Chin. Sci. Bull. DOI 10.1007/s11434-014-0461-9

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Article

Optoelectronics & Laser

Thermal diffusivity determination of solids by time-domain photoacoustic piezoelectric technique Binxing Zhao • Yafei Wang • Chunming Gao Qiming Sun

•

Received: 3 July 2013 / Accepted: 20 March 2014 Ó Science China Press and Springer-Verlag Berlin Heidelberg 2014

Abstract Thermal–physical property is one of the most important properties of materials. The conventional frequency-domain photoacoustic piezoelectric (PAPE) technique has been developed as an important method for thermal diffusivity determination. In this paper, the PAPE technique is extended to time domain (TD). First, based on a simplified thermoelastic model, the TD PAPE theory under square-wave-modulated excitation was developed, the dependence of the TD PAPE signal on modulation frequency as well as material parameters was obtained, and the determination of thermal diffusivity was simulated and theoretically analyzed. Second, the experimental system and the corresponding measurement method were established. Third, thermal diffusivities of various standard samples, such as copper, aluminum, and nickel, were measured, and the effectiveness of the technique was verified. The results show that the TD PAPE technique can provide a simple, fast and effective way for thermal diffusion study.

simplified thermoelastic model [6], thermal diffusivities of metals, composites, and biological tissues were successfully measured [6–12]. By introducing a two-layer model, the modified PAPE technique has higher measurement accuracy for thermal diffusivity determination [13–15]. Nevertheless, the present PAPE technique is in frequency domain (FD) where the frequency scan procedure is necessary. This leads to complex system, slow measurement, and expensive device which are not conducive to instrumentation and application. Hence, an effective, fast, simple and economical technique for thermal diffusivity measurement is in demand. Time-domain (TD) PAPE technique does not need lock-in amplifier and frequency scan, thus being able to be simple and fast. In this paper, the methodology of the TD PAPE determination of thermal diffusivity has been developed, and thermal diffusivities of various solid materials have been measured.

Keywords Thermal diffusivity  Time-domain photoacoustic piezoelectric technique  Thermoelastic model

2 Theory

1 Introduction The photoacoustic piezoelectric (PAPE) technique has been applied to study structure, composition, and thermal properties of various materials [1–15]. Based on Blonskij’s B. Zhao  Y. Wang  C. Gao (&)  Q. Sun School of Opto-Electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China e-mail: [email protected]

The principle of the PAPE technique is as follows: a periodic heat source excited by an intensity-modulated (square wave) laser generates thermal waves; the thermal waves propagate in the sample and cause thermoelastic strains; the periodic thermoelastic strains are detected by the PZT attached to the back side of the sample. The model is shown in Fig. 1. The sample is a thin plate, with PZT attached to the back side. l, R, k, q, c are the thickness, radius, thermal conductivity, density, and specific heat of the sample, respectively, and L is the thickness of the PZT. Based on Blonskij’s simplified thermoelastic model [6], the thermoelastic strains in the sample can be expressed as

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Chin. Sci. Bull.

z z

Modulated laser

z = l/2 Sample

l L

0

R, k, ρ, c

PZT PZT

r z = –l/2

Fig. 1 Theoretical model

Z

her þ eh i ¼ 2p T0 ¼

1 l

Z

12 s¼ 3 l

R

ðer þ eh Þrdr ¼ 2aðhT0 i þ zhsiÞ;

0 l=2

T ðr; zÞdz;

ð1Þ

l=2

Z

l=2

T ðr; zÞzdz;

l=2

where er is the strain in the r direction, eh is the strain in the h direction, a is the linear thermal expansion coefficient, hT0 i is the average thermal expansion due to the whole temperature rise, zhti is the bending due to the temperature gradient along z direction, and the final PAPE signal is the superposition of these two vibrations, T(r, z) is the complex amplitude of the temperature distribution. Considering the strong absorption case bl ? ?, where b is the optical absorption coefficient of the sample, we have   Ipb2 3Ipb2 2ð1  coshðrlÞÞ hT0 i ¼ ; zhsijz¼ l ¼  1þ : 2 rl sinhðrlÞ klr2 klr2 ð2Þ So the FD PAPE signal is [6]   eL l V ðxÞ ¼ 2a hT0 i  hsi eS 2   2Pa 3 1  cosh rl ¼ 2 1þ ; klr r l sinh rl

ð3Þ

where r = (ix/D)1/2, P = 2Ipb2eL/(eS), D is the thermal diffusivity, x is the angular modulation frequency, I is the laser intensity, b is the laser radius, S is the surface area of PZT, e is the piezoelectric modulus, and e is the permittivity. When the laser is square-wave modulated, the thermal source has the form 2 2 Q ¼ Iber =b ebðl=2zÞ Re½f ðtÞ;

where 1 1 2X ð1Þm1 ið2m1Þðxtp=2Þ e f ðt Þ ¼ þ 2 p m¼1 2m  1

 A0 þ

1 X m¼1

123

A2m1 ðxÞeið2m1Þxt :

ð4Þ

Combining Eqs. (3) and (4), the TD PAPE signal (neglecting the dc term) is " # 1 X A2m1 ðxÞV2m1 ðxÞeið2m1Þxt : ð5Þ V ðx; tÞ ¼ Re m¼1

Equation (5) shows the dependence of the TD PAPE signal on the modulation frequency and material parameters. Under the thermally thick condition, with known thickness and frequency, the TD PAPE signal is only determined by the thermal diffusivity and a normalized factor. In Fig. 2, the abscissa is the time normalized to the modulation period. The normalized theoretical curves are shown in Fig. 2a, with D = 100 mm2/s, f = 30 Hz, L = 2 mm. Figure 2a also shows the TD average expansion and the bending signal given by Eq. (2). The TD average expansion signal is always a near triangle wave and in-phase with the excitation, whatever the thermal diffusivity is large or small; however, the wave form of the TD bending signal varies with thermal diffusivity and is out of phase with the excitation. The reason for such out-of-phase feature is that PZT is attached on the back side of the sample, so when the vibration of the front surface (in phase) is expansion, the back surface will be compressed (out of phase). The theoretical foundation for thermal diffusivity determination by the TD PAPE technique is provided by virtue of the close correlation between the bending wave form and the thermal diffusivity. Figure 2b shows the theoretical TD PAPE signal with different thermal diffusivities. By fitting the theory to the experimental data (both normalized), the thermal diffusivity can be obtained. The variance of the difference between two curves, used for fitting, is defined as [16]: 2 N  P f Sm  S i i var ¼ i¼1 N   ; ð6Þ P f 2 Si i¼1

Sm i

where is the experimental data, Sfi is the theoretical value, and N is the sampling number per period. In the fitting procedure, within a certain error range, the fitting accuracy is usually influenced by the sensitivity of the variance. For theoretical analysis of the sensitivity of thermal diffusivity measurement, Fig. 3 shows the influence of thermal diffusivity on the variance, calculated by Eq. (6) with D = 2, 20, and 100 mm2/s, f = 30 Hz. As the thermal diffusivity varies, the variance has a V-shaped variation. The curve will have a larger slope when the thermal diffusivity is larger, which means the measurement can be more sensitive.

Chin. Sci. Bull.

Fig. 2 (Color online) Simulations: a average expansion and bending components, and their combined TD-PAPE signal; b TD-PAPE signals with different thermal diffusivities

Argon-ion laser

Acoustic-optic modulator

Lens

Modulator driver Function generator Computer

Silicon detector

Beam splitter

Sample PZT

Oscilloscope

Fig. 4 Schematic of the experimental system

Fig. 3 (Color online) Variance versus thermal diffusivity

3 Experiments The experimental setup of the TD PAPE detection is shown in Fig. 4. The sample and PZT are closely attached. An argon-ion laser (wavelength 514 nm, power 200 mW), modulated by an acousto-optic modulator driven by a function generator, induces periodic thermal wave and thermoelastic wave in the sample. The thermal wave is attenuated, and the thermoelastic wave is detected by the PZT (thickness 0.1 mm and diameter 18.2 mm) and converted to an electric signal. Oscilloscope records the signal and computer deals with the experimental data and fitting procedure. All measurements were in room temperature. For verifying the effectiveness of thermal diffusivity determination by the TD PAPE technique, three standard

samples copper, aluminum, and nickel were studied. The samples were made in disk form, with a diameter 17 mm, thicknesses 2.00, 1.90, and 1.98 mm, respectively. The TD PAPE experimental data and the best fitted curves are shown in Fig. 5, and the fitted thermal diffusivity values are shown in Table 1. The modulation frequencies are chosen to be 31 Hz for copper and aluminum measurements and 29 Hz for nickel, which satisfy the thermally thick conditions for all the samples. From Fig. 5 and Table 1, it can be seen that the measured thermal diffusivities of copper, aluminum and nickel are 114, 98, and 22.6 mm2/s, respectively; the relative errors are 1.3 %, 1.6 %, 2.2 %, respectively. The error sources may come from: (1) the sample used may not be exactly the same as the reference; (2) the simplified model neglects the influence of the PZT on the sample vibration; (3) the TD signal contains both amplitude and phase information, and the amplitude is vulnerable to the noise of the optical and electrical system; (4) the PAPE signal will decrease with increasing frequency, so the SNRs of the high frequency components are low, leading to the

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Chin. Sci. Bull. Acknowledgments This work was supported by the National Natural Science Foundation of China (50506006 and 61379013) and the Fundamental Research Funds for the Central Universities of China (E022050205 and ZYGX2012Z006). Conflict of interest of interest.

The authors declare that they have no conflict

References

Fig. 5 (Color online) Experimental data and theoretical fitting

Table 1 Thermal diffusivities of copper, aluminum and nickel Sample

Measured (mm2/s)

Reference (mm2/s)

Relative error (%)

Copper

114

115.5 [17]

1.3

Aluminum

98

96.5 [17]

1.6

Nickel

22.6

23.1 [17]

2.2

amplitude and phase error. However, the relative error is within 2.2 %, which means the TD PAPE determination of thermal diffusivity is effective.

4 Conclusions Based on the simplified PAPE model, the TD PAPE method is studied, and the thermal diffusivities of copper, aluminum and nickel samples are measured. Several conclusions are obtained: (1)

(2)

(3)

The TD PAPE theoretical model is developed, and the analytic expression of the TD signal on the modulation frequency and material parameters is obtained. Simulation results show that the bending vibration makes the TD PAPE signal wave form sensitive to the thermal diffusivity. Material with larger thermal diffusivity has higher measurement sensitivity and accuracy. The experimental system of the thermal diffusivity determination by TD PAPE method is set up, and three standard samples, copper, aluminum and nickel, are measured. The relative error is within 2.2 %.

The theoretical and experimental studies show that the TD PAPE technique is a fast, simple and effective method to measure the thermal diffusivities of various solid materials.

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