The nucleation and growth of liquidus phases in cooling magmas at constant rates are modeled taking into account homogeneous nucleation, diffusion-limited growth, and depletion of crystallizing component from melt, and the temperature-dependent diffusivity. The formulation of governing equations shows that four dimensionless parameters, whose physical meanings are the nucleation difficulty, the fusion enthalpy, the ratio of the growth rate to the cooling rate, and the activation energy of diffusion, control the crystallization phenomena. The nucleation behavior with time (or temperature) is determined primarily by the competition between increasing nucleation rate with cooling and the reduced supersaturation with depletion by progressive growth of crystals previously nucleated. The maximum nucleation rate and the number density of crystals increase with decreasing interfacial tension and diffusivity, and with increasing fusion enthalpy and cooling rate. Quantitative expressions of the time or temperature interval for which the nucleation remains appreciable, the peak nucleation rate, the number density of crystals and the mean crystal radius are derived as functions of controlling parameters, and can be used to estimate the cooling rate or other unknown parameters from the number density of crystals of a rock.
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