Bruce T. Murray, ACMD
Adam A. Wheeler, Southampton University
Martin E. Glicksman, Rensselaer Polytechnic Institute
The development and application of phase-field models for studying phase transitions under a variety of conditions has been the focus of a successful collaborative effort between researchers in CAML, MSEL, and some external to NIST. One component of this effort has been the investigation of first-order phase transitions, such as solidification of a pure material under conditions where the solid crystal grows in the form of a dendrite. The classical free boundary approach models solidification by treating the solid and liquid phases individually and by explicitly determining the moving interface between the two bulk phases. In contrast, the phase-field method introduces a continuous transition between the two phases across a thin layer of finite thickness. A phase-field variable (and corresponding field equation) is introduced and its value at a point identifies the phase. This additional field equation must be solved along with the equation that governs conservation of energy. The advantage of this approach is that the location of the interface does not have to be explicitly determined as part of the solution. Thus, the method is well suited for calculating the evolution of complicated solidification morphology (e.g., dendritic growth).
Computations of dendritic growth using the phase-field method have provided some of the most realistic simulations of this complicated phenomenon which involves an interplay between diffusion in the bulk phases and surface energy and kinetic effects at the solid/liquid interface. Simulations performed using numerical algorithms developed in the ACMD have provided a better understanding of the nature of the interaction between the various physical mechanisms and have allowed material scientists and physicists to test and modify existing simplified theories. Currently, it has been proposed to use the micro-scale study of dendritic growth via the phase-field computations to guide the development of meso-scale models which can then be employed in large-scale computations for the solidification of castings; casting simulations can reduce the development costs and improve the quality of cast parts.
As an example of the utility of the phase-field simulations, an attempt to understand a phenomenon observed in dendritic growth experiments under certain conditions has been the focus of a recent series of phase-field computations. Figure 5 shows an experimental photograph of a succinonitrile dendrite on the left. At an earlier time, the primary dendrite branch split into two main branches. After a short period of parallel growth, one of the branches becomes dominant and the growth of the other branch is suppressed. At the time shown, the suppressed branch appears as though it has been ``cleaved'' from the primary branch, and this phenomenon has been referred to as a cleaving event. Also included in the figure is a phase-field computation which is an attempt to simulate this cleaving behavior. While the growth conditions and the scale of the dendrite in the computations differ from those of the experiment, the simulation qualitatively represents the behavior observed in the experiment and will provide better understanding into the nature of such complex physical phenomena.
Figure 5: Experimental photograph of succinonitrile dendrite showing cleaving phenomenon (left); phase-field simulation exhibiting similar behavior (right). Note that the ``cleave'' occurs on the opposite side in the computation.