Andrew M. Mullis

Grain refinement and the stability of dendrites growing into undercooled pure metals and alloys

We present an analysis of the stability of a dendrite against a small perturbation to the tip velocity. We find that dendritic growth in pure metals and alloys will become unstable above some upper critical undercooling ΔT2*. In alloys above a critical concentration, dendritic growth may also become unstable below a lower critical undercooling, ΔT1*. In the example systems studied, Ni–Cu and Ag–O, the location of these unstable regions shows remarkably close agreement with the onset of spontaneous grain refinement. We obtain values of ΔT2* for Ni and Ag of 195 K and 160 K, respectively, in good agreement with the observed values of 170 K and 133–153 K.

Growth velocity-undercooling relationships and microstructural evolution in undercooled Ge and dilute Ge-Fe alloys

A melt encasement (fluxing) technique has been used to systematically study the velocity-undercooling relationship in samples of pure Ge and Ge doped with 0.01 at % Fe at undercoolings up to 300 K. The apparatus was designed such that it was possible to view the sample throughout the experiment, allowing solidification velocity measurements to be made. These velocity measurements were subsequently correlated with the as-solidified microstructure. From a combination of growth velocity measurements and microstructural characterisation it was possible to identify a change in growth morphology from faceted to non-faceted growth in both the pure metal and the dilute alloy. This transition occurred at a lower undercooling in the dilute alloy (ΔT > 150 K) than in the pure metal (ΔT > 170 K). Spontaneous grain refinement was also observed at ΔT > 210 K in Ge-Fe and at ΔT > 270 K in pure Ge. These transitions are discussed and a mechanisms for the change in growth morphology with small amounts of impurity is suggested.

Growth induced dendritic bending and rosette formation during solidification in a shearing flow

A free boundary model for dendritic growth in a non-stationary fluid is presented. The model is used to evaluate the effect on dendritic growth of fluid flow orthogonal to the principal growth direction. It is found that such a flow causes rotation of the tip due to thermal/solutal advection and that, for small angular deflections, the rotation is a constant per unit length of growth. This rotation is found to be a function of the cross flow and growth Peclet numbers. Results from the dendrite growth model are used in a Cellular Automaton (CA) model of rosette formation. It is found that the predicted bending rates could give rise to rosette formation without any need to invoke mechanical effects. Moreover, the magnitude of the bending parameter defines a maximum size to which a dendritic crystal could grow before being spheroidized.

Microstructural evolution and growth velocity-undercooling relationships in the systems Cu, Cu-O and Cu-Sn at high undercooling

A melt encasement (fluxing) technique has been used to systematically study the velocity-undercooling relationship in samples of Cu and Cu-O and Cu-3 wt% Sn at undercoolings up to 250 K. In pure Cu the solidification velocity increased smoothly with undercooling up to a maximum of 97 m s-1. No evidence of grain refinement was found in any of the as-solidified samples. However, in Cu doped with >200 ppm O we found that samples undercooled by more than 190 K had a grain refined microstructure and that this corresponded with a clear discontinuity in the velocity-undercooling curve. Microstructural evidence in these samples is indicative of dendritic fragmentation having occurred. In Cu-Sn grain refinement was observed at the highest undercoolings (greater than 190 K in Cu-3 wt% Sn) but without the spherical substructure seen to accompany grain refinement in Cu-O alloys. Microstructural analysis using light microscopy, texture analysis and microhardness measurements reveals that recrystallisation accompanies the grain refinement at high undercoolings. Furthermore, at undercoolings between 110 K and 190 K, a high density of subgrains are seen within the microstructure which indicate the occurrence of recovery, a phenomenon previously unreported in samples solidified from highly undercooled melts.

Mechanical deformation of dendrites by fluid flow during the solidification of undercooled melts

Abstract Mechanical interactions between growing dendrites and their parent melt are normally considered to be of little significance. During conventional solidification processing this is undoubtedly true. However, during the solidification of undercooled melts the twin conditions required to produce mechanical damage to dendrites, high flow velocities and very fine dendrites, may exist. This is most likely in strongly partitioning alloy systems where the tip radius experiences a local minimum at undercoolings in the range of 50–100 K. In this paper we present a model for the skin stress resulting from fluid flow around a family of realistically shaped dendrites. We find that within a narrow undercooling range about the minimum in the tip radius, mechanical deformation of the growing dendrite is likely. Experimental evidence is presented from the Cu–3wt%Sn and Cu–O alloy systems that appear to show evidence of extensively deformed dendritic structures consistent with mechanical damage. Other mechanisms for causing dendritic bending during growth are considered and shown to be unlikely in this case.

A phase field model for spontaneous grain refinement in deeply undercooled metallic melts

The origin of spontaneous grain refinement in deeply undercooled metallic melts has been of enduring interest within the solidification literature. Here we present the results of phase field simulations of dendritic growth into pure undercooled melts, at growth velocities up to 35 m s−1. We find that, at low growth velocities, dendrite morphologies are broadly self-similar with increasing growth velocity. However, above ≈15 m s−1 the initiation of side-branching moves closer to the dendrite tip with increasing growth velocity. This appears to be related to the level of kinetic undercooling at the tip. Once side-branch initiation begins to occur within 1–2 radii of the tip, profound morphological changes occur, leading to severe thinning of the dendrite trunk and ultimately repeated multiple tip-splitting. This process can be invoked to explain many of the observed features of spontaneous grain refinement in deeply undercooled metallic melts.

Rapid solidification within the framework of a hyperbolic conduction model

Abstract Utilising high speed pulsed lasers, crystal growth velocities in excess of 250 m s −1 can be achieved in metallic systems. At such high growth rates it becomes pertinent to question the assumptions implicit within the Fourier model of thermal conduction. Specifically, the Fourier model allows an infinite velocity of propagation for heat. The rapid solidification problem is posed within a hyperbolic conduction model, imposing a finite velocity of propagation on the heat transfer process. We find that dendritic solidification is only possible if the growth velocity is less than half the thermal wave velocity, C . The likely value of C is discussed.

Three dimensional thermal-solute phase field simulation of binary alloy solidification

We employ adaptive mesh refinement, implicit time stepping, a nonlinear multigrid solver and parallel computation to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt substantially cooled below its freezing point at the microscale. Underlying physical molecular forces are captured at this scale by a specification of the energy field. The time rate of change of the temperature, alloy concentration and an order parameter to govern the state of the material (liquid or solid) are controlled by the diffusion parameters and variational derivatives of the energy functional. The physical problem is important to material scientists for the development of solid metal alloys and, hitherto, this fully coupled thermal problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques this problem is now shown here to be tractable at appropriate resolution with relatively moderate computational resources.

Highly undercooled germanium: Growth velocity measurements and micro structural analysis

Abstract Growth velocities have been measured in samples of Ge (3–5 mm in diameter) which were undercooled using a fluxing technique. Samples of Ge were heated and undercooled in a viscous soda-lime glass flux contained within thin-walled, clean silica crucibles. Using this particular method, undercoolings of up to 250 K below equilibrium melting temperature were obtained. This maximum undercooling may have been restricted by sample size, experimental conditions or impurities present in the sample. It was possible to determine growth velocities up to undercoolings of 250 K The size of the samples used allowed growth velocities measurements to be made, by measuring recalescence times using a linear photo-diode array. These were measured at a variety of undercoolings, where nucleation was initiated using a thin alumina needle, and a maximum velocity of 7.6 ms −1 was recorded at an undercooling of 250 K. Microstructural analysis of samples revealed a range of transitions occurring as the level of undercooling increased. These include a gradual transition from faceted to non-faceted type growth at ΔT > 160 K, characterised by the observance of growth twins a small undercoolings which were not present in samples undercooled greater than 160 K. By transmission electron microscopy (TEM) examination, it was also confirmed that there was no microtwinning present in the highly undercooled samples. A general grain refinement at ΔT> 210 K from a coarse microstructure incorporating an extensive dendritic substructure to a fine structure with an impurity-rich, cross-shape dendrite fragment at the centre of the grains. These transitions could also be associated with trends seen between measured growth velocities and undercoolings.

Grain refinement and growth instability in undercooled alloys at low undercooling

In a previous paper [J. Appl. Phys. 82, 3783 (1997)] we presented an analysis of the stability of a dendrite against a small perturbation to the tip velocity. This model demonstrated that dendritic growth in pure metals and alloys would become unstable above some upper critical undercooling ΔT2*, while in alloys above a critical concentration, dendritic growth may also become unstable below a lower critical undercooling, ΔT1*. However, quantitative agreement between the example system studied, Ni–Cu, and the available data was poor. In this article we present an improved computational scheme which allows us to relax the assumption that dendrites propagate as isothermal paraboloids of revolution. This results in much improved agreement with the experimental data.