DIFFUSION ON THE PROGRADE LIMB: PROGRESS AND CHALLENGES

Many instances arise in the study of metamorphic rocks in which it is important to know the diffusive response of an element or isotope as its host phase progresses along the prograde limb of a temperature-time (T-t) path. Using a combination of numerical simulations and analytical mathematics, we have developed equations that describe diffusive loss from a spherical grain when T increases linearly with time, and also for "thermal pulse" scenarios in which the system of interest heats to a peak T and then falls back to the initial T along in two linear T-t segments (a "steeple" path), or along a parabolic T-t path. These equations assume no in-growth and constant surface concentration of diffusant, but they can be used for any plausible diffusion law and for any heating rate -- including those attending contact metamorphism and impacts. For the case of linear heating from an initially cold condition, we propose the use of a "center retention temperature" (T_CR) in reference to the temperature above which the concentration at the center of the spherical grain is compromised by diffusion. Interestingly, T_CR has values within a few degrees of Dodson's closure temperature for cooling (T_C), given the same magnitude of the instantaneous dT/dt, even though the assumed functional relationships between T and t are very different (T α t and T α 1/t). It must be noted, however, that the center retention criterion corresponds to ~50% diffusive exchange, so it should not be equated with initial diffusive opening, which occurs 10s to 100s of degrees below T_CR, depending upon the Arrhenius equation assumed. Because it is the integral of the diffusivity (D) over the T-t path that determines the diffusive response, steeple paths result in a smaller diffusive response than do parabolic paths of the same duration and peak temperature. Both paths result in substantially less diffusion than isothermal events of the same duration at a constant temperature corresponding to the peak value. For any symmetrical thermal pulse, ~75% of the diffusive response (as gauged by the fractional loss, F) occurs on the prograde limb, which might make it possible to estimate the prograde contribution from the total diffusive response of a natural sample. Efforts are underway to develop general equations for modeling daughter isotope diffusion with radiogenic in-growth.