Appendix 1 (9/10)

Riedel et al. / Grain-size evolution in subducted oceanic ...

Appendix 1 1.1. Kinetics of grain-boundary nucleated reactions

We give here a brief derivation of the kinetic equations used in the modelling section. The overall tranformation rate, (t), under constant P-T conditions for grain boundary nucleated reactions was derived by Cahn [20] as:

where ^vO^B is the grain boundary area per unit volume of the host phase; Y is the (constant) growth rate of new phase grains; and A_e is the increase in the extended area fraction of new phase material owing to all the grains nucleating at a time between t' and t. Eq. A1 has been generalized to non-isothermal, non-isobaric transformations by Rubie and Ross [10]. For this more general case, A_e is given by:

with I^B() as the grain boundary nucleation rate of spinel, and Eq. A1 is generalized to:

The numerical solution of Eq. A2 and Eq. A3 is rather difficult and possible only by employing a time-consuming iterative integration procedure for each time step (see appendix in [10] for details). Hence, it seems to be impossible to apply them to the whole transforming slab within reasonable cpu times. Furthermore, the necessary iteration steps are a possible source of numerical errors.

Here, we simplify the kinetic analysis on the basis of the following observation: At starting transformation (undersaturated nucleation), the overall transition kinetics are indistinguishable from a homogeneous nucleation and growth process. This can be shown to be valid as long as the kinetically determined Avrami length _Av is larger or in the same order as the initial grain size of the host phase, L_ini~1/^vO^B [6]. At later stages, site saturation along the preferred nucleation sites at grain boundaries sets in and leads to linear growth of new phase grains at the already transformed grain boundaries. It has been shown [20], that this change (site saturation and change from 3D->1D growth) occurs in a rather narrow time interval during the transformation, in the vicinity of the percolation point _Per of the new phase grains. As a consequence, it is possible to solve the two limiting cases separatedly instead of Eq. A2 Eq. A3 , and to switch between both cases numerically, depending on the calculated transformed volume degree (t)=1-exp(-X_3D(t)); X_3D, X_2D, X_1D, and X_0D denote in the following the total grain volume, grain area, grain diameter, and grain number of the product phase (as defined in [6], p. 398).

Case (1): Homogeneous nucleation, (t)_Per(exp[-A_e]1-A_e):

with I^V(t)=^vO^B I^B(t), and:

Case (2): Site-saturated nucleation, (t)_Per (exp[-A_e]0):

The percolation threshold for grain boundary nucleated reactions, _Per, is dependent on the L_ini/_Av ratio of the host phase and is calculated on the basis of the numerical simulation algorithm described in [6]. For a grain-size reduction of several orders in magnitude (i.e., _Av<<L_ini) it is well below ~1%, and Eq. A5 applies almost everywhere in the two-phase region of the slab.

2.2. kinetic phase boundary

The metastable overshoot, P, follows from Eq. 18 , combined with Eq. 2 Eq. 3 and Eq. 7 . After insertion, one obtains the following transcendental equation for P(T):

with the constants A₀=gv_slab/

, A₁=(Q_a+P₀V_a)/R, A₁^'=SV_a/VR, A₁^''=V_a/R, A₂=16³

/9k(V)², and A₃=V/R.

Eq. A6 is numerically solved with a Newton-Raphson iteration algorithm. It has a single solution for A₁^''=0; that is, for vanishing pressure effect on the activation energy. At finite A₁^'', a second solution appears for overcritical overpressures P>P_crit. The critical overpressure P_crit, where the kinetic phase boundary bifurcates in two branches (dP/dT->- infin ), is approximately given with:

where N_A is Avogadro's number. At small µ<<RT, we find:

to logarithmic accuracy.

The expected spinel grain size at µ<<RT is given with (Eq. 17 ):

Eq. A9 is an Arrhenius relationship with an apparent activation energy, Q_gs, close to the activation energy for diffusion of atomic units to the growing interface, Q_a. Neglecting the weak dependence of P on T, Q_gs has the order of magnitude:

where P₀ is the (extrapolated) intercept of the Clausius-Clapeyron equation with the pressure axis at T=0 K. In contrast, at large µ>>RT, we obtain (Eq. 19 ):

with:

The critical depth that separates both cases Eq. A9 Eq. A11 can be found from the condition µ~RT. This depth is for the olivine-spinel transformation at 850 K given by: