Thermal diffusivity measurement by lock-in

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The study of frequency-scan photothermal reflectance technique for thermal diffusivity ... narrow beam of light in a technique known as photothermal.
Thermal diffusivity measurement by lock-in photothermal shadowgraph method A. Cifuentes, S. Alvarado, H. Cabrera, A. Calderón, and E. Marín Citation: Journal of Applied Physics 119, 164902 (2016); doi: 10.1063/1.4947454 View online: http://dx.doi.org/10.1063/1.4947454 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/119/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The study of frequency-scan photothermal reflectance technique for thermal diffusivity measurement Rev. Sci. Instrum. 86, 054901 (2015); 10.1063/1.4919609 Measurement of thermal diffusivity based on the photothermal displacement technique using the minimum phase method J. Appl. Phys. 88, 588 (2000); 10.1063/1.373700 Thermal diffusivity measurement of polymeric thin films using the photothermal displacement technique. II. Onwafer measurement J. Appl. Phys. 86, 6028 (1999); 10.1063/1.371650 Thermal diffusivity measurement of polymeric thin films using the photothermal displacement technique. I. Freestanding film case J. Appl. Phys. 86, 6018 (1999); 10.1063/1.371649 New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers Rev. Sci. Instrum. 68, 1521 (1997); 10.1063/1.1147589

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JOURNAL OF APPLIED PHYSICS 119, 164902 (2016)

Thermal diffusivity measurement by lock-in photothermal shadowgraph method  n,1 and E. Marın1,a) A. Cifuentes,1,2 S. Alvarado,1,3 H. Cabrera,4 A. Caldero 1

Instituto Polit ecnico Nacional, Centro de Investigaci on en Ciencia Aplicada y Tecnologıa Avanzada, Unidad Legaria, Ciudad de M exico 11500, Mexico 2 Departamento de Fısica Aplicada I, Escuela T ecnica Superior de Ingenierıa, Universidad del Paıs Vasco UPV/EHU, Alameda Urquijo s/n, 48013 Bilbao, Spain 3 Laboratory for Soft Matter and Biophysics, Department of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, Heverlee B-3001, Belgium 4 Centro Multidisciplinario de Ciencias, Instituto Venezolano de Investigaciones Cientıficas, IVIC, M erida 5101, Venezuela and SPIE-ICTP Anchor Research in Optics Program Lab, International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, Italy

(Received 23 February 2016; accepted 12 April 2016; published online 27 April 2016) Here, we present a novel application of the shadowgraph technique for obtaining the thermal diffusivity of an opaque solid sample, inspired by the orthogonal skimming photothermal beam deflection technique. This new variant utilizes the shadow projected by the sample when put against a collimated light source. The sample is then heated periodically by another light beam, giving rise to thermal waves, which propagate across it and through its surroundings. Changes in the refractive index of the surrounding media due to the heating distort the shadow. This phenomenon is recorded and lockin amplified in order to determine the sample’s thermal diffusivity. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4947454]

I. INTRODUCTION 1

Thermal waves are an important tool for the thermal characterization of materials. They are generated when a material is subjected to periodic heating. Thermal waves can be detected through direct or indirect2 methods. One of the possibilities for indirect study is the observation of the induced changes in the refractive index of a medium (gas or liquid) surrounding a sample.3 The shadowgraph method4 is an optical technique that allows us to visualize disturbances of the refractive index of a transparent medium. It utilizes an expanded collimated beam of light that crosses a field of refractive index inhomogeneity. Light passing through this medium will suffer deflections due to these disturbances, thus when this light is projected onto a screen, a spatial modulation of the light intensity distribution, i.e., a shadow, is formed. Shadowgraphs have been used extensively in fluid mechanics and heat transfer research as an important visualization tool.5 In the area of photothermal techniques,2 the study of the refractive index variations in a fluid surrounding a sample has been undertaken by the observation of the deflection of a narrow beam of light in a technique known as photothermal beam deflection (PBD) or mirage effect.6,7 PBD is a well-established technique in which the course of a probe laser beam is altered by the change in the refractive index induced in the medium surrounding a sample undergoing the heating process. Although time resolved PBD techniques exist,8 this work is mainly interested in modulated PBD, and more specifically to the orthogonal beam surface skimming variant,9,10 which served as an a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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inspiration for the experimental configuration that will be presented in this work. In the orthogonal surface skimming variant, the technique requires both a probe laser beam and a pump laser beam whose trajectories are orthogonal. The intensity of the pump beam is periodically modulated and heats a small area of the sample. The continuous probe beam is directed as close to the surface of the sample as possible without touching it and is used for punctually sensing the refractive index changes in the vicinity of said surface. These changes produce a deflection on the probe beam’s trajectory. The beam’s deflection will convey information, such as phase and amplitude, of the thermal wave at each point. The probe laser beam deflection is usually monitored by means of a quadrant photodetector or similar device. A linear relationship that arises in this technique is that given by the phase lag of the thermal wave as a function of the distance to a punctual heat source when unidimensional heat diffusion can be guaranteed.11 This relationship is useful for the calculation of the sample’s thermal diffusivity, a, which can be obtained straightforwardly if the pump beam modulation frequency is well-known. The measurement procedure requires the experimenter to displace the probe beam a given distance from the heat source, measure the phase lag at that offset, and repeat for as many points as desired. This process can be quite lengthy in dependence of the number of points. In this work, it is demonstrated that the shadowgraph technique can be used successfully for measuring the thermal diffusivity of solids in a manner similar to that of PBD orthogonal surface skimming variant. Shadowgraph offers a much faster measurement than PBD as the diffusivity of a sample can be obtained from a single measurement.

119, 164902-1

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II. THEORY AND SIMULATION A. Theoretical background

Let us assume that a set of parallel rays of light propagating in the z-direction enter a plane located at coordinate zi and exit at a plane with coordinate ze, with coordinate pairs ðxi ; yi Þ describing the entrance position of each ray of light. Let L be the distance of the projection screen beyond the exit plane, n is the refractive index of the transparent medium, and sub-index “s” denotes the coordinates on said screen. It is well-known that a shadowgraph in which light rays are at normal incidence when entering the plane and undergo only infinitesimal deviations inside the inhomogeneous field but have a finite curvature, light ray coordinates will be mapped as follows:4 ð ze @ ðlogðnÞÞ xs  xi ¼ L dz; (1) @ ðx Þ zi ð ze @ ðlogðnÞÞ dz: (2) ys  yi ¼ L @ ðy Þ zi If the infinitesimal displacement assumption can be extended to be valid in the region between the exit plane and the projection screen, the intensity Is at point (xs, ys), which is the result of several beams moving from (xi, yi) and getting mapped onto (xs, ys), is given by I s ðx s ; y s Þ ¼

X I 0 ðx i ; y i Þ ;   @ ðx s ; y s Þ   ðxi ;yi Þ   @ ðx ; y Þ  i i

(3)

where I0 is the intensity at the region of study. Moreover, as long as the beam deflections dx and dy are small, the @ ðxs ;ys Þ can be further simplified as (4). Jacobian @ðx i ;yi Þ    @ ðx s ; y s Þ  @ ðx s  x i Þ @ ðy s  y i Þ   (4)  @ ð x ; y Þ   1 þ @ ð xÞ þ @ ð yÞ : i i Therefore, the governing equation for the shadowgraph under these assumptions and integrating for the dimension of the experimental apparatus, D, i.e., the distance in which the refraction index inhomogeneity is present,4 becomes I0  Is ¼ ð L  DÞr2 logðnðx; yÞÞ: Is

(5)

Further insight into the mathematical equations governing shadowgraphs can be found elsewhere.4,5,12 Suppose that the sample with a larger thermal conductivity than that of its surrounding medium (ks  kc ) is optically opaque and being periodically heated by a light source, such as a laser beam, within an area as small as possible, and that the plane of study is normal to the object’s surface. Let us also affirm that the refractive index inhomogeneity is directly proportional to a small variation in temperature of an object in thermal contact with a transparent media, an assumption that holds true for many liquids.13 The object is also assumed to be semi-infinite in the x dimension (see Figure 1). If such conditions are met, it has been shown14

FIG. 1. Problem geometry. Sample is semi-infinite in z and x directions. c denotes the surrounding medium.

that the spatial distribution of the thermal field generated in the surrounding medium, Tc , is given by   ð ðfaÞ2 1 P0 1 fJ0 ðfxÞexp  Tc ð x; yÞ ¼ 4pks 0 8 bs   1 þ expð2bs ls Þ (6)  exp ybc df; 1  expð2bs ls Þ where P0 is the excitation beam power, a is the beam radius defined at 1=e2 of the beam intensity, ks is the sample’s thermal conductivity, J0 denotes the Bessel function of zero order, sub-indexes c and s represent the medium and the sample, respectively, bi ¼ f2 þ ix ai ði ¼ c; sÞ, and a is the thermal diffusivity. The thermal diffusion length can be qffiffiffiffi ai defined as li ¼ pf . This temperature field, when analyzed at distances close to the object’s surface and when the sample has a larger thermal diffusion length (lc  ls ), displays a linear behavior between the phase-lag, /, and the horizontal offset to the heat source, x. The slope of this relationship sffiffiffiffiffi 1 pf m¼ ¼ ; (7) ls as is inversely proportional to the thermal diffusion length of the sample. This relation is of great interest as it provides a straightforward method for determining the thermal diffusivity of the sample. As mentioned above, the change in the refraction index, Dn, is assumed to be directly proportional to the change in temperature, DT, and both are related to one another by the temperature coefficient of the refractive dn index, dT , so that the refractive index field follows the same distribution as the thermal field shown in Equation (6).15 Under this assumption, Eqs. (5) and (6) can be used to synthetically generate a shadowgraph. It is important to note dn that a fluid with a higher dT will strengthen the signal. B. Simulation

A numerical simulation was carried out in order to show that Eq. (7) holds when a shadowgraph is used indirectly to assess the temperature field and to further understand on the technique limits. The optically opaque sample

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TABLE I. Simulation parameters. Parameter

Value

dnc dT

  m2 1:075  107 s   104 4:5 K

n a L D

1.3441 174 lm 0.01 ðmÞ 1  103 ðmÞ

ac

is supposed to be submerged in acetonitrile, a solvent for which thermal and optical properties are well known,16 and dn . Three different modulation which has a relatively high dT frequencies were arbitrarily chosen and two different sample thermal diffusivities were utilized, both are substantially higher than that of the acetonitrile so that heat diffusion through the sample governs over-diffusion through the medium. The simulations consider a pixel size of 12.3 lm, a value taken from a direct measurement from our experimental hardware (see Section III). Additional simulation parameters are shown in Table I. The simulation has been programmed to output a phase image. Figure 2 shows an example of a simulation output for a full 2:5 mm  0:3 mm shadowgraph as a colored contour plot for a sample thermal diffusivity of 1  105 m2/s and a modulation frequency of 8 Hz. The phase has been corrected for quadrant changes. The sample is parallel to the horizontal axis. The values of the pixels closest to the sample’s surface are given at y ¼ 12:3 lm, i.e., the pixel size. The exciting beam is positioned at x ¼ 0 lm. Simulations for both sample thermal diffusivities and the three arbitrary frequencies are shown in Figure 3. The simulation was carried out both for thermally thick, ls  ls , and thermally thin samples, ls  ls . Thermally thin samples will also show the same behavior on their back face. It is important to note that only the information given by the pixel line closest to the sample surface has been plotted. It is from these data points that the slope referred in Eq. (7) is obtained. The main objective is to estimate the sample’s thermal diffusivity, ^ a s , from the slope, utilizing Eqs. (5) and (6), and to obtain the estimation error, , by comparing it to the actual value, as , used for the simulation. The linear fit takes into account the 10 points furthest from the center in order to

avoid the influence of the nonlinear behavior seen near x ¼ 0. Fitting results are shown in Table II. The simulation confirms that the slope obtained from the intensity change phase lag versus heat source offset will be governed by the thermal diffusivity of the sample. This is the case for thermally thick and thermally thin samples. The shadowgraph mapping function, Eq. (5), faithfully reproduces the phase of the thermal waves generating the refraction index inhomogeneity. This entails that any sample geometry where the phase slope method is valid can be measured. It is also important to note that with lower frequencies, linearity occurs at a larger offset due to the diffusion of heat through the surrounding liquid. This effect is exacerbated due to the pump beam’s dimensions. Care should be taken when analyzing the experimental data, so that the fitting does not take place inside the pump beam influence zone, i.e., x > a, or where the medium diffusion effects are strong, i.e., the offset should be larger than the thermal diffusion length of the surrounding media, x > lc . Another important insight obtained from this simulation is that higher frequencies have to be favored as they translate into smaller estimation errors. Experimentally, the modulation frequency will be mostly limited by signal intensity since the amplitude of the signal falls drastically with frequency. The importance of the measurement taking place as close to the sample as possible is also an important factor to take into consideration as has been reported elsewhere.11,17,18 III. EXPERIMENTAL SETUP

The experimental setup is shown in Fig. 4. The probe light source is a laser beam collimated and widened. This is opposite to a PBD experiment, which requires a narrow probe beam. The sample has been immersed in acetonitrile, which provides amedium with high refractive index temper dn . ature coefficient dT A 445 nm diode laser with a 250 mW nominal output power and a spot size in the order of 200 lm, focused using proper optics and orthogonally positioned with respect to the probe light source, is used to heat the sample periodically generating the thermal waves, which will diffuse over the length of the sample. The laser’s output power is adjustable and is fine-tuned for each sample. The sample is placed within a 1 cm3 cell. For ease of measurement due to the cell’s dimensions, the samples

FIG. 2. Simulation results for as ¼ 1:0  105 ½m2 =s  and 8 Hz modulation frequency. (a) Thermally thin and (b) thermally thick sample.

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FIG. 3. Simulation results at pixel closest to the surface for two different thermal diffusivities. Thermally thin sample: (a) as ¼ 1  106 m2/s, (b) as ¼ 1  105 m2/s. Thermally thick sample: (c) as ¼ 1  106 m2/s, (d) as ¼ 1  105 m2/s. Modulation frequencies are shown with different symbols while the solid lines indicate the best least squares linear fit.

chosen were either filaments or rods, although plates, foil, or bulk material could have been used. The thermal diffusivities chosen are all above 1  105 m2/s. The cell is placed directly in the trajectory of the expanded probe beam generating the shadow. Said shadow can then be projected on a screen or directly on the sensing element. The camera used is a FLIR SC2500 working on the near infrared portion of the spectrum (0.9–1.7 lm) with a 320  256 pixel InGaAs sensor, a maximum frame rate of 340 Hz at full frame and on-board lock-in detection capabilities. This camera offers 14 bit pixel sensitivity. The wavelength of the probe laser used was 905 nm. TABLE II. Simulation results.  as Thermally thin

m2 s



1:00  106

1:00  105

Thermally thick 1:00  106

1:00  105

f ðHzÞ m 2 4 8 2 4 8 2 4 8 2 4 8

  rad R2 m

2500 3542 5012 810 1137 15848 2484 3538 5019 776 1114 1585

1 1 1 1 1 1 1 1 1 1 1 1

^ as

  m2 s

The pixels closest to the sample surface can be automatically identified using edge detection techniques. For this, a reference image with the pump beam turned off is used. As confirmed by the simulation, it is important for the pixels to be as close to the shadow horizon as possible, otherwise the effects of heat diffusion through the surrounding liquid would become more prominent and would alter the results. With the adequate optical elements, pixel resolutions equivalent to a few tens of microns at the sample surface have been obtained. Lock-in amplification is an important part of the information processing. In a lock-in amplifier (LIA), an amplitude modulated signal can be reconstructed in both phase and amplitude if a reference signal is known, even in extremely noisy environments. Measurements taken with the SC2500 camera directly output a phase image.

 ð%Þ

1:00  106 0:5 1:00  106 0:1 1:00  106