The kinetics of nucleation and crystal growth and scaling laws for magmatic crystallization

Magmatic crystallization depends on the kinetics of nucleation and crystal growth. It occurs over a region of finite thickness called the crystallization interval, which moves into uncrystallized magma. We present a dimensional analysis which allows a simple understanding of the crystallization characteristics. We use scales for the rates of nucleation and crystal growth, denoted by I _m and Y _m respectively. The crystallization time-scale _c and length-scale d _c are given by (Y _m ³ /I _m )^–1/4 and (·) _m ^1/2 respectively, where is thermal diffusivity. The thickness of the crystallization interval is proportional to this length-scale. The scale for crystal sizes is given by (Y _m /I _m )^1/4. We use numerical calculations to derive dimensionless relationships between all the parameters of interest: position of the crystallization front versus time, thickness of the crystallization interval versus time, crystal size versus distance to the margin, temperature versus time. We assess the sensitivity of the results to the form of the kinetic functions. The form of the growth function has little influence on the crystallization behaviour, contrary to that of the nucleation function. This shows that nucleation is the critical process. In natural cases, magmatic crystallization proceeds in continously evolving conditions. Local scaling laws apply, with time and size given by =(Y ³/I)^–1/4 and R=(Y/I)^1/4, where Y and I are the rates at which crystal are grown and nucleated locally. is the time to achieve crystallization and R the mean crystal size. We use these laws together with petrological observations to infer the in-situ values of the rates of nucleation and growth. Two crystallization regimes are defined. In the highly transient conditions prevailing at the margins of basaltic intrusions, undercoolings are high and the peak nucleation and growth rates must be close to 1cm^–3· ^–1 and 10^–7cm/s, in good agreement with laboratory measurements. In quasi-equilibrium conditions prevailing in the interior of large intrusions, undercoolings are small. We find ranges of 10^–7 to 10^–3 cm^–3 s^–1 and of 10^–10 to 10^–8cm/s for the local rates of nucleation and growth respectively.