Dendrites frequently form during solidification into an undercooled melt. These dendrites possess secondary and sometimes even tertiary arms. While the tip radius and tip velocity of the dendrite are set by the growth conditions, the sidebranches behind the tip undergo a coarsening process under nearly isothermal conditions. The resulting two-phase mixtures are topologically and morphologically complex with spatially varying mean and Gaussian curvatures. These dendritic two-phase mixtures are one example from a large class of morphologically and topologically complex structures found in nature that undergo coarsening. Included in this class are the bicontinuous two-phase mixtures produced following spinodal decomposition and ordering in crystals. Understanding the coarsening process in these systems requires theory, simulation, and experiments that capture their three-dimensional topology and morphology. Given the complicated morphology and topology of these systems it is not surprising that our understanding of this process is in its infancy.

At the core of the proposed investigation is the four-dimensional (4D) characterization and analysis approach, which follows the microstructure evolution in three dimensions and in time (an additional dimension). A combined theoretical and experimental program is proposed to examine the nature of the coarsening process in these highly complex, dendritic microstructures. The experiments will examine the time-dependent evolution of the structures in three dimensions in situ through X-ray microtomography. The results of these experiments will be used both to provide insights into the coarsening process that will guide the development of theory. The experiments will also serve as a test of phase field simulations and the theory, as well as to provide initial conditions for the simulations. The results of simulations of coarsening in bicontinuous mixtures will be used to develop a theory of coarsening in these systems that will elucidate the importance of spatially varying curvatures and complicated topology found in dendritic systems. The experiments, simulations, and theory will focus on the evolution of distributions of curvature and the topology of the structure. Given the various mechanisms that may be responsible for the evolution of the structure, we will examine coarsening in a

wide range of solid volume fractions and different initial states of the system at the beginning of coarsening. Through these experiments and theory, we aim to develop a comprehensive description of coarsening in complex and technologically important microstructures.