Fundamental Properties of Asphalts and Modified Asphalts, III

greater than 100 micrometers are considered “thermally thick”). PAS directly measures IR
absorption by sensing absorption induced heating of the sample within an experimentally
controllable sample depth below the samples surface [McClelland et al. 2002].
The PA sampling (thermal diffusion) depth for a homogeneous sample is conventionally
expressed as [Rosencwaig and Gersho 1976]
L π
1
2 = ( D / f )
(2-2.2.1)
where D is the thermal diffusivity, a measure of heat propagation speed, in cm 2 /sec, and f is the
infrared-intensity modulation frequency in hertz. The equation for thermal diffusivity (D) is
D = k/ρCp (2-2.2.2)
where k is the thermal conductivity in W/mK, ρ is density in kg/m 3 , and Cp is the specific heat
in J/kg K. From the literature it is estimated for an asphalt at 25˚C, typical values for ρ, k and Cp
are 1030 kg/m 3 , 0.75 W/m K and 920 J/kg K, respectively. Frequency modulation ( f ) can be
expressed as a function of V the optical path difference (OPD) velocity of the interferometer
mirror in cm/sec and ν the wave number in cm -1 .
f = Vν
(2-2.2.3)
Combining equations 2-2.2.1 and 2-2.2.3 results in
1
2
L = ( D / πVν
)
(2-2.2.4)
The sampling depth is inversely proportional to the square root of the modulation frequency and
therefore low modulation frequency or OPD velocity will result in PA signals from within the
sample, while high OPD velocities are nearer the surface region. The dependence of the
sampling depth on wavenumber (ν) means that it varies across a spectrum. For asphalt and the
thermal diffusivity estimated above, at an OPD velocity of 0.5 cm/s the thermal diffusion length
varies from 29 µm at 600 cm -1 to 11 µm at 4000 cm -1 . Thermal diffusion lengths at various
OPD velocities and selected wave numbers from 600 to 4000 cm -1 corresponding to areas of
significant asphalt absorbance peaks are listed in table 2-2.2.1.
Depth variation across the spectrum can be overcome by using a step-scan spectrometer
employing phase modulation photoacoustic measurements [Jones and McClelland 1996;
Drapcho et al. 1997]. Step-scan, as described by Drapcho et al. is the case where the “average
optical retardation is adjusted stepwise, with a secondary, single-frequency phase (optical path
difference) modulation applied. The data are collected when the average optical retardation is
kept constant; thus the velocity and wavenumber dependence of the Fourier frequency is
removed, and a constant probing depth across the spectrum for each phase modulation (PM)
frequency is achieved.”
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