Simple equations for diffusion in response to heating

Abstract

The diffusive closure temperature of minerals (TC) was originally conceived for application to systems undergoing 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∝time), and with constant surface concentration and no radiogenic in-growth, prograde “diffusive opening” is accurately described byTrt%=0.457⋅Ea/Rχh+logEa⋅D0R⋅dT/dt⋅a2where D0 (m2/s) and Ea (J/mol) are the Arrhenius parameters for the diffusant of interest, dT/dt is the heating rate (°/s), a is the radius (in meters) of the spherical domain under consideration, R is the gas constant (J/°-mol), and χh is a constant. For a given heating trajectory, Trt% is the temperature (in kelvins) at which a specific fractional retention (or loss) is reached, and where the constant χh has a specific value. For retention levels of 50%, 99% and 99.9%, χh 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 ~2° for ~90% of them. For noble gases specifically it is accurate to within 1° in almost all cases. There are essentially no restrictions on the grain size or heating rate (up to 2000°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 293K 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 bylogζ=logD0τa2+195Tpk−0.4416EaRTpk−1.35,where ζ=a−2∫t=0τD(t)dt, τ is the duration of the heating event (in seconds) and Tpk is the peak temperature in kelvins. The total fractional loss (F) is uniquely determined by the value of ζ; conversion of logζ to F is straightforward, as discussed in the text.Diffusive loss during parabolic T–t paths conforms to a similar relation:logζ=logD0τa2+140Tpk−0.437EaRTpk−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.