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Surface roughness influence on photothermal radiometry
Applied Surface Science 193 (2002) 156±166
Surface roughness in¯uence on photothermal radiometry H.G. Walther* Physikalisch-Astron. Fakultat, Institut fuÈr Optik und Quantenelektronik, Friedrich Schiller Universitat Jena, Max-Wien-Platz 1, D-07743 Jena, Germany Received 12 September 2001; accepted 2 April 2002
Abstract Frequency scanned radiometric measurements from rough absorbers are different from smooth samples. This effect has to be considered for accurate data interpretation, for instance, for photothermal depth pro®ling. Based on the equivalent layer model we discussed the effect of roughness induced thermal wave dispersion and derived a formula which correctly describes the observed experimental features: (i) appearance of phase difference extremum; (ii) rise of the relative signal magnitude at increased frequencies. The validity of the theory was successfully demonstrated by comparing calculations with radiometric measurements from rough steel samples. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface roughness; Photothermal radiometry; Thermal wave dispersion; Non-destructive evaluation
1. Introduction During the past decade both photothermal measuring techniques and related theoretical methods for data interpretation were improved decisively. In principle, now it is possible to retrieve depth pro®les of technologically relevant thermal properties from photothermal measurements. Photothermal depth pro®ling became an interesting and valuable tool for non-destructive evaluation. Among others, important applications are the estimation water migration pro®les in living tissues, the characterisation water content in textiles and packaging materials, the estimation of dopant distribution in implanted semiconductors, and ®nally the characterisation of hardness depth pro®les in surface hardened steel. Photothermal applications in industry have to be performed with industrial specimen surfaces of which * Tel.: 49-3641-947030; fax: 49-3641-947032. E-mail address: [email protected] (H.G. Walther).
are not ideal. They can be rough or contaminated. For accurate data interpretation it is necessary to understand the role the technologically processed surface under study plays for photothermal signal generation. To give a demonstration how surface texture can in¯uence photothermal measurements we refer to radiometric signals from differently rough steel samples [1]. The measured phase curves shown in Fig. 1 deviate characteristically from polished specimen. The difference against smooth surface behaviour are obviously connected with the level of roughness. Based on this fact it should be possible to correct photothermal data for surface roughness. This was proposed and for the ®rst time successfully demonstrated in [2]. Already some time ago photothermal signals from rough surfaces were investigated by Bein and coworkers. They performed photothermal measurements from rough graphite samples and was able to explain his results by a decreased subsurface thermal effusivity [3]. Bein and co-workers successfully ®tted the
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 2 ) 0 0 2 5 7 - X
H.G. Walther / Applied Surface Science 193 (2002) 156±166
Fig. 1. Radiometric phase measurements from rough steel samples and polished zirconia used as reference: ( ) reference; (*) polished steel; ( ) slightly rough; () very rough.
experimental date by supposing an effective surface layer with appropriately guessed thermal properties. Later on similar investigations were performed by Walther [4] from rough steel specimen. He also modelled the rough surface by an equivalent layer and explained its modi®ed thermal properties by a parallel connection of randomly distributed thermal resistors corresponding to the local height variations of the surface. This attempt was a step ahead to the quantitative relation between roughness and effective thermal properties. The frequency dependent photothermal signal obeys a simple power law if plane thermal waves are travelling from a smooth surface. Frequency dependent deviations from that behaviour occur for rough surface absorber. It can be qualitatively explained by the fractal dimension df of the rough heat source for which holds df > 2. First attempts to understand photothermal results by fractals were undertaken by Vandembroucq and Boccara [5] and Osiander and Korpuin [6]. A more extended approach to solve the heat diffusion equation in the vicinity of self-af®ne boundaries was published by Vandembroucq and Roux [7]. They used conformal mapping technique to image the rough boundary on a tailored co-ordinate transformations. The same transformation was also applied to heat diffusion equation and to the boundary conditions. A solution was found for constant heat ¯ux and conclusions about the roughness induced distortion of isotherms could be drawn.
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It is the aim of the paper to ®nd out how photothermal signals from rough surface absorbers depend on surface texture parameters. To reach this goal we develop an appropriate theoretical model which is based on effective layer assumption and compare it with measurements from known surfaces. Consequently the article is structured as follows: in Section 2 short mathematical introduction is given to de®ne surface texture parameters. In Section 3 a theoretical model describing signal generation from rough specimen is discussed and applied for numerical calculations in Section 4. Finally, in Section 5 the validity of the proposed model is tested by comparing radiometric measurements from rough specimen with calculations. 2. Mathematical description of rough surfaces Usually the surfaces of technical specimen are rough. Roughness expresses random deviations of the surface from the ideal smooth plane. In practice, surface texture parameters are de®ned by standards and commonly are measured by optical or stylus pro®lometers. From mathematical point of view the statistical approach to roughness parameters is quite instructive. For this reason let us start with the stochastic description of a rough surface whose pro®le is sketched in Fig. 2. z denotes the local height deviating from mean surface. For simplicity we consider only pro®les across the rough surface. The roughness pro®le z(x) is assumed to be generated by a random process with zero mean. Applying Fourier transform to z(x) we obtain ~z k Fz x (k is the spatial frequency). The frequency content of z(x) is evaluated by the power spectral density (PSD), de®ned by
Fig. 2. Pro®le of rough surface. De®nition of surface texture parameters.
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H.G. Walther / Applied Surface Science 193 (2002) 156±166
g k jF Z Xj2 , where g(k) is an important function involving main information on random roughness. For many technical applications two different features of surface texture have to be distinguished: waviness and roughness. Both effects can be separated by appropriate k-frequency-®ltering of the pro®le. PSD of roughness is de®ned by certain cut-off k-frequency. We speak of bandwidth limited PSD which is non-vanishing only between the lower and the upper frequency limits kmin and kmax. Another important surface parameter is random mean squared roughness s which can be derived from g(k) by relation Z kmax s2 2 g k dk: (1) kmin
In practice relevant surface texture is characterised by parameters which are strictly de®ned by standards [8]. Considering only roughness pro®le main parameters are as follows: Rp is the maximum profile deviation from mean plane. In technical literature the length 2Rp is called peak-to-valley distance. Rq denotes random mean squared height deviation q RL Rq 1=L 0 jz2 xj dx.
either along x-axis or z-axis. A self-af®ne surface is characterised by (i) roughness exponent z, (ii) amplitude parameter s and (iii) the bandwidth limitations kmin and kmax of the scaling invariant regime. Sophisticated pro®les with z < 0:5 are so-called anti-persistent pro®les and tend to become complex and rami®ed. Mechanically treated metal and ceramic surfaces have persistent pro®les with z > 0:5. As mentioned in [10] the fractal dimension df of self-af®ne pro®les can be estimated by df 2 z. Of course, the standardised and easy-to-measure parameters Rq and RSm are more convenient for practical use than sophisticated self-af®ne surface parameters. On the other hand, it is possible to move between both pictures. RSm can be approximated by 2p=kmin and Rq can be compared with s. This translation however is crude and not unambiguous. 3. Calculation of the frequency dependent photothermal signal
(2)
The target of this section is to ®nd a quantitative description of the photothermal signal from a rough surface absorber. The experimental situation under consideration is as follows: the sample is illuminated by a modulated heating beam. For simplicity we focus on such photothermal techniques like radiometry or gas cell photoacoustics [9] which give signals proportional to the spatially averaged surface temperature oscillation. As demonstrated in earlier papers [2±4] the measured signal curves can be modelled by an equivalent sample consisting of the homogeneous bulk material which is overcoated by a parallel and surface absorbing layer. The equivalency between the rough and the effective layer sample is sketched in Fig. 3. Non-stationary heat transport is controlled by thermal
within the range of spatial frequencies bound by kmin and kmax. b denotes the lateral stretching factor, which is an arbitrary number. The so-called roughness exponent z 0 < z < 1 controls the vertical structure of rough pro®le and estimates the exponent of power law of PSD according to 1 g k 12B : (3) k From Eq. (2) we see the roughness statistics is invariant to different scaling factors depending on orientation
Fig. 3. Thermal equivalence between rough surface and effective layer model.
RSm is the mean distance between dominant grooves which can be identified as the maximum spatial wavelength of the rough profile. The material distribution across the profile from bulk to surface is called the Abbott±Firestone curve M(z0 ) which is given by the summation over all profile lengths x for which holds z x > z0 . Many technical surfaces can be more appropriately described by self-af®ne pro®les. Such pro®les obey the scaling rule: x ! bx
and
z ! bB z
H.G. Walther / Applied Surface Science 193 (2002) 156±166
diffusivity a and thermal effusivity e which are connected with thermal conductivity k, mass density r and speci®c heat c by the relations a k=rc and p e krc. Obviously the thermal properties of the effective layer differ from the bulk. They are marked by primes (0 ) to distinguish them from the bulk properties. The layer thickness d is assumed to be of the order of the maximum vertical distance between pro®le peaks and valleys. We ask for the ratio S of the surface temperature T 0 of the effective layer-on-substrate system and the smooth half-space temperature TPLANE. In [4] we found that S
T0 TPLANE
e 1 R f exp 2s0 d e0 1 R f exp 2s0 d
(4)
with thermal re¯ection coef®cient R f e0 f e= e0 f e and thermal wave number s0 f p i 2pf =a0 . It is the goal of the following consideration to ®nd out how subsurface thermal properties depend on surface texture parameters, or, in other words, to estimate thermal parameters of the effective layer. Knowing thermal parameters and the surface characteristics should it make possible to forecast photothermal measurements from rough specimen. On the other hand, it would be useful to extract surface texture information from photothermal measurements. The sample under study is illuminated by modulated light. The diameter of the heating spot should be much wider than the characteristic lateral scale of the surface pro®le 2p/kmin. Thermal waves are generated at the absorber surface and propagate along different directions into the specimen. To illustrate the propagation of thermal waves arising from the rough surface let us have a look at Fig. 4. Due to thermal waves' noncollinear propagation and interference ``crumbled'' phase fronts are formed. They become smoothed out at suf®ciently long distances from the surface. The damping of thermal waves is controlled by the diffusion length m(f) which decreases for rising modulation frequency f. In the subsurface region and for diffusion lengths of the order of roughness s the propagation of thermal wave is three-dimensional. High frequency waves with short m(f) are strongly in¯uenced by roughness than low frequency ones. Quasi one-dimensional wave propagation (e.g. plane wave fronts) appears (i) for thermal diffusion lengths
159
Fig. 4. Isotherms behind a rough surface absorber. They become smoother with increasing distance from the surface. The equivalent smooth surface is shifted by the offset H with respect of the mean surface z 0.
large compared with roughness s and (ii) far apart from the rough surface. As a preliminary remark we consider localised (focussed) sample heating. We know that three-dimensional thermal diffusion length m3D generated by a heat source of spatial frequency k obeys the relation [11]: 1 m3D f ; k p m1D f k2 i 2pf =a
r a : pf (5)
The magnitude of m3D is smaller than plane wave diffusion length m1D. The quasi one-dimensional wave propagation (e.g. plane wave fronts) appears either far away from the rough surface or for thermal diffusion lengths larger than roughness s. Right next to the rough absorber surface the thermal waves propagate non-collinearly and form crumbled phase fronts. Due to this kind of propagation in the near ®eld the corresponding three-dimensional thermal diffusion length is shorter than m of the quasi-plane waves in the far ®eld. Now we discuss the connection between surface texture parameters and effective thermal properties in the subsurface region. We start the problem by considering stationary heat ¯ux through a rough interface. That topicÐthermal diffusion in the vicinity of a rough surfaceÐwas theoretically studied some time ago by Vandembroucq and Roux who applied the conformal mapping technique to rough boundaries [7]. By means of appropriately tailored co-ordinate
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H.G. Walther / Applied Surface Science 193 (2002) 156±166
transform they could stretch the rough surface to a smooth plane. Simultaneously heat diffusion equation and boundary conditions had to be subjected the same transform. Supposing constant surface temperature the ®eld of isotherms inside of the material was calculated both for smooth and rough surfaces. The behaviour of the isotherms can be seen qualitatively from Fig. 4. For this purpose now we interpret the crumbled lines as isotherms. Right next to the surface the isotherms replicate the rough pro®le, whereas most perturbations die away from the boundary very fast. The longest wavelength of the surface pro®le de®nes the distance from the surface at which isotherms are smoothed out. In the far ®eld these smoothed-out isotherms can be considered as being generated from an ``equivalent'' straight surface. It was found in [7] that this equivalent straight surface is offset from the geometrical average of the rough pro®le by the distance H resulting in shortened thermal propagation distance. This means physically reduced thermal conductivity. The offset H depends sensitively on the surface pro®le. To give an example offset H was estimated in [7] for a sinusoidal surface undulation of wavelength l and amplitude A: 2 2A pA H l arctan : (6) p 2l The negative sign expresses the distance reduction. Correspondingly, for a random rough surface the mean offset hHi has to be calculated by averaging H(l) over the corresponding bandwidth limited wavelength spectrum. Obviously H depends on roughness statistics. For persistent self-af®ne pro®les which are characterised by roughness exponent z > 0:5, RMS roughness s and lower bandwidth limit kmin the offset hHi was found in [7] hHi C Bkmin hs2 i:
(7)
We may identify kmin with the reciprocal maximum wavelength of the pro®le. The prefactor C(z) depending on the speci®c roughness exponent is de®ned by C B p
Z 2B ; Z 2B 1
(8)
and Z(z) denotes Riemann's zeta-function. Now let us turn from the constant to the alternating heat source which excites thermal waves. As mentioned
above heat diffusion in the vicinity of the surface is obstructed due to roughness. To illustrate the propagation of thermal waves isotherms in Fig. 4 have to be replaced by phase fronts of thermal waves. For thermal waves we also suppose an offset H between the average of the rough surface and the equivalent smooth boundary as it was found for constant heat ¯ux. This H has to be derived from of Eq. (7) by appropriate generalisation. There are three scales which are important for non-stationary heat transportation across the rough surface: characteristic vertical and lateral scales of the pro®le s and Lmax together with thermal diffusion length m(f). It is plausible that the distortion of thermal waves decreases for smoother surfaces (small s and large Lmax). Low frequency thermal waves with long diffusion lengths average over small and shortranging surface undulations given by the high frequency wing of g(k). Short thermal waves are of course much more sensitive to them. The scale of thermally smoothing out the roughness is of the order of m(f). That is why high frequency thermal waves become strongly distorted by the roughness than low frequency ones. The longer the thermal diffusion length (corresponding to decreased modulation frequencies f) the weaker is the in¯uence of high spatial frequency content on thermal wave propagation. We suppose that m(f) estimates the lower cut-off frequency kmin of PSD being responsible for thermal wave distortion. To express quantitatively this effect we have to ®nd out how H depends on modulation frequency. In the limit f ! 1 the offset H is already given by Eq. (7). This result is reasonable also for suf®ciently long diffusion lengths m Lmax . For shorter diffusion lengths we resort to the general de®nition of offset H given in [7] Z H
kmax
kmin
j~z2 kjk dk:
(9)
As m cuts off the PSD g(k) responsible for thermal wave distortion we have to replace kmin in Eq. (12) by m(f). The upper limit is approximated by 1, and we obtain Z 1 H f j~z2 kjk dk: (10) m f
H.G. Walther / Applied Surface Science 193 (2002) 156±166
For known PSD we can calculate H(f). Finally for selfaf®ne pro®les with z > 0:5 and lower bandwidth limit kmin 2p=Lmax we have 8 for m f Lmax < H0 2B 1 m f (11) H f : H0 for m f Lmax Lmax with H0 C B s2 =Lmax . The frequency depending offset H(f) describes the shortening of thermal propagation length in the nearsurface region. Thermal waves ``feel'' roughness in¯uenced thermal parameters a0 and e0 . The effective
161
thermal diffusion length m0 in the subsurface region is assumed to be shorter by distance H(f) than the smooth surface m: m0 f m f
H m f r f :
(12)
For abbreviation we introduced the reduction factor r f 1 H=m f . The ansatz of Eq. (12) is reasonable only for m larger than the maximum peak-to-valley distance of the surface pro®le. From Eq. (12) we obtain the thermal diffusivity a0 in the near-surface range a0 f a f r 2 f :
(13)
Fig. 5. Numeric calculations of photothermal frequency scans from self-af®ne rough samples with different roughness exponents z. Parameters are s 2 mm and Lmax 30 mm. The full line shows for comparison a sample with exponential roughness autocorrelation function (s 2 mm, decay length t 10 mm).
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H.G. Walther / Applied Surface Science 193 (2002) 156±166
Subsurface thermal effusivity e0 is also different from the bulk value e but cannot be unambiguously derived from m0 . Several possibilities exist to split up the roughness in¯uence in k0 and rc0 . We choose the simple de®nitions k0 kr 3 and rc0 rcr because they are consistent with (i) the vertical material distribution function (so-called Abbott±Firestone curve) and (ii) the existence of thermal re¯ection coef®cient R. Subsurface effusivity comes out as e0 f e f r 2 f :
(14)
Certain arbitrariness with the de®nition of k0 and rc0 can be overcome only by additional assumptions about the mode and by comparison with experiments.
4. Model calculations Numerical calculations have been performed to show that Eqs. (4) and (11)±(14) are able to describe adequately the observed experimental curves. We assumed steel samples (k 0:45 W/K cm, rc 3:5 W s/K cm2) with rough surfaces (s 1:5 3 mm, Lmax 10 30 mm) heated at modulation frequencies from 1 Hz to 100 kHz. At 100 kHz the thermal diffusion length is about 6.3 mm, which gives an estimate of the upper limit of guessed s-values. The in¯uence of the various layer and surface parameters on magnitude and phase of the surface temperature oscillation was studied. Layer thickness was assumed to be d 4 s.
Fig. 6. Numeric calculations of photothermal frequency scans from self-af®ne rough samples. Parameters are z 0:65 and Lmax 30 mm, while s was varied.
H.G. Walther / Applied Surface Science 193 (2002) 156±166
163
Fig. 7. Numeric calculations of photothermal frequency scans from self-af®ne rough samples. s and Lmax (denoted by t) were varied, while ®xed z 0:65.
Representative results are summarised in Figs. 5±7. For constant s and Lmax but different roughness exponents the frequency scans are shown in Fig. 5. For B ! 0:5 both amplitude ratios and phase differences rise. This result clearly illustrates the connection between the spatial frequency content of surface roughness and the thermally smoothing by thermal waves. The more rugged the surface is, the more subsurface thermal parameters distinguish from bulk properties. It is worth to mention that similar curves can be described by assuming exponential roughness autocorrelation function with appropriately chosen decay length. This ®nding underlines once more the
dif®culty to estimate unambiguously roughness parameters. The in¯uence of s at ®xed z and Lmax is shown in Fig. 6. For the same roughness exponent but increased roughness amplitudes the signals deviate strongly from the bulk, demonstrating the decisive in¯uence of surface texture on high frequency photothermal measurements. As shown in Fig. 7 the signal deviations increase for shorter Lmax corresponding to the more rugged pro®le. This is consistent with the result of Fig. 5. All the calculated phase curves exhibit the experimentally found high frequency extremum. The position and value of it sensitively depends on surface texture. From numerical ®tting the measured
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H.G. Walther / Applied Surface Science 193 (2002) 156±166
extremum conclusions about surface roughness can be drawn. 5. Experimental validation The proposed effective layer model with frequency depending thermal transport properties enables us to describe quantitatively the distortion of photothermal signals due to surface roughness. This was demonstrated by comparing radiometric measurements from surface processed steel specimens with numerical calculations. We prepared a set of three steel cubes of about 2 cm3 volume and different surface texture. The surface of sample R1 was polished while the surfaces of samples R2 and R3 were ground by means of sandpaper. The surface pro®les were pro®lometrically inspected, and standardised texture parameters Rp, Rq and RSm were estimated [12]. The results are summarised in Table 1.
We performed conventional photothermal radiometric measurements [13] using acousto-optically modulated laser radiation (frequency doubled 5 W Nd:YAG) for heating. The infrared radiation periodically emitted from the absorbing sample was picked up by a LN2 cooled MCT detector. The signals from rough samples R2 and R3 were normalised with respect of the smooth sample R1 and are presented in Fig. 8. Because of low absorptivity in the visible, low emissivity in the infrared and fairly high thermal conductivity the steel signals are weak, and due to the poor signal-to-noise ratio the data scatter strongly. In the same ®gure the comparison between measured frequency scans and model calculations is shown. A satisfactory agreement with the experiment can be achieved for appropriately guessed layer thickness d and surface pro®le parameters s, Lmax and z. Fit results for them are added to Table 1. The agreement between measured Rp, Rq and RSm with d, s and Lmax has to be evaluated on the base of the ``translation rules'' given
Fig. 8. Comparison between experiment and calculation for two rough steel samples (a) R2 and (b) R3. Sample parameters are given in Table 1. Fit parameters are written in the legend, and Lmax 35 mm (25 mm) for R2 (R3).
H.G. Walther / Applied Surface Science 193 (2002) 156±166
165
Table 1 Steel samplesÐcomparison between pro®lometrically estimated surface texture parameters Rp, Rq and RSm and photothermally ®tted parameters d, s, Lmax (distances in mm) and roughness exponents z Sample
Steel R1a Steel R2 Steel R3 a
Profilometrical
Photothermal
Rp
Rq
RSm
d
s
Lmax
z
0.87 3.06 5.19
0.11 0.37 0.79
32 29 37
± 4.5±6.0 7.5
± 1.5±2.0 2.5
± 35 25
± 0.60 0.05 0.55 0.02
Sample R1 was used for reference.
at the end of Section 2. Comparing peak-to-valley distance 2Rp with effective layer thickness d we found good agreement. Also the pro®lometrically estimated RSm and the ®tted Lmax correspond to each other. The ®tted RMS roughness s follows qualitatively the expected behaviour with R3 is rougher than R2. Quantitatively s and Rp differ from each other. This may be caused by the crude pro®lometric estimation of Rq, by relating the photothermal measurements to a reference surface which is assumed to be ideally smooth, and by the neglection of z-in¯uence on Rq. In spite of the indirect and rough estimation of s, d and Lmax the agreement between them and the pro®lometrical numbers Rp, Rq and RSm is quite reasonable. The experimental investigations were repeated with zirconia specimens having polished and rough surfaces. We also measured typical differences between smooth and rough samples. But in contradiction to steel the zirconia data could not be ®tted as well. There appeared a systematic deviation between experiment and calculation predominantly on the high frequency side of the extremum as shown in Fig. 9. We suspect it is due to the violation of essential prerequisites of the supposed model: above several kHz the thermal diffusion length becomes of the order of both the optical absorption length and the infrared emission length. At high modulation frequencies zirconia cannot be considered as surface absorber. This leads to decreased signal magnitude and additional phase shift. Moreover, microcracks resulting from the grinding process and being optically invisible may distort unpredictably the propagation of high frequency thermal waves. The only information about zirconia's surface texture that we can obtain from photothermal measurements is the approximate thickness of roughness layer. In the preceding example we roughly estimated s 5:5 mm and d 19 mm by ®tting the position of maximum phase
Fig. 9. Comparison between radiometric measurements from a rough zirconia sample Z3 (Rp 30 mm, Rq 3:5 mm, RSm 65 mm) with model calculations (s 5:5 mm, Lmax 60 mm, d 19 mm, z 0:65).
deviation. These numbers are close to the pro®lometrically measured Rq 3:5 mm and Rp 15 mm. 6. Summary and discussion Photothermal frequency scans from rough surface absorbers distinguish from smooth surface ones. This has to be taken into account if high frequency
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H.G. Walther / Applied Surface Science 193 (2002) 156±166
photothermal measurements have to be interpreted quantitatively, such as in case of subsurface depth pro®ling or microstructure inspection. It is the aim of this paper to quantify surface roughness in¯uence on photothermal measurements. To derive an adequate expression for modulated photothermal signals we combined the existing effective layer model [2±4] with the effect of roughness induced obstructions of thermal diffusion [7]. By faithful translating to thermal waves the results of [7] we could establish an equation describing frequency scans from rough surfaces which are characterised by RMS roughness s, roughness exponent z and bandwidth limit 2p/Lmax. To achieve proper ®ts to the measurements it was necessary to consider frequency depending modi®cation of effective thermal properties in the subsurface region. The thickness of the effective layer was equated with the peak-to-valley distance of the rough pro®le. The most important result of roughness in¯uence on thermal waves is the shortening of the thermal diffusion length described by Eq. (12). Together with Eq. (4), (11), (13) and (14) they re¯ect correctly the experimental features: (i) appearance of an extremum at phase difference curves; (ii) rise of the relative signal magnitude at increased frequencies. The performance of the proposed model was successfully demonstrated by comparing calculations with radiometric measurements from rough steel samples. It could be shown a reasonable agreement between measured and ®tted roughness parameters. Corresponding measurements from rough zirconia specimens could not be ®tted as well by our model. This is presumably due to the violation of model prerequisites: in zirconia at high modulation frequencies the optical penetration depth cannot be neglected, and after mechanical
surface treatment a lot of cracks are formed which cannot be detected pro®lometrically. The value of the proposed roughness model consists in the correct description of photothermal frequency scans. If samples under study fall within the scope of the model the reasonable orders of magnitude of surface texture parameters can be derived. Asking for the unambiguous and quantitatively exact characterisation of the surface roughness, however, would be too much for the presented effective layer model. References [1] H.G. Walther, M.C. Larciprete, P. Mayr, S. Paoloni, B. Schmitz, Anal. Sci. 17 (2001) 428, (special issue). [2] L. Nicolaides, A. Mandelis, C.J. Beingessner, Anal. Sci. 17 (2001) 383, (special issue). [3] W. Kiepert, B.K. Bein, J.H. Gu, J. Pelzl, H.G. Walther, Prog. Nat. Sci. 6 (Suppl.) (1996) 317. [4] H.G. Walther, J. Appl. Phys. 89 (2001) 2939. [5] D. Vandembroucq, C. Boccara, Prog. Nat. Sci. 6 (Suppl.) (1996) 313. [6] R. Osiander, P. Korpiun, Prog. Nat. Sci. 6 (Suppl.) (1996) 653±656. [7] D. Vandembroucq, S. Roux, Phys. Rev. E 55 (1997) 6171. [8] ISO 4287, Surface texture: pro®le methodÐterms, de®nition and texture parameters, 1997. [9] G. Busse, H.G. Walther, in: A. Mandelis (Ed.), Progress in Photothermal and Photoacoustic Sciences and Technology, Vol. 1, Elsevier, Amsterdam, 1992, pp. 205±298. [10] J. Schmittbuhl, J.P. Vilotte, S. Roux, Phys. Rev. E 51 (1995) 131±147. [11] L.C. Aamodt, J.C. Murphy, J. Appl. Phys. 52 (1982) 111. [12] Optisches TastschnittgeraÈt, Evaluation by Taylor±Hobson Method, Hommelwerke, Germany. [13] T.T.N. Lan, U. Seidel, H.G. Walther, G. Goch, B. Schmitz, J. Appl. Phys. 78 (1995) 4108±4111.
Surface roughness in¯uence on photothermal radiometry H.G. Walther* Physikalisch-Astron. Fakultat, Institut fuÈr Optik und Quantenelektronik, Friedrich Schiller Universitat Jena, Max-Wien-Platz 1, D-07743 Jena, Germany Received 12 September 2001; accepted 2 April 2002
Abstract Frequency scanned radiometric measurements from rough absorbers are different from smooth samples. This effect has to be considered for accurate data interpretation, for instance, for photothermal depth pro®ling. Based on the equivalent layer model we discussed the effect of roughness induced thermal wave dispersion and derived a formula which correctly describes the observed experimental features: (i) appearance of phase difference extremum; (ii) rise of the relative signal magnitude at increased frequencies. The validity of the theory was successfully demonstrated by comparing calculations with radiometric measurements from rough steel samples. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface roughness; Photothermal radiometry; Thermal wave dispersion; Non-destructive evaluation
1. Introduction During the past decade both photothermal measuring techniques and related theoretical methods for data interpretation were improved decisively. In principle, now it is possible to retrieve depth pro®les of technologically relevant thermal properties from photothermal measurements. Photothermal depth pro®ling became an interesting and valuable tool for non-destructive evaluation. Among others, important applications are the estimation water migration pro®les in living tissues, the characterisation water content in textiles and packaging materials, the estimation of dopant distribution in implanted semiconductors, and ®nally the characterisation of hardness depth pro®les in surface hardened steel. Photothermal applications in industry have to be performed with industrial specimen surfaces of which * Tel.: 49-3641-947030; fax: 49-3641-947032. E-mail address: [email protected] (H.G. Walther).
are not ideal. They can be rough or contaminated. For accurate data interpretation it is necessary to understand the role the technologically processed surface under study plays for photothermal signal generation. To give a demonstration how surface texture can in¯uence photothermal measurements we refer to radiometric signals from differently rough steel samples [1]. The measured phase curves shown in Fig. 1 deviate characteristically from polished specimen. The difference against smooth surface behaviour are obviously connected with the level of roughness. Based on this fact it should be possible to correct photothermal data for surface roughness. This was proposed and for the ®rst time successfully demonstrated in [2]. Already some time ago photothermal signals from rough surfaces were investigated by Bein and coworkers. They performed photothermal measurements from rough graphite samples and was able to explain his results by a decreased subsurface thermal effusivity [3]. Bein and co-workers successfully ®tted the
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 2 ) 0 0 2 5 7 - X
H.G. Walther / Applied Surface Science 193 (2002) 156±166
Fig. 1. Radiometric phase measurements from rough steel samples and polished zirconia used as reference: ( ) reference; (*) polished steel; ( ) slightly rough; () very rough.
experimental date by supposing an effective surface layer with appropriately guessed thermal properties. Later on similar investigations were performed by Walther [4] from rough steel specimen. He also modelled the rough surface by an equivalent layer and explained its modi®ed thermal properties by a parallel connection of randomly distributed thermal resistors corresponding to the local height variations of the surface. This attempt was a step ahead to the quantitative relation between roughness and effective thermal properties. The frequency dependent photothermal signal obeys a simple power law if plane thermal waves are travelling from a smooth surface. Frequency dependent deviations from that behaviour occur for rough surface absorber. It can be qualitatively explained by the fractal dimension df of the rough heat source for which holds df > 2. First attempts to understand photothermal results by fractals were undertaken by Vandembroucq and Boccara [5] and Osiander and Korpuin [6]. A more extended approach to solve the heat diffusion equation in the vicinity of self-af®ne boundaries was published by Vandembroucq and Roux [7]. They used conformal mapping technique to image the rough boundary on a tailored co-ordinate transformations. The same transformation was also applied to heat diffusion equation and to the boundary conditions. A solution was found for constant heat ¯ux and conclusions about the roughness induced distortion of isotherms could be drawn.
157
It is the aim of the paper to ®nd out how photothermal signals from rough surface absorbers depend on surface texture parameters. To reach this goal we develop an appropriate theoretical model which is based on effective layer assumption and compare it with measurements from known surfaces. Consequently the article is structured as follows: in Section 2 short mathematical introduction is given to de®ne surface texture parameters. In Section 3 a theoretical model describing signal generation from rough specimen is discussed and applied for numerical calculations in Section 4. Finally, in Section 5 the validity of the proposed model is tested by comparing radiometric measurements from rough specimen with calculations. 2. Mathematical description of rough surfaces Usually the surfaces of technical specimen are rough. Roughness expresses random deviations of the surface from the ideal smooth plane. In practice, surface texture parameters are de®ned by standards and commonly are measured by optical or stylus pro®lometers. From mathematical point of view the statistical approach to roughness parameters is quite instructive. For this reason let us start with the stochastic description of a rough surface whose pro®le is sketched in Fig. 2. z denotes the local height deviating from mean surface. For simplicity we consider only pro®les across the rough surface. The roughness pro®le z(x) is assumed to be generated by a random process with zero mean. Applying Fourier transform to z(x) we obtain ~z k Fz x (k is the spatial frequency). The frequency content of z(x) is evaluated by the power spectral density (PSD), de®ned by
Fig. 2. Pro®le of rough surface. De®nition of surface texture parameters.
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g k jF Z Xj2 , where g(k) is an important function involving main information on random roughness. For many technical applications two different features of surface texture have to be distinguished: waviness and roughness. Both effects can be separated by appropriate k-frequency-®ltering of the pro®le. PSD of roughness is de®ned by certain cut-off k-frequency. We speak of bandwidth limited PSD which is non-vanishing only between the lower and the upper frequency limits kmin and kmax. Another important surface parameter is random mean squared roughness s which can be derived from g(k) by relation Z kmax s2 2 g k dk: (1) kmin
In practice relevant surface texture is characterised by parameters which are strictly de®ned by standards [8]. Considering only roughness pro®le main parameters are as follows: Rp is the maximum profile deviation from mean plane. In technical literature the length 2Rp is called peak-to-valley distance. Rq denotes random mean squared height deviation q RL Rq 1=L 0 jz2 xj dx.
either along x-axis or z-axis. A self-af®ne surface is characterised by (i) roughness exponent z, (ii) amplitude parameter s and (iii) the bandwidth limitations kmin and kmax of the scaling invariant regime. Sophisticated pro®les with z < 0:5 are so-called anti-persistent pro®les and tend to become complex and rami®ed. Mechanically treated metal and ceramic surfaces have persistent pro®les with z > 0:5. As mentioned in [10] the fractal dimension df of self-af®ne pro®les can be estimated by df 2 z. Of course, the standardised and easy-to-measure parameters Rq and RSm are more convenient for practical use than sophisticated self-af®ne surface parameters. On the other hand, it is possible to move between both pictures. RSm can be approximated by 2p=kmin and Rq can be compared with s. This translation however is crude and not unambiguous. 3. Calculation of the frequency dependent photothermal signal
(2)
The target of this section is to ®nd a quantitative description of the photothermal signal from a rough surface absorber. The experimental situation under consideration is as follows: the sample is illuminated by a modulated heating beam. For simplicity we focus on such photothermal techniques like radiometry or gas cell photoacoustics [9] which give signals proportional to the spatially averaged surface temperature oscillation. As demonstrated in earlier papers [2±4] the measured signal curves can be modelled by an equivalent sample consisting of the homogeneous bulk material which is overcoated by a parallel and surface absorbing layer. The equivalency between the rough and the effective layer sample is sketched in Fig. 3. Non-stationary heat transport is controlled by thermal
within the range of spatial frequencies bound by kmin and kmax. b denotes the lateral stretching factor, which is an arbitrary number. The so-called roughness exponent z 0 < z < 1 controls the vertical structure of rough pro®le and estimates the exponent of power law of PSD according to 1 g k 12B : (3) k From Eq. (2) we see the roughness statistics is invariant to different scaling factors depending on orientation
Fig. 3. Thermal equivalence between rough surface and effective layer model.
RSm is the mean distance between dominant grooves which can be identified as the maximum spatial wavelength of the rough profile. The material distribution across the profile from bulk to surface is called the Abbott±Firestone curve M(z0 ) which is given by the summation over all profile lengths x for which holds z x > z0 . Many technical surfaces can be more appropriately described by self-af®ne pro®les. Such pro®les obey the scaling rule: x ! bx
and
z ! bB z
H.G. Walther / Applied Surface Science 193 (2002) 156±166
diffusivity a and thermal effusivity e which are connected with thermal conductivity k, mass density r and speci®c heat c by the relations a k=rc and p e krc. Obviously the thermal properties of the effective layer differ from the bulk. They are marked by primes (0 ) to distinguish them from the bulk properties. The layer thickness d is assumed to be of the order of the maximum vertical distance between pro®le peaks and valleys. We ask for the ratio S of the surface temperature T 0 of the effective layer-on-substrate system and the smooth half-space temperature TPLANE. In [4] we found that S
T0 TPLANE
e 1 R f exp 2s0 d e0 1 R f exp 2s0 d
(4)
with thermal re¯ection coef®cient R f e0 f e= e0 f e and thermal wave number s0 f p i 2pf =a0 . It is the goal of the following consideration to ®nd out how subsurface thermal properties depend on surface texture parameters, or, in other words, to estimate thermal parameters of the effective layer. Knowing thermal parameters and the surface characteristics should it make possible to forecast photothermal measurements from rough specimen. On the other hand, it would be useful to extract surface texture information from photothermal measurements. The sample under study is illuminated by modulated light. The diameter of the heating spot should be much wider than the characteristic lateral scale of the surface pro®le 2p/kmin. Thermal waves are generated at the absorber surface and propagate along different directions into the specimen. To illustrate the propagation of thermal waves arising from the rough surface let us have a look at Fig. 4. Due to thermal waves' noncollinear propagation and interference ``crumbled'' phase fronts are formed. They become smoothed out at suf®ciently long distances from the surface. The damping of thermal waves is controlled by the diffusion length m(f) which decreases for rising modulation frequency f. In the subsurface region and for diffusion lengths of the order of roughness s the propagation of thermal wave is three-dimensional. High frequency waves with short m(f) are strongly in¯uenced by roughness than low frequency ones. Quasi one-dimensional wave propagation (e.g. plane wave fronts) appears (i) for thermal diffusion lengths
159
Fig. 4. Isotherms behind a rough surface absorber. They become smoother with increasing distance from the surface. The equivalent smooth surface is shifted by the offset H with respect of the mean surface z 0.
large compared with roughness s and (ii) far apart from the rough surface. As a preliminary remark we consider localised (focussed) sample heating. We know that three-dimensional thermal diffusion length m3D generated by a heat source of spatial frequency k obeys the relation [11]: 1 m3D f ; k p m1D f k2 i 2pf =a
r a : pf (5)
The magnitude of m3D is smaller than plane wave diffusion length m1D. The quasi one-dimensional wave propagation (e.g. plane wave fronts) appears either far away from the rough surface or for thermal diffusion lengths larger than roughness s. Right next to the rough absorber surface the thermal waves propagate non-collinearly and form crumbled phase fronts. Due to this kind of propagation in the near ®eld the corresponding three-dimensional thermal diffusion length is shorter than m of the quasi-plane waves in the far ®eld. Now we discuss the connection between surface texture parameters and effective thermal properties in the subsurface region. We start the problem by considering stationary heat ¯ux through a rough interface. That topicÐthermal diffusion in the vicinity of a rough surfaceÐwas theoretically studied some time ago by Vandembroucq and Roux who applied the conformal mapping technique to rough boundaries [7]. By means of appropriately tailored co-ordinate
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transform they could stretch the rough surface to a smooth plane. Simultaneously heat diffusion equation and boundary conditions had to be subjected the same transform. Supposing constant surface temperature the ®eld of isotherms inside of the material was calculated both for smooth and rough surfaces. The behaviour of the isotherms can be seen qualitatively from Fig. 4. For this purpose now we interpret the crumbled lines as isotherms. Right next to the surface the isotherms replicate the rough pro®le, whereas most perturbations die away from the boundary very fast. The longest wavelength of the surface pro®le de®nes the distance from the surface at which isotherms are smoothed out. In the far ®eld these smoothed-out isotherms can be considered as being generated from an ``equivalent'' straight surface. It was found in [7] that this equivalent straight surface is offset from the geometrical average of the rough pro®le by the distance H resulting in shortened thermal propagation distance. This means physically reduced thermal conductivity. The offset H depends sensitively on the surface pro®le. To give an example offset H was estimated in [7] for a sinusoidal surface undulation of wavelength l and amplitude A: 2 2A pA H l arctan : (6) p 2l The negative sign expresses the distance reduction. Correspondingly, for a random rough surface the mean offset hHi has to be calculated by averaging H(l) over the corresponding bandwidth limited wavelength spectrum. Obviously H depends on roughness statistics. For persistent self-af®ne pro®les which are characterised by roughness exponent z > 0:5, RMS roughness s and lower bandwidth limit kmin the offset hHi was found in [7] hHi C Bkmin hs2 i:
(7)
We may identify kmin with the reciprocal maximum wavelength of the pro®le. The prefactor C(z) depending on the speci®c roughness exponent is de®ned by C B p
Z 2B ; Z 2B 1
(8)
and Z(z) denotes Riemann's zeta-function. Now let us turn from the constant to the alternating heat source which excites thermal waves. As mentioned
above heat diffusion in the vicinity of the surface is obstructed due to roughness. To illustrate the propagation of thermal waves isotherms in Fig. 4 have to be replaced by phase fronts of thermal waves. For thermal waves we also suppose an offset H between the average of the rough surface and the equivalent smooth boundary as it was found for constant heat ¯ux. This H has to be derived from of Eq. (7) by appropriate generalisation. There are three scales which are important for non-stationary heat transportation across the rough surface: characteristic vertical and lateral scales of the pro®le s and Lmax together with thermal diffusion length m(f). It is plausible that the distortion of thermal waves decreases for smoother surfaces (small s and large Lmax). Low frequency thermal waves with long diffusion lengths average over small and shortranging surface undulations given by the high frequency wing of g(k). Short thermal waves are of course much more sensitive to them. The scale of thermally smoothing out the roughness is of the order of m(f). That is why high frequency thermal waves become strongly distorted by the roughness than low frequency ones. The longer the thermal diffusion length (corresponding to decreased modulation frequencies f) the weaker is the in¯uence of high spatial frequency content on thermal wave propagation. We suppose that m(f) estimates the lower cut-off frequency kmin of PSD being responsible for thermal wave distortion. To express quantitatively this effect we have to ®nd out how H depends on modulation frequency. In the limit f ! 1 the offset H is already given by Eq. (7). This result is reasonable also for suf®ciently long diffusion lengths m Lmax . For shorter diffusion lengths we resort to the general de®nition of offset H given in [7] Z H
kmax
kmin
j~z2 kjk dk:
(9)
As m cuts off the PSD g(k) responsible for thermal wave distortion we have to replace kmin in Eq. (12) by m(f). The upper limit is approximated by 1, and we obtain Z 1 H f j~z2 kjk dk: (10) m f
H.G. Walther / Applied Surface Science 193 (2002) 156±166
For known PSD we can calculate H(f). Finally for selfaf®ne pro®les with z > 0:5 and lower bandwidth limit kmin 2p=Lmax we have 8 for m f Lmax < H0 2B 1 m f (11) H f : H0 for m f Lmax Lmax with H0 C B s2 =Lmax . The frequency depending offset H(f) describes the shortening of thermal propagation length in the nearsurface region. Thermal waves ``feel'' roughness in¯uenced thermal parameters a0 and e0 . The effective
161
thermal diffusion length m0 in the subsurface region is assumed to be shorter by distance H(f) than the smooth surface m: m0 f m f
H m f r f :
(12)
For abbreviation we introduced the reduction factor r f 1 H=m f . The ansatz of Eq. (12) is reasonable only for m larger than the maximum peak-to-valley distance of the surface pro®le. From Eq. (12) we obtain the thermal diffusivity a0 in the near-surface range a0 f a f r 2 f :
(13)
Fig. 5. Numeric calculations of photothermal frequency scans from self-af®ne rough samples with different roughness exponents z. Parameters are s 2 mm and Lmax 30 mm. The full line shows for comparison a sample with exponential roughness autocorrelation function (s 2 mm, decay length t 10 mm).
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Subsurface thermal effusivity e0 is also different from the bulk value e but cannot be unambiguously derived from m0 . Several possibilities exist to split up the roughness in¯uence in k0 and rc0 . We choose the simple de®nitions k0 kr 3 and rc0 rcr because they are consistent with (i) the vertical material distribution function (so-called Abbott±Firestone curve) and (ii) the existence of thermal re¯ection coef®cient R. Subsurface effusivity comes out as e0 f e f r 2 f :
(14)
Certain arbitrariness with the de®nition of k0 and rc0 can be overcome only by additional assumptions about the mode and by comparison with experiments.
4. Model calculations Numerical calculations have been performed to show that Eqs. (4) and (11)±(14) are able to describe adequately the observed experimental curves. We assumed steel samples (k 0:45 W/K cm, rc 3:5 W s/K cm2) with rough surfaces (s 1:5 3 mm, Lmax 10 30 mm) heated at modulation frequencies from 1 Hz to 100 kHz. At 100 kHz the thermal diffusion length is about 6.3 mm, which gives an estimate of the upper limit of guessed s-values. The in¯uence of the various layer and surface parameters on magnitude and phase of the surface temperature oscillation was studied. Layer thickness was assumed to be d 4 s.
Fig. 6. Numeric calculations of photothermal frequency scans from self-af®ne rough samples. Parameters are z 0:65 and Lmax 30 mm, while s was varied.
H.G. Walther / Applied Surface Science 193 (2002) 156±166
163
Fig. 7. Numeric calculations of photothermal frequency scans from self-af®ne rough samples. s and Lmax (denoted by t) were varied, while ®xed z 0:65.
Representative results are summarised in Figs. 5±7. For constant s and Lmax but different roughness exponents the frequency scans are shown in Fig. 5. For B ! 0:5 both amplitude ratios and phase differences rise. This result clearly illustrates the connection between the spatial frequency content of surface roughness and the thermally smoothing by thermal waves. The more rugged the surface is, the more subsurface thermal parameters distinguish from bulk properties. It is worth to mention that similar curves can be described by assuming exponential roughness autocorrelation function with appropriately chosen decay length. This ®nding underlines once more the
dif®culty to estimate unambiguously roughness parameters. The in¯uence of s at ®xed z and Lmax is shown in Fig. 6. For the same roughness exponent but increased roughness amplitudes the signals deviate strongly from the bulk, demonstrating the decisive in¯uence of surface texture on high frequency photothermal measurements. As shown in Fig. 7 the signal deviations increase for shorter Lmax corresponding to the more rugged pro®le. This is consistent with the result of Fig. 5. All the calculated phase curves exhibit the experimentally found high frequency extremum. The position and value of it sensitively depends on surface texture. From numerical ®tting the measured
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H.G. Walther / Applied Surface Science 193 (2002) 156±166
extremum conclusions about surface roughness can be drawn. 5. Experimental validation The proposed effective layer model with frequency depending thermal transport properties enables us to describe quantitatively the distortion of photothermal signals due to surface roughness. This was demonstrated by comparing radiometric measurements from surface processed steel specimens with numerical calculations. We prepared a set of three steel cubes of about 2 cm3 volume and different surface texture. The surface of sample R1 was polished while the surfaces of samples R2 and R3 were ground by means of sandpaper. The surface pro®les were pro®lometrically inspected, and standardised texture parameters Rp, Rq and RSm were estimated [12]. The results are summarised in Table 1.
We performed conventional photothermal radiometric measurements [13] using acousto-optically modulated laser radiation (frequency doubled 5 W Nd:YAG) for heating. The infrared radiation periodically emitted from the absorbing sample was picked up by a LN2 cooled MCT detector. The signals from rough samples R2 and R3 were normalised with respect of the smooth sample R1 and are presented in Fig. 8. Because of low absorptivity in the visible, low emissivity in the infrared and fairly high thermal conductivity the steel signals are weak, and due to the poor signal-to-noise ratio the data scatter strongly. In the same ®gure the comparison between measured frequency scans and model calculations is shown. A satisfactory agreement with the experiment can be achieved for appropriately guessed layer thickness d and surface pro®le parameters s, Lmax and z. Fit results for them are added to Table 1. The agreement between measured Rp, Rq and RSm with d, s and Lmax has to be evaluated on the base of the ``translation rules'' given
Fig. 8. Comparison between experiment and calculation for two rough steel samples (a) R2 and (b) R3. Sample parameters are given in Table 1. Fit parameters are written in the legend, and Lmax 35 mm (25 mm) for R2 (R3).
H.G. Walther / Applied Surface Science 193 (2002) 156±166
165
Table 1 Steel samplesÐcomparison between pro®lometrically estimated surface texture parameters Rp, Rq and RSm and photothermally ®tted parameters d, s, Lmax (distances in mm) and roughness exponents z Sample
Steel R1a Steel R2 Steel R3 a
Profilometrical
Photothermal
Rp
Rq
RSm
d
s
Lmax
z
0.87 3.06 5.19
0.11 0.37 0.79
32 29 37
± 4.5±6.0 7.5
± 1.5±2.0 2.5
± 35 25
± 0.60 0.05 0.55 0.02
Sample R1 was used for reference.
at the end of Section 2. Comparing peak-to-valley distance 2Rp with effective layer thickness d we found good agreement. Also the pro®lometrically estimated RSm and the ®tted Lmax correspond to each other. The ®tted RMS roughness s follows qualitatively the expected behaviour with R3 is rougher than R2. Quantitatively s and Rp differ from each other. This may be caused by the crude pro®lometric estimation of Rq, by relating the photothermal measurements to a reference surface which is assumed to be ideally smooth, and by the neglection of z-in¯uence on Rq. In spite of the indirect and rough estimation of s, d and Lmax the agreement between them and the pro®lometrical numbers Rp, Rq and RSm is quite reasonable. The experimental investigations were repeated with zirconia specimens having polished and rough surfaces. We also measured typical differences between smooth and rough samples. But in contradiction to steel the zirconia data could not be ®tted as well. There appeared a systematic deviation between experiment and calculation predominantly on the high frequency side of the extremum as shown in Fig. 9. We suspect it is due to the violation of essential prerequisites of the supposed model: above several kHz the thermal diffusion length becomes of the order of both the optical absorption length and the infrared emission length. At high modulation frequencies zirconia cannot be considered as surface absorber. This leads to decreased signal magnitude and additional phase shift. Moreover, microcracks resulting from the grinding process and being optically invisible may distort unpredictably the propagation of high frequency thermal waves. The only information about zirconia's surface texture that we can obtain from photothermal measurements is the approximate thickness of roughness layer. In the preceding example we roughly estimated s 5:5 mm and d 19 mm by ®tting the position of maximum phase
Fig. 9. Comparison between radiometric measurements from a rough zirconia sample Z3 (Rp 30 mm, Rq 3:5 mm, RSm 65 mm) with model calculations (s 5:5 mm, Lmax 60 mm, d 19 mm, z 0:65).
deviation. These numbers are close to the pro®lometrically measured Rq 3:5 mm and Rp 15 mm. 6. Summary and discussion Photothermal frequency scans from rough surface absorbers distinguish from smooth surface ones. This has to be taken into account if high frequency
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photothermal measurements have to be interpreted quantitatively, such as in case of subsurface depth pro®ling or microstructure inspection. It is the aim of this paper to quantify surface roughness in¯uence on photothermal measurements. To derive an adequate expression for modulated photothermal signals we combined the existing effective layer model [2±4] with the effect of roughness induced obstructions of thermal diffusion [7]. By faithful translating to thermal waves the results of [7] we could establish an equation describing frequency scans from rough surfaces which are characterised by RMS roughness s, roughness exponent z and bandwidth limit 2p/Lmax. To achieve proper ®ts to the measurements it was necessary to consider frequency depending modi®cation of effective thermal properties in the subsurface region. The thickness of the effective layer was equated with the peak-to-valley distance of the rough pro®le. The most important result of roughness in¯uence on thermal waves is the shortening of the thermal diffusion length described by Eq. (12). Together with Eq. (4), (11), (13) and (14) they re¯ect correctly the experimental features: (i) appearance of an extremum at phase difference curves; (ii) rise of the relative signal magnitude at increased frequencies. The performance of the proposed model was successfully demonstrated by comparing calculations with radiometric measurements from rough steel samples. It could be shown a reasonable agreement between measured and ®tted roughness parameters. Corresponding measurements from rough zirconia specimens could not be ®tted as well by our model. This is presumably due to the violation of model prerequisites: in zirconia at high modulation frequencies the optical penetration depth cannot be neglected, and after mechanical
surface treatment a lot of cracks are formed which cannot be detected pro®lometrically. The value of the proposed roughness model consists in the correct description of photothermal frequency scans. If samples under study fall within the scope of the model the reasonable orders of magnitude of surface texture parameters can be derived. Asking for the unambiguous and quantitatively exact characterisation of the surface roughness, however, would be too much for the presented effective layer model. References [1] H.G. Walther, M.C. Larciprete, P. Mayr, S. Paoloni, B. Schmitz, Anal. Sci. 17 (2001) 428, (special issue). [2] L. Nicolaides, A. Mandelis, C.J. Beingessner, Anal. Sci. 17 (2001) 383, (special issue). [3] W. Kiepert, B.K. Bein, J.H. Gu, J. Pelzl, H.G. Walther, Prog. Nat. Sci. 6 (Suppl.) (1996) 317. [4] H.G. Walther, J. Appl. Phys. 89 (2001) 2939. [5] D. Vandembroucq, C. Boccara, Prog. Nat. Sci. 6 (Suppl.) (1996) 313. [6] R. Osiander, P. Korpiun, Prog. Nat. Sci. 6 (Suppl.) (1996) 653±656. [7] D. Vandembroucq, S. Roux, Phys. Rev. E 55 (1997) 6171. [8] ISO 4287, Surface texture: pro®le methodÐterms, de®nition and texture parameters, 1997. [9] G. Busse, H.G. Walther, in: A. Mandelis (Ed.), Progress in Photothermal and Photoacoustic Sciences and Technology, Vol. 1, Elsevier, Amsterdam, 1992, pp. 205±298. [10] J. Schmittbuhl, J.P. Vilotte, S. Roux, Phys. Rev. E 51 (1995) 131±147. [11] L.C. Aamodt, J.C. Murphy, J. Appl. Phys. 52 (1982) 111. [12] Optisches TastschnittgeraÈt, Evaluation by Taylor±Hobson Method, Hommelwerke, Germany. [13] T.T.N. Lan, U. Seidel, H.G. Walther, G. Goch, B. Schmitz, J. Appl. Phys. 78 (1995) 4108±4111.