern, e.g.: Doe the coating fulfil the requirement of thermal barrier or alternatively of good conductor? I the coating opaque or emi-tranparent? What are then the thermal and phyical propertie of the unknown tructure? What about the coating thickne?, etc The Extremum Method give anwer to uch quetion by propoing general reult which help to get acce to the thermal tranport propertie of two-layer tructure Graphic of thermal reflection coefficient In Figure 4.11a, the thermal reflection coefficient i repreented a a function of the meaured phae extremum. A expected the thermal reflection coefficient i limited by the value, -1 and +1. The lower limit correpond to the ituation for which a coating of extremely poor thermal propertie ha been depoited on the ubtrate while the upper limit correpond to the ituation for which the thermal propertie of the ubtrate are extremely poor in comparion with the propertie of the depoited coating. One can oberve, that the phae extremum tanφ n extr i alo limited by the ame value, 1 and + 1. Mathematically, the origin of thi limitation can be clearly een in equ Depending on the value of the phae extremum, the graph of Figure 4.11a can be eparated into two part. For negative phae extrema phae minima, the thermal reflection coefficient i negative, -1 < R b < 0, and for poitive phae extremum phae maxima, the thermal reflection coefficient i poitive, 0 < R b < 1. A the meaurable quantity tanφ n extr approache the value Zero, the curve R b = R b tanφ n extr experience a dicontinuity, which i quite comprehenible ince the Extremum Method trictly applie to two-layer tructure. A can be een in equ.4.15, if the phae extremum take the value Zero, then the thermal reflection coefficient will be alo reduced to Zero, correponding to the cae of a homogenou ample, which i out the context in thi tudy. The two olution repreented in Fig. 4.11a are characteritic for the ample C6 and for the NiTi ample ee Table

72 4.5 General reult for on-line interpretation in indutrial application R b tan Φ n extr Fig. 4 11a: Thermal reflection coefficient R b a a function of the relative phae extremum tanφ n extr of the invere normalized phae lag e /e b 0.3 0,1 0,01-1,00-0,75-0,50-0,5 0,00 0,5 0,50 0,75 1, tanφ n extr Fig. 4 11b: Ratio of effuivitie coating-to-ubtrate e /e b a a function of the relative phae extremum tanφ n extr of the invere normalized phae lag. 6

73 4.5 General reult for on-line interpretation in indutrial application 4.5. Graphic of the ratio of effuivitie The graphic giving the ratio of effuivitie coationg to ubtrate a a function of the relative phae extremum tanφ n extr of the invere normalized phae lag i preented in Figure 4.11b. In analogy to the graphic of thermal reflection coefficient, two main domain can be ditinguihed: For negative relative phae extrema, the ratio of effuivitie i valid between 0 and 1, which mean that coating are depoited on ubtrate of better thermal propertie. For poitive relative extrema, the ratio of effuivitie i larger than the value 1, which repreent the ituation for which a coating of better thermal effuivity i depoited on a ubtrate. Here, the ratio ha been limited to 10 to account for the uual coating material K tan Φ n extr Figure 4.1: Key function K f a function of the meaured phae extremum Graphic of thermal diffuion time In comparion with the thermal reflection coefficient or the ratio of effuivitie which are eentially generated by different value of the relative phae extremum, the thermal 63

74 4.5 General reult for on-line interpretation in indutrial application diffuion time of the coating i rather determined by the value and the frequency of the phae extremum To make eaier the comprehenion and the readability of the graphic of thermal diffuion time, the thermal diffuion time can be calculated from equ K tan Φ n extr τ = 4.17 f extr with the key-value KtanΦ n extr {arcco[tanφ n extr ]} K tanφ n extr = π The key-function defined by equ.4.18 i repreented in Figure 4.1. In thi figure, one can oberve, that the key-function varie between 0 and 0.05 within the entire interval range of the poible phae extrema. The upper key-value i reached a tanφ n extr approache the value Zero. On the other hand, a the meaurable quantity tanφ n extr approache it lower or upper limit, tanφ n extr = 1, the key-function eem to adopt a value far below 0.005, which i already ten time maller than the maximal value, That mean, for a given frequency of the relative extremum, the thermal diffuion time decreae with increaing abolute value of the meaurable quantity tanφ n extr. An illutration of thi aertion i hown later on in Figure 4.14a. Thu, once a particular key-value i derived from a given value of the meaured phae maximum or minimum, the thermal diffuion time of the coating can be eaily determined, a equ.4.17 ugget. Figure 4.13 preent a graphic of thermal diffuion time repreented a a function of the frequency of the relative phae extremum for different value of the relative phae extremum. In thi Figure namely, the upper curve i generated by tanφ n extr = for different value of the modulation frequency compried between 0.1 khz and 10 khz. In the ame way, the lower curve i generated by tanφ n extr = for different value of the modulation frequency. Between the upper and the lower curve, and from top to bottom, the other curve are induced by tanφ n extr = 0.300, 0.45, 0.600, 0.750, repectively. For all curve and from top to bottom, the key-value are given in Table 4.3 a a function of the meaured phae extremum. Figure 4.13 alo how, that for a given value of the meaured phae extremum, a et of thermal diffuion time can be determined, depending on the modulation frequency of heating. In thi cae, the difference between the calculated thermal diffuion time relie on the variation of the coating thickne. 64

75 4.5 General reult for on-line interpretation in indutrial application tan Φ n extr = τ / µ f extr / Hz Figure 4.13: Thermal diffuion time a a function of the frequency meaured for the relative minima or maxima of the invere normalized phae lag, conidering different value of the relative extrema. tan Φ n extr K Table 4.3: Value K of the key function a a function of the relative extremum The theoretical reult preented in Figure 4.14a indicate that the thermal diffuion time of the thermal wave decreae with increaing abolute value of the relative phae extremum and are in good agreement with the repreentative curve of Figure 4.1 and of Figure In Figure 4.14b, value of f extr /Hz 1/ = 50, 75 and 100 for the frequency of the relative extremum induce repectively value of τ /µ = 14.83, 6.59, and 6.4 for the thermal diffuion time of the coating. For remind, the et of upper curve blue color refer to a coating of better thermal propertie with repect to the ubtrate and the et of lower curve red color refer to ubtrate of better thermal propertie with repect to the coating. 65

76 4.5 General reult for on-line interpretation in indutrial application tan ϕ n extr tan ϕ n extr f / Hz 1/ f / Hz 1/ Figure 4.14a: Evolution of the thermal diffuion time of the coating with the phae extremum for a contant value of the frequency of the relative extremum. tanφ n extr = 0.5, 0.65, 0.95 give repectively τ / µ = 8.05, 4.55, and 0.70 for f extr / Hz 1/ = 75 Figure 4.14b: Evolution of the thermal diffuion time of the coating with the frequency of the relative extremum, for a contant value of the phae extremum. f extr /Hz 1/ = 50, 75 and 100 give repectively τ / µ = 14.83, 6.59, and 6.4 for tanφ n extr = Example of on-line interpretation The aim of thi ection i to how briefly how experimental meaurement performed on two-layer tructure can be rapidly and methodically interpreted by uing the etablihed graphic and the above defined fundamental equation: tanφ n extr R b K f extr / khz τ / µ Example of comment Subtrate of better thermal tranport propertie. Coating: Micro- or nano-layer? Coating of better thermal tranport propertie. Thick urface layer? Table 4.4: Example of on-line report about the experimental meaurement on two-layer tructure. i If we uppoe, that the value of the relative minimum on the calibrated meaured phae i given by tanφ n extr = -0.55, then the graphic of thermal reflection coefficient Figure 4.11a will indicate R b = ubtrate of better thermal propertie a correponding thermal reflection coefficient. According to equ.4., from the obtained combined quantity 66

77 4.6 Interpretation of phae meaurement obcured by the background fluctuation one can immediately get acce to the coating effuivity if the ubtrate effuivity i known or alternatively to the ubtrate effuivity if there i any available information about the coating effuivity. On the other hand, the conidered relative minimum yield a key-value K = 0.03, which i then combined with the frequency of the relative minimum, f min /khz = 4.9, to determine the effective thermal diffuion time, τ /µ 6.5. The thickne or alternatively the thermal diffuivity of the coating can be extracted according to equ ii In the cae a relative maximum of tanφ n extr = 0.38 i rather oberved on the calibrated meaured phae excluively for a two-layer ytem!, then a value of R b = 0.55 coating of better thermal propertie will be identified on the graphic of thermal reflection coefficient. In the ame way, the effuivity of either the coating or the ubtrate i extracted from the available information about one of the two layer. In the meantime, the related keyvalue K = 0.04 help to derive a thermal diffuivity of the coating, τ /µ 9.6, if the modulation frequency at which the relative maximum occur i given for example by the value f min /khz = 4.5. For the two configuration, an example of hort on-line report about the experimental meaurement i given in the table Interpretation of phae meaurement obcured by the background fluctuation Since meaurement baed on modulated IR radiometry may be obcured at low frequencie due to 1/f noie and at high frequencie due to the background fluctuation limit [Bolte et al., 1997], an alternative interpretation method ha to be developed, which may be baed on any other value of the meaured invere normalized phae lag and the correponding value of the modulation frequency. The method preented in Sect. 4.., to determine the two relevant thermal tranport parameter τ and R b, the thermal diffuion time and the thermal reflection coefficient, from the meaured relative minimum or maximum of the invere normalized phae lag, i equivalent to a method baed on the value of the meaured invere normalized phae lag and of it lope, which at the poition of the relative minimum or maximum i Zero. Thu, an alternative method hould rely on the value of the invere normalized phae lag meaured in the neighbourhood of the relative minimum or maximum and on the lope of the invere normalized phae lag. In general the derivation of the lope from a curve of meaured value generate numerical error, leading to large error of interpretation. Alternative method which avoid uch numerical error of differentiation may rely on the olution of a problem-related integral equation, uch a propoed by Bein [Bein, 1986] for the derivation of heat fluxe from timereolved thermographical urface temperature meaurement. Another alternative may be the calculation of problem-related functional moment uch a the temporal moment method 67

78 4.6 Interpretation of phae meaurement obcured by the background fluctuation propoed by Balagea and co-worker [Balagea et al., 1987] for the interpretation of puled photothermal radiometry. Here a olution method i propoed which relie on a functional tranformation of the meaured invere normalized phae lag by applying a multiplication baed on the frequency dependence of the thermal diffuion length µ th 1/ f 1/ and which ubtitute the minimum or maximum of the invere normalized phae lag by the minimum or maximum of the tranformed phae lag function at a modulation frequency value in the neighborhood Theory of tranformation of the invere normalized phae The method ued in Sect. 4.. to determine the two combined thermal parameter, the thermal diffuion time τ and the thermal reflection coefficient R b, from the value and the frequency of the relative phae extremum i extended here to other frequency value, epecially to lower frequencie and the correponding phae value. To thi finality a multiplicative tranformation of the invere normalized phae lag i introduced tanφ exp in tan n f Rb πfτ πfτ Ψ n q f = = 4.19 q q f f {1 [ Rb exp πfτ ] } with the exponent q a mall real number in the neighbourhood of zero and the exponent q = 0 correponding to the not tranformed invere normalized phae tanψ n q= 0 f = tanφ n f 4.19a The derivative of the tranformed phae lag with repect to the quare root of the modulation frequency i calculated a tanψ f n q f = tanφ n f = q f f 1 f q tanφ n f + tanφ n f f f f q = = 1 f q 4R [1 exp 4 ]co b πτ exp πfτ Rb π fτ πfτ + [1 Rb exp 4 πfτ ] [1 + Rb exp 4 πfτ ] in πfτ + q f q+ 1 Rb exp πfτ in πfτ = [1 R exp 4 πfτ ] b 68

79 4.6 Interpretation of phae meaurement obcured by the background fluctuation 1 4Rb πτ exp πfτ = [ co πfτ q f [1 R exp 4 πfτ ] b 1+ Rb exp 4 π fτ q in πfτ in πfτ ] R exp 4 π fτ πfτ b At the relative extremum tan Ψ f / f = 0 of the tranformed phae lag function the condition n q tan πf extr q τ 1 + πf ha to be fulfilled, or alternatively q extr q τ 1 R 1+ R b b exp 4 exp 4 πf πf extr q extr q τ 1 R = τ 1+ R b b exp 4 exp 4 πf πf extr q extr q τ τ 4.1 tan πf extr q τ = 1+ q π f extr q 1 R b exp 4 extr q π f extr q τ q + [ 1 ] Rb exp 4 πfextr qτ τ π f τ 4.1a which ubtitute the condition 4.9 of the relative extremum of the original, not tranformed phae lag function. According to equ.4.19 the tranformed phae lag function at the new poition of the extremum i given by the equation tanψ n q f extr q Rb exp πfextr qτ in πfextr qτ = tanψ n q extr = 4. q f [1 R exp 4 πf τ ] extr q b extr q which ubtitute equation 4.10 of the not tranformed original phae lag. Similar to the equation 4.9 and 4.10 for the not tranformed phae lag, the two equation 4.1a and 4. depend on the thermal reflection coefficient and the thermal diffuion time and have to be reolved for thee two quantitie, to be given a function of the relative extremum and the correponding frequency of the tranformed phae lag function. To thi finality equ.4.1a i reolved for R b q 1 [ 1 + ] tan πf extr qτ π f extr qτ exp 4 π f extr qτ = 4.3 q 1 + [ 1 ] tan πf extr qτ π f τ extr q By inerting equ.4.3 in the quare of equ.4. 69

80 4.6 Interpretation of phae meaurement obcured by the background fluctuation tanψ n q extr b 4R exp 4 πf extr qτ [in πf extr qτ ] = 4.4 q f [1 R exp 4 πf τ ] extr q b extr q the thermal reflection coefficient R b i eliminated and an equation i obtained, which can be interpreted a an extenion of equ.4.1b f extr q q tanψ n q extr [ ] [in πf extr qτ π f extr qτ q = co4 πf extr qτ [ ]in4 πf extr qτ + π f τ extr q q + ] 4.5 and which by a numerical procedure i reolved for the thermal diffuion time τ = τ tan ψ n q extr, f extr q, given a a function of the relative extremum tan ψ n q f extr q = tan ψ n q extr of the tranformed phae lag and of the correponding value f extr q of the modulation frequency. Once, the thermal diffuion time i known, equ. 4.3 can be reolved for the thermal reflection coefficient R b q 1 [ 1 + ] tan πf extr qτ π f extr qτ = ± exp π f extr qτ 4.6 q 1 + [ 1 ] tan πf extr qτ π f τ extr q An alternative olution can be derived from equ.4. and 4.4 which allow an iterative olution and which in the cae of the exponent q=0 coincide with olution 4.13a of the not tranformed phae lag function. tan π f extr q τ = = q π f extr q + τ 1 [tanψ 1 n q extr q π f extr q f τ extr q q ] + tanψ + [tanψ n q extr n q extr f extr q q f extr q q ] q π f extr q τ 4.7 Although a reolution for τ in analytical form i not poible, equation 4.7 offer everal advantage: It can be reolved by a numerical procedure or by an iterative proce with the firt tep τ 1 calculated from the minimum of the tranformed phae lag function according to 70

81 4.6 Interpretation of phae meaurement obcured by the background fluctuation tan π f πf extr q τ 1 extr qτ 1 = = arctan 1 [tanψ 1+ tanψ 1 [tanψ 1+ tanψ n q extr n q extr n q extr n q extr f f extr q extr q f f q extr q extr q q ] q q = ] 1 [tanψ 1+ tanψ n q extr n q extr f f extr q extr q q q ] 4.8a 4.8b and the further tep with q 0 calculated according to tan π f i extr qτ q π f = = i 1 extr qτ + 1 [tanψ n q extr q 1 π f f extr q i 1 extr qτ q ] + tanψ + [tanψ n q extr n q extr f extr q f extr q q q ] π f q i 1 extr qτ 4.9 Additionally olution 4.7 can be interpreted a an extenion of olution 4.13 for the relative minimum of the not tranformed meaured phae lag function q= Application of the functional tranformation to meaurement: The problem of convergence In order to tet the tranformation method and to get information on the exponent q leading to converging olution for the thermal diffuion time and thermal reflection coefficient, the method i applied to the invere normalized phae lag meaured for a ample coniting of a diamond-like carbon coating C6 on tool teel which according to Figure 4.b perfectly agree with an opaque two-layer ytem in the meaured frequency interval. Figure 4.15 how that with increaing poitive q-value the minimum of the tranformed calibrated phae i hifted toward lower modulation frequencie with repect to the frequency of the minimum related to the not tranformed phae q = 0. One can ee in Table 4.5, which report on parameter and reult of thi tranformation, that only certain q- value lead to reult very cloe to the reult of the not tranformed cae. Thu, conidering q- value up to about yield a frequency range, in which the thermal diffuion time i determined with a relative error of le than 1%, % or 3 %. For q-value 0.175, there i a dicrepancy between the olution of the tranformed and the not tranformed phae q = 0. Figure 4.16 how, that with decreaing negative q-value the minimum of the tranformed calibrated meaured phae i rather hifted toward higher modulation frequencie with repect to the frequency of the minimum correponding to the not tranformed calibrated phae. The reult of the functional tranformation, reported in Table 71

82 4.6 Interpretation of phae meaurement obcured by the background fluctuation 4.6 indicate alo, that only certain negative q-value are appropriate to keep contant the thermal diffuion time and the ratio of effuivitie. In thi example, q-value -0.3 induce coniderable deviation while q-value reproduce the expected reult. Figure 4.17 and Figure 4.18 how the calibrated meaured phae and the ubequent tranformation obtained by operating on poitive Fig and negative Fig q- value. Parameter and reult of thi functional tranformation are reported in Table 4.7 and 4.8. One can ee in Table 4.7, that only a very mall and poitive q-value q = 0.01 keep approximately contant the value of the thermal diffuion time. The reult conigned in Table 4.8 indicate alo, that a very reduced number of negative q-value q contribute to reproduce a thermal diffuion time cloe to that obtained for q = 0. q tan ψ n q extr f extr q 1/ /Hz 1/ τ / µ R b g b Table 4.5: Parameter and reult of the invere normalized phae lag meaured for a diamondlike coating on tool teel C6 and after tranformation q > 0 of the phae lag. q tan ψ n q extr f extr q /Hz 1/ τ / µ R b g b Table 4.6: Parameter and reult of the invere normalized phae lag meaured for a diamond-like coating on tool teel C6 and after tranformation q < 0 of the phae lag. 7

83 4.6 Interpretation of phae meaurement obcured by the background fluctuation 0.0 f -1/ q tan Φ n q= f / Hz 1/ Figure 4.15: Tranformation of the invere normalized phae: Shifting of the phae extremum toward maller modulation frequencie f -1/ q tan Φ n q = f / Hz 1/ Figure 4.16: Tranformation q < 0 of the invere normalized phae lag, hifting the relative extremum of the tranformed function toward higher modulation frequencie, applied to the invere normalized phae lag meaured for a diamond-like coating on tool teel C6. 73

84 4.6 Interpretation of phae meaurement obcured by the background fluctuation q tan ψ n q extr f extr q /Hz 1/ τ / µ R b g b Table 4.7: Parameter and reult of the invere normalized phae lag meaured for ample F5 and after tranformation q > 0 of the phae lag. q tan ψ n q extr f extr q 1/ /Hz 1/ τ / µ R b g b Table 4.8: Parameter and reult of the invere normalized phae lag meaured for ample F5 and after tranformation q < 0of the phae lag. The diverity oberved in the reult of the functional tranformation of the invere normalized phae lag for the two different ample C6 and F5, implie the analyi of the convergence problem: i Limit of convergence in the cae of a coating-on-ubtrate ample, fulfilling the condition of an opaque two-layer ytem with one-dimenional thermal wave propagation, and ii Limit of convergence in the cae of a coating-on-ubtrate ample, preenting additionally reduced thermal propertie at the very urface of the coating and three-dimenional thermal wave propagation in the ubtrate. According to Figure 4.19, which repreent the thermal diffuion time and the ratio of effuivitie coating-to-ubtrate a a function of the q-value, the horizontal panel oberved in the range < q for the ample C6 mean that the four-point baed interpolation and the q-tranformation reproduce the reult, ince the thermal diffuion time and the ratio of effuivitie are kept contant. Thi reult ha a double ignificance: i It provide the proof that the tranformation applied to the phae correctly work over a certain range of q value. 74

85 4.6 Interpretation of phae meaurement obcured by the background fluctuation f -1/ q tan Φ n q = f / Hz 1/ Figure 4.17: Tranformation q > 0 of the invere normalized phae: Shifting of the phae extremum toward lower modulation frequencie ample F5 0.0 f -1/ q tan Φ n q = f / Hz 1/ Figure 4.18: Tranformation q < 0 of the invere normalized phae: Shifting of the phae extremum toward higher modulation frequencie F5 75

86 4.6 Interpretation of phae meaurement obcured by the background fluctuation ii Although not neceary for quantitative information with repect to τ and g b, which in the cae of ample C6 are ufficiently well determined at the original meaured poition of the minimum, the 1/f 1/ q -tranformation can give additional information about what range of frequencie the meaured ample can be ufficiently well approximated by an opaque twolayer model facing 1D heat tranport, and where deviation related to an effective three-layer tructure or due to emi-tranparency in the range of higher frequencie, and where deviation due to three-dimenional heat tranport in the range of lower frequencie affect the meaured ignal and therefore lead to a more complex cheme of thermal wave propagation in layered ytem. Thi i the cae for the ample F5 for which it can be oberved that the tranformed invere normalized phae lag produce non convergent olution in term of the thermal diffuion time, although a contant ratio of effuivitie i recorded for q-value within the interval < q In fact, a far a convergence i concerned, it may imply both the thermal diffuion time and the thermal reflection coefficient and not excluively one of the two quantitie C F τ /µ e /e b τ /µ e /e b q q Figure 4.19: Thermal diffuion time and ratio of effuivitie coating-to-ubtrate a a function of the tranformation exponent q Problem of convergence of the relevant thermal tranport parameter τ, R b, or g b. That the functional tranform applied to the ample F5 doe not produce convergent olution can be explained by the two phyical effect occurring at high modulation frequencie, namely the exitence of a very thin layer on top of the coating three-layer model and at low modulation frequencie, namely the 3-D thermal wave propagation in the uburface material, all apect which can contribute to reduce the frequency range of the convergence which i more extended in the cae of an opaque two-layer tructure with 1-D thermal wave propagation. 76

87 4.6 Interpretation of phae meaurement obcured by the background fluctuation Phae interpretation obcured by the background fluctuation Figure 4.0 how an example of meaurement on a coated cutting tool after expoure to friction wear. Due to a highly reflecting urface, the meaurement of the phae lag i already affected by the background noie before the minimum occur. A one can oberve in thi figure, the meaurement i already diturbed by the background fluctuation above heating modulation frequencie of about f /Hz 1/ > 55, o that the minimum can not eaily be localized. The four meaured value between 60 <f /Hz 1/ < 100 and all meaurement above f /Hz 1/ > 100 can not be interpreted. q tan ψ n q extr f extr q /Hz 1/ τ /µ R b g b Remark In thi range of q-value, no poible olution are found Minimum Minimum Table 4.9: Shifting of the tranformed normalized phae minimum a a function of the exponent q P1 q tan ψ n q extr f extr q /Hz 1/ τ /µ R b g b Remark In thi range of q-value, no poible olution are found Minimum Minimum Table 4.10: Shifting of the normalized phae minimum a a function of the exponent q 119P In order to detect the poition of the minimum, everal value of the exponent q have been teted. A indicated in Table 4.9, in the range of q-value < 0.3, no poible olution can be found. A relative minimum for the tranformed phae lag function ha been obtained for rather large q-exponent 0.3; The thermal diffuion time τ 8 µ and the thermal reflection coefficient R b = obtained from the functional tranformation baed on thee two value which lead to the ame reult of the exponent q are then ued to generate a theoretical olution according to equ. 4.10, which olution i correlated with the not tranfor- 77

88 4.6 Interpretation of phae meaurement obcured by the background fluctuation 0.00 f -1/ q tan Φ n q = f / Hz 1/ Figure 4.0: Invere normalized phae lag meaured for a friction wear affected coated cutting tool urface P1, in comparion with tranformed phae lag function and theoretical approximation f -1/ q tan Φ n q = f /Hz 1/ Figure 4.1: Invere normalized phae lag meaured for a friction wear affected coated cutting tool urface 119P, in comparion with tranformed phae lag function and theoretical approximation. 78

89 4.7 Concluion -med experimental invere normalized phae lag Figure 4.0. Through thi correlation, the frequency and value of the phae extremum that could be derived if the modulated IR ignal meaurement were not affected by the background fluctuation limit are etimated a follow: f extr /Hz 1/ = 71.3 and tanφ nextr = -0.16, repectively. Table 4.10 report on parameter and reult obtained for the functional tranformation in connection with another type of friction wear Fig A in the previou cae, there exit a range of q-value for which no poible olution can be retrieved. Mot intereting i however, that the interval range of q-value for which there i no recordable minimum, i enlarged. A indicated in Table 4.10, a relative minimum of the tranformed phae lag function ha been achieved only for q-value = 0.45 and Thee two q-exponent lead to contant value of the thermal diffuion time and the thermal reflection coefficient. Proceeding in the ame way a above, the combined parameter, τ 1.6 µ and R b = -0.33, obtained from the functional tranformation baed on q = 0.45 and q = 0.50 are ued to generate a theoretical olution according to equ. 4.10, which olution i correlated with the not tranformed experimental invere normalized phae lag Fig Thu, the probable frequency and value of the relative phae extremum that hould have been detected if the modulated IR ignal meaurement were not affected by the background fluctuation are repectively etimated a follow: f extr /Hz 1/ = and tanφ nextr = Concluion Thermal wave applied to control online-thicknee of coating or paint layer [Petry, 1998] require hort meaurement time and fat numerical routine. Meaurement baed on thermal wave method are known to give more reliable reult with the meaurement at only three modulation frequencie, one below the expected minimum frequency, one above the expected minimum frequency and another in the neighbourhood. In thi chapter, we have propoed and applied a new and fat procedure for the determination of thermal tranport propertie of two-layer tructure via two combined thermophyical quantitie, namely the thermal diffuion time and the thermal reflection coefficient, which are extracted from one fundamental point of the calibrated meaured phae, indexed a relative phae extremum. It ha been demontrated, that the frequency and value of the relative phae extremum of the calibrated meaured phae, if correctly identifed, lead to a more reliable information about the ample propertie. The Extremum Method ha alo provided general reult for on-line interpretation in indutrial application by allowing the etablihment of two main graphic, namely the graphic of thermal reflection coefficient a a unique function of the relative extremum and the graphic of thermal diffuion time a a function of the frequency at which the phae extremum occur, for different value of the relative extremum. Then a functional tranform 79

90 4.7 Concluion method, originating from the Extremum Method, ha alo been developed to enable the interpretation of meaurement baed on modulated IR radiometry, which may be obcured at low frequencie due to 1/f noie and at high frequencie due to the background fluctuation limit. In order to validate thi tranformation method and ubequently to tudy the convergence problem, everal tet have been performed on phae meaurement of two different ample, which meaurement are free of noie or of background fluctuation. For one ample C6, which fulfill the condition of an opaque two-layer ytem facing 1D thermal wave propagation and whoe minimum i eaily identifiable, the functional tranform method baed on both negative and poitive q-value ha reproduced reult converging to that obtained in the cae q = 0, by keeping contant the thermal diffuion time and the thermal reflection coefficient within a certain frequency range. For larger abolute q-value however, deviation between olution of tranformed and not tranformed phae meaurement were noted, providing the proof that the q-tranformation applied to the invere calibrated phae correctly work over a certain range of q-value. Although a relative minimum wa clearly identified on the calibrated meaured phae of another ample F5, ucceive q- tranformation of thee phae meaurement have not provided olution of τ and R b both convergent to the original olution q = 0. Such a ituation ha been explained by the fact, that the calibrated phae meaurement of the conidered ample F5 are affected by phyical effect uch a the 3-D thermal wave propagation at low frequencie and urface tructure additional thin layer on top of the coating or emi-tranparency at high modulation frequencie, thu giving a proof that the functional tranformation work very well for meaurement involving typical two-layer ytem facing 1-D thermal wave propagation. Finally, the functional tranform method ha been applied to ome example of meaurement which may be obcured by the background fluctuation o that no phae extremum can be oberved. Here, the q-value which led to tranformed invere normalized phae lag exhibiting a minimum, produced value of the thermal diffuion time and the thermal reflection coefficient, which were then ued to etimate the probable frequency and value of the relative phae extremum that hould have been detected if the modulated IR ignal meaurement were not affected by thee background fluctuation. In the next chapter, we alo demontrate that the relative extrema appearing on the calibrated meaured phae in the low modulation frequency range can contribute in thermal microcopy to get more reliable information about the lateral heat tranport and the identification and detection of hot pot in the tructure. 80

91 5. Detection of local Inhomogeneitie of Thermal Tranport and Localization of Heat Source in Micro-caled Sytem baed on Spot Diplacement 5.1 Motivation We have hown in chapter 3, that the relative phae maximum oberved on the normalized ignal phae of variou layered ytem, in the limit of very low modulation frequencie, i due to the effect of lateral heat propagation in the uburface material, and that thee thermal effect increae with decreaing heating pot radii and with increaing thermal diffuivitie of the uburface material. That mean, by conidering maller heating pot radii, the relative phae maximum can help to detect local variation of the thermal diffuivity of the ubtrate. In addition to the thermal tranport propertie, which play a key role in uch layer ytem ued e.g. a device in microelectronic or micromechanic, the localization of hot pot i particularly important in microelectronic where the reduction of ize i accompanied by a growing power diipation. In thi cope, numerical imulation of heating procee and of photothermal experiment become more and more important, both for the interpretation of meaurement and the development of meaurement method, applied for the deign and quality control of micro-tructured material. In the preent chapter, tarting from concrete example of photothermal meaurement of two-layer ytem with film thicknee in the range of 1 to 3 µm, we how with further imulation of photothermal experiment of two-layer ytem from macrocopic to microcopic cale, that by uing 3-D thermal wave propagation and conidering controlled diplacement ditance between the excitation and the detection pot [Ikari et al., 003], the relative extrema generated on the calibrated ignal phae in the low frequency range can contribute in thermal microcopy to get more reliable information about the lateral heat tranport and the identification of hot pot in the tructure. 5. Review of main reult of 3-D thermal wave propagation We have demontrated in chapter 3, that the modulated IR ignal meaured over the detection pot area of radiu r D comp. Figure 3.8, Chap. 3, reulting from the generation of a thermal wave at the urface of a two-layer ytem, can be calculated a 81

92 5.3 Diplacement between heating and detection pot δ M f, T, t = 4C f γ T σ T ε T * SB 3 rd η I o πrdr πk 0 0 exp rh, f / R 1 R b b f, exp d f, exp d J o r d exp iπft 5.1 * 3 The factor 4C f γ T σ SB T ε T in equ. 5.1 i a real quantity, which cannot affect the phae of the meaured thermal wave repone. For recall, both the meaured and the theoretical data are preented in the form of calibrated ignal phae, Φ n ref f = Φ f Φ f 5. where the ignal phae Φ ref f refer to a homogeneou reference body of known thermal and optical propertie, e.g. glay carbon, and the ignal phae Φ f refer to the ample, e.g. a two-layer ytem. Thi mean, the reult baed on the interpretation of the ignal phae, are independent of the pecific detection proce and thu alo apply to other detection technique, e.g. thermoreflectance. It ha been oberved comp. Section , chap. 3, that the deviation from 1-D thermal wave propagation increae with decreaing heating pot radii and with increaing thermal diffuivitie α b of the uburface material and that the normalized phae are rather enitive to variation of the thermal diffuivity α b of the uburface material. Thi mean, by working with maller heating pot radii in thermal microcopy, the normalized phae with their relative phae maximum in the low frequency range can be ued to detect local variation of the thermal diffuivity in the uburface material. 5.3 Diplacement between heating and detection pot In the imulation of 3-D thermal wave propagation, concentric heating and detection pot, a chematically hown in Figure 3.8 Chap. 3 have been conidered. If heating pot and detection pot are not well focued in the experiment, error of interpretation may occur due to deviation between the model of concentric excitation and detection and the experimental reality. On the other hand, a controlled pump-probe beam offet, intentionally introduced between heating and detection pot, a hown in Figure 5.1, can contribute in thermal microcopy to get more reliable information on the lateral heat tranport propertie and on the localization of heat ource hot pot, by comparing the data meaured at two or three neighbored poition. In uch experimental configuration, which can be ued in connection with both modulated thermoreflectance and modulated IR radiometry a detection technique [Milcent et al., 1995; Ikari et al., 1999] and which have already been applied to coating on metal 8

93 5.3 Diplacement between heating and detection pot ubtrate and emiconductor material [Milcent et al., 1998; Ikari et al., 003], the centre of the heating and detection pot are hifted againt each other by a ditance of d HD. The heating and the detection pot may have the different radii r H and r D. Figure 5.1: Schematic of the -layer ample and of the experimental configuration, baed on a finite diplacement d HD between the heating and the detection pot Theoretical background According to Figure 5., the maximal angle under which the upper or the lower border of the detector area of radiu r D can capture the modulated IR radiation induced by a thermal wave at the ample urface, localizable by it variable poition r from the center of the heating pot, i decribed by [ r ] BD r / r in θ max = 5.3 The term at the right hand ide of equ.5.3 can be determined by uing the Pythagore theorem of geometric analyi ee Fig. 5.: 83

94 5.3 Diplacement between heating and detection pot AD r = AB r + BD r 5.4a DC r = DB r + BC r 5.4b From equation 5.4, the following relation are extracted: AB r DB r r + d HD rd / d HD = 5.5a r + d r / 4d = r HD D HD 5.5b By introducing equ.5.5b into equ.5.3 one obtain, [ r ] = 1 d + r r / r d in θ 5.6 max HD D 4 HD and by conidering a urface element ee Fig. 5. on the upper hemiphere of the detection pot area, d = r dθ dr, and taking into account the contribution of the lower hemiphere to the detection hence the numerical factor to account for ymmetry, the ignal meaured over the detection pot area, e.g. by modulated IR radiometry, i then decribed by: δ M * 3 f, T, t = 4C f γ T σ SBT ε T dhd+ rd θmax dθ δt = r,z 0, f,t rdr dhd rd 0 5.7a Evaluating the angular integral in equ. 5.7a by taking into account the expreion of the poition-dependent angular upper bound given by 5.6, the modulated IR ignal take the following form: δm d * 3 f, T, t = 4C f γ T σ T ε T d HD HD + r D r D SB r dr arc in[ 1 d + r r /4r d ] δ T r, z = 0, t 5.7b HD D HD with the thermal wave [comp. equ.3.4 to equ.3.44, and ee alo equ.3.34, Chap. 3] given by δ T η Io r,0, t = d J0 exp rh /8 πk 0 1+ Rb, f exp d exp iπft 1 Rb, f exp d 5.8 at the variable poition r inide the detection pot area. One can remark, that the thermal wave carrie the characteritic of the excitation ource radiu of the heating pot, r H while the 84

95 5.3 Diplacement between heating and detection pot reulting ignal carrie in addition the characteritic of the detection ytem radiu of the detection pot, r D, and diplacement ditance between the heating and the detection pot, d HD. Figure 5.: Top view of the ample howing diplacement arrangement between excitation and detection pot Simulation of controlled diplacement between the two pot In Figure 5.3 the effect of defocuing, repectively controlled diplacement are imulated for a heating pot radiu of r H = 000 µm and a detection pot radiu of r D = 800 µm, uing increaing diplacement ditance between the centre of the two pot. The cae of well focued detection d HD = 0, which i in good agreement with the normalized phae meaured for a hard coating on a metallic alloy of high effuivity e b = W 1/ m - K -1 and thermal diffuivity α b = 10-6 m -1 i compared with theoretical approximation baed on d HD = {400 µm, 3000 µm, 3400 µm}. With increaing diplacement ditance d HD larger deviation of the phae are found at the very low frequencie, f /Hz 1/ 3, with the relative phae maximum reaching poitive value. 85

96 5.3 Diplacement between heating and detection pot d HD Φ n / deg f / Hz 1/ Figure 5.3: Effect of controlled diplacement on the normalized phae, imulated for contant heating and detection pot radii r H = 000 µm, r D = 800 µm. The diplacement ditance varie from bottom-to-top by d HD = {0 µm, 400 µm 3000 µm, 3400 µm}. The experimental data have been meaured for a hard coating on a metallic alloy of high effuivity at d HD = 0 µm d HD Φ n / deg α b = m / f / Hz 1/ Figure 5.4a: Relative maxima of the normalized phae calculated in the range of low modulation frequencie for different diplacement ditance d HD = {400 µm, 800 µm, 3000 µm, 300 µm, 3400 µm} and two thermal diffuivity value α b. r H = 000 µm, r D = 800 µm. 86

97 5.3 Diplacement between heating and detection pot Localization of heat ource Uing heating and detection pot of fixed ize, imulation are hown in Figure 5.4a and Figure 5.4b for a two-layer ytem with different thermal diffuivity value of the uburface material, α b. Conidering contant thermal diffuivity α b, the increaing diplacement ditance generate a et of relative phae maxima, which increae with the diplacement ditance and are found at nearly the ame modulation frequency. For a value of α b = 10-6 m / the relative phae maxima are found in Fig. 5.4a at about f /Hz 1/ 3 and for a higher value of the thermal diffuivity of the ubtrate, α b = m /, the relative phae maxima hift to higher modulation frequencie, f /Hz 1/ 5. Uing contant thermal diffuivity value of the ubtrate material in thi cae, α b = 10-6 m /, but maller heating pot radii, e.g. r H = 1000 µm in Fig. 5.4b, the relative phae maxima alo hift to higher modulation frequencie, f /Hz 1/ 6. By collecting and repreenting the data of the different relative phae maxima a a function of the poition of the detection pot, it become poible to localize the heat ource. A one can ee in Figure 5.5, which how the relative phae maxima a a function of the correponding diplacement ditance, extrapolation onto the heating pot i nearly independent of the thermal diffuivity of the uburface material. Practically, the reult of thee imulation indicate that when three or four meaurement are performed in the neighborhood of a heat ource the hot pot can be localized with good preciion d HD Φ n / deg α b = 10-6 m / f / Hz 1/ Figure 5.4b: Relative maxima of the normalized phae for different diplacement ditance from bottom-to-top d HD = {1400 µm, 1500 µm, 1600 µm 1700 µm} with the heating and detection pot radii given by r H = 1000 µm and r D = 400 µm, repectively, and contant thermal diffuivity of the uburface material, α b = 10-6 m /. 87

98 5.3 Diplacement between heating and detection pot Φ n max / deg Diplacement ditance / µm Figure 5.5: Phae maxima a a function of the diplacement ditance between detection pot and heating pot. The data point o, refer to the relative phae maxima found in Fig. 5.4a for different value of the thermal diffuivity α b of the ubtrate. The point refer to the relative phae maxima of Fig. 5.4b, found for maller radii of the heating and detection pot Comparion of experimental and theoretical reult Figure 5.6 and Figure 5.7 how repectively the phae and amplitude of the modulated IR ignal meaured for a diamond-like carbon on high effuivity metallic alloy and calibrated with the ignal meaured for a homogenou material glay carbon. The experimental meaurement are compared with the theoretical approximation. Globally, the experimental phae and amplitude reproduce the behavior of theoretical prediction. To perform thee meaurement, a gauian haped modulated laer beam Argon ion laer of radiu r H 000 µm heating pot wa ued for the excitation of thermal wave and detection of the thermal repone wa performed with the help of a MCT detector having a detection pot area of = mm. A for the meauring method, the detection pot wa maintained at the ame poition while the heating pot wa canned over the ample urface. Starting from a concentric configuration between the heating pot and detection pot, diplacement ditance of d HD = 1000, 000, 500 and 3000 µm were then conidered. However, correlation between experimental and theoretical approximation lead to the following data: d HD = 1000, 1900, 300 and 800 µm. The dicrepancie between theoretical and experimental value are not large, and it may be poible that ome unavoidable reading error occurred during experiment and o the experimental ditance could have been magnified. 88

99 5.3 Diplacement between heating and detection pot A can be een in Figure 5.6 and in agreement with the reult preented in Figure 5.3, both the experimental and theoretical relative phae maximum recorded in the limit of very low modulation frequencie, increae with increaing diplacement ditance. In the ame time, one can clearly ee that with increaing diplacement ditance between the two pot, the relative phae maximum hift to lower modulation frequencie. Thi experimental reult i in conformity with the theoretical prediction conigned in Figure 5.4a which attet, that the relative phae maximum hift to lower modulation frequencie with decreaing thermal diffuivity of the uburface material and hift to higher modulation frequencie with increaing thermal diffuivity of the uburface material. The reult of Figure 5.6 indicate therefore a local variation of the ubtrate thermal diffuivity. Correlation between experiment and theory yield different value of the ubtrate thermal diffuivity a a function of the diplacement ditance between the heating pot and the detection pot, e.g., for the ditance d HD = 0 concentric heating and detection pot α b 0 = m / and for d HD = 800 µm, α b 800 = m /. The calculated value of α b a a function of the diplacement ditance d HD are reported in Table 5.1. The modulation frequency at which the relative phae maximum occur i alo mentioned a a function of the diplacement ditance. Thee numerical value provide the proof, that the lateral inhomogeneitie of the uburface material can be pointed out when either the detection pot or the heating pot i canned over the ample urface Φ n /deg f/hz 1/ Figure 5.6: Normalized phae ymbol of the modulated IR ignal, meaured for a diamond-like carbon on high effuivity metallic alloy, in comparion with theoretical approximation. The diplacement ditance between the heating and the detection pot repectively of radii r H = 000 µm and r D = 800 µm varie from bottom-to-top by d HD = {0 µm, 1000 µm, 1900 µm, 300 µm, 800 µm}. 89

100 5.3 Diplacement between heating and detection pot d HD /µm f max /Hz 1/ α b 10-6 /m Table 5.1: Lateral variation of the ubtrate thermal diffuivity with the diplacement between the heating pot and the detection pot A careful obervation of Figure 5.6 in the range of high modulation frequencie 60 < f /Hz 1/ allow to etablih, that there are alo ome enitive variation of the calibrated meaured phae a the diplacement ditance between the two pot varie. Since the poition of the meaured relative phae minimum, localizable by it modulation frequency in the range 60 < f min /Hz 1/ < 80 i relatively table o that no change in the thickne of the thin film depoited on the ubtrate i of concern, thee variation of the meaured phae at mall penetration depth can be attributed to local variation of the coating thermal diffuivity a the heating pot i canned over the ample urface. Following thi, it i jutified to conclude, that by performing modulated excitation of the ample urface at one point and then detecting the thermal repone in the neighborhood, more reliable information about the thermal propertie of the invetigated ample can be eaily obtained. In Figure 5.6, the diagreement oberved between theory and experiment at modulation frequencie in the range 1 < f /Hz 1/ < and 5 < f /Hz 1/ < 30 can be explained by the internal tructuring of the uburface material. In Figure 5.7 the calibrated meaured amplitude are preented, which how, that although the deviation of the ignal i more pronounced in the limit of very low modulation frequencie, there exit a variation of the invere ignal in the entire frequency range with increaing diplacement ditance. Such a behavior can be explained by the fact, that on the contrary to the ignal phae the ignal amplitude i proportional to the optical aborptivity or photothermal converion efficiency η which i defined a the fraction of the laer intenity tranformed into heat and thu any variation of η induce necearily variation of the ignal amplitude. A correlation between experimental and theoretical invere ignal amplitude give in the cae of concentric heating and detection pot d HD = 0, an optical aborptivity of about η 0 = 0.37 while at d HD = 300 µm, η 300 = 0.4. Table 5. preent the value recorded for the photothermal converion efficiency η a a function of the diplacement ditance. A can be een in thi table, the optical aborptivity experience a random variation a the heating pot i canned over the ample urface, e.g., at d HD = 1900 µm, η 1900 = 0.43 and at d HD = 800 µm, η 800 = Thee reult are quite comprehenible ince the meaured ignal were obtained by canning the heating pot, intead of the detection pot, over the ample urface. Such a configuration alo provide the poibility to obtain more reliable information about the lateral inhomogeneitie of the optical aborptivity. Finally, by exciting the ample at one location of the ample urface and then detecting in the neighborhood, it i poible to obtain complete information about the local variation of the thermal and optical propertie of the ample. 90

101 5.4 Scaling of thermal localization of hot pot S n f/hz -1/ Figure 5.7: Invere normalized amplitude ymbol of the modulated IR ignal, meaured for a diamond-like carbon on high metallic alloy, in comparion with theoretical approximation. The diplacement ditance between the heating and the detection pot repectively of radii r H = 000 µm and r D = 800 µm varie from top-to-bottom d HD = {0 µm, 1000 µm, 1900 µm, 300 µm}. d HD /µm η Table 5.: Lateral variation of the photothermal converion efficiency with the diplacement between the heating and the detection pot. 5.4 Scaling of thermal localization of hot pot Starting from the reult obtained for diplacement ditance in the millimeter range, further imulation are performed while decreaing gradually the value of the heating and detection pot radii a well a the diplacement ditance between the center of the two pot. 91

102 5.4 Scaling of thermal localization of hot pot Furthermore, the ize of the ample i adapted to more realitic condition of thermal microcopy, with a thin film of a thickne of d = 0.8 µm at the urface and a ubtrate thickne of d b = 500 µm. In thi cae the thermal wave olution [comp. equ.5.8] i decribed by δ T η Io r,0, t = d J0 exp rh /8 πk 0 1+ R 1 R b b, f [exp d, f [exp d + exp d b exp d b b b ] + exp d ] exp d d b d b b b exp iπft 5.9 and the quantity b of equ.5.9 i given by b σ b = r D =800 µm Φ n / deg f / Hz 1/ Figure 5.8: Normalized phae for detection pot of different ize veru the quare root of the modulation frequency. From left to right, the detection pot decreae, r D = {800 µm, 00 µm, 50 µm, 5µm}. The ratio of diplacement ditance to detection pot radiu, d HD /r D =, and that of heating to detection pot radiu, r H /r D = 0.8, are contant. 9

103 5.4 Scaling of thermal localization of hot pot d HD / r D = Φ n / deg f / Hz 1/ Fig. 5.9: Effect of the diplacement ditance on the normalized phae. The diplacement ditance varie according to d HD /r D = {1.0, 1.5,.0}, while the radiu of the detection pot r D =800 µm and the ratio heating to detection pot radiu, r H /r D = 0.8, are contant r H / r D = ϕ n / deg f /Hz 1/ Fig. 5.10: Effect of the ratio heating to detection pot radiu r H /r D = {0.9, 0.8, 0.7} on the relative phae minima and maxima. The radiu of the detection pot r D = 800 µm and the diplacement ditance d HD /r D = are maintained contant. 93

104 5.5 Comparion with meaurement baed on Thermoreflectance The aim i to find out, at what ditance uch a localization of heat ource by mean of meaurement in the neighborhood i poible. According to the ratio of heating to detection pot radiu r H /r D = 0.8 and according to the ratio of diplacement ditance to detection pot radiu d HD /r D =, the theoretical olution in Figure 5.8 decribe experimental configuration, where the diplacement ditance between the two pot i larger than the two pot radii. Each phae olution i characterized by a large relative minimum followed by a large relative maximum, and with decreaing pot ize and diplacement ditance, the relative minima and maxima hift to higher modulation frequencie, e.g. for a heating pot of r H = 4µm, a detection pot of r D = 5µm, and a diplacement ditance of d HD = 10µm, the relative minimum would be found at rather high frequencie, e.g. at about f / Hz 1/ 400. Figure 5.9 preent the reult of imulation at given contant radii of the detection and heating pot, while the diplacement ditance between heating and detection pot varie ytematically to invetigate it influence on the phae ignal. A can be een, the difference between the relative phae minima and maxima increae coniderably with the diplacement ditance. Such a reult ha already been found in Figure 5.3 and confirmed by experimental meaurement in the millimetre range preented in Figure 5.6. In Figure 5.10, the reult of imulation are preented for a given radiu of the detection pot and a contant diplacement ditance. The radiu of the heating pot i varied ytematically. One can oberve, that the difference between the relative phae minima and phae maxima are rather large and that they enitively vary with the ize of the heating pot. The reult of Figure 5.9 and of Figure 5.10 allow to conclude that the more appropriate configuration in diplacement experiment i reached for larger diplacement ditance between heating and detection pot, and for radii of the heating pot maller than the detection pot. With decreaing pot radii Fig.5.8 and Fig.5.10, the relative phae minima and the phae maxima hift to higher modulation frequencie. 5.5 Comparion with meaurement baed on Thermoreflectance Meaurement baed on modulated thermo-reflectance [Dietzel et al., 003] with an increaing diplacement ditance between the contant excitation and detection pot are hown in Figure According to Dietzel et al., the deviation between meaured amplitude and theoretical approximation Fig. 5.11a at horter ditance, x < 0 µm, can be explained by lateral inhomogeneitie of the optical reflectance and do not affect the quality of the phae meaurement. Such a reult ha alo been found in ection 5.3.4, Figure 5.7. A can be een in Fig. 5.11b the ignal phae meaured at the modulation frequency of 0 khz are in very good agreement with the theoretical approximation up to a diplacement ditance of about 60 µm, and cover a detection range a calculated in Figure 5.6. According to the ratio 94

105 5.6 Concluion d HD /r D = Fig. 5.8, a diplacement ditance of about 60 µm at the modulation frequency of 0 khz correpond to a detection pot radiu in the range between 5 µm and 50 µm. Fig. 5.11: Amplitude a and phae b of the modulated optical reflectance meaured for a Cu-C interface ytem thin film of Cu of 0.8 µm on glay carbon with an additional thermal contact reitance between Cu film and C ubtrate at 0 khz a a function of diplacement ditance between heating and detection pot. In thee meaurement, the heating pot ha been canned over the ample urface, while detection pot and ample remained at a contant poition. Thi configuration i epecially enitive to local inhomogeneitie of the optical aborptivity. 5.6 Concluion Thermal wave imulation conidering controlled diplacement ditance between heating and detection pot have hown the potential of the method in detecting and identifying heat ource by meaurement in the neighborhood. Owing to the fact that only the thermal wave phae retardation ha mainly been conidered, the reult preented here hould apply to thermal microcopy, independent of the applied detection technique. The firt experimental meaurement performed in the millimeter range uing the modulated IR radiometric technique Fig.5.6 and Fig.5.7 have confirmed the reliability of the propoed photothermal method which provide better information on the lateral inhomogeneitie of the 95

106 5.6 Concluion thermal tranport propertie. On the other hand, correlation between meaured and theoretical ignal amplitude have alo revealed the lateral variation of the photothermal converion efficiency. In particular we have demontrated, that the relative maximum occuring in the limit of very low modulation frequencie on the calibrated phae and which ha been found to be a very ueful tool for the determination of the effective thermal diffuivity of the ample comp. Chap 3, can alo help to identify the heat ource in the tructure. Detecting and identifying the heat ource hot pot in micro-tructured electronic device remain one of the great challenge in thermal management ince the experimental technique find their meauring capabilitie limited with the dratic reduction of ize of device. Thi i why numerical method are more and more required to help predicting and undertanding for example the mechanim of thermal tranport which can occur in o tiny device. In chapter 7, we will ue the finite element method to calculate the temperature ocillation in everal micro-tructured device a well a the thermo-elatic diplacement of their urface and then compare the obtained reult with ome available experimental meaurement baed on canning thermal microcopy STM and canning thermo-elatic microcopy SThEM. However, going from the fact that the imulation of thermal wave problem i omewhat complex, chapter 6 i enrolled to how by mean of tip and example how thee type of cientific problem hould be efficiently handled with the ANSYS-aided finite element technique. 96

107 6. Efficient Simulation of Thermal Wave Problem with the ANSYS-aided Finite Element Technique 6.1 Introduction It i well known, that a numerical method might not reproduce the true olution if the cheme i not properly deigned. In general, numerical method are governed by many criteria including conitency, tability and convergence [Iaacon and Keller, 1966]. For example, in a numerical cheme baed on the Finite Difference Method FDM, the conitency i obtained when the global truncation error due to the approximation of derivative by difference varnihe, and the tability i oberved when the numerical error generated during the imulation are not magnified but rather minimized. A conitent and table numerical cheme i convergent. All thee criteria alo apply to the Finite Element Method FEM [Bathe, 1974]. Since the imulation of thermal wave problem i omewhat complex, the preent chapter i enrolled to how by mean of tip and example how uch type of problem hould be efficiently handled with the Finite Element technique. Another purpoe i to check the reliability of the numerical cheme that will be ued in the next chapter, by comparing the reult of the ANSYS-aided finite element imulation with ome ignificant theoretical reult, already preented in chapter 3 and in chapter 5 and applied to experimental meaurement. 6. Finite Element concept 6..1 Theoretical foundation The Finite Element Method FEM i a numerical technique for olving problem which are decribed by partial differential equation or can be formulated a functional minimization. If the phyical formulation of the problem i decribed a differential equation, then the olution method i the Weighted Reidual Method WRM [Ame, 1977]. If the phyical problem can be formulated a functional minimization, then the Variational Formulation VF i required [Norrie and de Vrie, 1973]. For example, the total potential 97

108 6. Finite Element concept energy of a ytem i minimal at equilibrium. Here, the potential energy contitute the functional. Conidering the variational formulation, the extremum condition impoed on the functional, e.g., the potential energy E p, { T } { T } E p = yield a ytem of equation from which the unknown primary variable at node are derived. The element equation can be aembled a follow: [ B ]{ T } = { q } 6. In equ.6., [ B ] i the matrix compriing the material propertie, the geometrical data, etc, { T } i for example the vector of nodal temperature, and { q } i the load vector. The nodal temperature are retrieved by numerical inverion of equ.6.: 1 { T } = [ B ] { q } 6.3 A for the weighted reidual method, one conider e.g. the following partial differential equation T LT t = 0 x D, t >0 6.4 where the unknown and exact olution, T x, t, i approximated by a trial olution T N u x, t = N x T t 6.5 j= 1 j j In equ.6.4, L denote a differential operator in the pace derivative of T, x i a vector of pace variable, and D i a pace domain. In equ.6.5, the N x are known baic function or hape function, and the T j t are the time-dependent! contant. By inerting the trial function 6.5 into equ.6.4, a reidual ubit, j R u t T ut = LuT 6.6 which can be reduced to Zero if the trial olution coincide with the exact olution. By chooing the optimal contant, T j t, the different reidual are Zero in ome average ene. 98

109 6.3 Finite Element Modeling Thi objective i reached by electing N weighting function W j, j = 1, N, then introducing the patial average or inner product A, B AB dv = V 6.7 and etting the weighted integral of the equation reidual to Zero. W R u = 0, j = 1, N j, 6.8 T Equation 6.8 repreent a et of equation for the T j t. If the contant T j are effectively time-dependent, then the equation are ordinary differential equation. Otherwie, the N equation are algebraic. In the Galerkin approach, the weighted function are the ame hape function which have been ued to define the trial olution 6.5 [Amè, 1977]. 6.. Fundamental tep In the Finite Element Method, ix fundamental tep can be mentioned: Step 1: The domain of interet i dicretized into imple geometric hape or element. Step : The individual element equation are developed by uing the weighted reidual method or the variational principle. Step 3: The generated element equation are aembled in a matrix form. Step 4: The boundary condition are impoed on the entire ytem, and thu modify the global equation. Step 5: The modified global equation are then olved for the primary unknown e.g. nodal temperature. Step 6: Other deired econdary variable are calculated by uing nodal value of the primary variable e.g., thermal gradient and thermal fluxe from nodal temperature. 6.3 Finite Element Modeling In order to improve the comprehenion and the ue of the Finite Element technique, a large variety of commercial oftware package are available e.g. ADINA, ANSYS, MARK, LARSTRAN, etc. The ANSYS oftware ha been ued in the frame of thi work. Three main component product derived from ANSYS/MULTIPHYSICS A/M are worth to be mentioned: A/M/MECHANICAL tructural and thermal capabilitie, A/M/Emag electromagnetic capabilitie, A/M/FLOTRAN Computational Fluid Dynamic capabilitie. 99

110 6.3 Finite Element Modeling The ued verion of ANSYS/MULTIPHYSICS, namely ANSYS 5.7 and ANSYS 6.1, differ with each other by the organization of their interface but are ubjected to the ame limitation concerning the maximum number of node that can be generated through the mehing. In general, all FEM oftware package are built up around three tandard phae Pre-proceing It i the mot important tep in the elaboration of the finite element calculation. Thi phae i aimed at preparing the conidered model by defining the geometry, the thermal and phyical propertie, the element type, and finally by performing the mehing. Mehing the model conit of ubdividing the conidered geometry into element connected at node: The phyical ytem i aid to be tranformed into a finite element model. Then, the boundary condition are applied to the model or directed impoed on the node and element. A poor choice of the element type a well a a poor mehing of the model lead unavoidably to coniderable inadequacie between the model and the actual phyical ytem. The preproceing phae involve the tep 1 and 4 defined above in ection Computation In thi phae, the et of equation generated by the mehed model i olved. The reult conit of a large et of numerical value repreenting the primary variable e.g., nodal temperature. To check the reliability of the numerical olution, a quantitative and qualitative aement i highly recommended. The computation phae regroup the tep, 3 and 5 defined above Pot-proceing In thi phae, the primary variable calculated in the computation phae are collected and ued for further proceing. For thi, ANSYS offer two type of pot-proceor: The general pot-proceor or POST1 allow to lit the reult of the imulation at a given time or frequency over the model while the time-hitory pot-proceor or POST6 allow to review the reult over time in a tranient analyi or over frequency in a harmonic analyi, at a particular location of the model. In thermal analyi, the pot-proceing conit of determining the thermal gradient, the thermal fluxe and of evaluating the heat loe or gain in the imulated model, from the calculated nodal temperature. In particular for thermal wave problem, the attention i focued on the imulated amplitude and phae of the temperature ocillation, which in their 100

111 6.4 Efficient imulation of thermal wave problem turn are exploited to ae the amplitude and phae of the photothermal ignal e.g., IR radiometry, Thermoreflectance, Thermoelatic deformation. The pot-proceing phae refer to tep 6 of ection 6... However, one hould keep in mind that uing a oftware package to perform thee numerical imulation doe not automatically prevent the olver from obtaining upiciou reult. Thi i why it i alway of great interet to check the convergence of the numerical olution. For thermal wave problem epecially, we make ome propoition which can help to obtain more accurate numerical reult. 6.4 Efficient imulation of thermal wave problem Convergence of the numerical olution Non-uniform mehing and meh refinement Mot often, all reult of the numerical imulation are not neceary. There are particular region of the imulated model where information i needed and therefore a meh refinement i required at thee location to generate more accurate reult. For example, in a imple thermal wave problem where the frequency-dependent amplitude and the frequencydependent phae of the urface temperature are of interet, the model mut be non-uniformly mehed from the bottom to the top of the ample with more meh refinement beneath the ample urface. Non-uniform mehing i ueful for two reaon: i It allow to avoid the computation of too much data which are uele. ii It contribute to peed the imulation by minimizing the number of node involved in the differential equation. Concretely, tarting from the fact that the diffuion length of the thermal wave, µ = α / πf, i proportional to the patial coordinate, e.g. z, in the main direction of thermal wave propagation, and that it would be aburd to have a meh ize larger than the diffuion length, the meh denity mut be relatively high at large penetration depth correponding to low modulation frequencie, and very mall at low penetration depth correponding to high modulation frequencie. The meh refinement i aid optimal when there are no or little change in the olution. In thi cae the tability of the numerical olution i etablihed and the convergence criteria are fulfilled Control of convergence with the reference phae hift In addition to the principle of meh refinement, we propoe another powerful tool to control the convergence of the numerical olution. Thi method i pecific for thermal wave problem and baed on the phae hift of the thermal wave at the urface of a homogeneou 101

112 6.4 Efficient imulation of thermal wave problem material, according to the theory of 1-D thermal wave propagation in olid. The value of thi phae hift reference phae hift i given by Φ ref = -45 or tanφ ref = -1. Thu, to imulate the propagation of thermal wave in a layered tructure, it i advied to affect the pooret thermal propertie all belonging to one of the layer! to all other layer of the ytem in order to fulfill the requirement of a homogenou ample. In the conidered interval of modulation frequencie, the numerical data recorded for the phae hift of the thermal wave at the ample urface mut be equal or at leat very cloe to the reference value Φ ref = -45. If it i not the cae, then the mehing i poor and a further meh refinement i neceary until thi value i reached. A can be een in Figure 6.1a, which how the frequency-dependent phae of a Sibaed homogenou ample, imulated by the FEM, the reference value Φ ref = -45 i reached in the interval of modulation frequencie, 1.00 Hz < f < 100 khz. Thi i the proof, that the mehing ha been very well performed and that the convergence criteria are fulfilled ince the numerical and the analytical olution are in very good agreement. To obtain thi reult, a value of d = 6000 µm ha been affected to the ample thickne. The mehed model i preented in Figure 6.1b, where a meh refinement i obervable near the top of the ample Φ ref [deg] f / Hz 1/ Figure 6.1a: Phae hift thermal wave at the urface of a mooth and homogenou olid, imulated by finite element: Figure 6.1b: Portion of the model howing a non-uniform mehing from the bottom to the top, with meh refinement near the urface According to Fig. 6.1b, the lateral meh ize i not of great importance ince the thermal wave propagate excluively in the vertical direction due application of an uniform thermal load on top of the ample e.g. excitation of the entire urface by an intenity- 10

113 6.4 Efficient imulation of thermal wave problem modulated laer beam. On the contrary, the vertical meh denity i non-uniform, going from large value at large penetration depth low modulation frequencie to very mall value at mall penetration depth high modulation frequencie. To obtain the above reult, a ratio of Rt = 500 between the larget and the mallet meh ize ha been conidered. However uch a reult i not hazardou. The modeller hould be aware of the fact that the value Φ ref = -45 i typical for a model regarded a emi-infinite. In Figure 6.a, the effect of ample thickne on the phae hift i invetigated. With decreaing value of the ample thickne, e.g. d = 500 µm, the phae hift deviate from Φ ref = -45 to Φ ref = 90 in the limit of very low modulation frequencie or at very large penetration depth while no change i recorded at the intermediate and high modulation frequencie d = 500 µm α = m / Φ ref [deg] Φ ref [deg] f / Hz 1/ f / Hz 1/ Figure 6.a: Phae hift of thermal wave at the urface of a mooth and homogenou olid, imulated by the FEM a a function of the modulation frequency for different value of the thickne. Thermal diffuivity α = m / Figure 6.b: Phae hift of thermal wave at the urface of a homogenou olid, imulated by the FEM a a function of the modulation frequency for different value of the thermal diffuivity. Sample thickne: d = 500 µm In Figure 6.b, the ample thickne i maintained at d = 500 µm and the thermal diffuivity α = m / for GaA material, and α = m / for SiO material i varied ytematically. One can remark, that with maller value of the thermal diffuivity e.g., α = m /, the phae hift recover from the deviation oberved in Figure 6.a for the thickne d = 500 µm and the thermal diffuivity α = m /. Thee deviation of the phae hift from the value Φ ref = -45 a the ample thickne decreae or a the thermal diffuivity of the material varie can be better undertood if one recall the analytical expreion of the complex modulated urface temperature for a homogenou ample, 103

114 6.4 Efficient imulation of thermal wave problem η I exp iπ / 4 1 exp[ 1 i d πf / α ] o + + δt f, t = 6.9 e πf 1 exp[ 1 + i d πf / α ] In equ.6.9, a o = d π f / α repreent the thermal thickne, which i a meaure of the ability of the thermal wave to propagate throughout the entire ample. Table 6.1 indicate the value calculated for the thermal thickne, at the modulation frequency of f = 1.9 Hz and for the thermal diffuivity of α = m / Si-material. A can be een in thi table, the ample i thermally thick for a thickne of d = 6000 µm. That mean, the thermal diffuion length, calculated according to µ = α / πf and which ha the value 3900 µm, i maller than the ample thickne. In thi cae, the thermal wave can be completely damped before reaching the rear face of the ample and therefore the aumption of emi-infinite ample i jutified [Bein et al., 1989]. For a maller ample thickne, e.g. d = 500 µm, the obtained value for the thermal thickne indicate a very thermally thin ample ince the thermal diffuion length i rather larger than the ample thickne. In thi cae, the thermal wave can eaily reach the rear face of the ample before been completely damped. Once the convergence criteria are fulfilled, the actual thermal and phyical propertie of each layer of the modelled tructure are reinerted into the program to perform the deired numerical imulation. Sample thickne: d /µm Thermal thickne: a o Remark Thermally thick Thermally thin Very thermally thin Table 6.1: Evaluation of the thermal thickne at a given frequency for different value of the ample thickne Methodology of imulation Thermal filtering and phyical unit The firt thing to do before tarting to build up a program i to filter the thermal entrie from other entrie e.g., tructural, electromagnetic and fluid menu topic propoed by the oftware. Thi tak i achieved by turning on the pecified dialog box of the Graphical Uer Interface GUI. Such a filtering enable for example the dicrimination of thermal problem from tructural problem. On the other hand, it hould be noted, that ANSYS doe not pecify the unit for any variable and o it i at the charge of the modeller to enure that the unit of the variable are conitent. A good tip conit of firt defining the unit of the primary variable o that the unit of other variable are automatically identified in the conidered ytem of unit. For 104

115 6.4 Efficient imulation of thermal wave problem example, by defining the unit of the thermal conductivity k, the ma denity ρ, and the pecific heat capacity c, the unit of the thermal diffuivity, α α = k/ρ.c, and the unit of the thermal effuivity e e= ρ c k, are automatically defined. If for example the International Sytem of Unit SI i preferred, then one hould uually write the following command at the beginning of the program: /Unit, SI Parameter, material propertie and element type A good organization of the numerical work conit of defining a group of parameter in order to avoid writing the ame phyical quantitie everal time along the program. Although parameter can be defined anywhere, the readability of the program i more facilitated by entering all neceary parameter jut before the prep-proceing phae PREP7. Parameter are entered in the following way: *SET, I o, 5. Thi mean, the value 5 i affected to the parameter I o, which will be later on recalled at one or many location in the program a intenity of the heat ource. The material propertie are alo entered in the program by uing appropriate code. For example, mp, k xx, 1, k 1, mean that the material property mp i the thermal conductivity k xx of the material numbered 1, whoe value i given by k 1 uppoed to have been defined earlier a parameter. Selecting the appropriate element type ecure the olver from obtaining poor reult. Each element type i identified by a category name, e.g. D olid element have the category name PLANE. PLANE 4 i a four-node quadrilateral element with each node having two degree of freedom [Moaveni, 1999] Geometry definition, mehing The geometry of the model and that of the real phyical domain can be imilar. In uch a cae, there are no obervable centre or axe of ymmetry in the phyical domain and o all geometrical parameter of the domain are tranferred to the model. If the phyical domain preent ome point or axe of ymmetry, then the model will be the capture of a part of the domain, generally the half part. Then the mehed model will be imulated and the obtained reult will be interpreted while taking into conideration the entire phyical domain. Example are hown in Figure 6.5 and in Figure 6.8. The exitence or not of ymmetrie depend alo on the way the load i applied to the phyical domain. In order to meh the model, the ANSYS oftware offer two poibilitie to the olver: i Free mehing: Yield a random ditribution of element in the model. In thi cae, element with different morphologie can cohabit, e.g., when mehing the model with the element type PLANE 75 both triangular three-node and tetrahedral four-node element are generated. 105

116 6.5 Finite Element control of theoretical reult ii Mapped mehing: Yield a regular ditribution of element in the model. In thi cae, the morphology of the element i uniform, e.g., when mehing the model with the element type PLANE 75, either triangular or excluively tetrahedral element are generated. We have eentially ued the mapped mehing ince it allowed u to eaily control the meh ize and to correctly define the input of the heat generation rate a well a the vertical thermal expanion Solution and interpretation The olution phae begin with the decription of the analyi method. Modulated thermal wave problem mut be olved by pecifying the harmonic analyi. Thi i done by iuing the following command: ANTYPE, HARMIC analyi type, harmonic. Thi intruction i directly followed by the command, HROPT, method,, which pecifie the option. The full method i generally choen a option. Then, another command, HROUT, key,, i written which indicate the output printing option. Here, two poibilitie are offered: If the complex modulated temperature are to be printed a real and imaginary component, then the key ON i choen. If the output reult mut be printed a amplitude and phae angle, then the key OFF i uited. The deired harmonic frequency range i etablihed by iuing the command, harfrq, f min, f max and the number of ub-tep between the minimal and the maximal frequencie i defined by writing, nubt, n max. In the cae of an unique harmonic frequency, e.g. for the imulation of the amplitude and phae of the thermo-elatic ignal, f min = f max and n max = 1. After having calculated the primary variable in the olution phae, namely the real and the imaginary component or the amplitude and phae of the nodal temperature temperature ocillation, the cloe tak i the calculation and collection of the econdary variable in thi work: amplitude and phae of the thermo-elatic diplacement. 6.5 Finite Element control of theoretical reult In order to check the reliability of the finite element cheme which conit of uing the reference phae hift to control the convergence of the numerical olution, we compare the reult of FE imulation with ome theoretical reult, which have been found to be in good agreement with the experimental meaurement cf. Chap. 3 and Chap. 5, and which are baed on the 3-D thermal wave propagation in opaque two-layer ytem where concentric heating and detection pot are firt conidered followed by a diplacement ditance between the two pot. 106

117 6.5 Finite Element control of theoretical reult D thermal wave propagation Figure 6.3 how the frequency-dependent normalized amplitude a and phae b of the thermal wave at the top urface of a two two-layer ytem. One can oberve in Figure 6.3a, that the invere normalized ignal amplitude increae the ignal decreae! in the range of low modulation frequencie with increaing value of the thermal diffuivity of the uburface material. In Figure 6.3b, the invere normalized phae experience a relative maximum in the limit of very low modulation frequencie, which varnihe for relative mall value of the thermal diffuivity and adopt the general behaviour oberved in the cae of 1-D thermal propagation compare, Fig. 3.13a, Chap. 3. Such a reult ha already been found and applied to experimental meaurement compare, Fig. 3.15, Chap. 3. One can oberve, that the analytical and the numerical reult are in good agreement. To obtain uch an agreement between analytical and numerical olution, the model wa mehed with the element type PLANE 75 4-node quadrilateral element, axiymmetry, harmonic analyi. In the direction of high thermal gradient, along the depth, a very mall meh denity wa ued in the top layer 5 5 S -1 n olid line: Theory ymbol: FE α b = m -1 6 Φ n / deg olid line: Theory ymbol: FE α b = m f / Hz -1/ f / Hz 1/ -40 Figure 6.3 a + b: from left to right Normalized amplitude a and phae b of the thermal wave at the top urface of a two two-layer ytem, according to 3-D thermal wave propagation, for different value of the thermal diffuivity of the uburface material. There i a very good agreement between the reult of FE imulation ymbol and the analytical prediction olid line. coating and the meh ize wa progreively enlarged from the top to the bottom of the ubtrate. Firt, by conidering a laer pot very large at the top urface of of the ample to 107

118 6.5 Finite Element control of theoretical reult enure 1D heat propagation and by affecting the ame thermal propertie to the two layer of the model, value of the phae hift very cloe to Φ ref = 45 were obtained in the entire frequency range, atteting therefore that the meh refinement i optimal and that the convergence of the numerical olution i etablihed. Thu, change in the numerical olution were no more oberved when the meh ize along the depth in the urface layer reached the value 10-7 and the ratio of the larget to the mallet meh ize in the ubtrate wa adjuted to 500. Then conidering the realitic thermal load heating of the top urface of the ample by a Gauian-haped laer beam of radiu r H, time modulated at the frequency f, the real and imaginary component of the nodal temperature along the ample urface ee the mehing in Figure 6.5b were collected a function of the modulation frequency, and tranferred into the graphical commercial program ORIGIN 6.0 for further proceing. The real and imaginary component of the modulated IR ignal were then obtained with the help of a numerical integration [Davi and Rabinowitz, 1967]. For thi, the expreion [comp. equ. 3.45, chap.3], r rdr δt r, f = Cont D r D δ M f = Cont dr δu r, f wa decompoed by uing e.g. the trapezoid rule a follow: δ M Num r N = r D r = 0 f = Cont drδu 1 r, f δu r, f = Cont hr + δu r, f + + δu rn rn, 1 δu f 1, f In equ.6.11, h r = r N r 1 /N-1 i the meh ize along the radial lateral axi at the ample urface, thi mean the eparation between two conecutive nodal temperature. N repreent the number of node and δ U r, f = r δt r, f, k = 1 N. The r k are the variable radial k k k poition inide the detection area. Then the ignal amplitude and the ignal phae were obtained by operating on the real and the imaginary component a follow: S Num Num Num [ δ M f ] + [ δm ] f = f 6.1a Re al Im Num Num δm Im f 180 Φ f = atan in degree 6.1b Num δm Re al f π In equ.6.1, the lower ubcript Real and Im refer to the real and the imaginary component, repectively while the upper ubcript mean: numerical value. To eliminate the contant coefficient carried by the modulated ignal a hown e.g. in equ.6.10, the ame 108

119 6.5 Finite Element control of theoretical reult operation were performed to obtain the amplitude and phae of the reference material and the obtained numerical reult were normalized with the reult of the two-layer ample Theory Symbol: FE e b =7000 W 1/ m - K Theory Symbol: FE e b =7000 W 1/ m - K ϕ n / deg -0 ϕ n / deg f / Hz 1/ f / Hz 1/ Figure 6.4 a + b: from left to right Frequency-dependent normalized phae of the thermal wave at the urface of a two two-layer ytem, according to 3-D thermal wave propagation, for different value of the thermal effuivity of the uburface material. Comparion between the reult of FE imulation ymbol and the theoretical reult olid line indicate a good agreement. Fig. 6.5 a: 3D view of the phyical two-layer ytem howing a thin film depoited on a emiinfinite ubtrate. Fig. 6.5 b: Finite Element model howing the repartition of the meh denity in the urface layer and in the ubtrate 109