The analytical limits of modeling short diffusion timescales

Diffusion modeling of chemical zonation in crystals and glasses is a widely used method of quantifying the timescales of a variety of geological processes. Obtaining timescale information from diffusion modeling relies on fitting modeled diffusion profiles to measured compositional gradients. Thus analytical factors, such as spatial resolution, analysis location and analytical uncertainty have the potential to limit the accuracy and precision of calculated diffusion timescales, especially when the resolution of the individual analyses approaches the width of the observed diffusion gradient. Herein we use a probabilistic modeling approach to assess the accuracy of timescales based on diffusion modeling at various spatial resolutions and diffusivities. We have calculated synthetic 1D diffusion profiles produced from simple step function geometries at various diffusivities and timescales and then resampled these at common analytical spatial resolutions. Standard diffusion modeling techniques were used to estimate apparent diffusion timescales from the resampled profiles to compare with the "true" synthetic times. Results confirm that for a given diffusivity, higher analytical spatial resolution gives access to shorter timescales, and that for each analytical resolution there is a minimum threshold timescale, below which diffusion modeling significantly overestimates the true timescale. The accuracy of short diffusion timescales depends on the width of the diffusion gradient (the portion of a profile that has been modified by diffusion) relative to the analytical spatial resolution, with models becoming accurate (within 20%) at a gradient width/spot size ratio >2. The precision of diffusion timescales depends on the gradient width to spot size ratio, the magnitude of analytical uncertainty and the magnitude of the compositional difference across the step function boundary. Precision is improved with a large gradient width/spot size ratio, and where the magnitude of the compositional difference across the step function boundary is large relative to the analytical uncertainty. We present a generalized method to quantify these threshold timescales for any given spatial resolution and diffusivity, and present simple guidelines to aid in selecting compositional profiles that will produce more accurate diffusion timescale estimates.