Theory of photothermal depth profiling in time domain R. Li Voti, G. L. Liakhou, S. Paoloni, E. Scotto, S. Sibilia, and M. Bertolotti Citation: AIP Conference Proceedings 463, 37 (1999); View online: https://doi.org/10.1063/1.58183 View Table of Contents: http://aip.scitation.org/toc/apc/463/1 Published by the American Institute of Physics THEORY OF PHOTOTHERMAL DEPTH PROFILING IN TIME DOMAIN R. Li Voti, G.L. Liakhou*, S. Paoloni, E. Scotto, C. Sibilia and M.Bertolotti INFM - Dipartimento di Energetica, Universith degli Studi di Roma "La Sapienza", Via Scarpa 16, 00161, Roma, ltaly *Technical University of Moldova, Stefan Cel Mare 168, 277012 KTshinau, Moldova In this paper we want to introduce the theory of depth profiling in time domain. We present a new theoretical model for the heat diffusion in media with variable thermal parameters and show the retrieval procedure to reconstruct the thermal effusivity depth profile from the surface temperature .dynamic. The effect of the noise on the quality of the reconstruction is also briefly discussed. INTRODUCTION The theory of photothermal depth profiling to characterize samples with thermal condactivity k(z) and diffnsivity D(z) both functions of depth, has been studied and applied in frequency domain [13]. In a typical experiment a pump laser beam, modulated at the frequency f is absorbed at the sample surface. The temperature rise at the surface as a function o f f contains all information about the conductivity and diffusivity depth profile, which can be successfully reconstructed by following different procedures. Since a clear relationship between frequency domain and time domain exists, an important question comes out: is it possible to develop an analogous theory for photothermal depth profiling in time domain? In other words, if the sample is illuminated by a single-pulsed pump laser beam, does the dynamic of the temperature rise at the surface Ts contain again the useful information to reconstruct the depth profiles k(z) and D(z)? This paper want to introduce the theory of depth profiling in time domain, showing its chief advantages as for example the possibility to get an instantaneous (real time) reconstruction [4]. THEORY The starling point is the solution of the 1D heat Fourier diffusion equation for a sample which exhibits a depth dependence of both diffnsivity and conductivity, and is subjected to an ideal laser pulse absorbed et the surface, at the time t=O 0[k(z)0Vtz")l ---g-3- at ' tl) where pc(z) is the heat capacity defined as pc=k/D. By using the Laplace transform Eq.(1) becomes subjected to the surface condition dT(O,s) _ - Q -k(oi where Q is the heat deposited at the surface during the pulse. Eq.(2) is formally analogous to the 1D diffusion equation in frequency domain which, for slowly varying depth profiles, has been already solved elsewhere [3,5]. The solution for the Laplace transform of the surface temperature is CP463, Photoacoustic and Photothermal Phenomena: l Oth International Conference edited by F. Scudieri and M. Bertolotti © 1999 The American Institute of Physics 1-56396-805-3/99/$15.00 37 Q l+R(o,s) e(O)irs 1- R(O,s) INVERSE PROCEDURE (3) Eqs.(7) and (8) show a direct relationship between T(z=O,t) and e(z) (direct problem) and represent the starting point of the retrieval problem to reconstruct e(z). One simple procedure is given by dividing the sample in a number j of sublayers with constant effusivity ej so to replace the integral in Eq.(7) and (8) with the summation. Since the surface temperature T(z=O,O may be measured only for a set of t~, Eq.(7) becomes an algebraic linear system, unfortunately ill-posed, which may anyway be inverted by using different procedures such as SVD. or Tikhonov method. where e(O) is the surface sample effusivity and R is the thermal refiectivity which must fulfill the Riccati nonlinear differential equation dR dz 2~_~) R l dlr~e(z))(_R2) 1 =0 2 dz (4) If the sample exhibits a slowly varying depth profile [RI ~ 1, and, as a consequence, the nonlinear term in Eq.(4) may be neglect. With this simplification the surface thermal reflectivity assumes the form NUMERICAL RESULTS -2z~s 0 R(O,s): ~-dln(e(z)) ~ 2dz e As an e~mple we report in Fig. 1 the numerical simulations on steel sample where the heat capacity is assumed equal to pc=3.6 J/cm3 K. In such case only one independent thermal parameter exists: let us say the thermal diffusivity. The dynamic of the temperature rise at the surface is reported in Fig. 1 for the four different diffusivity depth profiles shown in Fig2 which describes possible surface hardening processes. Note that when a steel sample is hardened its diffusivity decreases. The differences in the temperature fall between homogeneous samples (c,d), graded hardened steel (b) and step hardened steel (a) which are clear already 100msec after the pulse. (5) where D~v is the average diffusivity inside the material. By combining Eq.(3) with Eq.(5), and by I+R replacing the term - with 1+21L one obtains 1-R T(O,S)=e~sIl-!dl~z(Z))e~D~S~dz 1 (6) which is the Laplace transform of the following surface temperature in time domain [6] I T(O't):e(O)-4~ - "o az e 10 z2] dz. (7) o e~ Note that the term Q/eqrn7 represents the surface temperature in case the sample were homogeneous. Let us consider now the temperature deviation AT from the homogeneous case; this quantity is directly connected to the integrated effusivity profile as follows o 0.1 0.I 0 ~z e dz 1o 100 Time, m s e c FIGURE 1. Numerical simulation for the surface t~nperature rise vs the time after the heating pulse. The different curves refer to the diffusivity profiles in Fi~2: a step hardened steel (a), a graded hardened steel (b) and homogeneous saraples with differ~t diffusivities (c,d). 22 Th - - ~ I (8) 38 3. Li Voti,R, Be~tolotti, M., Sibilia,C., Proc. Advances in Signal 13)' inverting the data in Fig.1 for the graded hardened steel (b) by using SVD procedure, one obtains a reconstructed depth profile (0, Fig.2) which weakly differs by the original one 00). .~ 0.2 ->- 0.15 Processing for Non Destructive Evaluation of Materials, Quebec City (1997). 4. Li Voti, R., Liakhou, G.L, Paoloni, S., Sibilia,C., Bertoletti, M., submilled on J Appl. Phys. 5. Be~tolotti, M., Liakhou., G.L., Li Voti, R., Paoloni, S., Sibilia., C., Violante, V., Thermal conductWity depth profihng of inhomogeneous compounds by photothermal techniques, (c) . Proc. Eurotherm 57 Microscale heat transfe~ (1998). (a) o I (d) 0.l 0.5 1 1.5 Depth, mm FIGURE 2. Diffusivity depth profiles: (a,b,c,d) theoretical profile, (0) recon~mcted profile with SVD Of course the quali.ty of the reconstruction profile is limited by the noise in the surface temperature dynamic. In order to discuss this effect, in Fig.3 we have reconstructed the graded profile (b) of Fig.2, starting from surface temperature dynamics affected by different noise levels. The larger is the noise level the worse is the reconstructed profile. 0.2 ¥. -~ 0.15 • °o~l,oo 0.5 Depth, mm FIGURE 3. Reconstucted diffusivity depth profiles through SVD technique for diffexent Gaussian noise levels: (~) 0.1%, (+) 1%, (A) 5%(0) 10%. REFERENCES 1. Glorieux,C., Thoea,J.,d.Appl.Phys. 80, 6510-15 (1996). 2. Lan, T.T.N, Seidel, U., Walther, H.G., J. Appl. Phys. 77 p.4739 (1995); 39

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