Impeding Nucleation for More Significant Grain Refinement

Grain refinement has been a topic of extensive research due to its scientific and technological importance as a common industrial practice for over seven decades. The traditional approach to grain refinement has been to reduce nucleation undercooling by the addition of potent nucleant particles. Here we show both theoretically and experimentally that more significant grain refinement can be achieved through increasing nucleation undercooling by using impotent nucleant particles. Based on the concept of explosive grain initiation, this new approach is illustrated by grain initiation maps and grain refinement maps and validated by experiments. It is anticipated that this new approach may lead to a profound change in both nucleation research and industrial practice well beyond metal casting.


Solidification processes
Based on the recent advances in the understanding of heterogeneous nucleation [1][2][3][4][5] and our current knowledge of grain growth 6 , solidification of a single phase alloy may involve a number of individual stages, as schematically illustrated in Supplementary Fig. 1. Here we analyze briefly these stages to establish a physical model of solidification to facilitate numerical modelling.
Supplementary Fig. 1. A schematic cooling curve illustrating the stages of the solidification process by following a specific nucleant particle (in blue) that initiates a grain in the solidified microstructure. Please note that the length scale of the sketches increases with time during the solidification process.
Prenucleation refers to the phenomenon of pronounced atomic ordering in the liquid adjacent to the substrate/liquid interface at temperatures above the nucleation temperature (T n ) 2 . Prenucleation manifests itself by substantial atomic layering along the normal of the substrate surface and in-plane ordering parallel to the substrate surface 1,2 . Such atomic ordering is affected by the lattice misfit between the substrate and the new solid phase 2 , chemical interaction between atoms of the substrate and the liquid 3 and atomic-level surface roughness of the substrate 5 . Therefore, prenucleation produces a precursor for the subsequent heterogeneous nucleation process.
Heterogeneous nucleation creates a template on a substrate in the liquid, from which the new solid phase (either stable or metastable) can grow. It occurs at a nucleation temperature, , through structural templating by the substrate and proceeds by a layer-by-layer growth mechanism, as described in the epitaxial nucleation model 3 . Epitaxial nucleation is independent of the size of nucleant particles; once the nucleation undercooling is achieved, heterogeneous nucleation should take place on all nucleant particles 3 . However, a nucleus formed at this stage may not necessarily lead to a grain in the solidified microstructure.
Constrained cap growth refers to the initial growth process experienced by a nucleus before grain initiation. After nucleation, the solid phase cannot grow at a constant temperature due to the constraint of the curvature effect. Further growth can only be possible by increasing the undercooling to overcome the curvature constraint. Such constrained cap growth of a solid particle continues until the grain initiation undercooling is reached 7 .
Grain initiation refers to the process of formation of a grain of radius (r), which can grow freely with a grain initiation undercooling (∆ ). r can be understood to be the radius of the smallest sphere of the solid phase that engulfs the entire nucleant particle, and thus r equals the maximum dimension of the nucleant particle. Grain initiation is governed by the grain initiation criterion 7 : where Г is the Gibbs-Thompson coefficient. Eq. (1) suggests that grain initiation starts with the largest solid particle and the on progressively smaller ones. Grain initiation is a deterministic process and is dependent on the size of the solid particle 7 . It is clear that not every nucleus formed at can grow freely to initiate a grain. Those that fail to grow freely will become thermodynamically unstable and will simply dissolve back into the liquid.
Spherical growth occurs immediately after grain initiation with a growth velocity of 8 : where r' is the radius of the solid particle, is the diffusion coefficient of the solute, and is the supersaturation defined by Zener 9 , which can be calculated by the following equation 8 : where is the liquidus slope, 0 is the solute concentration, is the solute partition coefficient and ∆ is the growth undercooling.
Dendritic growth: At a critical size, the spherical morphology becomes unstable 10 and a grain starts to grow dendritically with a growth velocity of 11 : where 0 = 2.08 − 1.534k − 0.434 2 , and 1 = 0.648 + 0.124 .
Recalescence occurs at a temperature, (the recalescence temperature), at which the heat released by solidification equals the heat dissipated into the environment. After recalescence the rising temperature (i.e., decreased undercooling) will stifle any further grain initiation. Therefore, it is usually assumed that the number of grains in the final solidified microstructure equals that already initiated at recalescence 7,12-14 .

Method for numerical simulation
We have numerically modelled the key solidification processes described in the previous section. At each stage, the heat extracted from the melt to the environment ( ) equals to the heat released due to temperature drop and/or the latent heat ( ): When < where is the time at which nucleation occurs, the heat released is only due to the temperature drop and one has where is the specific heat capacity of the melt per unit volume.
When ≤ ≤ where is the time at which recalescence takes place, the heat released is due to both temperature drop and latent heat and one has where ∆ is the latent heat of fusion per unit volume, and is the fraction of the solid phase.
When > , no further grain initiation occurs, and the heat balance also follows Eq. (7).
As revealed by experimental observation 15,16 , nucleant particles have a log-normal size distribution: where 0 is the total number density of the particles in the melt, is the number density of the particles of size d, 0 is the geometrical mean particle size, and is the standard deviation of the particle size distribution. Eq. (8) is the probability distribution function. Using the cumulative distribution function, the number of nucleant particles that have diameters larger than can be calculated: where is the errorfunction.
To conduct a numerical simulation we need to determine the size of the largest particle, (1 st ), which has a cumulative distribution function of 1 and a grain initiation undercooling, ∆ (1 st ). The results for the Mg-1Al/MgO system are presented in Supplementary Fig. 2.
It should be pointed out that both (1 st ) and ∆ (1 st ) are functions of 0 , , 0 and the total volume (assumed here to be 1 cm 3 ). For a given type of nucleant particle with lognormal size distribution, (1 st ) increases and ∆ (1 st ) decreases with increasing particle number density. The average grain size ( ) is calculated from the number of grains per unit volume which is assumed to be equal to the total number density of initiated grains ( ) 7 : To simplify numerical modelling, we have made the following basic assumptions:  The alloy melt is in an isothermal condition.  The heat extraction rate by the environment is constant.  Morphological instability occurs at = .  The size of the nucleant particles has a log-normal distribution.
A matlab code was compiled to perform the numerical simulation. All the parameters used as input for the numerical simulations are summarized in Supplementary Table 1. The total volume used for all the simulations is fixed at 1 cm 3 . The time step used is usually 10 -3 s but 10 -5 s in cases of explosive grain initiation. In addition, we have used other dendrite growth models, such as the LKT model 17 , to calculate , and found that the simulation results are insensitive to the dendritic growth model used.