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Phase-field modeling of rapid alloy solidification, in which the rejection of latent heat from the growing solid cannot be ignored, has lagged significantly behind the modeling of conventional casting practices which can be approximated as isothermal. This is in large part due to the fact that if realistic materials properties are adopted, the ratio of the thermal to solute diffusivity (the Lewis number) is typically 10^{3}–10^{4}, leading to severe multi-scale problems. However, use of state-of-the-art numerical techniques, such as local mesh adaptivity, implicit time-stepping and a non-linear multi-grid solver, allow these difficulties to be overcome. Here we describe how the application of such a model, formulated in the thin-interface limit, can help to explain the long-standing phenomenon of spontaneous grain refinement in deeply undercooled melts. We find that at intermediate undercoolings the operating point parameter, σ*, may collapse to zero, resulting in the growth of non-dendritic morphologies such as doublons and ‘dendritic seaweed’. Further increases in undercooling then lead to the re-establishment of stable dendritic growth. We postulate that remelting of such seaweed structures gives rise to the low undercooling instance of grain refinement observed in alloys.

Dendritic solidification is a subject of enduring interest within the scientific community, both because dendrites are a prime example of spontaneous pattern formation and due to their pervasive influence on the engineering properties of metals. One long-standing problem with regard to the dendritic solidification of metals has been that of spontaneous grain refinement in undercooled pure melts, first reported to occur in Ni by Walker [^{*} = 140–150 K, Walker observed an abrupt transition from a coarse columnar grain structure to a fine equiaxed structure, with a reduction in grain size of at least one order of magnitude. Similar behavior was found in Co, with a value for Δ^{*} of ≈180 K. This effect has subsequently been identified in other pure metals [_{1}^{*}. At yet higher undercooling a second region of columnar growth is observed which, in most of systems, is replaced by a second region of equiaxed growth at still higher undercooling, with the critical undercooling for this second grain refinement transition being Δ_{2}^{*}.

In many of the systems in which spontaneous grain refinement is observed simultaneous measurement of the dendrite growth velocity has also been undertaken. As first demonstrated by Willnecker ^{*}, growth velocity can be adequately represented by current dendrite growth models, with ^{β}, β > 1. Above Δ^{*} the velocity-undercooling relationship is approximately linear. In alloy systems this transition is usually observed to occur coincident with the Δ_{2}^{*} transition. In most cases the transition to a grain refined microstructure at the lower undercooling, Δ_{1}^{*}, does not appear to have any obvious signature in the velocity undercooling curve. Moreover, differences between the as-solidified microstructures of the two grain refined regions indicate that there may be subtle differences in the underlying grain refinement mechanism. In the grain refined structures, found at high undercooling, the solute segregation pattern is spheroidal about a small dendrite fragment, whereas in grain refined materials formed at low undercooling some of the grains appear to contain small, but well developed, equiaxed dendritic structures [

The origins of the effect are controversial. Early theories suggested a range of mechanisms which included nucleation ahead of the solidification front induced by the pressure pulse associated with solidification [

However, the dendritic fragmentation model of Schwarz _{bu}, which is the time required for fragmentation of the side branches due to remelting and Rayleigh instability and is a monotonic function of the dendrite trunk radius, with small radii giving short breakup times. If, as seems likely from observations of dendritic growth in transparent systems such as xenon [_{bu} being a monotonic function of ρ. The second timescale is the plateau time, τ_{pl}, which is the time the melt remains at, or around, the melting temperature during recalescence and is determined by the macroscopic heat extraction rate. The theory postulates that grain refinement occurs when τ_{bu} < τ_{pl}, which corresponds to the tip radius being below some critical value, ρ*, determined by the heat extraction rate. Consequently, if it is assumed that ρ varies with undercooling as predicted by marginal stability theory [

The dependence of the dendrite tip radius upon undercooling as predicted by marginal stability theory for alloy systems, with its characteristic local minimum followed by a local maximum, has very much been a cornerstone of rapid solidification theory for the past 20 years. However, since the advent of microscopic solvability theory [

Schematic illustration of the dendrite tip radius, ρ, as a function of undercooling, Δ

Recently though it has become possible to use quantitative phase-field modeling to simulate the growth of dendrites under coupled thermo-solutal control [

These calculations of the dendrite tip radius in undercooled alloy systems present a potentially serious problem for the accepted model of spontaneous grain refinement in alloy systems. If the radius does not display a local maximum as the undercooling is increased, then it is difficult to reconcile how a break-up time that scales monotonically with tip radius can predict two region of grain refined microstructure, nor indeed why the growth of grain refined structures above Δ_{1}^{*} should give way to columnar growth as the undercooling is increased. However, these calculations also revealed that the tip selection parameter, σ*, did show this pattern of local minimum followed by local maximum as the undercooling were increased. In fact, qualitatively, the similarities between the behavior of σ* as predicted by the phase-field model and the form of the curve shown in

The model adopted here is based upon that of [_{0} and τ_{0}, the evolution of the phase-field, φ, and the dimensionless concentration and temperature fields _{0}_{E}_{p}_{2} = _{1}_{0}/_{0} with _{1} and _{2} taking the values 5√2/8 and 0.6267 respectively [

The governing equations are discretized using a finite difference approximation based upon a quadrilateral, non-uniform, locally-refined mesh with equal grid spacing in both directions. This allows the application of standard second order central difference stencils for the calculation of first and second differentials, while a compact 9-point scheme has been used for Laplacian terms, in order to reduce the mesh induced [

It has been shown elsewhere that if explicit temporal discretization schemes are used for this problem the maximum stable time-step is given by Δ^{2}, where

When using implicit time discretization methods it is necessary to solve a very large, but sparse, system of non-linear algebraic equations at each time-step. Multigrid methods are among the fastest available solvers for such systems and in this work we apply the non‑linear generalization known as FAS (full approximation scheme [

_{∞} = 0.05, _{E}^{2} using a maximum of 12 levels of refinement, giving a minimum mesh size, ^{12} × 2^{12}. For each parameter set simulations were run over the undercooling range Δ = 0.2–0.8. Below Δ = 0.2 growth is very slow leading to excessive computation times while above Δ = 0.8 the requirement that _{0}

At ε = 0.020 the results are as previously reported, with the velocity increasing monotonically with undercooling and displaying, to a very good approximation, a power-law dependence with exponent ≈2.3. The radius displays a minimum at intermediate undercooling and subsequently increases at high undercooling. This results in an operating point parameter, σ*, which initially decreases with increasing undercooling, before passing through a local minimum to increase with undercooling. At yet higher undercooling σ* passes through a local maximum so at the highest undercoolings studied σ* is decreasing rapidly with increasing undercooling. We have previously argued [

(

Calculated operating point parameter, σ*, as predicted by the phase field model as a function of undercooling and anisotropy strength. At ε = 0.01 stable dendritic growth is not observed in the undercooling range 0.4625 ≤ Δ ≤ 0.5875. Instead a tip-splitting instability leads to the growth of ‘dendritic-seaweed’. It is suggested that remelting of this seaweed structure gives rise to the grain-refined microstructures observed at low undercoolings.

If we now consider reducing the anisotropy strength from ε = 0.020 to ε = 0.015 the effect on the radius and velocity is broadly in line with what we might expect. The radius follows the same trend as at the higher anisotropy level, but with a larger radius being observed at all undercoolings. This is to be expected as σ* is a monotonically decreasing function of ε. The velocity is correspondingly reduced, in line with the expectation that the Peclet number is only very weakly dependent upon ε. σ* is reduced at all undercoolings and, like the tip radius, displays the same general form as at the higher anisotropy level, although in this case we note that the difference between the minimum and maximum values is also significantly reduced.

Finally we consider a further reduction in the anisotropy strength to ε = 0.010, wherein a significant change in behavior is observed. If we consider the behavior of σ* first we observe that, in line with expectation, the value for Δ→0 is reduced, and as with the curves for ε = 0.020 and 0.015, the value of σ* initially decreases as Δ is increased. However, for undercooling between Δ = 0.4625 and 0.5875 stable dendritic growth was not observed, with the solid nuclei used to seed solidification initially developing a preferred four-fold growth morphology but then experiencing a bifurcation at the tip which ultimately leads to a tip-splitting instability in the growing crystal (doublon formation), mediating a transition from a dendritic to ‘dendritic seaweed’ morphology. As a consequence of this tip splitting instability, in the undercooling range Δ = 0.4625–0.5875 neither a value for ρ nor σ* could be obtained. At undercoolings above Δ = 0.5875 stable dendritic growth is once again established. From the values of σ* either side of this undercooling range it appears that this growth instability is consistent with a collapse to zero in the value of σ*. For undercoolings either side of this unstable range the measured tip radius appears to be abnormally large, which is also consistent with the hypothesized collapse of σ*, with small σ* giving rise to large values of ρ. In the unstable growth regime we have not attempted to determine a characteristic growth velocity as the instantaneous growth rate is subject to significant fluctuations as the morphology of the tip changes. However, once stable dendritic growth is re-established in the high undercooling regime, we note that the measured velocities are consistent with a power-law relationship with the same exponent as in the low undercooling regime.

The breakdown of dendritic growth in phase-field simulations of solidification at high undercooling has been noted by a number of groups and has generally been attributed to a competition between capillary and kinetic anisotropies, with capillary effects dominating at low undercooling and kinetic effects dominating at high undercooling. In the case where these anisotropies are oppositely directed, doublon or dendritic seaweed morphologies, which are characteristic of growth at low anisotropy, may be observed when the competing effects are of similar magnitude. However, we do not believe that this is the case here. Firstly, the model has been constructed in the thin-interface limit, wherein the choice of parameters adopted here should eliminate all kinetics from the model and, secondly, the stable dendritic growth observed in the high undercooling regime has the same orientation as in the low undercooling regime (

For the ε = 0.010 case given in ^{4} while in [

Finally, we illustrate how the formation of doublons or dendritic seaweed might give rise to grain refined microstructures. An example of a dendritic seaweed morphology is shown in

(

A phase-field model of coupled thermo-solute dendritic growth formulated in the quantitative thin-interface limit has been used to study the solidification of undercooled binary alloys. By using a range of advanced numerical techniques such as adaptive mesh refinement, implicit time-stepping and a non-linear multigrid solver this model can be extended to high undercoolings and large Lewis numbers. We find that the dendrite operating point parameter, σ*, is a strongly non-monotonic function of the undercooling, and specifically it can pass through a local minimum at intermediate values of the undercooling, before passing through a local maximum, such that at high undercooling it is a decreasing function of undercooling again. Moreover, a parameter space exists wherein σ* collapse to zero, which is manifest by the dendritic structure being replaced by a ‘dendritic seaweed’. We show that subsequent remelting of this dendritic seaweed can give a highly grain refined microstructure and the occurrence of such a ‘seaweed’ structure at both intermediate and high undercoolings offers a plausible explanation for grain refinement in some alloy systems.

The authors declare no conflict of interest.

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