This contribution presents the application of nonlinear model predictive control to the Vertical Gradient Freeze crystal growth process. Due to the time-varying spatial extent of the crystal and melt during growth, this process is characterised by two coupled free boundary problems that form a so called two-phase Stefan problem which is of nonlinear nature. To apply model predictive control to this process, a simplified, spatially distributed representation of the system is derived and transferred into a spatially lumped form by means of the finite element method. For this model, a nonlinear control problem is formulated, that takes process limitations into account and tries to satisfy different quality objectives by formulating demands on the systems spatiotemproal temperature distribution. This provides the foundation for the presented predictive control design. Finally, the approximated model and the controller are verified for different real-world scenarios that include model errors and parameter uncertainties.