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Photothermal investigations of thermal and optical properties of liquids by mirage effect
Sensors and Actuators A 138 (2007) 335–340
Photothermal investigations of thermal and optical properties of liquids by mirage effect Faycel Saadallah ∗ , Leila Attia, Sameh Abroug, Noureddine Yacoubi Equipe de Photothermique de Nabeul, Institut Pr´eparatoire aux Etudes d’Ing´enieur de Nabeul, BP. 62, Merazka 8000, Nabeul, Tunisia Received 13 September 2006; received in revised form 12 April 2007; accepted 27 May 2007 Available online 2 June 2007
Abstract In this paper, the “mirage effect” detection technique was used in order to determine the thermal diffusivity and the absorption spectrum of liquids. Experiments were more particularly realized with paraffin oil. Generally, since the thermal diffusion length is relatively short in liquids, the different parts of the probe beam are differently deflected by the heat flow. So, a new mathematical expression of the deflection related to the probe beam dimensions was established to calculate the “effective” deflection in liquids. Such an expression was used to measure the thermal diffusivity of paraffin oil. Then, the “mirage effect” technique was also used to obtain indirectly the optical spectrum of paraffin oil through the measurement of the absorption of a carbon black layer immersed in this liquid. © 2007 Elsevier B.V. All rights reserved. Keywords: Photothermal deflection; Liquids; Thermal diffusivity; Optical absorption
1. Introduction The mirage effect technique had been widely used [1–6] in the investigation of thermal and optical properties of solid materials. However, few photothermal investigations were concerned with liquids [7–13]. Some studies have used mirage detection in its collinear [12] or rear [13] configurations. In collinear scheme, a three-dimensional model must be used, and convection is not negligible. Rear detection has been only used for highly absorbent liquids. In this paper, we have improved the frontal-scheme of mirage detection in order to determine the optical absorption spectrum and the thermal diffusivity of liquids. The liquid absorption spectrum is indirectly determined from the measurement of the absorption of a black carbon layer immersed in the liquid. The deflection of probe beam is mainly caused by the heat flow induced by the carbon layer which is heated by a modulated and uniform pump beam. The heat propagation and the temperature gradient are in one direction perpendicular to the layer surface. Therefore, we have to use simple theoretical calculations, based on the resolution of the one-dimension heat equation ∗
Corresponding author. Tel.: +216 98248252; fax: +216 72220181. E-mail address: [email protected] (F. Saadallah).
0924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2007.05.022
[2–6], from which we have deduced the probe beam deflection. For thermal diffusivity measurement we have to vary the modulation frequency. In liquids, when the frequency increase the thermal diffusion length becomes small inducing the deflection of only a part of the probe beam. Therefore, the theoretical model will be modified so as to consider the dimensions of the probe beam. 2. Theoretical model 2.1. Principle of the mirage effect technique The absorption of a modulated and uniform light beam by a solid sample induces a heat flow in both the surrounding liquid and the sample. The liquid refractive index, induced by temperature variations near the surface, causes the deflection of a probe laser beam skimming the surface (Fig. 1). This deflection could be related to thermal and optical properties of the sample and deflecting liquid. In a glass cell fulfilled with paraffin oil (Fig. 2) a plexiglass plate covered with a thin black carbon layer is introduced. This black layer is a good absorbent in a wide wavelength scale. The cell is illuminated with a modulated and uniform heating beam whose intensity could be written as: I = I0 [1 + exp(jωt)].
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Fig. 2. Schema of the studied sample.
The amplitude is Fig. 1. Schema of the probe beam deflection near the sample surface.
|Ψ (x)| = − 2.2. Classical expression of the probe beam deflection ψ
1 dn √ L 2|Ts | exp n dT μf
−x μf
(7)
and the phase is When the dimension of the probe beam is neglected, and for small temperature variations, the beam deflection could be expressed as a function of the liquid temperature Tf [1]: 1 dn L/2 dTf dx =− Ψ= dy (1) dy n dTf −L/2 dx when heat propagates in one direction and the fluid is nonabsorbent, the expression of the fluid temperature could be written as [3]: Tf (x, t) = Ts exp(−σf x) exp(jωt)
(2)
where Ts is the temperature at the solid surface in contact with the transparent fluid, x the distance between probe beam and solid surface, n the liquid refractive index and L is the width of the solid sample. 1/2 πF jω 1+j = (1 + j) = (3) σf = Df Df μf
φ=−
x π +θ+ μf 4
2.3. Determination of the surface temperature Ts In order to determine the temperature distribution described by Eq. (2), we have to calculate the surface temperature Ts . The phenomena of thermal convection and thermal emission are neglected in mirage technique. So, only the thermal conduction in the fluid and the black sample will be considered. Therefore, we can express the heat diffusion in the pump beam direction, by the following equations [2–6]: ∂ 2 Tg 1 ∂Tg = , 2 ∂x Dg ∂t
0 ≤ x ≤ h1
Ts = |Ts |exp(jθ)
1 ∂Tb ∂ 2 Tb = , ∂x2 Db ∂t
Finally we obtain
1 dn (1 + j) x Ψ (x, t) = − L|Ts |exp − n dT μf μf x π × exp j − + θ + + ωt μf 4
(6)
The amplitude and phase (argument) of the deflection, which correspond to the amplitude and phase of the theoretical photothermal signal. Ts = − −
h1 ≤ x ≤ lv + h1
(9)
∂ 2 Tf 1 ∂Tf = − Af exp[αf (x − h1 )][1 + exp(jωt)], Df ∂t ∂x2
F = ω/2π is the modulation frequency, Df the thermal diffusivity and μf is the thermal diffusion length in the liquid. Then deflection becomes: 1 dn Ψ∼ (4) Ts σf L exp(−σf x) exp(jωt) =− n dT The surface temperature could be written as (5)
(8)
(10)
1 ∂Tc ∂ 2 Tc = − Ac exp(αc x)[1 + exp(jωt)], ∂ 2 x2 Dc ∂t −1 ≤ x ≤ 0
(11) − lb − 1 ≤ x ≤ −1
(12)
where Ti , Di and αi are, respectively, temperature, thermal diffusivity and optical absorption coefficient in each medium i. The abbreviations g, f, c and b correspond, respectively, to glass, deflecting fluid (paraffin oil), carbon layer, and backing (plexiglass) (Fig. 2). The terms: Af exp[αf (x − h1 )] and Ac exp(αc x) describe the optical absorption contributions of the fluid and the black layer. By using the continuity condition equations for temperature and heat flux at each interface, we obtain the expression of the temperature Ts in the fluid medium given by:
{Ccf S0 (L0 + 2d0 M0 )Z0 + (1/2)[S0 (L1 + 2M1 d1 ) + F0 ]Zf } exp[σf (h1 − x1 )] B1 {Ccf S1 (L0 + 2d0 M0 )Z0 + (1/2)[S1 (L1 + 2M1 d1 ) + F1 ]Zf } exp[−σf (h1 − x1 ) B1
+ [αf I0 /2Kf (α2f − σf2 )] exp[αf (x − h1 )]
(13)
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where Ts is the temperature at the solid–fluid interface (x = 0), Af , Ac , Ag , B, U,√ V, Ts , Zf , Z0 , W are√complex constants given below. σi = (1 + j) πf/Di and μi = Di /πf is the thermal diffusion length. Z0 = αs I0 exp(−αf h1 )/2Kc (α2c − σc2 ); Zf = αf I0 / 2Kf (α2f − σf2 ), M0 = Cbc − rc , M1 = b0 Q0 (rf − Ccf ) − P0 a0 (rf + Ccf ), Q0 = −(1 + Cbc ), Q1 = a0 P0 (Ccf − 1) + b0 Q0 (Ccf + 1), P0 = 1 + Cbc , P1 = a0 P0 (Ccf + 1) + b0 Q0 (Ccf − 1), S0 = 1 + Cfg , S1 = 1/Cfg , F0 = B1 (rf − 1), F1 = −B1 (rf + 1), B1 = a0 P0 (1 + rc ) + b0 Q0 (1 − rc ), L0 = a0 P0 (1 + rc ) + b0 Q0 (1 − rc ), L1 = a1 P1 (rf + 1) − b1 Q1 (rf − 1), Ccf = kc σ c /kf σ f , Cfg = kf σ f /kg σ g , Cbc = kb σ b /kc σ c , a0 = exp(−σ c l), a1 = exp (−σ f h1 ), b0 = exp(σ c l), b1 = exp(σ f h1 ), d0 = exp(−αc h), d1 = exp(αc h), rf = αf σ f , rs = αc σ c . 3. Experimental results In this work, we have studied the experimental variations of the amplitude and phase of the photothermal signal. From the amplitude variations versus the heating beam wavelength, we have obtained the absorption spectrum of the paraffin oil. However, during the measurement of its thermal diffusivity we should vary the modulation frequency, which affects the thermal diffusion length in liquid. 3.1. Experimental setup Experimental apparatus is well described in reference [6]. As it is shown in Fig. 1, a probe laser beam (630 nm) is deflected by the heat flow from a black layer heated by a uniform, modulated and monochromatic light beam. This black layer is placed in a glass cell and immersed in a deflecting liquid (paraffin oil or CCl4 ). 3.2. Determination of the thermal diffusivity of the paraffin oil The above calculations (Eqs. (1) and (13)) of the probe beam deflection are done without considering the beam dimensions. The obtained deflection expression could be only used for the determination of the liquid optical properties. However, for the determination of its thermal diffusivity, we need to vary the modulation frequency. Therefore, the beam dimensions must be considered, because the thermal diffusion length μf become lower than the beam diameter for high frequencies, so only one part of the beam is deflected. Therefore, we must divide the probe beam into many small beams [14], each of them will reach one of the tow photodetector cells. The difference between the light powers absorbed by each cell corresponds to the measured photothermal signal which should be proportional to the theoretical effective deflection. 3.2.1. Measurement of the distance x0 between probe beam axis and solid surface When the dimensions of the probe beam are neglected and the fluid is transparent, the photothermal signal could be expressed as a function of the modulation frequency and the distance x
Fig. 3. Phase of the photothermal signal, vs. f1/2 , air is used as the deflecting medium.
between the deflected beam and the sample surface: 1 dn (1 + j) x Ψ (x, t) = − L|Ts |exp − n dT μf μf x π × exp j − + θ + + ωt , μf 4 1/2 Π 1/2 1/2 Ψ = Cf exp −(1 + j) f x Df Then we can write the logarithm of the amplitude as 1/2 Π xf 1/2 log|Ψ | = C1 + log(f 1/2 ) − Df and the phase as 1/2 Π xf 1/2 Φ = K1 − Df
(14)
(15)
(16)
√ where C1 = log(|Ts |(L/n)(∂n/∂Tf ) 2Π/Df ), K1 = θ + Π/4, θ = arg (Ts ) The above equations show linear variations of the amplitude logarithm log |Ψ | and the phase with square root frequency f1/2 . The curves in Figs. 3 and 4 represent, respectively, the phase and amplitude of the experimental signal versus f1/2 . These curves are obtained when air is used as the deflecting fluid. Air is transparent and its thermal diffusion length is higher than the beam diameter, so the entire beam is uniformly deflected. The x0 value is determined from the slope of the phase or the amplitude logarithm curves for high frequencies, we find x0 = 108 m. 3.2.2. Calculation of the effective deflection Without changing the glass cell position, we replace air by paraffin oil. This liquid is used in its transparency region in order to exclude the absorption contribution to the signal. Before heating, the probe beam that has a cylindrical form is not deflected, and its impact on the photodetector is a disc with a radius R. However, after deflection this form is not conserved. So we should divide the beam into infinity of small beams. The
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is the sum of all the small beams power that falls on its surface. Furthermore, the probe beam has a Gaussian power distribution with intensity given by −2[(x − x0 )2 + y2 ] I(x, y) = I0 exp R2 I0 represents the maximum at the probe beam axis (x = x0 ). Then we can express the tow powers by X0 Ymax P1 = I(X, y)dXdy (20) Xmin
Fig. 4. Amplitude of the photothermal signal vs. f1/2 , air is used as the deflecting medium.
distance between a small beam having a thickness dr, and the probe beam axis is r, and so we can write y2 + (x − x0 )2 = r2
(17)
here x and y are the coordinates of the small beam. In addition, x represents the distance between the sample surface and the small beam. After deflection, the beam impact is not a disk [15], but it has another shape (Fig. 5) limited in the X direction by Xmin and Xmax . Since each small beam has different deflection from the others, its abscise on the photodetector will change from x into X given by X = D|Ψ (x)| + x
(18)
where D is the distance between the sample and the photodetector. The small beam deflection amplitude |ψ(x)| is given in Eq. (7). Moreover, there is no deflection in the y axis direction because the heating beam intensity is uniform, so the y abscise of the ray does not change, then from Eq. (17) we can write 1/2
y = ±[r 2 − (x − x0 )]
Ymin
on the first cell. Xmax Ymax P2 = I(X, y)dXdy X0
(21)
Ymin
on the second cell Where the limits of the probe beam impact
on the photodetector are given by Ymax = −Ymin = R2 − (x − x0 )2 , Xmin = D|Ψ (x0 − R)| + x0 − R and Xmax = D|Ψ (x0 + R)| + x0 + R.The effective deflection amplitude is then Ψeff = P2 − P1
(22)
This expression is in good agreement with the classical deflection expression if the deflecting medium has high thermal diffusivity such as air. However, there is no agreement when liquids are used. 3.2.3. Determination of the thermal diffusivity In the first experience air is used as the deflecting medium. In the second, air is replaced with paraffin oil, without changing the distance x0 . In order to determine the thermal diffusivity of the paraffin oil we plot the theoretical effective deflection amplitude as a function of the square root frequency for different values of the thermal diffusivity (Fig. 6). The best diffusivity value
(19)
The effective deflection corresponds to the absorbed light power difference between the tow cells. The power absorbed by a cell
Fig. 5. Deformation of the probe beam section after deflection.
Fig. 6. Experimental photothermal amplitude (open dots) and the effective deflection amplitude calculated for three different values of the thermal diffusivity: 10−8 (dashed line), 4 × 10−8 (solid line) and 7 × 10−8 m2 s−1 (dotted line).
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Fig. 9. Photothermal spectrum of paraffin oil. Fig. 7. Photothermal spectrum of the heating lamp (measured amplitude in CCl4 ).
obtained while fitting theoretical curve to experimental one, is 4 × 10−8 m2 s−1 . 3.3. Determination of the paraffin oil absorption spectrum When using a low modulation frequency (8 Hz) the probe beam is considered to be uniformly deflected. So, the liquid absorption spectrum can be deduced from Eq. (1). 3.3.1. The lamp spectrum In a first step, as a reference signal, the photothermal spectrum of the lamp is deduced. The glass cell (Fig. 2) is fulfilled with CCl4 which is a transparent liquid, so all the pump power will be absorbed by the carbon layer. Thereby, the measured signal amplitude will correspond to the photothermal spectrum of the lamp shown in Fig. 7. 3.3.2. Photothermal spectrum of paraffin oil When we replace CCl4 with paraffin oil, we obtain the curve given in Fig. 8. This curve presents three absorption regions near the following wavelengths: 1.2, 1.4 and 1.7 m. The normalized amplitude shown in Fig. 8 represents the ratio of the photothermal amplitude of the paraffin oil to the corresponding amplitude of CCl4 (Fig. 9).
Fig. 8. Measured amplitude of the photothermal signal obtained in paraffin oil.
3.3.3. Theoretical amplitude Complex expression of the theoretical deflection is obtained from Eqs. (1) and (13). The variations of its normalized amplitude with oil absorption coefficient (αf ) are given in Fig. 10. For weak absorption coefficient values (αf < 0.1 cm−1 ), the liquid is totally transparent and all the heating power is absorbed by the black layer, inducing a saturation of the amplitude. When αf increases, the liquid absorption reduces the amount of light that reaches the black layer, leading to the decrease of the amplitude. When the light power is entirely absorbed by the liquid, the amplitude saturates once again. Indeed, as the absorption occur near the liquid–glass interface which is 4 mm (Fig. 1) distant from the probe beam, the heat flow coming from this absorption has not any contribution to the photothermal signal. 3.3.4. Optical absorption spectrum of the paraffin oil The optical absorption spectrum is obtained by fitting the experimental amplitude to the theoretical one. As the amplitude saturates for high and low values of αf , we can only determine the medium values between 0.1 and 10 cm−1 . While comparing, the experimental and theoretical amplitudes, we deduce the optical absorption spectrum of the paraffin oil αf (λ) shown in Fig. 11. This curve is in a good agreement with the transmittance spectrum (Fig. 12) obtained by FTNIR system. We notice that each maximum of absorption corresponds to a minimum of FTNIR transmittance.
Fig. 10. Theoretical amplitude obtained with paraffin oil.
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deflection was established and permits to determine the correct values of thermal diffusivity of paraffin oil. Then, in a second step, the absorption spectrum of paraffin oil was indirectly obtained from “mirage effect” deflection measurement when comparing the experimental amplitude of the photothermal signal with the theoretical one. The obtained spectrum is in good agreement with the transmittance obtained from FTNIR measurement. References
Fig. 11. Absorption spectrum of the paraffin oil.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14] [15] Fig. 12. FTNIR transmittance of paraffin oil.
4. Conclusion In this work, the ability of a new method based on “mirage effect” detection technique in order to determine the thermal diffusivity and the absorption spectrum of liquids was demonstrated. This new method was successfully applied to the paraffin oil. In a first step, the thermal diffusivity of such a liquid was evidenced using the amplitude variation of the deflected beam when modulation frequency varies. However, in this case, since the thermal diffusion length in liquids becomes lower than the probe beam diameter, the “classical” expression of deflection beam gives erroneous thermal diffusivity values and cannot be used anymore. So, a new expression representing the “effective”
J.C. Murphy, L.C. Aamodt, J. Appl. Phys. 51 (1980) 4580. A. Mandelis, J. Appl. Phys. 54 (1983) 3404. N.C. Fernelius, J. Appl. Phys. 51 (1980) 650. N. Yacoubi, B. Girault, J. Fesquet, Appl. Opt. 25 (1986) 15. N. Yacoubi, A. Hafaiedh, A. Joulli´e, Appl. Opt. 33 (1994) 7171. F. Saadallah, N. Yacoubi, F. Genty, C. Alibert, J. Appl. Phys. 94 (2003) 5041. R.L. Swofford, J.A. Morrell, J. Appl. Phys. 49 (1978) 3667. P.E. Nordal, S.O. Kanstad, Phys. Spectra 20 (1979) 659. A.C. Boccara, D. Fournier, J. Badoz, J. Appl. Phys. Lett. 36 (1980) 130. D. Fournier, A.C. Boccara, N.M. Amer, R. Gerlach, Appl. Phys. Lett. 37 (1980) 519. D. Bicanic, M. Chirtoc, I. Chirtoc, P. Helander, J.P. Favier, Appl. Spectrosc. 49 (1995) 1954. M. Bertolotti, R. Li Voti, G. Liakhou, C. Sibilia, Rev. Sci. Instrum. 64 (1993) 1576. M.J. Smith, R.A. Palmer, Photoacoustic and photothermal phenomena III, in: D. Bicanic (Ed.), Springer Series in Optical Sciences, vol. 69, 1992, p. 211. E. Legal Lasalle, F. Lepoutre, J.P. Roger, J. Appl. Phys. 64 (1988) 1. A. Mandelis, B.S.H. Royce, Appl. Opt. 23 (1984) 2892.
Biographies Faycel Saadallah, PhD in physics, professor of physics at the University “Institut Pr´eparatoire aux Etudes d’Ing´enieurs de Nabeul (IPEIN)” since 1993. He is a member in the laboratory of “Equipe de Photothermique de Nabeul”. His research interest includes photothermal and optical characterizations of semiconductors, metal alloys and liquids. Leila Attia, master degree, prepare her PhD in “Equipe de Photothermique de Nabeul”. Her research interest includes spectroscopic investigations of liquids. Sameh Abroug, PhD in physics, member in “Equipe de Photothermique de Nabeul”. His research interest includes photothermal and ellipsom´etric characterization of semiconductors. Noureddine Yacoubi, PhD in physics, professor of physics at the University “Institut Pr´eparatoire aux Etudes d’Ing´enieurs de Nabeul (IPEIN)”. He is a member in the laboratory of “Equipe de Photothermique de Nabeul”. His research interest includes photothermal and spectroscopic characterizations of materials: semiconductors, metal alloys, insolent and liquids.
Photothermal investigations of thermal and optical properties of liquids by mirage effect Faycel Saadallah ∗ , Leila Attia, Sameh Abroug, Noureddine Yacoubi Equipe de Photothermique de Nabeul, Institut Pr´eparatoire aux Etudes d’Ing´enieur de Nabeul, BP. 62, Merazka 8000, Nabeul, Tunisia Received 13 September 2006; received in revised form 12 April 2007; accepted 27 May 2007 Available online 2 June 2007
Abstract In this paper, the “mirage effect” detection technique was used in order to determine the thermal diffusivity and the absorption spectrum of liquids. Experiments were more particularly realized with paraffin oil. Generally, since the thermal diffusion length is relatively short in liquids, the different parts of the probe beam are differently deflected by the heat flow. So, a new mathematical expression of the deflection related to the probe beam dimensions was established to calculate the “effective” deflection in liquids. Such an expression was used to measure the thermal diffusivity of paraffin oil. Then, the “mirage effect” technique was also used to obtain indirectly the optical spectrum of paraffin oil through the measurement of the absorption of a carbon black layer immersed in this liquid. © 2007 Elsevier B.V. All rights reserved. Keywords: Photothermal deflection; Liquids; Thermal diffusivity; Optical absorption
1. Introduction The mirage effect technique had been widely used [1–6] in the investigation of thermal and optical properties of solid materials. However, few photothermal investigations were concerned with liquids [7–13]. Some studies have used mirage detection in its collinear [12] or rear [13] configurations. In collinear scheme, a three-dimensional model must be used, and convection is not negligible. Rear detection has been only used for highly absorbent liquids. In this paper, we have improved the frontal-scheme of mirage detection in order to determine the optical absorption spectrum and the thermal diffusivity of liquids. The liquid absorption spectrum is indirectly determined from the measurement of the absorption of a black carbon layer immersed in the liquid. The deflection of probe beam is mainly caused by the heat flow induced by the carbon layer which is heated by a modulated and uniform pump beam. The heat propagation and the temperature gradient are in one direction perpendicular to the layer surface. Therefore, we have to use simple theoretical calculations, based on the resolution of the one-dimension heat equation ∗
Corresponding author. Tel.: +216 98248252; fax: +216 72220181. E-mail address: [email protected] (F. Saadallah).
0924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2007.05.022
[2–6], from which we have deduced the probe beam deflection. For thermal diffusivity measurement we have to vary the modulation frequency. In liquids, when the frequency increase the thermal diffusion length becomes small inducing the deflection of only a part of the probe beam. Therefore, the theoretical model will be modified so as to consider the dimensions of the probe beam. 2. Theoretical model 2.1. Principle of the mirage effect technique The absorption of a modulated and uniform light beam by a solid sample induces a heat flow in both the surrounding liquid and the sample. The liquid refractive index, induced by temperature variations near the surface, causes the deflection of a probe laser beam skimming the surface (Fig. 1). This deflection could be related to thermal and optical properties of the sample and deflecting liquid. In a glass cell fulfilled with paraffin oil (Fig. 2) a plexiglass plate covered with a thin black carbon layer is introduced. This black layer is a good absorbent in a wide wavelength scale. The cell is illuminated with a modulated and uniform heating beam whose intensity could be written as: I = I0 [1 + exp(jωt)].
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Fig. 2. Schema of the studied sample.
The amplitude is Fig. 1. Schema of the probe beam deflection near the sample surface.
|Ψ (x)| = − 2.2. Classical expression of the probe beam deflection ψ
1 dn √ L 2|Ts | exp n dT μf
−x μf
(7)
and the phase is When the dimension of the probe beam is neglected, and for small temperature variations, the beam deflection could be expressed as a function of the liquid temperature Tf [1]: 1 dn L/2 dTf dx =− Ψ= dy (1) dy n dTf −L/2 dx when heat propagates in one direction and the fluid is nonabsorbent, the expression of the fluid temperature could be written as [3]: Tf (x, t) = Ts exp(−σf x) exp(jωt)
(2)
where Ts is the temperature at the solid surface in contact with the transparent fluid, x the distance between probe beam and solid surface, n the liquid refractive index and L is the width of the solid sample. 1/2 πF jω 1+j = (1 + j) = (3) σf = Df Df μf
φ=−
x π +θ+ μf 4
2.3. Determination of the surface temperature Ts In order to determine the temperature distribution described by Eq. (2), we have to calculate the surface temperature Ts . The phenomena of thermal convection and thermal emission are neglected in mirage technique. So, only the thermal conduction in the fluid and the black sample will be considered. Therefore, we can express the heat diffusion in the pump beam direction, by the following equations [2–6]: ∂ 2 Tg 1 ∂Tg = , 2 ∂x Dg ∂t
0 ≤ x ≤ h1
Ts = |Ts |exp(jθ)
1 ∂Tb ∂ 2 Tb = , ∂x2 Db ∂t
Finally we obtain
1 dn (1 + j) x Ψ (x, t) = − L|Ts |exp − n dT μf μf x π × exp j − + θ + + ωt μf 4
(6)
The amplitude and phase (argument) of the deflection, which correspond to the amplitude and phase of the theoretical photothermal signal. Ts = − −
h1 ≤ x ≤ lv + h1
(9)
∂ 2 Tf 1 ∂Tf = − Af exp[αf (x − h1 )][1 + exp(jωt)], Df ∂t ∂x2
F = ω/2π is the modulation frequency, Df the thermal diffusivity and μf is the thermal diffusion length in the liquid. Then deflection becomes: 1 dn Ψ∼ (4) Ts σf L exp(−σf x) exp(jωt) =− n dT The surface temperature could be written as (5)
(8)
(10)
1 ∂Tc ∂ 2 Tc = − Ac exp(αc x)[1 + exp(jωt)], ∂ 2 x2 Dc ∂t −1 ≤ x ≤ 0
(11) − lb − 1 ≤ x ≤ −1
(12)
where Ti , Di and αi are, respectively, temperature, thermal diffusivity and optical absorption coefficient in each medium i. The abbreviations g, f, c and b correspond, respectively, to glass, deflecting fluid (paraffin oil), carbon layer, and backing (plexiglass) (Fig. 2). The terms: Af exp[αf (x − h1 )] and Ac exp(αc x) describe the optical absorption contributions of the fluid and the black layer. By using the continuity condition equations for temperature and heat flux at each interface, we obtain the expression of the temperature Ts in the fluid medium given by:
{Ccf S0 (L0 + 2d0 M0 )Z0 + (1/2)[S0 (L1 + 2M1 d1 ) + F0 ]Zf } exp[σf (h1 − x1 )] B1 {Ccf S1 (L0 + 2d0 M0 )Z0 + (1/2)[S1 (L1 + 2M1 d1 ) + F1 ]Zf } exp[−σf (h1 − x1 ) B1
+ [αf I0 /2Kf (α2f − σf2 )] exp[αf (x − h1 )]
(13)
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where Ts is the temperature at the solid–fluid interface (x = 0), Af , Ac , Ag , B, U,√ V, Ts , Zf , Z0 , W are√complex constants given below. σi = (1 + j) πf/Di and μi = Di /πf is the thermal diffusion length. Z0 = αs I0 exp(−αf h1 )/2Kc (α2c − σc2 ); Zf = αf I0 / 2Kf (α2f − σf2 ), M0 = Cbc − rc , M1 = b0 Q0 (rf − Ccf ) − P0 a0 (rf + Ccf ), Q0 = −(1 + Cbc ), Q1 = a0 P0 (Ccf − 1) + b0 Q0 (Ccf + 1), P0 = 1 + Cbc , P1 = a0 P0 (Ccf + 1) + b0 Q0 (Ccf − 1), S0 = 1 + Cfg , S1 = 1/Cfg , F0 = B1 (rf − 1), F1 = −B1 (rf + 1), B1 = a0 P0 (1 + rc ) + b0 Q0 (1 − rc ), L0 = a0 P0 (1 + rc ) + b0 Q0 (1 − rc ), L1 = a1 P1 (rf + 1) − b1 Q1 (rf − 1), Ccf = kc σ c /kf σ f , Cfg = kf σ f /kg σ g , Cbc = kb σ b /kc σ c , a0 = exp(−σ c l), a1 = exp (−σ f h1 ), b0 = exp(σ c l), b1 = exp(σ f h1 ), d0 = exp(−αc h), d1 = exp(αc h), rf = αf σ f , rs = αc σ c . 3. Experimental results In this work, we have studied the experimental variations of the amplitude and phase of the photothermal signal. From the amplitude variations versus the heating beam wavelength, we have obtained the absorption spectrum of the paraffin oil. However, during the measurement of its thermal diffusivity we should vary the modulation frequency, which affects the thermal diffusion length in liquid. 3.1. Experimental setup Experimental apparatus is well described in reference [6]. As it is shown in Fig. 1, a probe laser beam (630 nm) is deflected by the heat flow from a black layer heated by a uniform, modulated and monochromatic light beam. This black layer is placed in a glass cell and immersed in a deflecting liquid (paraffin oil or CCl4 ). 3.2. Determination of the thermal diffusivity of the paraffin oil The above calculations (Eqs. (1) and (13)) of the probe beam deflection are done without considering the beam dimensions. The obtained deflection expression could be only used for the determination of the liquid optical properties. However, for the determination of its thermal diffusivity, we need to vary the modulation frequency. Therefore, the beam dimensions must be considered, because the thermal diffusion length μf become lower than the beam diameter for high frequencies, so only one part of the beam is deflected. Therefore, we must divide the probe beam into many small beams [14], each of them will reach one of the tow photodetector cells. The difference between the light powers absorbed by each cell corresponds to the measured photothermal signal which should be proportional to the theoretical effective deflection. 3.2.1. Measurement of the distance x0 between probe beam axis and solid surface When the dimensions of the probe beam are neglected and the fluid is transparent, the photothermal signal could be expressed as a function of the modulation frequency and the distance x
Fig. 3. Phase of the photothermal signal, vs. f1/2 , air is used as the deflecting medium.
between the deflected beam and the sample surface: 1 dn (1 + j) x Ψ (x, t) = − L|Ts |exp − n dT μf μf x π × exp j − + θ + + ωt , μf 4 1/2 Π 1/2 1/2 Ψ = Cf exp −(1 + j) f x Df Then we can write the logarithm of the amplitude as 1/2 Π xf 1/2 log|Ψ | = C1 + log(f 1/2 ) − Df and the phase as 1/2 Π xf 1/2 Φ = K1 − Df
(14)
(15)
(16)
√ where C1 = log(|Ts |(L/n)(∂n/∂Tf ) 2Π/Df ), K1 = θ + Π/4, θ = arg (Ts ) The above equations show linear variations of the amplitude logarithm log |Ψ | and the phase with square root frequency f1/2 . The curves in Figs. 3 and 4 represent, respectively, the phase and amplitude of the experimental signal versus f1/2 . These curves are obtained when air is used as the deflecting fluid. Air is transparent and its thermal diffusion length is higher than the beam diameter, so the entire beam is uniformly deflected. The x0 value is determined from the slope of the phase or the amplitude logarithm curves for high frequencies, we find x0 = 108 m. 3.2.2. Calculation of the effective deflection Without changing the glass cell position, we replace air by paraffin oil. This liquid is used in its transparency region in order to exclude the absorption contribution to the signal. Before heating, the probe beam that has a cylindrical form is not deflected, and its impact on the photodetector is a disc with a radius R. However, after deflection this form is not conserved. So we should divide the beam into infinity of small beams. The
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is the sum of all the small beams power that falls on its surface. Furthermore, the probe beam has a Gaussian power distribution with intensity given by −2[(x − x0 )2 + y2 ] I(x, y) = I0 exp R2 I0 represents the maximum at the probe beam axis (x = x0 ). Then we can express the tow powers by X0 Ymax P1 = I(X, y)dXdy (20) Xmin
Fig. 4. Amplitude of the photothermal signal vs. f1/2 , air is used as the deflecting medium.
distance between a small beam having a thickness dr, and the probe beam axis is r, and so we can write y2 + (x − x0 )2 = r2
(17)
here x and y are the coordinates of the small beam. In addition, x represents the distance between the sample surface and the small beam. After deflection, the beam impact is not a disk [15], but it has another shape (Fig. 5) limited in the X direction by Xmin and Xmax . Since each small beam has different deflection from the others, its abscise on the photodetector will change from x into X given by X = D|Ψ (x)| + x
(18)
where D is the distance between the sample and the photodetector. The small beam deflection amplitude |ψ(x)| is given in Eq. (7). Moreover, there is no deflection in the y axis direction because the heating beam intensity is uniform, so the y abscise of the ray does not change, then from Eq. (17) we can write 1/2
y = ±[r 2 − (x − x0 )]
Ymin
on the first cell. Xmax Ymax P2 = I(X, y)dXdy X0
(21)
Ymin
on the second cell Where the limits of the probe beam impact
on the photodetector are given by Ymax = −Ymin = R2 − (x − x0 )2 , Xmin = D|Ψ (x0 − R)| + x0 − R and Xmax = D|Ψ (x0 + R)| + x0 + R.The effective deflection amplitude is then Ψeff = P2 − P1
(22)
This expression is in good agreement with the classical deflection expression if the deflecting medium has high thermal diffusivity such as air. However, there is no agreement when liquids are used. 3.2.3. Determination of the thermal diffusivity In the first experience air is used as the deflecting medium. In the second, air is replaced with paraffin oil, without changing the distance x0 . In order to determine the thermal diffusivity of the paraffin oil we plot the theoretical effective deflection amplitude as a function of the square root frequency for different values of the thermal diffusivity (Fig. 6). The best diffusivity value
(19)
The effective deflection corresponds to the absorbed light power difference between the tow cells. The power absorbed by a cell
Fig. 5. Deformation of the probe beam section after deflection.
Fig. 6. Experimental photothermal amplitude (open dots) and the effective deflection amplitude calculated for three different values of the thermal diffusivity: 10−8 (dashed line), 4 × 10−8 (solid line) and 7 × 10−8 m2 s−1 (dotted line).
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Fig. 9. Photothermal spectrum of paraffin oil. Fig. 7. Photothermal spectrum of the heating lamp (measured amplitude in CCl4 ).
obtained while fitting theoretical curve to experimental one, is 4 × 10−8 m2 s−1 . 3.3. Determination of the paraffin oil absorption spectrum When using a low modulation frequency (8 Hz) the probe beam is considered to be uniformly deflected. So, the liquid absorption spectrum can be deduced from Eq. (1). 3.3.1. The lamp spectrum In a first step, as a reference signal, the photothermal spectrum of the lamp is deduced. The glass cell (Fig. 2) is fulfilled with CCl4 which is a transparent liquid, so all the pump power will be absorbed by the carbon layer. Thereby, the measured signal amplitude will correspond to the photothermal spectrum of the lamp shown in Fig. 7. 3.3.2. Photothermal spectrum of paraffin oil When we replace CCl4 with paraffin oil, we obtain the curve given in Fig. 8. This curve presents three absorption regions near the following wavelengths: 1.2, 1.4 and 1.7 m. The normalized amplitude shown in Fig. 8 represents the ratio of the photothermal amplitude of the paraffin oil to the corresponding amplitude of CCl4 (Fig. 9).
Fig. 8. Measured amplitude of the photothermal signal obtained in paraffin oil.
3.3.3. Theoretical amplitude Complex expression of the theoretical deflection is obtained from Eqs. (1) and (13). The variations of its normalized amplitude with oil absorption coefficient (αf ) are given in Fig. 10. For weak absorption coefficient values (αf < 0.1 cm−1 ), the liquid is totally transparent and all the heating power is absorbed by the black layer, inducing a saturation of the amplitude. When αf increases, the liquid absorption reduces the amount of light that reaches the black layer, leading to the decrease of the amplitude. When the light power is entirely absorbed by the liquid, the amplitude saturates once again. Indeed, as the absorption occur near the liquid–glass interface which is 4 mm (Fig. 1) distant from the probe beam, the heat flow coming from this absorption has not any contribution to the photothermal signal. 3.3.4. Optical absorption spectrum of the paraffin oil The optical absorption spectrum is obtained by fitting the experimental amplitude to the theoretical one. As the amplitude saturates for high and low values of αf , we can only determine the medium values between 0.1 and 10 cm−1 . While comparing, the experimental and theoretical amplitudes, we deduce the optical absorption spectrum of the paraffin oil αf (λ) shown in Fig. 11. This curve is in a good agreement with the transmittance spectrum (Fig. 12) obtained by FTNIR system. We notice that each maximum of absorption corresponds to a minimum of FTNIR transmittance.
Fig. 10. Theoretical amplitude obtained with paraffin oil.
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deflection was established and permits to determine the correct values of thermal diffusivity of paraffin oil. Then, in a second step, the absorption spectrum of paraffin oil was indirectly obtained from “mirage effect” deflection measurement when comparing the experimental amplitude of the photothermal signal with the theoretical one. The obtained spectrum is in good agreement with the transmittance obtained from FTNIR measurement. References
Fig. 11. Absorption spectrum of the paraffin oil.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14] [15] Fig. 12. FTNIR transmittance of paraffin oil.
4. Conclusion In this work, the ability of a new method based on “mirage effect” detection technique in order to determine the thermal diffusivity and the absorption spectrum of liquids was demonstrated. This new method was successfully applied to the paraffin oil. In a first step, the thermal diffusivity of such a liquid was evidenced using the amplitude variation of the deflected beam when modulation frequency varies. However, in this case, since the thermal diffusion length in liquids becomes lower than the probe beam diameter, the “classical” expression of deflection beam gives erroneous thermal diffusivity values and cannot be used anymore. So, a new expression representing the “effective”
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Biographies Faycel Saadallah, PhD in physics, professor of physics at the University “Institut Pr´eparatoire aux Etudes d’Ing´enieurs de Nabeul (IPEIN)” since 1993. He is a member in the laboratory of “Equipe de Photothermique de Nabeul”. His research interest includes photothermal and optical characterizations of semiconductors, metal alloys and liquids. Leila Attia, master degree, prepare her PhD in “Equipe de Photothermique de Nabeul”. Her research interest includes spectroscopic investigations of liquids. Sameh Abroug, PhD in physics, member in “Equipe de Photothermique de Nabeul”. His research interest includes photothermal and ellipsom´etric characterization of semiconductors. Noureddine Yacoubi, PhD in physics, professor of physics at the University “Institut Pr´eparatoire aux Etudes d’Ing´enieurs de Nabeul (IPEIN)”. He is a member in the laboratory of “Equipe de Photothermique de Nabeul”. His research interest includes photothermal and spectroscopic characterizations of materials: semiconductors, metal alloys, insolent and liquids.