Infrared Physics & Technology 45 (2004) 315–321 www.elsevier.com/locate/infrared Normalisation procedure in thermal wave approach of thermal diffusivity measurement of solids using pyroelectric sensor B.Z. Azmi *, H.S. Liaw, W.M.M. Yunus, M. Hashim, M.M. Moksin, W.M.D.W. Yusoff Photoacoustic Laboratory, Department of Physics, Universiti Putra Malaysia, 43400 Serdang, Selangor Darul Ehsan, Malaysia Received 7 May 2003 Abstract The theory of thermal wave interference in generation of photoacoustic signal suggested by Bennett and Patty [Appl. Opt. 21 (1982) 49] was applied to measure thermal diffusivity of optically opaque sample using photopyroelectric technique. To determine thermal diffusivity with precision, one needs to work with frequencies high enough in order to be in a thermally thick regime. However, this condition cannot always easily be fulfilled due to the great attenuation of the amplitude signal at high frequency resulting in decrease in signal-to-noise ratio. In this work, we proposed thermal diffusivity measurements to be carried out with the thermal wave probing starting from the sample–pyroelectric interface towards the thermally thick region of the sample. Reflection of thermal wave in the pyroelectric coatings, which was ignored in previous models, was considered in the generation of photopyroelectric signal. Normalisation procedure was used to eliminate the number of media parameters of photopyroelectric cell that otherwise need to be known before one can determine the thermal diffusivity of the sample. With the appropriate sample–pyroelectric detector dimension, the thermal diffusivity of any solid sample is readily being determined. The method was experimentally tested for aluminum, copper, and nickel, and the values obtained were close to the literature values.
2004 Elsevier B.V. All rights reserved. PACS: 77.70.+a; 66.10.Cb; 66.70.+f; 78.20.Nv Keywords: Photothermal; Photopyroelectric; Thermal wave; Thermal diffusivity; PVDF 1. Introduction * Corresponding author. Tel.: +60-3894-66650; fax: +603894-54454. E-mail address: azmizak@fsas.upm.edu.my (B.Z. Azmi). Recently, there has been considerable interest in polyvinylidene diflouride (PVDF) polymer since the discovery of its strong pyroelectric (PE) effect coefficient by Bergman et al. in 1971 [1]. The low cost, light weight, flexible, insensitive to acoustics and mechanical noise, and good electrical prop- 1350-4495/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2004.01.002 316 B.Z. Azmi et al. / Infrared Physics & Technology 45 (2004) 315–321 erties of PVDF film make the film an excellent choice for signal detection in photopyroelectric (PPE) technique. In PPE technique, the test sample is usually deposited on or placed in good thermal contact with the PE sensor and is exposed to intensity modulated light. The theory of this configuration in PPE signal generation has been presented by Mandelis et al. [2] and thermal diffusivity measurement of optically opaque sample has been carried out by Marinelli et al. [3]. It is inevitable to work at high frequencies or to operate in a thermally thick regime as being predicted by the theory [2]. However, this condition cannot always easily be fulfilled due to the great attenuation of the amplitude signal at high frequency resulting in decrease in the signal-to-noise ratio [4]. On the other hand, in the case of thermally very thin sample, the PPE signal is independent of sample thermal property [2]. Therefore, we examine the suitability of a new approach in thermal diffusivity measurement that has capability to probe from the near sample–PE interface and continuously through the thermally thick region of the sample. In this paper, the theory of thermal wave interference in photoacoustic (PA) signal generation suggested by Bennett and Patty [5] has been adopted and applied to thermal diffusivity measurement of optically opaque solids using a standard PPE configuration. In the PPE signal generation, we have considered the transmitted thermal waves from the PVDF film coatings into the film. The thermal diffusivity value of Al, Cu, and Ni samples are then obtained by fitting the theory to the experimental data. mitted to the PE detector, instead of the transmitted terms that transmitted through the gas to the detector in PA technique. If the test sample is highly opaque, in which the optical absorption length is much smaller than sample thickness, this approach is further simplified by considering only the heat source generated on the sample surface [3]. Then, the resulting thermal wave is independent of the optical properties of the sample. A one-dimensional configuration of PPE cell is shown in Fig. 1. To understand the route and effect of the thermal wave, consider now the thermal wave generated on sample surface and propagates to the left. It will be partially transmitted and reflected upon striking the sample–coating interface, A. Generally for a thermal wave propagating in medium 1 towards medium 2, the thermal wave reflection coefficient R12 and the thermal wave transmission coefficient T12 at the interface are respectively given by [5,6] R12 ¼ 1  b12 ; 1 þ b12 T12 ¼ 2 ; 1 þ b12 ð1Þ where b12 is the ratio of the thermal diffusivity of the medium 2 to medium 1. The wave partially reflected back into sample Rsc A ers Ls , where A ¼ Qo =2ks rs , Qo is the source intensity, ks is thermal conductivity of the sample, rs ¼ ð1 þ iÞ=ls , ls ¼ 1=2 ðas =pf Þ is the thermal diffusion length of sample at light modulation frequency f , Ls is the sample thickness, and as is the thermal diffusivity of the sample. Thermal diffusivity is defined by as ¼ ks = qs cs , where qs is the density and cs is the specific heat. This reflected thermal wave will eventually 2. Theory The thermal wave generated within the sample will be partially reflected and transmitted upon striking at the interface of two media. The effect of infinite multiple reflections of the thermal wave within the sample was extensively discussed by Bennett and Patty in the PA signal generation [5]. In this work, a similar approach was used to obtain the complex PPE signal by adding all the transmitted terms of thermal waves that trans- Fig. 1. Thermal wave presence in each region of PPE cell. The thermal waves are partially reflected and transmitted upon striking the boundaries. Subscript g, s, c, p, and b stand for gas, sample, coating, PE detector, and backing, respectively. B.Z. Azmi et al. / Infrared Physics & Technology 45 (2004) 315–321 died off in short thermal diffusion length operations. However, the transmitted wave at A, hs ¼ Tsc A ers Ls ; ð2Þ will travel across the coating [5,6]. Since the coating thickness is thermally much more thinner compare to that of PVDF film, there will be a series of partially reflected waves hs Rcp erc Lc , hs R2cp Rcs e3rc Lc , hs R3cp R2cs e5rc Lc , etc. and of partially transmitted waves hs Tcp erc Lc , hs Rcp Rcs Tcp e3rc Lc , hs R2cp R2cs Tcp e5rc Lc , etc. at the coating–PVDF interface, x ¼ 0. The resultant of the transmitted wave in PVDF emerging from the right coating is given by, hpr ¼ hs Tcp erc Lc ; 1  Rcp Rcs e2rc Lc ð3Þ and it contributes to the temperature distribution in PVDF film. Then, if we stepped backward and trace the first transmitted wave hs Tcp erc Lc from point B after traveling through a distance Lp in the PVDF film, see Fig. 1, it will be partially transmitted into left coating hs Tcp erc Lc Tpc erp Lp and partially reflected back into PVDF hs Tcp erc Lc Rpc erp Lp at the PVDF– coating interface, C. Similarly at the interface points D and C0 and so on, there will be many partially transmitted and reflected waves. As a result of these, the resultant wave emerges from the left coating at x ¼ Lp into PVDF film is given by 
Tpc Tcp Rcb e2rc Lc 1 rc Lc rp Lp hpl ¼ hs Tcp e : e Rpc þ 1  Rcb Rcp e2rc Lc ð4Þ Similarly as the first transmitted wave, the transmitted waves from the right that are of factors Rcp Rcs e2rc Lc , R2cp R2cs e4rc Lc and so on, of the first wave hs Tcp erc Lc will produce multiple transmissions from the left coating. The resultant wave hpl in PVDF is therefore given by hpl ¼ hs Tcp e   rc Lc e Tpc Tcp Rcb e2rc Lc Rpc þ 1  Rcb Rcp e2rc Lc  : 2rc Lc rp Lp 1 1  Rcp Rcs e
 ð5Þ 317 This amplitude of thermal wave is equivalent to the amplitude of thermal wave transmitted to the gas treated by Bennett and Patty [5]. In the analysis of thermal wave propagation in a sample, Opsal and Rosencwaig [7] suggested a multi-layer model that provides expressions for temperature at the surface of each layer. The model is applicable both to the measurement of surface temperature variations, as in the cases of gasmicrophone photoacoustics and photothermal detection, and to the measurement of subsurface thermoelastic response, as in the cases of piezoelectric and thermal-wave experiments. Minamide et al. [8] then applied the model in their 3-layer PPE cell, which was the layer of sample, air gap and PVDF film, to obtain the surface temperature of the film. The averaged temperature hp in the film with no reflection in PVDF–coating interface is given by Z Lp 1 hc erp x dx; ð6Þ hp ðxÞ ¼ Lp 0 where hc is the surface temperature of the PVDF film. In our case, by considering multiple thermal wave transmission from the left coating of expression stated in Eq. (5), as well as that from the right, Eq. (3), the averaged temperature hp in the PVDF film is given by Z Lp 1 hp ðxÞ ¼ ½hpr erp x þ hpl erp Lp þrp x dx: ð7Þ Lp 0 The average PE voltage produced by the PVDF film is given by [2] V ðxÞ ¼ pLp hp ; ee0 ð8Þ where p is the PE coefficient, e is the dielectric constant of PE detector, and e0 is the permittivity constant of vacuum. Substituting the result of (7) into Eq. (8) leads to V ðxÞ ¼  Qo p Tcp Tsc eðrs Ls þrc Lc Þ  ð1  erp Lp Þ 2eeo ks rs rp 1  Rcp Rcs e2rc Lc   Tpc Tcp Rcb e2rc Lc : þ erp Lp ðerp Lp  1Þ Rpc þ 1  Rcp Rcb e2rc Lc ð9Þ The a values of the samples can be determined from either the amplitude or the phase of Eq. (9). 318 B.Z. Azmi et al. / Infrared Physics & Technology 45 (2004) 315–321 However, a number of pertinent values for gas, PE coating, PE, and backing need not to be used by normalising the signal of the test sample to that of the reference sample. This simplifies Eq. (9) to
 V1 T1c er1 L1 1  Rcp Rc2 e2rc Lc ðxÞ ¼ b12 ; ð10Þ V2 T2c er2 L2 1  Rcp Rc1 e2rc Lc where subscripts 1 and 2 represent the test and reference samples, respectively. A similar normalisation procedure was reported by Delenclos et al. [4], in which the measured PPE signal was normalised to the one obtained with the sensor alone, or to the signal obtained with a reference sample. However, laborious work, which involved solving 10 · 10 matrix by CramerÕs rule, was needed in obtaining the average PE voltage, and it still constraints to work at high frequency in thermal diffusivity measurement. In this paper, an easier approach of thermal wave interferometry presented here has greatly reduced the work in getting the average PE voltage. To further simplify Eq. (10), the term in the square bracket can be neglected due to its small attenuation of phase signal. Therefore the phase signal of Eq. (10) can be written as /1  /2 ¼  pf a2 1=2 L2   pf a1 1=2 L1 ; ð11Þ where subscripts 1 and 2 refer to the test sample and the reference sample respectively. From Eq. (11), we should expect a linear dependence of the normalised phase signal with f 1=2 . Hence, if m1 and m2 are the gradients for the test and the reference samples respectively, it can be written as  1=2  1=2 p p L2  L1 : ð12Þ m1  m2 ¼ a2 a1 This means that by obtaining gradients of plot phase signal versus f 1=2 , that is m1 of test and m2 of reference sample, a1 can be calculated by substituting the known value a2 of the reference sample. In this model, multiple reflections in the sample are neglected because the chopping frequencies are made for thermally thick operation for the sample. Meaning the thermal wave is very short and dies off after one reflection at sample–coating interface. Similarly single reflection is only considered in PVDF film due to its extremely small a ( 104 cm2 s1 ) compare to metal sample ( 100 cm2 s1 ). The thermal diffusion length in Al sample with a of 0.979 cm2 s1 and in PVDF film with a of 5.4 · 104 cm2 s1 at 30 Hz are 1020 and 24 lm, respectively. 3. Experimental procedures The schematic diagram of PPE experimental setup system is shown in Fig. 2. The beam of 30mW He–Ne laser (05-LHR-991) modulated by variable frequency optical chopper (SR540) is impinged onto the sample surface which is in thermal contact with 36-lm-thick PVDF film supported by 1.2-cm-thick Perspex backing. A very thin layer of thermal conductive grease was used as a coupling fluid to optimize the sample– detector thermal contact. The grease layer does affect the measured signal; moreover, it can be eliminated via the process of normalisation. The output from the PE detector is fed into a low-noise preamplifier for signal amplification and then into a lock-in amplifier (SR530) for signal analysis. The effects of the wrap of PVDF film and piezoelectricity due to expansion and contraction of sample were effectively eliminated by clamping the sample and PVDF film between two Perspex plates [9]. The front plate has a circular opening to allow uninterrupted illumination of the laser beam to the sample. The measurement was carried out at room temperature on aluminum, copper, and nickel samples and the PPE signal was recorded over a frequency range that suits the model proposed in the theory. Fig. 2. Schematic diagram of PPE experimental setup. B.Z. Azmi et al. / Infrared Physics & Technology 45 (2004) 315–321 4. Results and discussion 319 In the experiment care has been taken, firstly, to produce a reasonably short thermal diffusion length so that the reflected thermal wave in the sample will die off quickly in the sample after first or before second reflection at sample–coating interface. As evidence at starting frequency for Al sample, for thermal diffusion length at even slightly larger than sample thickness Ls the plot still produces a good straight line. This is because the reflected thermal wave will diminish in sample as long as the thermal diffusion length is shorter than 1:5Ls . The scanning in the thermally thick regime or towards the shorter thermal diffusion lengths will ensure the reflection of the thermal wave in these samples is only due to the weak exponential tail of the wave. Secondly, the gradient difference between the test and the reference samples has to be reasonably large. If it is small the normalised gradient will close to zero and thus will introduce greater errors in calculating the thermal diffusivity of the test sample. Fig. 3 shows the plot for the phase of PE voltage versus the same range of square-root frequency generated by the infinite terms of thermal wave in the right coating, which is in contact with Cu sample, that is the term 1  Rcp Rc2 e2rc Lc of Eq. (10). The gradient m of the plot is in the order of 104 , which can be neglected in our following calculation of a. Figs. 4–5 show the plot of linear dependant of unnormalised PE phase signal as a function of f 1=2 for Al (Ls ¼ 670 lm), Cu (Ls ¼ 940 lm), and Ni (Ls ¼ 500 lm) recorded over the same range of frequency, where m is the slope of linearly fit line. The phase signal was chosen in present study because it does not fall off as quickly as the magnitude signal over the frequency range. From these figures, the Ni sample gives the highest magnitude of slope among the three samples. Thus, one can expect that Ni gives the lowest value of a. By using Eq. (12), the calculated values of a with respect to different reference sample were shown in Table 1, and the results obtained are reasonably close to the literature values [10], which is less than 4%. The experimental error is only due to errors in the sample thickness measurement and the plot gradient. The effect of gradient difference on a values can be seen in Table 1. The gradient for Cu is in between gradients for Ni and Al. By using Ni as reference, a of Al is closer to the literature values compared to the case of using Cu as reference. Similarly by using Al as reference, a of Ni is closer to the literature value compared to when Cu is Fig. 3. Theoretical PPE phase signal caused by the infinite terms of thermal wave in the left PVDF coating in contact with Cu sample of 940 lm thick. The solid line represents the best-fit line with slope m. Fig. 4. PPE phase signal of aluminum as a function of square root chopping frequency. The solid line represents the best-fit line with slope m. 320 B.Z. Azmi et al. / Infrared Physics & Technology 45 (2004) 315–321 Fig. 5. PPE phase signal of (a) copper and (b) nickel as a function of square root chopping frequency. The solid line represents the best-fit line with slope m. Table 1 The a values for materials obtained using phase signal Sample Al Cu Ni Calculated thermal diffusivity a (cm2 s1 ) Al (as reference) Cu (as reference) Ni (as reference) – 1.138 ± 0.046 0.233 ± 0.006 1.006 ± 0.040 – 0.237 ± 0.008 0.956 ± 0.027 1.117 ± 0.042 – used as reference. The effect of thin thermal grease layer even though is not included in the present analysis this technique still produces reasonably good result. This is because it can be excluded after normalisation similarly as the considered PVDF coating layer. Application of this normalisation procedure to amplitude signal does not give the thermal diffusivity value as can be seen in Table 2. This is be- Table 2 The a values for materials obtained using amplitude signal Sample Al Cu Ni Calculated thermal diffusivity a (cm2 s1 ) Al (as reference) Cu (as reference) Ni (as reference) – 3.0170 1.4412 0.4892 – 0.9628 0.3508 0.8540 – Average (cm2 s1 ) Literature value (cm2 s1 ) Deviation (%) Reference 0.981 1.128 0.235 0.979 1.163 0.229 0.2 3.0 2.6 [10] [10] [10] cause larger noise to signal ratio in the system as such after normalisation, the data were too scattered hence the thermal diffusivity value are completely off from the actual value. 5. Conclusion In short thermal diffusion length operations of samples, a series of transmitted waves from right and left coatings, and hence resultants of transmitted waves from right and left coatings into PVDF film were taken into account in the generation of PPE signal. A normalisation procedure was used as a convenient way to eliminate the effect of a number of unknown parameters in other media of PPE cell. The obtained value of thermal diffusivity for aluminum, copper, and nickel samples were reasonably close to the literature values. B.Z. Azmi et al. / Infrared Physics & Technology 45 (2004) 315–321 Acknowledgements The authors are grateful to the Ministry of Science, Technology and the Environment of Malaysia for supporting this work under IRPA grant no. 02-02-04-0132-EA001. References [1] J.B. Bergman, J.H. McFee, G.R. Crane, Appl. Phys. Lett. 18 (5) (1971) 203. [2] A. Mandelis, M.M. Zver, J. Appl. Phys. 57 (9) (1985) 4421. 321 [3] M. Marinelli, U. Zammit, F. Mercuri, R. Pizzoferrato, J. Appl. Phys. 72 (3) (1992) 1096. [4] S. Delenclos, M. Chirtoc, A.H. Sahraoui, C. Kolinsky, J.M. Buisine, Anal. Sci. 17 (2001) 161. [5] C.A. Bennett, R.R. Patty, Appl. Opt. 21 (1982) 49. [6] D.P. Almond, P.M. Patel, Photothermal Science and Techniques, Chapman & Hall, London, 1996. [7] J. Opsal, A. Rosencwaig, J. Appl. Phys. 53 (6) (1982) 4240. [8] A. Minamide, M. Shimaguchi, Y. Tokunaga, Jpn. J. Appl. Phys. 37 (1998) 3144. [9] M. Aravind, P.C.W. Fung, Meas. Sci. Technol. 10 (1999) 979. [10] R.D. Lide, CRC Handbook of Chemistry and Physics, 78th ed., CRC Press Inc., Florida, 1997.