Garnet Diffusion
Diffusion Kinetics of Divalent Cations in Garnet
Advantages of the Methodology
We present a statistical framework for the cation diffusion coefficients of garnet and derive a new set of diffusion parameters, based on modeling experimental data by a composite Arrhenius relationship and exploring the model space using the efficient Markov-Chain Monte Carlo (MCMC) approach and Gibbs sampler. The three major advantages of this approach are:
1) All the experimental uncertainties are incorporated; the experimental results of higher precision are more heavily weighted.
2) An additional "inter-experiment bias" term is introduced to account for the unquantified experimental conditions (e.g., water content, defect density).
3) A new statistical framework is available to estimate the uncertainties on the diffusion coefficients at specific conditions.
A similar approach was applied by Korenaga and Karato (2008) to calibrate the flow law parameter of olivine deformation. With appropriate modifications, the methods used in this study are applicable to diffusion data for other components in garnet, diffusion in other minerals, or any other Arrhenius relationships.
Results
The MCMC simulation consists of about 4.5 × 108 Gibbs samplings, and consumes 240 hours on Yale High-Performance Computation clusters. The recommended diffusion parameters are presented, together with the parameters from previous studies for comparison.
Units: k: nm-1; Q: kJ; V: kJ mol-1 kbar-1. Uncertainties are as ± one standard deviation.
The diffusion parameters calibrated in this study generally predict faster Mn diffusion and slower Ca diffusion than those of Carlson (2006) at low temperatures (< 800 °C). Fe, Mn and Ca diffusion rates with respect to Mg for a representative almandine-pyrope-rich garnet composition at crustal P-T conditions are shown in the figure below.
The diffusion coefficient of Fe is large than Mg at low pressures, and almost identical to DMg at high pressures. Mn diffuses more than one order of magnitude faster than Mg at low temperatures; the diffusion rate approaches that of Mg as temperature increases. Ca is the slowest-diffusion cation, consistently more than one order of magnitude slower than Mg. Our diffusion coefficient predictions are qualitatively consistent with the common observation that Mn profiles appear the most smoothed by diffusion, and Ca profiles are the least smoothed.
Application to Barrovian metamorphic zones
We model the compositional profiles of garnets from the Dalradian metapelites of the Grampian Highlands Terrane, Scotland (Ague and Baxter, 2007; Vorhies and Ague 2011), to illustrate the timescale estimate with newly-derived diffusion coefficients. The diffusion timescales are estimated using 1000 sets of randomly generated diffusion coefficients; each set of Arrhenius parameters is randomly drawn based on their distributions, and the pre-exponential factors are normalized using the framework provided in this study. In this way the covariance factors are accounted for. A best-fit diffusion timescale is determined for each of the 1000 sets of diffusion coefficients. Accordingly, the uncertainties on the timescale estimate are mapped to the model uncertainties and, ultimately, the uncertainties of the experimental conditions.
The application of the new diffusion coefficients indicates that the total duration of peak metamorphism was of the order of one million years. This is a factor of roughly 8-10 times longer than estimates made with earlier coefficient calibrations (Ague and Baxter, 2007), but is still geologically very brief. Peak thermal conditions of one million years likely occurred as several shorter duration thermal pulses that were superimposed on the longer history of prograde metamorphism lasting about ten million years. The brevity of the thermal peak requires advective heat transport; the heat was supplied mostly by syn-orogenic mafic magmatism (Baxter et al. 2002; Reverdatto and Polyansky 2004; Dewey 2005; Ague and Baxter 2007; Lyubetskaya and Ague 2010a,b; Vorhies and Ague 2011; Viete et al. 2013).