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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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