Thermal-Wave Probing at Various Spatial Scales
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that: B0/A0 e0 e1/e0 e1, the analogue of the refractive index being the thermal effusivity e (/D1/2). A strong thermal mismatch can be useful for the photothermal investigation of a layered sample. To carry out a thermal-diffusivity measurement, a focused heating spot has to be used. Therefore, in order to interpret the results, the heat-diffusion equation has to be solved in a cylindrical geometry [T(r, z, t)]. Fourier and Hankel transforms allow easy calculations in the case of modulated heat sources. In the case of a heat point source, the result in spherical geometry is given by
Thermal-Wave
Probing at Various Spatial Scales Danièle Fournier
T(R, t) T0 /R exp(R/) exp( j[t R/]).
Introduction
Definition of Thermal Waves
In recent years, high thermal conductivity has been found in materials with heterogeneous microstructures, that is, ceramics and films with granular microstructures having different phases. Understanding the thermal conductivities and microstructures of these materials is more difficult, however, than in the case of single-crystal materials because they consist of grains and grain boundaries. Since the macroscopic thermal properties of these materials are strongly dependent upon the microscopic properties of their grains and grain boundaries, knowledge of thermal parameters at various spatial scales is necessary. Thermal-wave probing has turned out to be a very interesting way to study the thermal diffusivity of materials at various scales without much metallographic preparation of the samples. In this article, we describe this technique for characterizing the thermal properties of materials with heterogeneous microstructures.
In order to explain the concept of a thermal wave, we have to solve the heatdiffusion equation in a semi-infinite medium with a plane-modulated heat source located at x 0:
Determining Thermal Parameters with a Photothermal Experiment Photothermal Experiment Schemes A photothermal experiment consists of illuminating the sample with a pulsed or modulated pump beam and detecting the surface or volume temperature variation related to the transformation of this radiant energy into heat. This detection can be done with different experimental setups, depending upon the physical parameters under investigation: refractive index, infrared emissivity, acoustic waves, or local deformation. For this discussion, we will mainly use two kinds of optical detection: the mirage effect for one-dimensional (1D) or three-dimensional (3D) experiments and photoreflectance for 3D configurations, both of which allow the detection of the surface or the bulk temperature variation with a high degree of sensitivity.
MRS BULLETIN/JUNE 2001
T(x, t) T0 exp(x/) exp( j[t x/]) , (1) where T(x, t) is the temperature at a distance x from the source and T0 is related to the amplitude of the heat source; is the thermal-diffusion length, which characterizes the heat propagation in the medium and which can be controlled by the modulation frequency f /2 of the heat source.
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