Due to its technological importance, modeling of dendrite growth in pure metals and alloys remains a significant challenge in the field of materials science. In this review recent achievements in the dendrite modeling problem, using two distinct length scale approaches, are summarized. At the nanometer scale, molecular dynamics and Monte Carlo techniques have been developed to extract two important properties of the solid–liquid interface: the kinetic coefficient and the solid–liquid interfacial free energy. Perhaps more importantly the atomistic simulation methods are capable of accurately determining the small, yet crucially important, anisotropies of these parameters. At the mesoscopic scale, advances in phase field modeling have largely overcome the numerical problem associated with the large disparity in length scales typically found in dendrite growth. It is demonstrated that, when the atomistic and continuum level approaches are combined, accurate and parameter free predictions of dendrite growth velocities are possible. In addition, extensions of atomistic and phase field modeling to the case of binary alloys are described.

Atomistic and continuum modeling of dendritic solidification

Even though our theoretical understanding of dendritic solidification is relatively well developed, our current ability to model this process quantitatively remains extremely limited. This is due to the fact that the morphological development of dendrites depends sensitively on the degree of anisotropy of capillary and/or kinetic properties of the solid-liquid interface, which is not precisely known for materials of metallurgical interest. Here we simulate the crystallization of highly undercooled nickel melts using a computationally efficient phase-field model together with anisotropic properties recently predicted by molecular dynamics simulations. The results are compared to experimental data and to the predictions of a linearized solvability theory that includes both capillary and kinetic effects at the interface.

Linking Phase-Field and Atomistic Simulations to Model Dendritic Solidification in Highly Undercooled Melts

The kinetics of isothermal crystallization and melting are studied for elemental Ni employing non-equilibrium molecular-dynamics simulations based on interatomic potentials of the embedded-atom-method form. These simulations form the basis for calculations of the magnitude and crystalline anisotropy of the kinetic coefficient $\ensuremath{\mu},$ defined as the constant of proportionality between interface velocity and undercooling. We obtain highly symmetric rates for crystallization and melting, from which we extract the following values of $\ensuremath{\mu}$ for low index ${100},$ ${110},$ and ${111}$ interfaces: ${\ensuremath{\mu}}_{100}=35.8\ifmmode\pm\else\textpm\fi{}22,$ ${\ensuremath{\mu}}_{110}=25.5\ifmmode\pm\else\textpm\fi{}1.6,$ and ${\ensuremath{\mu}}_{111}=24.1\ifmmode\pm\else\textpm\fi{}4.0$ in units of cm/s K. The results of the present study are discussed in the context of previous molecular-dynamics simulations for related systems, and kinetic models based upon transition-state and density-functional theories.

Kinetic coefficient of Ni solid-liquid interfaces from molecular-dynamics simulations

New evidence for the dual origin of anomalous eutectic structures in undercooled Ni–Sn alloys: In situ observations and EBSD characterization

Ni-18.7 at. pct Sn eutectic melt was undercooled and spontaneously solidified in the encasement of a glass flux. Structure morphologies of the sample surface as well as inside the sample were systematically examined, and a critical undercooling of 130 K was clearly revealed for the alloy. Below the critical undercooling, coupled lamellar eutectics (α-Ni and β-Ni3Sn) dendritically grow in the undercooled melt. Beyond the critical undercooling, α-Ni dendrites first form during the early recalescence. β-Ni3Sn nucleates uniformly in the remaining liquid and then separately grows with α-Ni. Solidification of the remaining liquid into lamellar eutectics only occurs at the places in the sample surface layer where the space between the α-Ni dendrite arms is large enough. The finding that all the solidification structures at undercooling above 20 K comprise anomalous eutectics indicates that both coupled eutectic growth and decoupled dendritic growth in the rapid solidification can result in the anomalous eutectic formation. The results also indicate that it is feasible to measure the crystal growth velocity by observation of the recalescence front when undercooling exceeds 50 K for this alloy.

Structural Evidence for the Transition from Coupled to Decoupled Growth in the Solidification of Undercooled Ni-Sn Eutectic Melt

Rapid solidification of undercooled pure nickel has been imaged at sufficiently high spatial resolution (64 ×X 64 pixels) and temporal resolution (40,500 frames/s) to observe interface shape and motion at solidification velocities exceeding 45 m/s. Imaging was of 8 g, quartz-enclosed melts at undercoolings of 70 to 300 K. Dendrite velocities within the melt were calculated from the surface velocities observed employing a simple geometric model of growth. Solidification was found to proceed invariably from a single nucleation point; growth velocity then followed an approximate power-law relationship with respect to undercooling up to some critical value ΔT*, where 150 K < ΔT* < 180 K. At higher undercoolings, velocity increased less rapidly than predicted by the power-law relationship and the interface morphology changed in appearance from angular to macroscopically smooth.

John W. Lum

Papers

High-Speed imaging and analysis of the solidification of undercooled nickel melts