New contactless method for thermal diffusivity measurements using modulated photothermal radiometry S. Pham Tu Quoc, G. Cheymol, and A. Semerok Citation: Review of Scientific Instruments 85, 054903 (2014); doi: 10.1063/1.4874195 View online: http://dx.doi.org/10.1063/1.4874195 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simultaneous measurement of thermal diffusivity and optical absorption coefficient using photothermal radiometry. I. Homogeneous solids J. Appl. Phys. 110, 033515 (2011); 10.1063/1.3614524 Two-detector measurement system of pulse photothermal radiometry for the investigation of the thermal properties of thin films J. Appl. Phys. 102, 064903 (2007); 10.1063/1.2778642 Thermal-wave nondestructive evaluation of cylindrical composite structures using frequency-domain photothermal radiometry J. Appl. Phys. 97, 014911 (2005); 10.1063/1.1819999 Erratum: “A modification of Ångström’s method that employs photothermal radiometry to measure thermal diffusivity: Application to chemical vapor deposited diamond” [Rev. Sci. Instrum. 69, 237 (1998)] Rev. Sci. Instrum. 69, 2576 (1998); 10.1063/1.1148977 A modification of Ångström’s method that employs photothermal radiometry to measure thermal diffusivity: Application to chemical vapor deposited diamond Rev. Sci. Instrum. 69, 237 (1998); 10.1063/1.1148873
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 054903 (2014)
New contactless method for thermal diffusivity measurements using modulated photothermal radiometry S. Pham Tu Quoc,a) G. Cheymol, and A. Semerok French Alternative Energies and Atomic Energy Commission, Division of Nuclear Energy, DEN/DANS/DPC/SEARS/LISL, 91191 Gif/Yvette, France
(Received 17 January 2014; accepted 18 April 2014; published online 9 May 2014) Modulated photothermal radiometry is a non-destructive and contactless technique for the characterization of materials. It has two major advantages: a good signal-to-noise ratio through a synchronous detection and a low dependence on the heating power and the optical properties of the sample surface. This paper presents a new method for characterizing the thermal diffusivity of a material when the phase shift between a modulated laser power signal and the thermal signal of a plate sample is known at different frequencies. The method is based on a three-dimensional analytical model which is used to determine the temperature amplitude and the phase in the laser heating of the plate. A new simple formula was developed through multi-parametric analysis to determine the thermal diffusivity of the plate with knowledge of the frequency at the minimum phase shift, the laser beam radius r0 and the sample thickness L. This method was developed to control the variation of the thermal diffusivity of nuclear components and it was first applied to determine the thermal diffusivity of different metals: 304 L stainless steel, nickel, titanium, tungsten, molybdenum, zinc, and iron. The experimental results were obtained with 5%–10% accuracy and corresponded well with the reference values. The present paper also demonstrates the limit of application of this method for plate with thickness r0 /100 ≤ L ≤ r0 /2. The technique is deemed interesting for the characterization of barely accessible components that require a contactless measurement. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4874195] I. INTRODUCTION
The principle of the photothermal radiometry (PTR)1 consists in heating a sample with a modulated light power and then observing the emitted infrared radiation. In the stationary regime of the heating, the thermal response can be described as a modulated signal giving the amplitude and the phase shift of the temperature. With knowledge of the amplitude and the phase shift at different modulated frequencies, it is possible to characterize the thickness and the thermal properties of the sample. While the temperature amplitude depends on the sample’s optical properties and the light power,2 the phase shift is of great interest since it is independent of these parameters.3, 4 Among thermal properties, the thermal diffusivity plays an important role in the heat transfer process; it measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy. Nowadays, there are several methods to measure these parameters but each one has its disadvantages: can only be applied to bulk materials,5–7 requires use of a small focused beam,8 and requires the variation of the sample surface temperature which can be influenced by the heat loss effect.9 Our concern is to obtain, first, remote and contactless measurements with one face access; second, simple methods with acceptable performance in order to measure in situ the variation of the thermal properties of nuclear components. This paper presents a new type of measurement for the thermal diffusivity that involves determining the phase shift between the laser power and the thermal signal. The method is a) Electronic mail: [email protected].
0034-6748/2014/85(5)/054903/6/$30.00
based on a model of the laser heating of a sample10 that gives an analytical expression for the amplitudes and phases of the sample surface temperature with constant optical and thermophysical parameters during the laser heating. The variation of the temperature amplitude is very small in the stationary regime (of the order of a few Kelvins), and through multiparameter analysis, a simple formula has been obtained to determine the thermal diffusivity of a plate when the following parameters are known: fmin , i.e., the frequency at the minimum phase shift of the curve plotting the phase shift versus the frequency, the laser beam radius r0 and the sample thickness L. This formula was verified by the experimental setup. II. 3D THERMAL MODEL OF THE MODULATED LASER HEATING FOR PLATE IN AIR
Below is presented the thermal model for the laser heating of homogeneous and isotropic plate with infinite dimensions. The variation of optical and thermal properties with temperature, the surface roughness, and the heat exchange by convection between the sample surfaces and the air were supposed to be negligible. By applying the Fourier series analysis to the intensity of the laser beam and the temperature in the stationary regime of the laser heating,10 the complex temperature amplitude of the front face for plate sample can be written as +∞ α(1 − R)iI0 (ξ ) ˜ T (r) = − 2 k ξ − α 2 − 2πifk Cv 0 α 2α(e−L − e−αL ) + − 1 J0 (ξ r)dξ, (1) × (eL − e−L )
85, 054903-1
© 2014 AIP Publishing LLC
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054903-2
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
ξ r2 ξ 2r2 with: = ξ 2 − 2πifk Cv ; (ξ ) = 20 exp(− 4 0 ) for a Gaussian beam, where r is the radial distance from the center of the heated zone at the sample surface; Cv , k, and L are, respectively, the volumetric specific heat, the thermal conductivity, and the thickness of the sample; α and R are, respectively, the laser absorption coefficient and the reflectivity of the sample surface; r0 and I0 are, respectively, the laser beam radius at 1/e intensity and the mean intensity of the Gaussian laser beam; f and i are, respectively, the repetitive rate frequency of the laser and the complex unity; and ξ and J0 are, respectively, the variables of integration and the zero order Bessel functions of the first kind. The phase shift between the laser and the thermal response of the front surface can be described as = a tan(Real(T˜ (r)) / I mag(T˜ (r))).
(2)
This model can be used to fit the theoretical phase-shift to the experimental results in order to characterize the sample material and its thickness. Our objective is to develop a simple characterization method with good precision. We thus performed multi-parametric modeling to determine the dependence of particular points in the phase-shift vs. frequency curve which can provide us with simple relations between different parameters. III. MULTI-PARAMETRIC MODELING OF THE PHASE SHIFT FOR PLATE IN AIR
We successively analyzed, one at a time, the influence of different interactions and plate parameters on the phase shift, including: intensity of the laser beam I0 , surface reflectivity R, absorption coefficient α, thermal conductivity k, volumetric specific heat Cv , laser beam radius r0 , plate thickness L, and measured point position r. According to the modeling results, the following was concluded with regard to the influence of the different parameters on the phase shift: –The phase shift does not depend on the intensity of the laser beam and the surface reflectivity. –The influence of the absorption coefficient α is negligible for opaque sample which signifies that the laser photon is only absorbed at a shallow depth (αL 1, αμ 1, μ is the thermal diffusion length). –The influence of the measured point position r is negligible for r < r0 /2. –The thermal conductivity and volumetric specific heat influence the phase shift by their ratio k/Cv (thermal diffusivity D) and this influence is presented in Figure 1. It was observed that the minimal phase shift min is independent of the thermal diffusivity D. A formula was found to determine the thickness (L) when the minimum phase shift (min ) and the Gaussian laser beam radius (r0 )11 were known: r0 0.2273min + 57.8562min L= 100 +5688.2min + 208620) . (3)
FIG. 1. Phase shift as a function of frequency for different values of thermal diffusivity D of the plate. Laser beam radius r0 = 1 mm and plate thickness L = 100 μm.
–The influence of the plate thickness and the laser beam radius of the phase shift curve are presented in Figures 2 and 3. We can see that the level of the minimum phase shift depended on L and r0 . When the laser beam radius was much larger than the sample thickness, the thermal model became a 1D-heating, the heating propagation took the same direction as the propagation of the laser beam, and the minimum phase shift approaches the asymptote −90◦ . Inversely, when the laser beam radius is much smaller than the sample
FIG. 2. Phase shift as a function of frequency for different values of plate thickness L. Laser beam radius r0 = 1 mm and thermal diffusivity D = 0.1 cm2 /s.
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054903-3
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
FIG. 3. Phase shift as a function of frequency for different values of laser beam radius. Plate thickness L = 100 μm and thermal diffusivity D = 0.1 cm2 /s.
thickness and the diffusion length, the thermal model becomes a 3D-heating with heating on a dimensionless surface point. Subsequently, the maximum phase shift approaches the asymptote 0◦ and the phase shift curve does not have a local minimum. While the minimum phase shift does not render it is possible to determine the thermal properties, since it only varies with the thickness and the beam radius, according to (3), the frequency of the minimum phase shift seems vary linearly with the thermal diffusivity, cf. Figure 1, and could provide us with information about the thermal diffusivity. Now, we will focus on the frequency fmin of the minimum phase shift.
FIG. 4. Frequency fmin as a function of the thermal diffusivity. Laser beam radius r0 = 1 mm and plate thickness L = 100 μm.
We thus have an expression to determine the thermal diffusivity: D = αr0 /L fmin Lr0 ,
(5)
2
with fmin in Hz, L in m, D in m /s, and r0 in m, and where αr0 /L depends on the ratio r0 /L. Numerical calculations were performed for αr0 /L and the results of αr0 /L are presented in Figure 7. Equation (4) requires the phase shift curve to have a minimum phase shift that depends on the plate thickness and the laser beam radius. The minimum phase shift was observed to be in the range −90◦ to −45◦ . By using (3), we could determine the application limit of Eq. (5): 2 ≤ r0 /L ≤ 100.
IV. VARIATIONS OF THE FREQUENCY fmin OF THE MINIMUM PHASE SHIFT
According to the modeling results of Figures 1–3, the frequency fmin depends on three parameters: the thermal diffusivity (D), the plate thickness (L), and the laser beam radius (r0 ): fmin = f(D, L, r0 ). By using the results from Figures 1–3, we plotted in Figures 4–6 the curves of the frequency fmin versus the thermal diffusivity D, the plate thickness L, and the laser beam radius r0 . Then, linear regression was employed to determine the relation between these parameters. Figures 4–6 point to the frequency fmin varying linearly with the thermal diffusivity and almost linearly with the inverse of the plate thickness and the inverse of the laser beam radius. It was thus concluded that the expression of fmin can be generally written as fmin = f (L, r0 )DL−1 r0−1 , where f(L, r0 ) depends on L and r0 .
(4) FIG. 5. Frequency fmin as a function of the inverse of plate thickness. Laser beam radius r0 = 1 mm and thermal diffusivity D = 0.1 cm2 /s.
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054903-4
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
FIG. 8. Scheme of the experimental setup.
FIG. 6. Frequency fmin as a function of the inverse of laser beam radius. The plate thickness L = 100 μm and thermal diffusivity D = 0.1 cm2 /s.
V. EXPERIMENTAL RESULTS
The schema for the experimental set up is presented in Figure 8. We used the fiber laser MANLIGHT ML50-CWR-TKS to heat the sample with an average power from 5 W to 10 W. The beam radius r0 was 1740 μm (radius at 1/e intensity) and the wavelength λ was 1080 nm. The laser beam was guided to the sample by a beam splitter with 99% power transmission, and the 1% reflected laser power was collected by a photodiode. The thermal flux of the sample was measured by an IR detector VIGO PVMI-3TE-10.6 with a useful spectral range λ from 2 μm to 10 μm and a sensitive area of 1 mm2 . The thermal flux was focused on the IR detector by a ZnSe convex lens. In order to avoid that the diffuse reflection
FIG. 7. Values of αr0 /L for different values of the r0 /L ratio.
of the laser entered the detector center (there is low sensitivity on 1 μm wavelength), a Germanium filter was positioned in front of the detector. Thus, only wavelengths ranging between 2 μm and 14 μm were transmitted. We used the lock-in amplifier SR530 to determine the phase shift between the laser power signal and the thermal signal. Our detector best performs in the spectral range from 6 μm to 8 μm, therefore we chose to heat the sample around 400 K ± 50 K. The temperature amplitude in the stationary regime was inferior to 10 K and the thermal properties could thus be considered constant during the heating. The IR detector was set to receive the thermal flux coming from the center of the heating on the sample surface. In our method, the accuracy of the measurement of the phase shift and modulation frequency is important to determine the frequency at the minimum phase shift with high accuracy. In order to measure the phase shift ϕ thermal − laser between the laser power and the thermal signal, we measured first the phase-shift ϕ 1 between the signal of the electric generator and the thermal signal detected by the IR detector ϕ 1 = ϕ thermal − laser + ϕ laser + ϕ IR detector + ϕ lock-in , where ϕ laser , ϕ IR detector , and ϕ lock-in are, respectively, the internal phase shifts of the laser, the IR detector, and the lock-in amplifier. Second, the phase-shift ϕ 2 between the signal of the electric generator and the laser power detected by the photodiode ϕ 2 = ϕ laser + ϕ photodiode + ϕ lock-in , where ϕ photodiode is the internal phase shift of the photodiode. The phase shift ϕ thermal − laser was ϕ thermal − laser = ϕ 1 − ϕ 2 − ϕ IR detector + ϕ photodiode . The phase shift ϕ photodiode − ϕ IR detector was measured by using a chopper (Stanford SR540) and the laser in continuous regime. We measured the phase shift ϕ 3 between the chopper and the photodiode ϕ 3 = ϕ chopper + ϕ photodiode + ϕ lock-in and the phase shift ϕ 4 between the chopper and the IR detector (without germanium filter) ϕ 4 = ϕ chopper + ϕ IR detector + ϕ lock-in , where ϕ chopper is the internal phase shift of the chopper. Finally, we had ϕ thermal − laser = ϕ 1 − ϕ 2 + ϕ 3 − ϕ 4 with all internal phase shifts of the instrument included. The final accuracy of the phase shift measurement was ≈98%. The frequency of the electric
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054903-5
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
FIG. 9. Phase shift as a function of frequency for nickel, tungsten, zinc, 304 L stainless steel, titanium, molybdenum, and iron. The incertitude of the phase shift was ±1◦ ; the incertitude of the frequency was ±0.5%. Laser beam radius r0 = 1740 μm.
generator was measured by the lock-in amplifier with ≈99.5% accuracy. Figures 9(a)–9(d) present the experimental results of the phase shift as a function of frequency for different materials: 304 L stainless steel, nickel, titanium, tungsten, molybdenum, iron, and zinc. From the figures, the minimum phase shift min and its frequency fmin were determined. Subsequently, the thickness and the thermal diffusivity were calculated by using Eqs. (3) and (5) and are presented in Table I. VI. DISCUSSION AND CONCLUSION
Table I shows that the samples thickness Lm obtained by Eq. (3) corresponded to the values Lc measured by the height gauge TESA-HITE 400. The accuracy of Lm was higher than that of Lc but the advantage of Lc measurements is the fact that the measurements were contactless. The measured values of both Dm and Dc corresponded to the reference values Dref . The incertitude of Dm ranges
from 5% to 10% and that of Dc from 10% to 15%. The incertitude of Dc and Dm depended mainly on the precision of the determination of the frequency fmin at the minimum phase-shift. The accuracy of Dm was higher than that of Dc because of the precision of the sample thickness measurement. While Dc was obtained by a contactless measurement of L, the values of Dm had to be measured with a contact measurement of L. Most of the thermal diffusivity results were underestimated with the reference values; this could be caused by an underestimation of fmin . It should also be noted that the reference values depend on the temperature and the chemical composition or purity of the sample. We are working on the optimization of the determination of fmin and the temperature measurement to increase the accuracy of the results. We have herein proposed and validated a new method for measuring the thermal diffusivity of a plate. When the plate thickness and the minimum phase shift between the laser and
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054903-6
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
TABLE I. Heating parameters and thermal diffusivity measured for nickel, tungsten, zinc, 304 L stainless steel, titanium, molybdenum, and iron. Comparison with reference values.12 Titanium 127 μm T (K)a r0 (μm)b min (deg)c fmin (Hz)d Lm (μm)e Lc (μm)f Dm (cm2 /s)g Dc (cm2 /s)h Dref (cm2 /s)i
Titanium 456 μm
Iron 496 μm
Molybdenum 210 μm
Nickel 100 μm
Nickel 506 μm
Tungsten 315 μm
Zinc 508 μm
304 L stainless steel 400 μm
400 ± 5 410 ± 5 370 ± 5 360 ± 5 410 ± 5 350 ± 5 350 ± 5 370 ± 5 450 ± 5 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 −81 ± 1 −60.3 ± 1 −58.8± 1 −74.8 ± 1 −82.5 ± 1 −58.8 ± 1 −69.5 ± 1 −59.2 ± 1 −64.8 ± 1 18.5 ± 2 4.25 ± 0.5 9±1 77.5 ± 10 50 ± 5 9±1 55 ± 5 19 ± 2 2.5 ± 0.3 127 ± 2.5 456 ± 2.5 496 ± 2.5 210 ± 2.5 100 ± 2.5 506 ± 2.5 315 ± 2.5 508 ± 2.5 400 ± 2.5 119 ± 14 456 ± 22 488 ± 23 206 ± 15 98 ± 13 487 ± 23 296 ± 17 494 ± 21 385 ± 19 0.076 ± 0.008 0.073 ± 0.007 0.171 ± 0.016 0.538 ± 0.068 0.177 ± 0.016 0.174 ± 0.016 0.618 ± 0.052 0.369 ± 0.033 0.037 ± 0.004 0.071 ± 0.011 0.073 ± 0.008 0.168 ± 0.017 0.528 ± 0.077 0.159 ± 0.027 0.168 ± 0.017 0.538 ± 0.057 0.359 ± 0.034 0.034 ± 0.004 0.082 ± 0.008 0.081 ± 0.008 0.191 ± 0.019 0.520 ± 0.052 0.185 ± 0.018 0.204 ± 0.020 0.626 ± 0.062 0.395 ± 0.039 0.039 ± 0.004
a
Temperature measured by thermocouple. Laser beam radius measured by CCD camera. c Minimum phase shift ϕ min measured on the phase shift curve. d Frequency fmin measured by a numerical algorithm. e Plate thickness measured by height gauge TESA-HITE 400. f Plate thickness calculated with Eq. (3). g Thermal diffusivity measured by Eq. (5) with Lm . h Thermal diffusivity measured by Eq. (5) with Lc . i Thermal diffusivity reference 12 at temperature T(K). b
the thermal response of the front face are known, the thermal diffusivity can be determined. This method is applicable to opaque plate with a thickness L much larger than the laser absorption length 1/α and L much smaller than the plate dimensions. The laser beam radius must be in the range 2 L ≤ r0 ≤ 100 L. This method has several advantages: the process and experimental setup are simple; the phase shift does not depend on the laser power or the optical properties of the plate surface. Such PTR method can be used to determine the thicknesses and thermo-physical properties of plates, to detect the defects of the components and also to measure their variations in time. The understanding of the thermal diffusivity behavior in time, at various temperatures and in severe environments helps to determine the lifetime of components and astutely choose materials for design and construction.
1 O.
Breitenstein and M. Langenkamp, Lock-in Thermography (Springer, Berlin, 2003). 2 G. Busse, D. Wu, and W. Karpen, J. Appl. Phys. 71, 3962 (1992). 3 A. Rosencwaig and J. B. Willis, J. Appl. Phys. 51, 4361 (1980). 4 G. Busse, Appl. Opt. 21, 107 (1982). 5 G. Rousset and F. Lepoutre, Rev. Phys. Appl. 17, 201 (1982). 6 A. Salazar, A. Stinchez-Lavega, and J. Fernández, J. Appl. Phys. 69, 1216 (1991). 7 L. Fabbri et al., Rev. Sci. Instrum. 66, 3593 (1995). 8 A. Mendioroz, R. Fuente-Dacal, E. Apiñaniz, and A. Salazar, Rev. Sci. Instrum. 80, 074904 (2009). 9 A. Salazar, A. Mendioroz, and R. Fuente, Appl. Phys. Lett. 95, 121905 (2009). 10 A. Semerok, F. Jaubert, S. Fomichev, P.-Y. Thro, X. Courtois, and Ch. Grisolia, Nucl. Instrum. Methods Phys. Res., Sect. A 693, 98 (2012). 11 D. Melyukov, Ph.D. thesis, French Atomic Energy and Alternative Energy Commission, 2011. 12 Y. S. Touloukian, R. W. Powell, C. Y. Ho, and M. C. Nicolaou, Thermal Diffusivity (IFI/Plenum, New York, 1973).
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 054903 (2014)
New contactless method for thermal diffusivity measurements using modulated photothermal radiometry S. Pham Tu Quoc,a) G. Cheymol, and A. Semerok French Alternative Energies and Atomic Energy Commission, Division of Nuclear Energy, DEN/DANS/DPC/SEARS/LISL, 91191 Gif/Yvette, France
(Received 17 January 2014; accepted 18 April 2014; published online 9 May 2014) Modulated photothermal radiometry is a non-destructive and contactless technique for the characterization of materials. It has two major advantages: a good signal-to-noise ratio through a synchronous detection and a low dependence on the heating power and the optical properties of the sample surface. This paper presents a new method for characterizing the thermal diffusivity of a material when the phase shift between a modulated laser power signal and the thermal signal of a plate sample is known at different frequencies. The method is based on a three-dimensional analytical model which is used to determine the temperature amplitude and the phase in the laser heating of the plate. A new simple formula was developed through multi-parametric analysis to determine the thermal diffusivity of the plate with knowledge of the frequency at the minimum phase shift, the laser beam radius r0 and the sample thickness L. This method was developed to control the variation of the thermal diffusivity of nuclear components and it was first applied to determine the thermal diffusivity of different metals: 304 L stainless steel, nickel, titanium, tungsten, molybdenum, zinc, and iron. The experimental results were obtained with 5%–10% accuracy and corresponded well with the reference values. The present paper also demonstrates the limit of application of this method for plate with thickness r0 /100 ≤ L ≤ r0 /2. The technique is deemed interesting for the characterization of barely accessible components that require a contactless measurement. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4874195] I. INTRODUCTION
The principle of the photothermal radiometry (PTR)1 consists in heating a sample with a modulated light power and then observing the emitted infrared radiation. In the stationary regime of the heating, the thermal response can be described as a modulated signal giving the amplitude and the phase shift of the temperature. With knowledge of the amplitude and the phase shift at different modulated frequencies, it is possible to characterize the thickness and the thermal properties of the sample. While the temperature amplitude depends on the sample’s optical properties and the light power,2 the phase shift is of great interest since it is independent of these parameters.3, 4 Among thermal properties, the thermal diffusivity plays an important role in the heat transfer process; it measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy. Nowadays, there are several methods to measure these parameters but each one has its disadvantages: can only be applied to bulk materials,5–7 requires use of a small focused beam,8 and requires the variation of the sample surface temperature which can be influenced by the heat loss effect.9 Our concern is to obtain, first, remote and contactless measurements with one face access; second, simple methods with acceptable performance in order to measure in situ the variation of the thermal properties of nuclear components. This paper presents a new type of measurement for the thermal diffusivity that involves determining the phase shift between the laser power and the thermal signal. The method is a) Electronic mail: [email protected].
0034-6748/2014/85(5)/054903/6/$30.00
based on a model of the laser heating of a sample10 that gives an analytical expression for the amplitudes and phases of the sample surface temperature with constant optical and thermophysical parameters during the laser heating. The variation of the temperature amplitude is very small in the stationary regime (of the order of a few Kelvins), and through multiparameter analysis, a simple formula has been obtained to determine the thermal diffusivity of a plate when the following parameters are known: fmin , i.e., the frequency at the minimum phase shift of the curve plotting the phase shift versus the frequency, the laser beam radius r0 and the sample thickness L. This formula was verified by the experimental setup. II. 3D THERMAL MODEL OF THE MODULATED LASER HEATING FOR PLATE IN AIR
Below is presented the thermal model for the laser heating of homogeneous and isotropic plate with infinite dimensions. The variation of optical and thermal properties with temperature, the surface roughness, and the heat exchange by convection between the sample surfaces and the air were supposed to be negligible. By applying the Fourier series analysis to the intensity of the laser beam and the temperature in the stationary regime of the laser heating,10 the complex temperature amplitude of the front face for plate sample can be written as +∞ α(1 − R)iI0 (ξ ) ˜ T (r) = − 2 k ξ − α 2 − 2πifk Cv 0 α 2α(e−L − e−αL ) + − 1 J0 (ξ r)dξ, (1) × (eL − e−L )
85, 054903-1
© 2014 AIP Publishing LLC
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054903-2
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
ξ r2 ξ 2r2 with: = ξ 2 − 2πifk Cv ; (ξ ) = 20 exp(− 4 0 ) for a Gaussian beam, where r is the radial distance from the center of the heated zone at the sample surface; Cv , k, and L are, respectively, the volumetric specific heat, the thermal conductivity, and the thickness of the sample; α and R are, respectively, the laser absorption coefficient and the reflectivity of the sample surface; r0 and I0 are, respectively, the laser beam radius at 1/e intensity and the mean intensity of the Gaussian laser beam; f and i are, respectively, the repetitive rate frequency of the laser and the complex unity; and ξ and J0 are, respectively, the variables of integration and the zero order Bessel functions of the first kind. The phase shift between the laser and the thermal response of the front surface can be described as = a tan(Real(T˜ (r)) / I mag(T˜ (r))).
(2)
This model can be used to fit the theoretical phase-shift to the experimental results in order to characterize the sample material and its thickness. Our objective is to develop a simple characterization method with good precision. We thus performed multi-parametric modeling to determine the dependence of particular points in the phase-shift vs. frequency curve which can provide us with simple relations between different parameters. III. MULTI-PARAMETRIC MODELING OF THE PHASE SHIFT FOR PLATE IN AIR
We successively analyzed, one at a time, the influence of different interactions and plate parameters on the phase shift, including: intensity of the laser beam I0 , surface reflectivity R, absorption coefficient α, thermal conductivity k, volumetric specific heat Cv , laser beam radius r0 , plate thickness L, and measured point position r. According to the modeling results, the following was concluded with regard to the influence of the different parameters on the phase shift: –The phase shift does not depend on the intensity of the laser beam and the surface reflectivity. –The influence of the absorption coefficient α is negligible for opaque sample which signifies that the laser photon is only absorbed at a shallow depth (αL 1, αμ 1, μ is the thermal diffusion length). –The influence of the measured point position r is negligible for r < r0 /2. –The thermal conductivity and volumetric specific heat influence the phase shift by their ratio k/Cv (thermal diffusivity D) and this influence is presented in Figure 1. It was observed that the minimal phase shift min is independent of the thermal diffusivity D. A formula was found to determine the thickness (L) when the minimum phase shift (min ) and the Gaussian laser beam radius (r0 )11 were known: r0 0.2273min + 57.8562min L= 100 +5688.2min + 208620) . (3)
FIG. 1. Phase shift as a function of frequency for different values of thermal diffusivity D of the plate. Laser beam radius r0 = 1 mm and plate thickness L = 100 μm.
–The influence of the plate thickness and the laser beam radius of the phase shift curve are presented in Figures 2 and 3. We can see that the level of the minimum phase shift depended on L and r0 . When the laser beam radius was much larger than the sample thickness, the thermal model became a 1D-heating, the heating propagation took the same direction as the propagation of the laser beam, and the minimum phase shift approaches the asymptote −90◦ . Inversely, when the laser beam radius is much smaller than the sample
FIG. 2. Phase shift as a function of frequency for different values of plate thickness L. Laser beam radius r0 = 1 mm and thermal diffusivity D = 0.1 cm2 /s.
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054903-3
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
FIG. 3. Phase shift as a function of frequency for different values of laser beam radius. Plate thickness L = 100 μm and thermal diffusivity D = 0.1 cm2 /s.
thickness and the diffusion length, the thermal model becomes a 3D-heating with heating on a dimensionless surface point. Subsequently, the maximum phase shift approaches the asymptote 0◦ and the phase shift curve does not have a local minimum. While the minimum phase shift does not render it is possible to determine the thermal properties, since it only varies with the thickness and the beam radius, according to (3), the frequency of the minimum phase shift seems vary linearly with the thermal diffusivity, cf. Figure 1, and could provide us with information about the thermal diffusivity. Now, we will focus on the frequency fmin of the minimum phase shift.
FIG. 4. Frequency fmin as a function of the thermal diffusivity. Laser beam radius r0 = 1 mm and plate thickness L = 100 μm.
We thus have an expression to determine the thermal diffusivity: D = αr0 /L fmin Lr0 ,
(5)
2
with fmin in Hz, L in m, D in m /s, and r0 in m, and where αr0 /L depends on the ratio r0 /L. Numerical calculations were performed for αr0 /L and the results of αr0 /L are presented in Figure 7. Equation (4) requires the phase shift curve to have a minimum phase shift that depends on the plate thickness and the laser beam radius. The minimum phase shift was observed to be in the range −90◦ to −45◦ . By using (3), we could determine the application limit of Eq. (5): 2 ≤ r0 /L ≤ 100.
IV. VARIATIONS OF THE FREQUENCY fmin OF THE MINIMUM PHASE SHIFT
According to the modeling results of Figures 1–3, the frequency fmin depends on three parameters: the thermal diffusivity (D), the plate thickness (L), and the laser beam radius (r0 ): fmin = f(D, L, r0 ). By using the results from Figures 1–3, we plotted in Figures 4–6 the curves of the frequency fmin versus the thermal diffusivity D, the plate thickness L, and the laser beam radius r0 . Then, linear regression was employed to determine the relation between these parameters. Figures 4–6 point to the frequency fmin varying linearly with the thermal diffusivity and almost linearly with the inverse of the plate thickness and the inverse of the laser beam radius. It was thus concluded that the expression of fmin can be generally written as fmin = f (L, r0 )DL−1 r0−1 , where f(L, r0 ) depends on L and r0 .
(4) FIG. 5. Frequency fmin as a function of the inverse of plate thickness. Laser beam radius r0 = 1 mm and thermal diffusivity D = 0.1 cm2 /s.
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054903-4
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
FIG. 8. Scheme of the experimental setup.
FIG. 6. Frequency fmin as a function of the inverse of laser beam radius. The plate thickness L = 100 μm and thermal diffusivity D = 0.1 cm2 /s.
V. EXPERIMENTAL RESULTS
The schema for the experimental set up is presented in Figure 8. We used the fiber laser MANLIGHT ML50-CWR-TKS to heat the sample with an average power from 5 W to 10 W. The beam radius r0 was 1740 μm (radius at 1/e intensity) and the wavelength λ was 1080 nm. The laser beam was guided to the sample by a beam splitter with 99% power transmission, and the 1% reflected laser power was collected by a photodiode. The thermal flux of the sample was measured by an IR detector VIGO PVMI-3TE-10.6 with a useful spectral range λ from 2 μm to 10 μm and a sensitive area of 1 mm2 . The thermal flux was focused on the IR detector by a ZnSe convex lens. In order to avoid that the diffuse reflection
FIG. 7. Values of αr0 /L for different values of the r0 /L ratio.
of the laser entered the detector center (there is low sensitivity on 1 μm wavelength), a Germanium filter was positioned in front of the detector. Thus, only wavelengths ranging between 2 μm and 14 μm were transmitted. We used the lock-in amplifier SR530 to determine the phase shift between the laser power signal and the thermal signal. Our detector best performs in the spectral range from 6 μm to 8 μm, therefore we chose to heat the sample around 400 K ± 50 K. The temperature amplitude in the stationary regime was inferior to 10 K and the thermal properties could thus be considered constant during the heating. The IR detector was set to receive the thermal flux coming from the center of the heating on the sample surface. In our method, the accuracy of the measurement of the phase shift and modulation frequency is important to determine the frequency at the minimum phase shift with high accuracy. In order to measure the phase shift ϕ thermal − laser between the laser power and the thermal signal, we measured first the phase-shift ϕ 1 between the signal of the electric generator and the thermal signal detected by the IR detector ϕ 1 = ϕ thermal − laser + ϕ laser + ϕ IR detector + ϕ lock-in , where ϕ laser , ϕ IR detector , and ϕ lock-in are, respectively, the internal phase shifts of the laser, the IR detector, and the lock-in amplifier. Second, the phase-shift ϕ 2 between the signal of the electric generator and the laser power detected by the photodiode ϕ 2 = ϕ laser + ϕ photodiode + ϕ lock-in , where ϕ photodiode is the internal phase shift of the photodiode. The phase shift ϕ thermal − laser was ϕ thermal − laser = ϕ 1 − ϕ 2 − ϕ IR detector + ϕ photodiode . The phase shift ϕ photodiode − ϕ IR detector was measured by using a chopper (Stanford SR540) and the laser in continuous regime. We measured the phase shift ϕ 3 between the chopper and the photodiode ϕ 3 = ϕ chopper + ϕ photodiode + ϕ lock-in and the phase shift ϕ 4 between the chopper and the IR detector (without germanium filter) ϕ 4 = ϕ chopper + ϕ IR detector + ϕ lock-in , where ϕ chopper is the internal phase shift of the chopper. Finally, we had ϕ thermal − laser = ϕ 1 − ϕ 2 + ϕ 3 − ϕ 4 with all internal phase shifts of the instrument included. The final accuracy of the phase shift measurement was ≈98%. The frequency of the electric
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054903-5
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
FIG. 9. Phase shift as a function of frequency for nickel, tungsten, zinc, 304 L stainless steel, titanium, molybdenum, and iron. The incertitude of the phase shift was ±1◦ ; the incertitude of the frequency was ±0.5%. Laser beam radius r0 = 1740 μm.
generator was measured by the lock-in amplifier with ≈99.5% accuracy. Figures 9(a)–9(d) present the experimental results of the phase shift as a function of frequency for different materials: 304 L stainless steel, nickel, titanium, tungsten, molybdenum, iron, and zinc. From the figures, the minimum phase shift min and its frequency fmin were determined. Subsequently, the thickness and the thermal diffusivity were calculated by using Eqs. (3) and (5) and are presented in Table I. VI. DISCUSSION AND CONCLUSION
Table I shows that the samples thickness Lm obtained by Eq. (3) corresponded to the values Lc measured by the height gauge TESA-HITE 400. The accuracy of Lm was higher than that of Lc but the advantage of Lc measurements is the fact that the measurements were contactless. The measured values of both Dm and Dc corresponded to the reference values Dref . The incertitude of Dm ranges
from 5% to 10% and that of Dc from 10% to 15%. The incertitude of Dc and Dm depended mainly on the precision of the determination of the frequency fmin at the minimum phase-shift. The accuracy of Dm was higher than that of Dc because of the precision of the sample thickness measurement. While Dc was obtained by a contactless measurement of L, the values of Dm had to be measured with a contact measurement of L. Most of the thermal diffusivity results were underestimated with the reference values; this could be caused by an underestimation of fmin . It should also be noted that the reference values depend on the temperature and the chemical composition or purity of the sample. We are working on the optimization of the determination of fmin and the temperature measurement to increase the accuracy of the results. We have herein proposed and validated a new method for measuring the thermal diffusivity of a plate. When the plate thickness and the minimum phase shift between the laser and
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054903-6
Pham Tu Quoc, Cheymol, and Semerok
Rev. Sci. Instrum. 85, 054903 (2014)
TABLE I. Heating parameters and thermal diffusivity measured for nickel, tungsten, zinc, 304 L stainless steel, titanium, molybdenum, and iron. Comparison with reference values.12 Titanium 127 μm T (K)a r0 (μm)b min (deg)c fmin (Hz)d Lm (μm)e Lc (μm)f Dm (cm2 /s)g Dc (cm2 /s)h Dref (cm2 /s)i
Titanium 456 μm
Iron 496 μm
Molybdenum 210 μm
Nickel 100 μm
Nickel 506 μm
Tungsten 315 μm
Zinc 508 μm
304 L stainless steel 400 μm
400 ± 5 410 ± 5 370 ± 5 360 ± 5 410 ± 5 350 ± 5 350 ± 5 370 ± 5 450 ± 5 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 1740 ± 30 −81 ± 1 −60.3 ± 1 −58.8± 1 −74.8 ± 1 −82.5 ± 1 −58.8 ± 1 −69.5 ± 1 −59.2 ± 1 −64.8 ± 1 18.5 ± 2 4.25 ± 0.5 9±1 77.5 ± 10 50 ± 5 9±1 55 ± 5 19 ± 2 2.5 ± 0.3 127 ± 2.5 456 ± 2.5 496 ± 2.5 210 ± 2.5 100 ± 2.5 506 ± 2.5 315 ± 2.5 508 ± 2.5 400 ± 2.5 119 ± 14 456 ± 22 488 ± 23 206 ± 15 98 ± 13 487 ± 23 296 ± 17 494 ± 21 385 ± 19 0.076 ± 0.008 0.073 ± 0.007 0.171 ± 0.016 0.538 ± 0.068 0.177 ± 0.016 0.174 ± 0.016 0.618 ± 0.052 0.369 ± 0.033 0.037 ± 0.004 0.071 ± 0.011 0.073 ± 0.008 0.168 ± 0.017 0.528 ± 0.077 0.159 ± 0.027 0.168 ± 0.017 0.538 ± 0.057 0.359 ± 0.034 0.034 ± 0.004 0.082 ± 0.008 0.081 ± 0.008 0.191 ± 0.019 0.520 ± 0.052 0.185 ± 0.018 0.204 ± 0.020 0.626 ± 0.062 0.395 ± 0.039 0.039 ± 0.004
a
Temperature measured by thermocouple. Laser beam radius measured by CCD camera. c Minimum phase shift ϕ min measured on the phase shift curve. d Frequency fmin measured by a numerical algorithm. e Plate thickness measured by height gauge TESA-HITE 400. f Plate thickness calculated with Eq. (3). g Thermal diffusivity measured by Eq. (5) with Lm . h Thermal diffusivity measured by Eq. (5) with Lc . i Thermal diffusivity reference 12 at temperature T(K). b
the thermal response of the front face are known, the thermal diffusivity can be determined. This method is applicable to opaque plate with a thickness L much larger than the laser absorption length 1/α and L much smaller than the plate dimensions. The laser beam radius must be in the range 2 L ≤ r0 ≤ 100 L. This method has several advantages: the process and experimental setup are simple; the phase shift does not depend on the laser power or the optical properties of the plate surface. Such PTR method can be used to determine the thicknesses and thermo-physical properties of plates, to detect the defects of the components and also to measure their variations in time. The understanding of the thermal diffusivity behavior in time, at various temperatures and in severe environments helps to determine the lifetime of components and astutely choose materials for design and construction.
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