Axisymmetric modelling of transient thermal response in solids for application to infrared photothermal radiometry technique

Abstract

To induce temperature changes on the sample surface by the incidence of a monochromatic modulated light beam and detect the changes produced in the thermal radiation emission is the basic principle of the infrared photothermal radiometry technique. Until now, in order to analyze the thermal response mathematical models based in an one-dimensional model were used considering a sample with a finite thickness and an infinite incidence surface, as well as, the linear approximation of the Stefan-Boltzmann Law in the calculus of the heat losses due to thermal radiation. In this work, analytical and numerical models for the 2D heat diffusion in homogenous finite solid samples, are presented. These models were obtained by solving the heat diffusion equation, under cylindrical symmetry, considering mixed boundary conditions to include radiation and convection heat losses through the surfaces of the sample, and a monochromatic Gaussian excitation beam impinging on the front of the sample. The analytical models were obtained by solving the governing equations, considering the well-known linear approximation of the Stefan-Boltzmann law in the calculus of the heat losses due to thermal radiation. To analyse the effects of the non-linearity of the heat losses by thermal radiation on the thermal transient response, in the numerical model it was taken into account the full expression of the Stefan-Boltzmann law, and the transport equation was solved numerically by means of the Finite Element Method (FEM). The analytical solution for the oscillatory thermal response reveals the close dependence of the thermal response on the ratio of thickness to the radius of the sample, represented by the form factor sf. Both, the analytical and the numerical solutions were employed to simulate the thermal response of homogenous materials, and compared with experimental results reported elsewhere by part of our same research group. Finally, the difference between the thermal response predictions, from the analytical and numerical models, were analyzed.