Time-resolved, photothermal-deflection spectrometry with step optical excitation

Transcription

1 Zhao et al. Vol. 1, No. 5/May 004/J. Opt. Soc. Am. B 1065 Time-resolved, photothermal-deflection spectrometry with step optical ecitation Jianhua Zhao, Jun Shen, Cheng Hu, and Jianqin Zhou National Research Council of Canada, Institute for Fuel Cell Innovation, 350 East Mall, Vancouver, British Columbia V6T 1W5, Canada Mauro Luciano Baesso Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790, Maringá, Paraná Brazil Received September 19, 003; revised manuscript received December 15, 003; accepted January 1, 004 A theoretical model of a novel time-resolved, photothermal-deflection spectrometry with step optical ecitation (a rectangular pulse of a finite duration) is presented. To make the theory practicable, i.e., able to be used to determine the values of optical and thermal properties of the sample by fitting eperimental data to the theory, approimations are introduced. The numericaimulations show that when the value of the thermal conductivity of the deflecting medium (e.g., air) is small, the adiabatic approimation for the heat conduction at both front and rear surfaces of a sample is effective for both transparent and opaque samples, while the adiabatic and isothermal approimations for the heat conduction at the front and rear surfaces, respectively, are valid only for an opaque sample. With these approimations, the optical and thermal properties such as optical absorption coefficient and thermal diffusivity of solid materials (optically transparent or opaque) can be directly measured by this novel method. 004 Optical Society of America OCIS codes: , , , , , INTRODUCTION Since 1970, photothermal techniques have been developed for nondestructive material characterization, such as measuring optical and thermophysical properties of different materials and depth profiling, that are important because of their high sensitivity and versatility. 1 3 These techniques can be classified into two main categories by the criterion whether the samples contact or do not contact the detection system. 3 Photoacoustic and photopyroelectric methods are eamples of the contact techniques. Remote (noncontact) techniques, such as photothermal deflection (namely, optical beam deflection) and thermal lens methods, are suitable for studying samples in a severe environment. In transverse photothermal-deflection (PD) (or mirage effect) spectrometry, a probe beam senses the gradient of the optical refractive inde in the ambient medium adjacent to a sample surface, resulting in deflection of the probe beam. The gradient of the optical refractive inde is induced by the temperature gradient from the sample surface to the medium, and the temperature rise in the sample is the result of conversion of the absorbed electromagnetic ecitation radiation into heat. The optical and thermophysical properties of the sample, therefore, can be measured by monitoring the frequency dependence (for frequency-modulated ecitation 4,5 ) or the time dependence (for impulse ecitation 6,7 ) of the PD signal. The theory, and applications of PD spectrometry can be found in Refs. 1 7, and the reported thermophysical property measurement with the PD technique were achieved mainly by use of frequency-modulated ecitation. 3 One may measure thermal diffusivity in a frequencymodulated eperiment by using the linear relationship between two measurable parameters whose slope can be used to retrieve the thermal diffusivity value. However, a careful management of the linear relations is necessary since some of them are strictly valid only under ideal conditions that are very difficult to achieve in practice. 8 With no requirement to find any linear relations, timeresolved, thermal-lens spectrometry by means of step optical ecitation (i.e., the rectangular-pulse optical ecitation) has enjoyed great success in the determination of the thermal diffusivities of transparent samples The results are absolute measurements, and the thermal diffusivity values can be obtained directly by a convenient procedure of fitting eperimental data to a theoretical model. In this paper, we propose theoretically a novel time-resolved, photothermal-deflection spectrometry with step optical ecitation. Similar to the thermal-lens technique, this PD technique can be used not only for measuring optical properties of materials, but can be employed for determining thermophysical properties. Furthermore, this technique can be employed to study both transparent and opaque samples, while thermal-lens spectrometry can be applied only to transparent samples. In this work, we start with the calculation of the onedimensional temperature rise due to the absorbed energy of step optical ecitation in a thin, solid slab sample of finite thickness with boundary conditions. 1 Then we work out the temperature-rise distribution in the deflecting medium owing to the heat conduction from the front surface of the sample to the medium. Finally, we derive /004/ $ Optical Society of America

2 1066 J. Opt. Soc. Am. B/ Vol. 1, No. 5/ May 004 Zhao et al. the time-resolved PD signal based on the temperature gradient in the deflecting medium. To calculate the temperature rise in the sample and its surrounding medium after the absorption of the step optical ecitation by the sample, scientists have used Laplace transforms, such as in the theoretical work in Ref. 13. Only in some special cases could the temperature rise (in the time domain, not in s space) in the gas medium adjacent to the sample s front surface be obtained with an analytical epression. Green s functions have also been employed to solve the equation of heat conduction and find the temperature rise in the sample; one may find an eample in the theoretical work of Ref. 14. Although the Green s function method might be an approach to calculate the temperature increases after the step optical ecitation in both the sample and the surrounding media, as a complete treatment (not just for special cases), to the best of our knowledge, there is little work reported in which this approach was used. In using Green s functions to solve the heat conduction equations, we present here a complete theoretical treatment of photothermal-deflection spectrometry with step optical ecitation, aiming to obtain a practicable theory, i.e., one that can be used to determine the values of optical and thermal properties of the sample by fitting eperimental data to the theory. However, there is an obstacle to the theory s being practicable: To use this theory practicably to measure optical and thermal properties, one needs to know the thermal property to be measured to solve a transcendental equation. To remove this obstacle, we introduce approimations. Numericaimulations show that when the thermal conductivity of the deflecting medium is small (e.g., air), the adiabatic approimation for the heat conduction at both front and rear surfaces of the sample is effective for both transparent and opaque samples, while the adiabatic and isothermal approimations for heat conduction at the front and rear surfaces, respectively, are valid only for an opaque sample. Consequently, these approimations make this novel time-resolved, photothermal-deflection spectrometry practicable.. THEORETICAL MODEL In a PD eperiment as shown in Fig. 1 an isotropic, homogenous, thin slab sample is placed in a transparent and homogeneous deflecting medium. The thickness of the sample is, and the optical absorption coefficient of the sample is. The distance between the probe beam and the sample surface is a. A heating beam of spot size d is uniformly incident on the sample surface at 0 with intensity I 0 and a pulse duration p (step heating): It I 0, t p, active heating interval. 0, t p, relaation interval (1) For the convenience of later discussion, all variables with subscript a are for the active heating interval (t p ), while those with subscript r are for the relaation interval (t p ). If we assume the absorbed optical energy is converted into heat through a nonradiative process completely and instantaneously, the heat due to the optical absorption per unit volume per unit time at in the sample is Q, t Itep. () The differential equations of heat conduction in the sample and the deflecting medium are given by T s, t 1 T s, t Q, t, (3) s t k s T g, t 1 T g, t 0. (4) g t Here, we define T i (, t) as the temperature rise for both the active interval and relaation interval; the actual temperature is the sum of the temperature rise T i (, t) and the ambient temperature T. k i is the thermal conductivity, i k i / i c i is the thermal diffusivity, and i and c i are the density and the specific heat, respectively. i s or g stands for the sample or the deflecting medium, respectively. The initial condition is T i, 0 0. (5) The boundary conditions are the continuities of the temperature and heat flues at the boundaries of 0 and : T s 0, t T g 0, t, T s, t T g, t, (6a) (6b) T s, t k s 0 T g, t k g, (6c) 0 T s, t T g, t k s k g. (6d) Fig. 1. Schematic diagram of the photothermal-deflection spectrometry of a thin-slab solid sample in a one-dimensional treatment. These conditions assume that the heat transfer from the sample to the deflecting medium adjacent to the sample surfaces is dominated by heat conduction. The heat conduction from the sample surface to a location l g in Fig. 1 in the deflecting medium takes time g l g /4 g (Ref., p. 106). We may call this thin layer of medium in contact with a sample surface with thickness l g a thin-skin layer (Ref. 15, p. 0). When t g the

3 Zhao et al. Vol. 1, No. 5/May 004/J. Opt. Soc. Am. B 1067 temperature increases T g (l g, t) and T g ( l g, t) are zero, i.e., the actual temperature at l g and l g is the ambient temperature T. By neglecting the heat capacities of this thin skin layer on the front sample surface 0 and following Refs. 9 and 16, as a first approimation we have T g, t T g 0, t T g l g, t k g k g l 0 g k g T g 0, t. (7) l g Relation (7) is the rate of heat flow through the thin-skin layer (Ref. 15, p. 0) when t g, and at that time T g (l g, t) 0. By substituting the boundary condition Eq. (6a) of the temperature continuity at the boundary 0 into Eq. (7), the heat flu continuity of Eq. (6c) can then be replaced by the boundary condition for the linear heat transfer through the thin-skin layer into a medium at zero temperature rise 16 T s, t k s h 1 T s 0, t 0. 0 (6c) Similarly, one can find the boundary condition at as T s, t k s h T s, t 0. (6d) Here, h 1 h k g /l g are the heat transfer coefficients for the boundary condition of the third kind. 1 One can find a similar epression in Ref. 15, p. 0. When t g, Eqs. (6a), (6b), (6c), and (6d) are the boundary conditions of the differential Eqs. (3) and (4) of heat conduction in the sample and the deflecting medium. The Green s function for the heat conduction Eq. (3) with the initial condition of Eq. (5) and the boundary conditions of Eqs. (6c) and (6d) is given by G,, t, m1 A m Z, ep s m t, (8) (not valid for both h 1 0 and h 0; Ref. 15, p. 360), where T s,a, t 0 t 0 Q, G,, t, dd, s c s t p, (1a) Q, T s,r, t G,, t, dd, 0 p0 s c s t p. (1b) T s,a (, t) stands for the temperature rise in the active heating interval and T s,r (, t) the temperature rise in the relaation interval. Let u m m. By substituting Eqs. () and (8) into Eqs. (1), we have T s,a, t m1 T s,r, t m1 where B m k su m cos u m 1 ep u m t B m k su m cos u m ep u m p s I 0 A m B m k s u m u m l k su m s h 1 sin u m s, t p, h 1 sin u m 1 ep u m (13) t s, t p, (14) ep u m sin u m cos u m h 1 ep u m cos u m sin u m u m. (15) In Eqs. (13) and (14), s l s / s is the characteristic diffusion time constant of the sample. 17 The physical meaning of s is that for a semi-infinite solid of thermal diffusivity s heated by a Dirac delta heating impulse on its surface plane 0att0, the temperature rise at in the sample due to heat conduction is less than k s m h A m, (9) k s m h 1 k s m h k s h k s h 1 k s m h Z, k s m cos m h 1 sin m k s m cos m h 1 sin m. (10) m is the root of the following transcendental equation: tan m mk s h 1 h k s m h 1 h. (11) The temperature rise in the sample due to the absorption of the step optical ecitation is then given by 10% of the surface magnitude when t s. From Eqs. (13) and (14), we have the temperature rise on the sample surface 0as B m k s u m T s,a 0, t m1 1 ep u m t s, t p, (16)

4 1068 J. Opt. Soc. Am. B/ Vol. 1, No. 5/ May 004 Zhao et al. T s,r 0, t m 1 B m k s u m ep u m 1 ep u m p s t t p. (17) s, Equations (16) and (17) will be used as the boundary conditions in calculating the temperature rise in the deflecting medium. The deflecting medium can be regarded as semiinfinite, so the temperature rise with the above initial and boundary conditions Eqs. (5) and (6a) is given by T g, t / g t t 4 g ep d (18) (Ref. 15, p. 63), where (t) T s,a (0, t) in the active heating interval and (t) T s,r (0, t) in the relaation interval. By substituting Eqs. (16) and (17) into Eq. (18), we can obtain the temperature rise in the deflecting medium as B m k s u m T g,a, t m1 erfc g t ep u m t s ep j s u m g j u m t s ep j s u m g erfc g t erfc g t j u m t s, t p, (19) T g,r, t T g,a, t T g,a, t p, t p. (0) Here erfc(z) is the complementary error function. From the geometric optics theory of photothermaldeflection spectroscopy, the probe beam deflection angle in one dimension can be written as 18, t 1 n T, t d, (1) n T where n is the optical refractive inde of the deflecting medium. n/t is the temperature coefficient of the refractive inde of the medium, which can be regarded as constant (usually negative) when the temperature rise is small. With Eqs. (19) to (1), the deflection angle can be epressed as a, t 1 n jks n T d B m u m m1 ep j s u m g erfc g t ep j s u m g erfc g t s g ep u m j u m t s j u m t s, t s t p, () r, t a, t a, t p, t p. (3) 3. DISCUSSION To use the above theory to measure the optical and thermal properties of the sample, one needs to solve transcendental Eq. (11) that is associated with the thermal conductivity k s of the sample. This in turn means that one needs to know the thermal conductivity to be measured to apply the theory for measuring the thermal properties of the sample, which renders the theory impracticable for eperimental measurements. It is, therefore, necessary to find an approach to simplify Eq. (11) so it can be easily solved without knowledge of the sample s thermal properties. Furthermore, with the real boundary conditions Eqs. (6c) and (6d), the mathematical epressions of the temperature rise in the sample and the deflecting medium, as well as the probe-beam deflection angle (PD signal), are complicated. It will be helpful for processing the eperimental data if the mathematical epressions can be simplified. In the discussion below, we will try to develop an approach and determine its validity and the conditions for the approach and simplification based on some reasonable approimations. A. Boundary Conditions For optically opaque samples, the temperature rise is produced mainly at the front surface 0 in Fig. 1. With the physical meaning of s, one can infer that when t s, the heat has not reached the rear surface of the sample, 17 i.e., the temperature rise at the rear surface is then zero; that is, the actual temperature is the ambient temperature, which is equivalent to isothermal contact with the deflecting medium, i.e., h. At the front surface of the sample 0, if the thermal conductivity value of the deflecting medium is low, the heat flow crossing the boundary at 0 will be insignificant. So the boundary condition at the front surface of the sample may be approimated by adiabatic insulation h 1 0 when t is short and the thickness l g 4 g t of the thinskin layer in the deflecting medium adjacent to the sample front surface is thin enough that the thermal ca-

5 Zhao et al. Vol. 1, No. 5/May 004/J. Opt. Soc. Am. B 1069 Table 1. Optical and Thermal Properties of Glassy Carbon and Lime Glass Sample Density (kg/m 3 ) Specific Heat (J/kg K) Thermal Conductivity (W/m K) Absorption Coefficient (m 1 ) Thermal Diffusivity (m /s) Thermal Effusivity (Ws 1/ /m K) Glassy Carbon a Lime Glass b c a See Ref. 19. b See Ref. 10; c J/m 3 K. c At nm. pacity of the thin-layer medium is negligible and the approimation of Eq. (7) is applicable. For optically transparent samples, the optical absorption occurs through the sample from 0to, and so does the temperature rise. The adiabatic approimation might be applied to both front and rear surfaces of the transparent sample if t were short. Air is the common deflecting medium in PD eperiments. In this medium, one may epect that the adiabatic isothermal (AI) approimation for opaque samples and the adiabatic adiabatic (AA) approimation for transparent samples will be effective when t is short because of the low conductivity and consequent small value of the heat transfer coefficient h with air (Ref. 15, p. 0). With the AI approimation, we can easily solve Eq. (11) without knowledge of sample properties: m m 1/, m 1,, 3,.... (4) Similarly, we can determine the solution for Eq. (11) with the AA approimation: m m/, m 1,, 3,.... (5) The mathematical epressions of the temperature rise in the sample and the medium, as well as the PD signal with these two important approimations, are also simplified. The analytical epressions with the AA approimation can be found in Appendi A. B. Simulations In the following simulations, we will discuss the conditions and validity of the AI and AA approimations. Glassy carbon 19 will be used as an opaque sample and lime glass (7% SiO, 18% Na O, and 10% CaO in weight) 10 as a transparent sample in the simulations. The optical and thermal properties of theses two samples can be found in Table 1. In the simulations, the total time, i.e., the sum of the active heating interval (t p ) and the relaation interval (t p ), does not eceed the characteristic diffusion time constant s of the sample. The thickness l g of the thin-skin layer in the deflection medium is estimated by l g 4 g s. The parameters we used in the simulations with air and water as the deflecting media are listed in Table. The thermal properties of the media can be found in Ref. 0. In our simulations, we will discuss three possible boundary condition cases: (1) the normal (real) boundary case (h 1 h k g /l g ), () the AA approimation (h 1 h 0), and (3) the AI approimation (h 1 0 and h ). In the simulation, the criterion for ceasing the summation in Table. Sample Parameters Used in the Simulations Deflection Medium (mm) h (W/m K) s (s) Glassy Carbon Air Glassy Carbon Water Lime Glass Air Lime Glass Water Eqs. (13), (14), (19), (0), (), and (3) will be the ratio mth term:first term Temperature Rise in the Samples The temperature rise in the samples in the active heating interval ( p 1 s ) for different times t is shown in Fig.. For glassy carbon in air, the temperature rises in the three cases at the front surface of the sample are almost the same [see Fig. (a)]. The relative difference between case (1) and case () and between case (1) and case (3) is less than 0.5%. However, at the rear surface of the sample, the AI approimation keeps the temperature rise always at zero, accounting for the difference between case (1) and case (3) when t 0.5 s. For an opaque sample such as glassy carbon, the heat source produced after absorption of the heating beam energy is located very close to the front surface of the sample. When t s, the heat flow does not reach the rear surface; as a result there is no temperature rise at the rear surface. Thus the AI approimation is valid only when t s. For the AA approimation, the temperature rise in the sample is almost the same as in case (1) with small relative difference 1% because of the low value of the thermal conductivity k g of air and consequent low heat flu from the sample to air, which is close to the adiabatic situation. Figure (b) shows the temperature rise in the glassy carbon sample in water. There is no big difference between case () and case (3) at the front surface of the sample. However, there are differences of 4%, 9%, and 13%, between case (1) and case () or case (3) when t 0.1 s, 0.5 s, and 1.0 s, respectively, because of the high value of the thermal conductivity of water. One may notice that at the rear surface the difference between case (1) and case () is not as great as the differences at the front surface. This smaller difference may be due to the temperature rise at the rear surface and the consequent heat flu from the sample to the medium, which are much lower than that at the front surface.

6 1070 J. Opt. Soc. Am. B/ Vol. 1, No. 5/ May 004 Zhao et al. As opposed to the glassy carbon sample, lime glass is optically transparent, and the heat source is spread over the sample in the direction of its thickness. The low value of k g and the small difference in the temperature rises at the front and rear surfaces are the reasons the temperature rises calculated in case (1) and in the AA approimation [case ()] are almost the same (difference 0.5%) for lime glass in air, as shown in Fig. (c). There is a big difference at the rear surface between case (1) and case (3) because the AI approimation keeps the temperature rise at zero at the rear surface of the sample, which is unrealistic for the transparent sample, even with short t. For lime glass in water, Fig. (d), the temperature rise at the middle of the sample along the thickness direction is higher than that at its surfaces, and all the approimations deviate from case (1) because of the high thermal conductivity of water and the consequent high heat flu from the glass to water. For t 0.1 s, the difference between case (1) and case () is smaller than 8%.. Temperature Rise and Photothermal-Deflection Angle in the Deflecting Media The temperature rise in the deflecting medium is dependent on the temperature rise at the front surface of the sample, as predicted by Eq. (18). The photothermaldeflection signal is proportional to the derivative of the temperature rise in the medium, as predicted by Eq. (1). One may epect that if there is good agreement between case (1) and an approimation [case () or case (3)] in the temperature rise at the front surface, there will be good agreement in the prediction of the temperature rise in the medium and the PD signal. Figures 3 and 4, in general, meet this epectation. Figure 3 shows the temperature rise in the deflecting media with p 0.5 s for the active and relaation intervals. The temperature rises at the front surfaces of the samples in Figs. 3(a) and 3(c) reach a maimum when t 0.5 s and start to fall thereafter. The temperature rise in air falls faster with glassy carbon than it does with lime glass because of the different heat Fig. 3. Temperature rise in the deflecting media with the heating beam pulse duration p 0.5 s for different time t; l g 4 g s. (a) Glassy carbon in air, (b) glassy carbon in water, (c) lime glass in air, and (d) lime glass in water. Solid curves, case (1); diamonds, case (); squares, case (3). Fig. 4. Time-resolved, photothermal-deflection signals with the heating beam pulse duration p 0.5 s for different positions in the deflecting media; l g 4 g s. All results are normalized to d/nn/t. (a) Glassy carbon in air, (b) glassy carbon in water, (c) lime glass in air, (d) lime glass in water. Solid curves, case (1); diamonds, case (); squares, case (3). Fig.. Temperature rise in the samples during the active heating interval with the heating beam pulse duration p 1 s for different time t. (a) Glassy carbon in air, (b) glassy carbon in water, (c) lime glass in air, (d) lime glass in water. Solid curves, case (1); diamonds, case (); squares, case (3). source distributions. It is worth noting that for both samples, the temperature rises at 4 g s in the deflecting medium (air) deviate more from zero with increase of time t, which means that Eqs. (7), (6c), and (6d) are not applicable when t s. Figure 4 shows the time-resolved PD signals with p 0.5 s at the different positions in the deflecting media. The prediction by AA approimation [case ()] is in good agreement with that by AI for both samples in air, as shown in Figs. 4(a) and 4(c). It is interesting to note that when the heating beam shuts off, the PD signal does not decrease instantaneously, and the maima of the signals with different positions in the deflecting media take place at different times. This phenomenon is probably due to

7 Zhao et al. Vol. 1, No. 5/May 004/J. Opt. Soc. Am. B 1071 the time delay for the probe beam to sense the effect of the termination of the heating beam, depending on the distance a between the probe beam and the sample front surface in Fig. 1. When the distance a is small, e.g., 0.14 g s, the PD signatarts to fall almost immediately. For a larger distance a, the time delay is longer because it takes more time for the heat to diffuse through a longer distance. For the lime glass sample, the timeresolved PD signal decays slower after the termination of the heating beam than the PD signal does with the glassy carbon sample. For the transparent lime glass, the heat source is distributed throughout the sample, while for the opaque glassy carbon it is only within a thin layer close to the front surface. Heat conduction from the inside of the lime glass sample to the deflecting media after termination of the heating beam slows down the PD signal decay. Figure 4(c) shows that the prediction by the AI approimation is not as good as that by the AA approimation for the lime glass sample; this is because of the unrealistic isothermal approimation for the transparent sample. Concerning the temperature rise in water, Figs. 3(b) and 3(d), neither approimation is very close to case (1) for both samples because of the high value of the thermal conductivity of water and the consequent high heat flu from the samples to water, especially when t 0.1 s. Correspondingly, for the PD signals in Figs. 4(b) and 4(d), relative differences of more than 8% can be found at 0.54 g s and t 0.5 s for both approimations compared with case (1). The difference increases with the decrease of in the medium. 4. CONCLUSIONS A novel time-resolved, photothermal-deflection spectrometry with step optical ecitation has been proposed with a set of closed-form analyticaolutions. For the purpose of the practical application and the simplification of the theory, the adiabatic adiabatic (AA) and adiabatic isothermal (AI) approimations have been fully eplored to determine their validity. The numericaimulations reveal that for deflecting media with low values of thermal conductivity, such as air, the AA approimation is weluited for both opaque and transparent samples used in the simulations if t is short, while the AI approimation is applicable only to the opaque sample. For media with high thermal conductivity, e.g., water, neither approimation is very effective. We conclude that these approimations can be effectively used for characterizing solid materials, such as measuring their optical and thermal properties, with this novepectrometry. APPENDIX A With the AA approimation h 1 0 and h 0 m m/, m 1,, 3,.... (A1) With these boundary conditions, the Green s function is (Ref. 15, p. 361) G,, t, 1 1 m1 ep m t s cos m m cos. (A) By using Eq. (1), the temperature rise in the sample is found to be T a, t qt m1 E m 1 epm t/ s cos m, T r, t q p m1 E m epm p / s 1 Here epm t/ s cos m (A3). (A4) E m I m ep, (A5) k s m/ m/ q I 0 1 ep. s c s The temperature rise in the deflection medium is T g,a, t qt 1 g t erfc g t g ep t 4 g t m1 E m erfc ep m t s ep jm s g jm t s g t erfc ep jm s g erfc jm t g t g t (A6) s, t p (A7) T g,r, t T g,a, t T g,a, t p, t p. (A8) The deflection angles in the active and relaation intervals are, respectively,

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