Kinetics of Crystal Growth in a Terrestrial Magma Ocean

The problem of crystal sizes is one of the central problems of differentiation of a terrestrial magma ocean and it has been an arbitrary parameter in previous models. The crystal sizes are controlled by kinetics of nucleation and crystal growth in a convective magma ocean. In contrast with crystallization in magma chambers, volcanic lavas, dikes, and other relatively well studied systems, nucleation and crystallization of solid phases occur due to the adiabatic compression in downward moving magma (adiabatic "cooling"). This problem is solved analytically for an arbitrary crystal growth law, using the following assumptions: convection is not influenced by the kinetics, interface kinetics is the rate controlling mechanism of crystal growth, and the adiabatic cooling is sufficiently slow for the asymptotic solution to be valid. The problems of nucleation and crystal growth at constant heat flux from the system and at constant temperature drop rate are shown to be described with similar equations. This allows comparison with numerical and experimental data available for these cases. A good agreement was found. When, during the cooling, the temperature drops below the temperature of the expected solid phase appearance, the subsequent evolution consists of three basic periods: cooling without any nucleation and crystallization, a short time interval of nucleation and initial crystallization (relaxation to equilibrium), and slow crystallization due to crystal growth controlled by quasi-equilibrium cooling. In contrast to previously discussed problems, nucleation is not as important as the crystal growth rate function and the rate of cooling. The physics of this unusual behavior is that both the characteristic nucleation rate and the time interval during which the nucleation takes place are now controlled by a competition between the cooling and crystallization rates. A probable size range for the magma ocean is found to be 10 -2 1 cm, which is close to the upper bound for the critical crystal size dividing fractional and nonfractional crystallization discussed elsewhere in this issue. Both the volatile content and pressure are important and can influence the estimate by 1-2 orders of magnitude. Different kinds of Ostwald ripening take place in the final stage of the crystal growth. If the surface nucleation is the rate-controlling mechanism of crystal growth at small supercooling, then the Ostwald ripening is negligibly slow. In the case of other mechanisms of crystal gro•vth, the crystal radius can reach the critical value required to start the fractional crystallization. It can happen in the latest stages of the evolution when the crystals do not dissolve completely and the time for the ripenlug is large.