Precise measurements of Thermal Diffusivity by Photothermal Radiometry for Semi-infinite Targets using Accurately Determined Boundary Conditions (original) (raw)

Precise measurements of Thermal Diffusivity by Photothermal Radiometry for Semi-infinite Targets using Accurately Determined Boundary Conditions.

Christophe Martinsons, Andrew Levick* and Gordon Edwards
Centre for Basic, Thermal and Length Metrology
National Physical Laboratory
Teddington, Middlesex, TW11 0LW, England

Abstract

An experimental set-up was developed at the NPL with the goal of improving the accuracy of photothermal measurements. The principle is to perform accurate measurements of the experimental boundary conditions during a photothermal radiometry experiment. A numerical 3D heat diffusion model based on thermal transfer functions has been developed to use the measured boundary conditions. To prove the validity of this methodology, the thermal diffusivities of pure metal samples were estimated from experimental thermal radiance data by a least-squares fitting to the model. Experiments carried out at about 900 K demonstrate good agreement between the theoretical and experimental data and accuracies of about 1.5%1.5 \% for thermal diffusivity measurements.

(Received on June 21, 2000; Accepted on October 22, 2000)

Abstract

Keywords: Photothermal radiometry, infrared, laser, thermal diffusivity.

Photothermal techniques are being developed at the NPL for the simultaneous measurement of temperature 1{ }^{1} and thermophysical properties such as emissivity and thermal diffusivity. This paper describes the results of recent investigations directed particularly at improving the measurement accuracies. Well-established techniques for the thermal characterization of solids such as the flash technique 2,3{ }^{2,3} or thermal wave techniques 4{ }^{4} often rely on simplified geometries and ideal boundary conditions used a priori in a mathematical model. Systematic errors inevitably occur because of experimental deviations from initial assumptions 4{ }^{4}. In the laser flash technique for instance, a uniform 1D excitation and a simple temporal pulse shape are assumed 2,3{ }^{2,3}. Pulsed laser beams are in essence difficult to measure and the desired characteristics are not always met. This may partly explain why the uncertainties reported for thermal diffusivity values are rarely better than a few percent, even with high purity materials. In order to avoid such biases, we propose a methodology based on the accurate measurement of experimental boundary conditions. We applied it to a photothermal radiometry 5{ }^{5} experiment designed to measure the thermal diffusivity of solids. In the case of a sufficiently large sample opaque at the pump wavelength, the relevant boundary conditions are the laser beam profile at the surface and the time dependence of the incident intensity. Experimental data on platinum and titanium heated to about 900 K in air are presented and discussed in the last section of this paper.

Measurement Principle

The excitation of the sample is achieved using a periodically scanned cw laser beam. By absorption of a fraction of the incident light, a periodic temperature field is induced which in turn is responsible for a modulation of the thermal radiance emitted by the stimulated surface. Visible and infrared optical detection are performed in the time domain to measure the modulated thermal radiance and the back-scattered laser light. The modulated thermal radiance is proportional to a local average of the induced temperature field whereas the back-
scattered signal gives the time dependence of the incident laser intensity. The beam profile is accurately measured using a beam analysing device made of a scanning slit attached to the moving mirror of a Michelson interferometer.

The theoretical waveform of the periodic temperature field is given by a linear relationship between the thermal source term and a heat diffusion operator. A mathematical model was developed to solve this 3D time-dependent heat conduction problem. The thermal diffusivity is estimated from the experimental data sets by a non-linear least-squares fitting to the model.

Experimental Set-Up

A schematic of the photothermal set-up is shown in Fig. 1. The sample’s front surface is optically excited by an argon ion laser beam delivering a maximum power of 2 W at 514.5 nm . A laser scanner, driven by a programmable function generator, is used to periodically scan the beam across the sample’s surface. A CaF2\mathrm{CaF}_{2} plano-convex doublet forms an image of a fixed point of the surface on to two optical detectors. The doublet is used in 1 to 1 imaging ratio and has a numerical aperture of about 0.1 . The optical signal is the superimposition of the back-scattered laser light and the thermal radiance emitted by sample surface. A longwave-pass germanium beamsplitter is used to transmit wavelengths greater than 1.85μ m1.85 \mu \mathrm{~m} towards an InSb infrared detector (dimensions 0.1 mm×0.1 mm0.1 \mathrm{~mm} \times 0.1 \mathrm{~mm} ) and to reflect shorter wavelengths towards a silicon detector which has a suppressed infrared sensitivity. The infrared thermal radiance is filtered using a narrow bandwidth interference filter centred at 4.05μ m4.05 \mu \mathrm{~m}. A narrow bandwidth detection has the advantage of minimising the effect of chromatic aberrations induced by the collecting lenses.

When a square wave is used to drive the scanner, the beam is periodically deflected on and off the targeted spot, producing a chopped laser excitation. Unlike an optical chopper, the laser scanner can be operated reproducibly at very low frequencies with a short step time (<10μ s)(<10 \mu \mathrm{~s}).

After preamplification, both the back-scattered signal and the thermal radiance signal are digitised using a 16 bits A/D card. The data acquisition is triggered by a 100 kHz clock signal generated by

[1]

1. • To whom correspondence should be addressed
     ↩︎

a frequency multiplier from the low-frequency signal that drives the laser scanner.

Fig. 1 Photothermal radiometry set-up
This synchronous sampling allows averaging of the signal over a great number of periods without significant drift or waveform distortion. The averaging process is realised in real time by an infinite impulse response digital filter implemented with LabView. The laser scanner driving signal is usually a 10 Hz square wave and typically the sampled sets of back-scattered and thermal radiance data consist of N=10000N=10000 points.

During this data acquisition, a small fraction of the excitation laser beam is sampled using a polarising beamsplitter cube and sent to a beam analysing device. In order to measure the beam profile at the position of the sample’s surface, the beamsplitter is placed at identical optical path lengths from the sample and the beam analyser.

The beam analyser (Fig. 2) is a scanning slit whose displacement is monitored by a Michelson interferometer. A large area silicon photodiode placed behind a 10μ m10 \mu \mathrm{~m} wide slit is mounted on a motorised translation stage together with one of the interferometer mirrors. Using a 5 mW HeNe laser source, the movement of the assembly produces highly contrasted sinusoidal interference fringes which are detected by another silicon photodiode. Each fringe corresponds to a displacement equal to half a wavelength of the HeNe laser (0.3164μ m)(0.3164 \mu \mathrm{~m}). An electronic trigger circuit generates a reference TTL signal from the fringes signal. Both the reference signal and the scanning slit signal are fed in an A/D card and a LabView program is used to acquire one sample for every 10 fringes detected, equivalent to a slit displacement of 3.164μ m3.164 \mu \mathrm{~m}. The accuracy of the beam profile measurement was assessed by fitting a Gaussian profile to the data. The uncertainty on the beam diameter is typically 0.2%0.2 \%.

Mathematical Modeling

With a laser spot of a diameter of about 2 mm and a scanning frequency greater than a few tenths of Hertz, the thermal perturbation is confined within a volume of about 1 mm3\mathrm{mm}^{3} for most materials. Therefore, the heat conduction problem solved for a semi-infinite medium is appropriate.

Fig. 2 Beam analyser device
Let us assume a periodically modulated laser beam incident on the surface of an opaque semi-infinite homogeneous material. The absorbed laser light is converted into heat which diffuses in the material. In the case of a chopped-like modulation, the heat source created by laser excitation can be expressed as the product of a spatial term by a temporal term:

q(x,y,z,t)=f(x,y)δ(z)h(t)q(x, y, z, t)=f(x, y) \delta(z) h(t)

where f(x,y)f(x, y) is the 2D beam profile, δ\delta is the Dirac distribution and h(t)h(t) is the time dependence of the laser intensity. h(t)h(t), a periodic function with a frequency γ\gamma, can be expanded in a Fourier series:

h(t)=∑n=−∞n=∞hˉne2iπn⋅γh(t)=\sum_{n=-\infty}^{n=\infty} \bar{h}_{n} e^{2 i \pi n \cdot \gamma}

where hˉn\bar{h}_{n} is the nn-th component of the Fourier spectrum of h(t)h(t). After a transient regime, a periodic regime is established in the material. The temperature field can therefore be Fourier expanded:

T(x,y,z,t)=∑n=−∞n=∞Tˉn(x,y,z)e2iπn⋅γT(x, y, z, t)=\sum_{n=-\infty}^{n=\infty} \bar{T}_{n}(x, y, z) e^{2 i \pi n \cdot \gamma}

Each component Tˉn\bar{T}_{n} is the solution of the heat conduction equation in the harmonic regime 6{ }^{6} with a Neumann type boundary condition at z=0\mathrm{z}=0 :

∇2Tˉn−2iπn⋅γDTˉn=0(−KdTˉndz)z=0=f(x,y)hˉn\begin{aligned} & \nabla^{2} \bar{T}_{n}-\frac{2 i \pi n \cdot \gamma}{D} \bar{T}_{n}=0 \\ & \left(-K \frac{d \bar{T}_{n}}{d z}\right)_{z=0}=f(x, y) \bar{h}_{n} \end{aligned}

KK and DD are respectively the thermal conductivity and the thermal diffusivity of the material. The other boundary conditions are those of a semi-infinite medium (null temperature and null flux at infinity).
Let v~n(ux,uy,z)\widetilde{v}_{n}\left(u_{x}, u_{y}, z\right) be the 2D Fourier transform of Tˉn(x,y,z)\bar{T}_{n}(x, y, z) with respect to xx and yy :

v~n(ux,uy,z)=∫−∞+∞∫−∞+∞Tˉn(x,y,z)e−2iπ(ux+uy)3dxdy\widetilde{v}_{n}\left(u_{x}, u_{y}, z\right)=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \bar{T}_{n}(x, y, z) e^{-2 i \pi\left(u_{x}+u_{y}\right) 3} d x d y

Replacing Tˉn\bar{T}_{n} by its Fourier decomposition leads to a onedimensional differential equation for v~n\widetilde{v}_{n} as a function of the space variable zz :

[(2iπux2)+(2iπuy2)]v~n+d2v~ndz2−2iπγDv~n=0(−Kdv~ndz)z=0=fˉ(ux,uy)hˉn\begin{aligned} & {\left[\left(2 i \pi u_{x}^{2}\right)+\left(2 i \pi u_{y}^{2}\right)\right] \widetilde{v}_{n}+\frac{d^{2} \widetilde{v}_{n}}{d z^{2}}-\frac{2 i \pi \gamma}{D} \widetilde{v}_{n}=0} \\ & \left(-K \frac{d \widetilde{v}_{n}}{d z}\right)_{z=0}=\bar{f}\left(u_{x}, u_{y}\right) \bar{h}_{n} \end{aligned}

where f~(ux,uy)\widetilde{f}\left(u_{x}, u_{y}\right) is the 2D Fourier transform of the beam profile. The solution of the differential equation (7), expressed at z=0z=0, is:

v~n(ux,uy,z)=DKf~(ux,uy)h~n2iπnγ+4π2D(ux2+uy2)\widetilde{v}_{n}\left(u_{x}, u_{y}, z\right)=\frac{\sqrt{D}}{K} \frac{\widetilde{f}\left(u_{x}, u_{y}\right) \widetilde{h}_{n}}{\sqrt{2 i \pi n \gamma+4 \pi^{2} D\left(u_{x}^{2}+u_{y}^{2}\right)}}

Let us define the thermal transfer function Hn(ux,uy)H_{n}\left(u_{x}, u_{y}\right) of the problem:

Hn(ux,uy)=DK12iπnγ+4π2D(ux2+uy2)H_{n}\left(u_{x}, u_{y}\right)=\frac{\sqrt{D}}{K} \frac{1}{\sqrt{2 i \pi n \gamma+4 \pi^{2} D\left(u_{x}^{2}+u_{y}^{2}\right)}}

The surface temperature field in the Fourier domain is given by the following algebraic product:

v~n(ux,uy,z=0)=Hn(ux,uy)f~(ux,uy)h~n\widetilde{v}_{n}\left(u_{x}, u_{y}, z=0\right)=H_{n}\left(u_{x}, u_{y}\right) \widetilde{f}\left(u_{x}, u_{y}\right) \widetilde{h}_{n}

In the experimental set-up, a photodetector of dimensions xdx_{d} and ydy_{d} is used to detect the thermal radiance emitted by the surface in a small spectral band centred around the detection wavelength. In the limit of small laser heating, the signal S(t)S(t) induced by the laser periodic heating is proportional to the periodic surface temperature, averaged over the detection zone. The proportionality factor, which includes the spectral emissivity, the first derivative of the Planck function, the numerical aperture of the optics and the instrumental gain, is omitted for clarity:

S(t)=1xdyd∫xd/2−1/2xd/2∫yd/2yd/2T(x,y,z=0,t)dxdyS(t)=\frac{1}{x_{d} y_{d}} \int_{x_{d} / 2-1 / 2}^{x_{d} / 2} \int_{y_{d} / 2}^{y_{d} / 2} T(x, y, z=0, t) d x d y

S(t)S(t) is also a periodic function and therefore can be expressed as a Fourier series with coefficients S~n\widetilde{S}_{n}. After replacing T~n\widetilde{T}_{n} by its 2D Fourier expansion and performing the double integration over xx and y,S~ny, \widetilde{S}_{n} is given by:

S~n=∫−∞+∞∫−∞+∞v~n(ux,uy,z=0)sinc⁡(πxduy)sinc⁡(πyduy)duxduy\widetilde{S}_{n}=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \widetilde{v}_{n}\left(u_{x}, u_{y}, z=0\right) \operatorname{sinc}\left(\pi x_{d} u_{y}\right) \operatorname{sinc}\left(\pi y_{d} u_{y}\right) d u_{x} d u_{y}

where sinc⁡(x)=sin⁡(x)/x\operatorname{sinc}(x)=\sin (x) / x. Let us define the detector transfer function T(ux,uy)T\left(u_{x}, u_{y}\right) :

T(ux,uy)=sinc⁡(πxduy)sinc⁡(πyduy)T\left(u_{x}, u_{y}\right)=\operatorname{sinc}\left(\pi x_{d} u_{y}\right) \operatorname{sinc}\left(\pi y_{d} u_{y}\right)

The Fourier components S~n\widetilde{S}_{n} of the periodic signal can now be expressed as:

S~n=h~n∫−∞+∞Hn(ux,uy)T(ux,uy)f~(ux,uy)duxduy\widetilde{S}_{n}=\widetilde{h}_{n} \int_{-\infty}^{+\infty} H_{n}\left(u_{x}, u_{y}\right) T\left(u_{x}, u_{y}\right) \widetilde{f}\left(u_{x}, u_{y}\right) d u_{x} d u_{y}

Assuming a rotational symmetry of the laser beam, Eq. (15) becomes:

S~n=h~n∫−∞+∞Hn(ux,uy)T(ux,uy)f~(uy)f~(uy)duxduy\widetilde{S}_{n}=\widetilde{h}_{n} \int_{-\infty}^{+\infty} H_{n}\left(u_{x}, u_{y}\right) T\left(u_{x}, u_{y}\right) \widetilde{f}\left(u_{y}\right) \widetilde{f}\left(u_{y}\right) d u_{x} d u_{y}

The periodic signal S(t)S(t) is given by the following Fourier series:

S(t)=∑n=−∞∞S~ne2iπnγS(t)=\sum_{n=-\infty}^{\infty} \widetilde{S}_{n} e^{2 i \pi n \gamma}

The numerical implementation of the model was done using MATLAB. The Fourier spectra f~\widetilde{f} and h~n\widetilde{h}_{n} are computed using the Fast Fourier Transform algorithm (FFT) applied to the beam profile and the backscattered light data. The double integral in Eq. (16) is approximated by a double sum over the positive
spatial frequencies. S(t)S(t) is computed according to Eq. (17) by inverse FFT. The computation of a set of 10000 values of S(t)S(t) takes about 10 seconds on a 350 MHz PC.

Parameter Estimation

Although the temperature and the photothermal signal are given by linear relationships, the model described below is not linear with respect to the thermal diffusivity. This parameter must be therefore estimated by a non-linear parameter estimation procedure aiming to fit the model to the thermal radiance data. Assuming Gaussian white noise with zero-mean and uniform variance, the least-squares technique provides minimal variance estimators 7{ }^{7}. In order to avoid any bias, the experimental data are not normalised. Instead, instrumental parameters (offset, gain and arbitrary time origin) are estimated together with the diffusivity in the form a four-parameter vector (D, offset, gain, origin). The Gauss-Newton iterative algorithm is chosen to perform the minimisation of the least-squares cost function. This parameter estimation problem is well-conditioned and the algorithm usually converges after 3 or 4 iterations when realistic initial values are used. An estimate σ^2\hat{\sigma}^{2} of the variance of the noise is given by:

σ^2=1N−4∑i=1N(Sexp (ti)−Stheo (ti))2\hat{\sigma}^{2}=\frac{1}{N-4} \sum_{i=1}^{N}\left(S_{\text {exp }}\left(t_{i}\right)-S_{\text {theo }}\left(t_{i}\right)\right)^{2}

where Sexp (ti)S_{\text {exp }}\left(t_{i}\right) and Stheo (ti)S_{\text {theo }}\left(t_{i}\right) are respectively the measured and the fitted theoretical data. The main source of error in the joint estimation of the diffusivity and the 3 other parameters is the noise in the thermal radiance data, which is essentially due to the background photon noise.

The covariance matrix CmC_{m} of the estimated parameters is evaluated using the linear least-squares theory 7{ }^{7} :

Cm=σ^2(G′G)−1C_{m}=\hat{\sigma}^{2}\left(G^{\prime} G\right)^{-1}

where GG is the Jacobian matrix of the partial derivatives of the data with respect to the parameters to estimate 7.Cm{ }^{7} . C_{m} is evaluated using the optimal values of the parameters.

Fig. 3. Boundary conditions used for the analysis of titanium. (a) Beam profile. (b) Laser intensity time dependence. The smaller graphs shows the finite rise time of the laser excitation ( ∼0.2 ms\sim 0.2 \mathrm{~ms} ).

Table 1. Experimental results on the thermal diffusivity of pure metals

The number of degrees of freedom of this uncertainty term is N−4N-4, which typically is 9996 . Because of the very good accuracy on the measurement of the beam profile, the corresponding uncertainty term is considered to be negligible compared to the uncertainty induced by the thermal radiance noise.

Experimental Results

The thermal diffusivities of platinum and titanium have been measured. Each sample was placed in a tube furnace and heated in air to about 900 K . A beam diameter of about 2 mm and a scanning frequency of 10 Hz were used. The measured boundary conditions are shown in Fig. 3. The laser power was set at about 100 mW in order to minimise non-linear effects in the thermal radiance signal. A temperature perturbation of about 1 K was induced in the samples. To compensate for the very low infrared signal level, a large number of periods, typically 1000, were averaged according to the procedure previously described. The thermal radiance signal measured on titanium and the best fit are shown in Fig. 4. Table 1 presents the results obtained with the set of samples.

Fig. 4. Thermal radiance signal measured on titanium.
(a) Experimental data (noisy signal) and best fit (continuous line).
(b) Data residuals after fitting.

The thermal diffusivities of platinum and titanium were measured with an accuracy of about ±1.5%\pm 1.5 \% at a confidence level of 95%95 \%. These results are in good agreement with the
literature values 8{ }^{8}. Platinum and titanium gave excellent agreement between experimental and calculated data.

By a precise control and measurement of the boundary conditions in photothermal measurements, a reduction in systematic errors induced by the use of simplified models and erroneous assumptions can be expected. The accuracy of the measurements can therefore be significantly improved. This methodology was applied to the measurement of the thermal diffusivity of pure metals by photothermal radiometry. The thermal diffusivities of platinum and titanium were measured with an accuracy of about ±1.5%\pm 1.5 \%. Although the accuracies of the literature values are not generally as good, they agree well with our measurements.

Acknowledgement

This work was carried out as part of the Thermal Metrology Programme funded by the UK Department of Trade and Industry.

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