The effects of collimator geometry and self-absorption on in-situ mass diffusion measurements

We consider a real-time methodology for determining self-diffusion coefficients at several temperatures utilizing a single sample, based on a radiotracer technique put forth by Codastefano, Di Zanza and Russo (Codastefano, P., Di Russo, A., Zanza, V. 1977. A new apparatus for accurate diffusion measurements in fluids. Reviews of Science Instrumentation 48, 1650-1653 (CDZ)) and recently optimized by our group (Jalbert, L.B., Banish, R.M., Rosenberger, F.E. 1997. Real-time diffusivity measurements in liquids at several temperatures with one sample. Physical Review E 57, 1727-1736 (JBR)). In this technique the evolution of the concentration distribution of the radiotracer is followed by shielding the source except for a pair of collimated apertures placed at mathematically advantageous positions along the diffusion axis. Radiation detectors at each collimator measure the intensity of radioactive tracer emission at each aperture, and the counting rate difference versus time leads to a determination of the diffusion coefficient. The effects of self-absorption of radiation in the source were previously treated by us using a one-dimensional approximation, which is a typical procedure. However, in order to obtain meaningful counting statistics in the detectors, the size of the collimator apertures must be large enough that the measured counting rate is actually an integral over a three-dimensional region of the source volume rather than an ideal point measurement. Here we consider, by a ray-tracing algorithm, the effects of this finite extent of the observed volume on the experimentally-determined diffusivity. Self-absorption effects are considered both for photon energies that have essentially no absorption in the sample (so that the entire thickness of the sample is effectively viewed) and for those that suffer sufficient self-absorption that the measured radiation comes from only a surface "skin-depth" (approximately 100 microm in extent). Collimation holes from approximately 1 to approximately 100% of the sample diameter are investigated. We find that neither the effects of self-absorption nor of the intrinsic three-dimensional geometry involved have a significant effect on the experimentally determined value of the diffusivity. In addition, linear concentration gradients along the diffusion axis do not influence the diffusivity measurement. A (quadratic) correction formula to determine the actual diffusion coefficient from the experimentally-obtained value is presented.