Simple equations for diffusion in response to heating

The diffusive closure temperature of minerals (T-c) was originally conceived for application to systems under-going cooling (Dodson, 1973) and is of limited use for cases of diffusive opening during heating or for complete heating-cooling cycles. Here we use a combination of numerical simulations and mathematics to arrive at general equations for progressive diffusive loss from a sphere when temperature increases linearly with time, and also for discrete thermal pulses. For linear heating (T proportional to time), and with constant surface concentration and no radiogenic in-growth, prograde diffusive opening is accurately described by T-rt% = 0.457.(E-a/R)/chi(h) + log[E-a.D-0/R.dT/dt.a(2)] where D-0 (m(2)/s) and E-a (J/mol) are the Arrhenius parameters for the diffusant of interest, dT/dt is the heating rate (degrees/s), a is the radius (in meters) of the spherical domain under consideration, R is the gas constant (J/degrees-mol), and chi(h) is a constant. For a given heating trajectory, T-rt% is the temperature (in kelvins) at which a specific fractional retention (or loss) is reached, and where the constant chi(h) has a specific value. For retention levels of 50%, 99% and 99.9%, chi(h) has values of -0.785,2.756 and 4.751, respectively. The equation is accurate to within 5 for the vast majority of measured diffusion laws, and to within similar to 2 degrees for similar to 90% of them. For noble gases specifically it is accurate to within 1 degrees in almost all cases. There are essentially no restrictions on the grain size or heating rate (up to 2000 degrees C/Myr) that can be assumed without loss of accuracy. For thermal pulses in which the temperature of the spherical grain of interest rises at a constant rate from 293 K to a maximum value and then falls back linearly to the starting temperature (i.e., a steeple T-t path), the diffusive response for the thermal cycle is given by log zeta = log[D-0 tau/a(2)] + 195/T-pk - 0.4416E(a)/RTpk - 1.35, where zeta = a(-2)integral D-tau(t=0)(t)dt, tau is the duration of the heating event (in seconds) and T-pk is the peak temperature in kelvins. The total fractional loss (F) is uniquely determined by the value of zeta; conversion of log zeta to F is straightforward, as discussed in the text. Diffusive loss during parabolic T-t paths conforms to a similar relation: log zeta = log[D-0 tau/a(2)] + 140/T-pk - 0.437E(a)/RTpk - 0.8. Given knowledge of the Arrhenius law for the diffusant of interest, these equations provide accurate estimates of the total diffusive loss for a steeple- or parabola-shaped T-t path of any duration and intensity-including asymmetrical paths. (C) 2012 Elsevier B.V. All rights reserved.