Computer Simulations Of Depth Profiling By Photoacoustic Infrared Spectroscopy

Abstract

In photoacoustic (PA) spectrometry of solids, the sample is placed in a sealed cell in contact with a gas. Modulated radiation passes through the cell window and illuminates the sample. Absorbed radiation is converted to heat which is conductively transferred to the cell gas. The modulated heating of the gas produces pressure variations that are detected by a microphone, so that the signal is proportional to the sample absorptivity. As described by Rosencwaig and Gersho,1 the signal results from photons absorbed near the surface of the sample. The depth of this layer is called the thermal diffusion length, ps, and is determined by ps = (lc/ pC f ) 2 (1) where k is the thermal conductivity, p is the density, C is the heat capacity and f is the modulation frequency (Hz). For low-density polyethylene (LDPE), a modulation frequency of 10 Hz corresponds to a ps of 70 pm, and a frequency of 1000 Hz corresponds to a ps of 7 pm. This range of depths is appropriate for depth profiling of polymer laminates. One could imagine a PA depth profiling scheme where the laminate sample spectrum is measured at several modulation frequencies with spectral subtraction being applied to obtain the spectrum of each layer. One must include a valid scaling factor for spectral subtraction to yield meaningful results. Furthermore, the thermal diffusion length is not a distinct boundary, but rather is the inverse of the decay constant of an exponential attenuation function, A(x) A(x) = exp(-x/ps) (2) This function represents the attenuation of the heat from a modulated heat source at depth x reaching the surface of the sample and producing the signal. Because there is no simple solution that transforms the variation of PA signal with frequency to absorptivity versus depth, an iterative estimation approach was envisioned. In this iterative approach, a model laminate structure is guessed and the expected PA signal is estimated from a computer simulation. By iteratively adjusting and checking the model, one can derive an estimate of the laminate structure. In order for the iterative method to work, one must have a quick and accurate method for predicting the PA signal from a given laminate model. We have been investigating methods for doing these calculations.