3.3.2 PA Effect, Signal Generation and Detection The PA effect was discovered by Alexander Graham Bell in 1880. In a condensed-phase PA experiment, PA signal generation includes absorption of modulation optical illumination (modulation frequency within an acoustic range) by the sample, thermal diffusion from within the sample to adjacent medium (usually helium) and pressure oscillation of helium. The pressure wave (sound) is then detected using a very sensitive microphone. This process is illustrated in Figure 3.2.
The PA signals depend on many factors, including the intensity and modulation frequency of light, thermal, optical and geometric properties of the sample, cell and media. The one-dimensional thermal piston model proposed by Rosencwaig and Gersho for the PA effect in condensed-phase matter, i.e. the RG theory, has been most widely referenced in literature. In the RG theory, the PA signal originating from a homogeneous solid sample is described as the pressure variation of the gas in the PA cell, △P(t), i.e.
where ω is the angular modulation frequency of the incident light, γ is the ratio of the specific heats (Cp/Cv) of the sample, P0 and T0 are the static pressure and the average temperature of the PA cell gas respectively, Ts(0, ω) is the complex temperature at the solid-gas boundary (surface), lg is the distance from the surface of the sample to the cell window and μg the thermal diffusion depth of the cell gas. It can be seen that the PA signal has both magnitude and phase, and it depends primarily on the temperature of the surface of the sample. The magnitude represents the strength of a PA signal and the phase is a signature of its spatial origin. Simplifications of equation (3-1) can be made on different optical and thermal conditions of a sample under study. In general, as a conceptual understanding of the relationship, PA signal is approximately proportional to log10(βμ), over the region of –1< log10(βμ)<+1, where β is optical absorptivity and μis thermal diffusion depth. Since thermal diffusion is relatively slow and thermal waves damp out quickly, only those generated within a certain sampling depth will be primarily detected. Thus, the thermal diffusion length (μ) also represents the sampling depth is given by,
where f is the modulation frequency (or Fourier frequency in continuous-scan mode) and α is the thermal diffusivity [α=k/(ρCp), where k, ρ and Cp are thermal conductivity, density and specific heat of the sample, respectively].
3.3.3 PA Signal Phase and Phase Difference Models As briefly mentioned above, PA signal phase directly contains spatial information about the signal origin, and thus it is of great importance in spectral depth profiling analysis. Even for homogeneous samples, PA signals would exhibit phase lags with respect to the optical modulation because thermal diffusion is much slower (on the order of 10-6—10-3 second). The phase lags depend on the modulation frequency, the instrument, and the spatial origin of the signals. A finite time delay (△t) for a PA signal generated from a deeper layer of a sample to reach the microphone with respect to the surface PA signal can be related to the phase difference by Equation(3-3):
In general, as depicted in Figure 3.3, PA signals originating from deeper parts of a sample have greater phase lags than those from the shallower parts, and vice versa. In addition, smaller phase lags ate associated with stronger bands from the same layer when this layer is thermally thick ( the layer is thicker than the thermal diffusion depth) or optically opaque (the layer is thicker than the optical penetration depth). A quantitative expression of PA signal phase for multi-layered materials, as an extension to the classic RG theory for homogeneous solids, has been developed. For layer j of any thermally thick or optically opaque material, the total PA signal phase lag Фj,total relative to the phase of the light modulation can be expressed by Equation(3-4):
where d, β, μ are the thickness, optical absorption coefficient, and thermal diffusion depth for layer j or n, respectively. It can be seen from Equation (3-4) that this total PA phase lag is due to the thermal transport within this absorbing layer, the thermal transport within all above transparent layers (from 1 to j-1) and the total relative phase angle shift due to all external factors (positioning of the sample, PA cell resonance, spectrometer, etc.), Φ0 (constant for a given sample at particular modulation frequency of a given instrument). When layer j is both thermally thin and optically transparent, Equation (3-4) becomes:
Using Equation (3-4) and (3-5), phase difference models for various scenarios can be derived. Assuming a multi-layer sample with layer j above layer k, Equation (3-6) and (3-7) show two useful cases for practical PA phase analysis of multi-layered samples. Equation (3-6) holds for the case in which both layers j and k are thermally thin (with respect to μj and μk) and optically transparent [with respect to βj(σj) and βk(σk)]. Equation (3-7) describes the case in which layer j is thermally thin (with respect toμj) and optically transparent [with respect toβj(σj) and layer k is thermally thick (with respect toμk) or optically opaque with respect toβk(σk)]:
Further simplification can be made when dealing with two-layer samples. Applications of these models for thickness determination will be further demonstrated in Section 3.3.8.
3.3.4 Continuous-Scan FT-IR PAS The coupling of a PA detector to a continuous scan FT-IR, as first demonstrated in the late 1970's and early 1980's combines all the interferometric advantages (throughput, multiplexing and registration) with those unique features of PAS as dectribed in Section 3.3.1. Thus FT-IR PAS can be used for convenient qualitative identification or nondestructive depth profiling of a variety of samples. Although PAS can be used in any region of the electromagnetic spectrum, it is particularly effective in the infrared. Especially for organic materials, distinctive infrared functional group absorption bands afford layer, and/or component identification not often possible in the UV/Vis region. In the infrared, the broad spectral range and high throughput of FT-IR make it the most versatile spectral technique for use with PA detection.
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