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Quantitative phase field modeling of solute trapping and continuous growth kinetics in rapid solidification

Solute trapping is an important phenomenon in rapid solidification of alloys, for which the continuous growth model (CGM) is a popular sharp interface theory. Using matched asymptotic analysis, we show how to quantitatively map the sharp interface behavior of a binary alloy phase field model onto the CGM kinetics of Aziz et al. [1], with a controllable partition coefficient k(V ). We demonstrate the parameterizations that allow the phase field model to map onto the corresponding CGM or classical sharp interface models. We also demonstrate that the mapping is convergent for different interface widths. Finally we present the effect that solute trapping can have on cellular growth in a directional solidification simulation. The treatment presented for solute trapping can be easily implemented in different phase field models, and is expected to be an important feature in future studies of quantitative phase field modeling in rapid solidification regimes, such as those relevant to additive manufacturing.


Introduction
Rapid solidification of metallic alloys is a common feature in advanced industrial manufacturing processes such as additive manufacturing, laser welding, and thermal spray coatings.
The rapid solidification is often accompanied with incomplete solute partitioning at the solid-liquid interface, which is called solute trapping. This affects the solidification microstructure by influencing the growth morphology, length scale, microsegregation and the resulting precipitation of secondary phases.
These microstructural features determine, to a large extent, the properties and performance of the material. Moreover, these features can be related back to the controllable process details through computer modeling.
Classic sharp interface models (hereafter SIM) can well be used to describe traditional casting processes. operate at low to moderate cooling rates, which are well described by the classic sharp interface model (hereafter SIM), typically with a vanishing kinetic coefficient (β = 0). The classical SIM assumes zero interface width, and that the interface is near equilibrium dur-ing solidification. This practically means that the solid-liquid interface is much smaller than the capillary length, which is the smallest characteristic length in the solidification problem.
During rapid solidification, in contrast, the equilibrium conditions that prevail in the classic SIM break down, and the atomic attachment kinetics and other non-equilibrium effects that emerge in the physically non-zero solid-liquid interface.
These include a velocity-dependent solute partition coefficient k(V ) and a velocity-dependent interface undercooling, or concentration. These effects become dominant at rapid solidification rates and can strongly affect the microstructure kinetics, morphology and phase formation.
A convenient method for modeling microstructure problems in solidification and solid-state transformation is phase field method. This is due to its fundamental origins, connections with non-equilibrium thermodynamics, and numerical efficiency compared to interface tracking approaches. Phase field modeling has been used in the study of solidification in a range of materials, from ideal dilute binary alloys [2,3] to more complex binary alloys [4] and multi-component or multi-phase al- loys [5,6,7].
For a special class of models called multi-phase field models, Steinbach et al. presented a finite dissipation model [8] for simulating solute trapping in rapid solidification. A specific solute partitioning k(V ) is achieved by coupling a kinetic equation between the phase concentration, and by adjusting a so-called rate constant together with the numerical interface width to control k(V ) through the interface dissipation term. Their results for k(V ) are are consistent with a different approach of Danilov and co-workers [9] in a similar range of interface velocities, and both are consistent with experiments. These studies did not report phase field model predictions of kinetic interface concentration or undercooling, and how these compare to predictions of a non-equilibrium solidification model, such as the continuous growth mode (CGM) of Aziz and Boettinger [1].
This paper examines the continuous growth limit of another class of phase field models based on order parameter fields [3].
In the limit of low undercooling (or low supersaturation), a robust set of results derived from a matched asymptotic boundary layer analysis of this model, for an ideal binary alloy [3], can be used to map the model's behaviour quantitatively onto the classical sharp interface model; these results can also be essentially used to recover the classical sharp interface limit of most of the above-cited phase field models [4,6,7].
Previous order parameter-based phase field model studying rapid solidification have used the aforementioned classical sharp interface limit (i.e. k(V ) = k e in most studies [10,11,12]. For the same classical SIM parametrization, Ghosh et al. [13] includes solute trapping by using a combination of large W and V , such that incomplete anti-trapping leads to some emergent k(V ) > k e , which depends on the chosen interface width W , and is also different in 1D, 2D, and 3D simulations. Currently no quantitative phase field model parameterization exists which consistently maps a phase field model onto the appropriate non-equilibrium sharp interface limit described by a specific k(V ) and interface undercooling/concentration.
There is presently no generally accepted SIM to describe the rapid solidification regime. Several sharp interface mod-els for this regime have been proposed. The two most popular paradigms are the continuous growth model (CGM) of Aziz and co-workers [14,1] and that of Sobolev and co-workers [15,16]. The former assumes standard diffusion accompanied by attachment-limited kinetics at the interface, while the latter further incorporates two-time-scale dynamics to describe both inertial and diffusive dynamics of solute atoms near and through a rapidly advancing interface. The two approaches give similar results at low velocities (although still large enough to be in the rapid solidification regime). In this work, we will focus on the former, however, we expect our results to be straightforwardly generalizable to the latter.
al [19] showed that a phase field model of alloy solidification,

Methods
This section briefly reviews the continuous growth model in the sharp interface limit, and the ideal dilute binary alloy phase field model used in this work, and its extension to the CGM regime.

Review of continuous growth model
In continuous growth model for dilute binary alloys, the nonequilibrium partition coefficient has the form [1] where V is the interface velocity, k e is the equilibrium partition coefficient, V CGM D is the so-called diffusive velocity which is typically fit to velocity -partition coefficient experiments. The continuous growth model also predicts a kinetic undercooling that has a velocity-dependent liquidus slope [1]. Assuming an externally imposed temperature at the interface, T , the kinetic undercooling expression can be inverted to give the liquid-side concentration as where c o l is the average solute concentration in the alloy, T l is the liquidus temperature, d o is the solutal capillary length, κ is the local interface curvature, β is the kinetic coefficient, and f (k(V )) is the velocity-dependent correction to the liquidus slope, given by where D is a parameter that can be tuned to represent complete solute drag (D = 1) or no solute drag (D = 0) [1]. For sufficiently small interface velocities, solute partitioning can be assumed to be at equilibrium, i.e. k(V ) → k e . In this limit It is thus expected that simulating the CGM limit in phase field simulations should also allow for independent control of the partition coefficient and kinetic undercooling.

Phase field model of an ideal binary alloy
Phase field modeling of solidification of a dilute binary alloy is described by an order parameter φ (using here the limits −1 ≤ φ ≤ 1) and concentration field c, whose dynamics are governed by where τ = τ (n) is the anisotropic interface attachment time scale, W = W (n) is the anisotropic interface width and W 0 is its magnitude, λ is the coupling constant, m e l is the equilibrium liquidus slope, D L/S is the liquid/solid diffusion coefficient, and a t is the antitrapping coefficient.

Classic sharp interface limit of phase field model
The sharp interface limit of a phase field model is achieved where n := ∇φ/|∇φ| is the interface normal, and a 1 , a 2 , and a t are asymptotic analysis constants that depend on the chosen interpolation functions. For h(φ) and q(φ) given by Eq. (6), they are given by The capillary length d o (n) and kinetic coefficient β(n) are typically anisotropic in 2D and 3D. For example, for cubic crys-tal lattices with weak anisotropy, this anisotropy is expressed as where d mag o is the magnitude of the anisotropic capillary length d o (n), and c is the capillary anisotropy strength. Analogously, β 0 is the magnitude of the anisotropic kinetic coefficient β(n), and k is the kinetic anisotropy strength.

CGM sharp interface limit of phase field model
In this section we will show how the above standard binary phase field model can be modified to model the kinetics of the continuous growth model, described in particular by a particular partition coefficient k(V ) and kinetic undercooling given by the CGM model. This will be done by modifying the original form of the antitrapping current a t , which leads to a correction to the asymptotic constant a 2 .
To show how to achieve controlled solute trapping in the phase field equations in Eqs. (4) and (5), the antitrapping coefficient a t in Eq. (11) is modified as follows: where A is trapping parameter, introduced to control the amount of solute trapping. As shown in Supplementary material, the modified antitrapping coefficient a t in Eq. (14)) leads to a modification to the asymptotic analysis constant a 2 used to set β in Eq. (8), given by where a + 2 corresponds to zero solute drag, a − 2 corresponds to 4 full solute drag, and the constants in Eq. (15) are given bȳ For A = 0 the modified antitrapping coefficient a t reverts back to a t in Eq. (11) and a 2 reverts back to a 2 in Eq. (10) It is noteworthy that the form of a t is a convenient choice that makes the integrals arising from the asymptotic analysis easily tractable. Other similar forms are possible, each leading to a different specific value of the constants appearing in Eq. (16).
As shown in supplementary material, when the constants F + =F − in Eqs. (16), there is a chemical potential jump across the effective sharp interface. It is well documented that this leads to solute trapping as the interface is no longer able to maintain local equilibrium [3,20]. To second order in the perturbation theory used to analyze the phase field equations, the solute partition coefficient is given implicitly by a transcendental relationship between interface velocity and non-equilibrium partition coefficient: where is a characteristic solute trapping velocity, W 0 is the magnitude of anisotropic interface width W (n), and A is the trap-ping parameter for a t introduced in Eq. (14). Equation (17) where ϑ = gives zero solute drag, ϑ = α gives complete solute drag, andf α (f ) is the free energy density of the solid (liquid). The where β is given by the following modified sharp interface relation Evaluating Equation (19) Fig. 1, where order parameter is the red solid This gave the solid-side estimation of concentration the biggest error, at approximately 5% relative error -this estimation error, however did not have a large influence on the evaluation of the partition coefficient.
We also tested a simpler approach that considered the liquidside concentration as the phase field profile maximum. With this approach the estimated liquid-side concentration was systematically underestimated compared to the extrapolation ap-      Table 1. The data were obtained using two small computational interface widths W to demonstrate that the phase field model converges well to the aforementioned sharp interface models at higher interface velocities. The dark red data corresponds to smaller interface width W = 0.2 nm, and dark blue to W = 0.5 nm. As expected, the smaller interface width data (dark red) converge to the corresponding theory at a higher interface velocity than the larger interface width data (dark blue). The dimensionless undercooling for these runs was   It should also be noted that estimating the solid-side concentration has the most scatter, when computed with the method described in section 2.5. This is because for the transient concentration profiles, solid-side concentration has a complicated shape particularly in the initial stages of the simulation. We also     Table 1, except ke is increased from 0.15 to 0.8. Left graph shows convergence to ke and k P F (V ) from Eq. (17). The right graph shows the convergence to liquid-side concentration c CGM L to Eq. (2) for the different CGM cases indicated. Dimensionless undercooling is set to ∆ = 0.75. 9 Figure 5: Convergence of different phase field simulations (red and blue scatter points) to corresponding sharp interface models (red and blue scatter points) for Si-Al alloy for two diffuse interface widths. The left graph shows convergence of the partition coefficient to ke and k P F (V ) from Eq. (17). The right graph shows the convergence of the liquid-side concentration c CGM L to Eq. (2) for the different non-equilibrium cases indicated in the text. Dimensionless undercooling is set to ∆ = 0.55. growth of an Si-As alloy was simulated with parameters from Table 1. Steady state patterns of cellular growth fingers are shown in Fig. 6. The figures plot a snapshot in time of the concentration field. The upper contour corresponds to the case of no solute trapping (i.e., following the SIM with k(V ) = k e ), while the the lower contour corresponds to the case of solute trapping with k(V ) from Eq. (17), and with full solute drag (D = 1) according to Eq. (2). In both cases, the thermal gradient was set to 400 000 K/m, and the pulling speed was 0.5 cm/s. The simulations were done in a co-moving reference frame, with periodic boundary conditions in the vertical direction. The systems size was set to 12 µm x 46 µm. We chose the pulling speed to be clearly smaller than the interface velocity where the transient 1D runs for Si-As converge in Fig. 5.
The data of Fig. 6 shows that for the two solidification conditions shown, including solute trapping (with full solute drag) leads to a thicker cell than for the case of no solute trapping. This can be motivated by the rough rule that microstructural length scale is inversely proportional to the material freezing  Table 1 in a thermal gradient of 400 000 K/m and pulling speed 0.5 cm/s. System size 12 µm x 46 µm in a co-moving reference frame. range [24]; when solute trapping is active, the freezing range decreases. It is also noted that since the thicker cell in lower contour of Figure 6 leaves less space to distribute the rejected solute in the liquid, this leads to a higher concentration levels in the remaining liquid.
The dendritic cell in the bottom simulation in Fig. 6 (which contains solute trapping) is seen to be growing side branches along the length of the trunk. This indicates that the cellular finger is becoming unstable, in contrast to the top frame of Fig. 6, which remains cellular throughout the simulation. This instability is consistent with the fact that in directional solidification solute trapping can decrease the velocity where the growth mode changes from cellular to dendritic [24].

Conclusion
We presented a methodology, based on asymptotic analysis, for conducting quantitative phase field simulations of an alloy with controllable solute partitioning (k(V )) and a control- The phase field results were extracted by measuring the instantaneous interface velocity and solid-and liquid-side concentrations from a transient concentration profile under a fixed dimensionless undercooling. Similar results were found when extracting these measured quantities from a steady state moving interface pulled by a thermal gradient at a constant speed.
The considered phase field model was mapped onto the CGM limit with a matched asymptotic analysis for a general class of phase field models. This asymptotic analysis can be readily implemented to non-dilute and multicomponent alloys by using low supersaturation limit of a grand potential model, which can directly use the sharp interface relations as presented in this paper.
The presented phase field model with controllable solute trapping and CGM kinetics can be used to create more accurate process-microstructure maps for rapid solidification in order to, for example, determine morphological transition between dendritic, cellular, and planar growth in directional solidification.
To properly model solute trapping and kinetic undercooling in simulations of industrially relevant applications, solute trapping measurements should be conducted for these respective alloys, for example for different grades of steels and nickel superalloys.