Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al – 4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography

Abstract

Dendritic equiaxed growth from the melt by continuous cooling-down is investigated by quantitative 2D phase-field simulations. The results are compared with detailed data from solidification experiments on Al–4 wt.% Cu alloy with in situ X-ray monitoring. In a first step, the simulation of an isolated equiaxed alloy dendrite growing freely in <100>-direction in the melt is performed. Then, the impingement between two grains is considered by simulating two dendritic crystals growing towards each other in <100>-direction. From the phase-field simulations, the time evolution of the equiaxed crystals is characterized by measuring the lengths and tip velocities of the primary dendrite arms in free growth and in the presence of neighbor interaction, which enables the analysis of growth dynamics. In a second part, the results of the phase-field simulations are compared to data extracted from an experiment on Al – 4 wt.% Cu alloys carried out at the European Synchrotron Radiation Facility (ESRF), with in situ and real-time characterization by means of X-ray radiography, and to analytical relationship for dendrite tip growth. The limitations of 2D-phase-field simulations to fully describe the dynamic formation and interaction of dendritic equiaxed grains are briefly discussed.

1. Introduction

Dendritic morphology is the most common feature observed in solidification processing of metallic alloys from the melt, and the as-cast grain structure can be either columnar or equiaxed depending of process conditions. In most cases, a dendritic equiaxed microstructure is preferred as it gives uniform and isotropic mechanical properties at macroscopic scale. A thorough understanding of dendritic equiaxed growth is thus of uppermost importance to optimize the casting processes and improve the quality of products. In recent years, the rapid development of numerical computer simulations of the solidification microstructures has enabled significant advances on the comprehension of the physical phenomena underlying solidification, especially regarding the dynamic formation of dendritic crystals in alloys.^1,2,3,4,5,6) However, quantitative validation of these simulations, necessary to discriminate and improve the numerical models, remains limited, mainly due to the lack of precise experimental data that can be used for reliable comparison. In the case of metallic alloys, experimental characterization of dendritic morphology has been conventionally performed by post-mortem analysis of etched sections of samples, which gives only a “frozen” picture of the solidification microstructure, with disturbing artifacts of quenching, coarsening or solid-state diffusion. Fortunately, monitoring of solidification process by synchrotron radiation X-ray radiography has currently reached high time and spatial resolutions, which enables investigating the dynamics of equiaxed growth in metallic alloys in situ and in real-time.^7,8,9) The precise data obtained from such in situ experiments, which enlighten the dynamic phenomena taking place during solidification, are timely to benchmark theoretical models of crystal growth. Accordingly, coupling numerical simulations and in situ experiments on metallic alloy solidification has become the method of choice to analyze growth microstructure formation. By clarifying pending issues in the evolution dynamics in experiment, this approach is a major driving force in the advancement of theoretical modeling.

Combining quantitative phase-field simulation with in situ observation during the directional solidification of dilute Al–Cu alloy, the initial transient planar solidification and the growth dynamics following morphological instability were analyzed thoroughly.^10,11) The effects of natural convection in the melt on the evolution of the microstructure morphology were evidenced by analyzing the differences between experiments on ground and simulations in the limit of diffusion transport. The thermosolutal convection effects were then confirmed by coupling the phase-field simulations with fluid flow dynamics in the liquid.¹²⁾ The present studies focus on simulations of dendritic equiaxed growth from the melt by continuously cooling down of an Al–4 wt.% Cu sample. First, the in situ and real-time experimental observation is analyzed to determine the initial input parameters and boundary conditions for the numerical simulations. Then, quantitative phase-field simulations for the conditions of the solidification experiment are performed. The simulated dendritic growth dynamics and morphology are compared with the experimental data. Finally, the discrepancies between phase-field simulation predictions and grain growth and interaction in experiment are discussed.

2. Experiment, Phase-field Model and Numerical Method

2.1. Experiment

The solidification experiment with in situ and real-time observation used to calibrate the phase-field simulations was carried out at ID19 beamline of the European Synchrotron Radiation Facility (ESRF) in Grenoble (France). The sheet-like thin sample (40 × 6 × 0.2 mm³) of Al–4 wt.% Cu alloy was put vertically in a Bridgman furnace. Two heating elements, placed on the top and bottom sides of the sample, were used to impose and control the sample temperature. During the experiment, the sample was first melted and then maintained to a temperature above the liquidus temperature, i.e. 920 K for Al–4 wt.% Cu alloy, for several hours to homogenize the solute concentration in liquid before initiating solidification. To achieve equiaxed dendritic solidification, the temperature of the two heating elements was precisely adjusted to achieve almost isothermal conditions, so that temperature gradient was eliminated throughout the sample. The power down method, used to solidify the molten alloy, was applied by switching the same cooling rate, R = 0.5 K/min, simultaneously on both heating elements of the furnace. The in situ and real-time observations were realized by setting the main surface of the sample perpendicular to the incident monochromatic X-ray beam (field-of-view of 15 mm × 6 mm). Detailed description of the solidification experiments can be found elsewhere.¹³⁾ A sequence of four recorded radiographs is shown in Fig. 1. Assuming uniform thickness, the image contrast is mainly related to the phase (solid or liquid), and in each phase to the solute content. In Fig. 1, the equiaxed grains, mostly composed of aluminum, appear in white in the radiographs while the Cu-enriched liquid is dark. The spatial resolution (pixel size is 7.46 μm × 7.46 μm) and the acquisition rate (typically 1 frame per second) are sufficient to measure the length of the dendrite arms on equiaxed grains as a function of time, and then characterize the growth dynamics. The reference time was taken at the beginning of the cooling phase.

Fig. 1.

Sequence of four radiographs recorded in situ by means of synchrotron X-ray radiography during equiaxed dendritic growth from undercooled melt of Al – 4 wt.% Cu alloy. Continuous cooling-down at rate R = 0.5 K/min. The equiaxed dendrite at the center with fourfold symmetry, selected for measurement of dendrite growth dynamics, is marked by a yellow circle. Image width = 5.04 mm.

As seen in Fig. 1, a great number of grains nucleated and grew from the melt during the continuous cooling. Due to the spatial constraint, the dendritic grains rapidly interacted with their neighbors. A thorough analysis of the equiaxed grain growth with the same experimental procedure but on Al–4 wt.% Cu alloys has been reported in a previous paper.¹³⁾ The main conclusion was that the evolution of the dendrite tip velocity with applied thermal undercooling showed two growth regimes: an accelerating regime corresponding to the period when the growing dendrite arm was isolated from neighboring grains and a decelerating regime where solutal interaction with the grains around had a dominant effect. Qualitatively, this conclusion remains valid for the case of Al – 4 wt.% Cu alloy presented in this paper. Therefore, in order to achieve a better understanding of crystal growth dynamics, two growth conditions are considered in following simulations: (1) a fixed (= no sedimentation) dendritic equiaxed grain, resembling that encircled in Fig. 1(c), which can be assumed to grow freely from the bulk liquid; (2) two dendritic grains which will gradually interact with each other.

2.2. Phase-field Model

The quantitative phase-field model proposed by Karma for directional solidification^14,15) is modified and adapted to simulate equiaxed dendrite growth from the melt by continuous cooling-down at rate R, where the relaxation time and the free energy density are functions of temperature and thus of the cooling rate in present model. Taking ψ = 1 and ψ = −1 for solid and liquid respectively, the governing equations are as follows:

• phase-field equation   

$$\begin{array}{l}\left[1+\frac{Rt}{\left|m\right|c_0}\right]{\eta }^2{\tau }_0\frac{\partial \psi }{\partial t}=\overrightarrow{\nabla }\cdot \left(W_0^2{\eta }^2\overrightarrow{\nabla }\psi \right)-\frac{\partial }{\partial x}\left(W_0^2\eta \eta \text{'}\frac{\partial \psi }{\partial y}\right)\\ +\frac{\partial }{\partial y}\left(W_0^2\eta \eta \text{'}\frac{\partial \psi }{\partial x}\right)+\left[\psi -\lambda \left(U-\frac{Rt}{\text{(}1-k\text{)}\left|m\right|c_0}\right)\left(1-{\psi }^2\right)\right]\left(1-{\psi }^2\right)\end{array}$$(1)

• solute conservation equation   

$$\begin{array}{r}\frac{\left[\left(1+k\right)-\left(1-k\right)\psi \right]}{2}\frac{\partial U}{\partial t}-\frac{1}{2}\left[1+\left(1-k\right)U\right]\frac{\partial \psi }{\partial t}\\ =\overrightarrow{\nabla }\cdot \left[Dq\text{(}\psi \text{)}\overrightarrow{\nabla }U-{\overrightarrow{j}}_{at}\right]\end{array}$$(2)

where x, y are the horizontal and vertical axes, respectively and c₀ the initial concentration of the alloy, W₀ the interface width, ${\overrightarrow{j}}_{at}=-\frac{1}{2\sqrt{2}}\left[1+\left(1-k\right)U\right]\frac{\partial \psi }{\partial t}\frac{\overrightarrow{\nabla }\psi }{\left|\overrightarrow{\nabla }\psi \right|}$ is the solute anti-trapping current. U is the dimensionless solute concentration, $U=\frac{2c/c_0-\left[1+k-\left(1-k\right)\psi \right]}{\text{(}1-k\text{)}\left[1+k-\left(1-k\right)\psi \right]}$ . The parameters λ, ξ and q(ψ) are set as λ = a₁ξ, $\xi \text{=}\frac{W_0}{d_0}$ , $q\left(\psi \right)=\frac{1}{2}\left[\left(1-\psi \right)+k\left(1+\psi \right)\frac{D_s}{D_l}\right]$ , where D_s and D_l are the solute diffusion coefficients in solid and liquid, respectively. The chemical capillary length is $d_0=\frac{\mathrm{\Gamma }}{\left|m\right|c_0\text{(}1-k\text{)}}$ with m the liquidus slope and k the solute partition coefficient, which are taken constant, Γ the Gibbs-Thomson coefficient. For simplicity, the usual choice of fourfold anisotropy of surface energy η, which is suitable for cubic primary-Al phase, is adopted in the phase-field model, η = 1 + ε₄cos4θ, η′ = dη/dθ where θ = arctan(ψ_y/ψ_x) is the angle between the normal to the interface pointing to the liquid and x the horizontal direction. The coefficient ε₄ is the fourfold anisotropy strength of the solid-liquid interfacial energy. As the growth rate of solidification front is not high, the interface kinetic effects are avoided in the model, so that the additional constraint is imposed in the solute conservation equation: D = a₁a₂ξW₀²/τ₀, a₁ = 0.8839 and a₂ = 0.6267 as derived by Karma and Rappel.¹⁶⁾ Here, it should be stressed that for the non-isothermal solidification of alloys the chemical capillary length d₀ is temperature dependent.¹³⁾ As the interface width and the dissipation time of the interface in the phase-field model, as well as the referred time and spatial units in simulations, are depending on the capillary length, such complexity of d₀, which would make the model very intricate and hardly be feasible in practical simulations, will nevertheless be neglected. As dendrite growth from the melt occurs in time-dependent conditions during continuous cooling down,¹³⁾ the solute concentration in liquid at the dendrite tip is not constant but much lower than the equilibrium concentration c₀/k at the solidus temperature, rather close to the initial concentration. Therefore, the capillary length is assumed constant at the nominal concentration of the alloy. Since heat diffusion is several orders of magnitude faster than solute diffusion, thermal diffusion is ignored here. Indeed, for metallic alloy solidification, it is long recognized that the influence of latent heat release is negligible and growth is controlled by solute diffusion. Although some fluid flow often exists in the melt in experiments on ground, it is not taken into account in the present study.

2.3. Numerical Implementation

The parameters used in this study are compiled in Table 1, the cooling rate R is set equal to that applied in the experiment. Using the finite element method, the governing equations for phase-field and mass conservation, Eqs. (1) and (2), were discretized over a whole computational domain that was divided into small triangular elements. To decrease computational time and increase accuracy, the adaptive procedure for grid discretization using the sequential program described in Refs. 10)¹⁰⁾ and 21)²¹⁾ was further developed for multi-processor parallel computing. The communication between processors is handled by the standard MPI message-passing procedures. In the calculations, the domain was first partitioned into N (the number of CPU cores used in simulations) subdomains based on the initial most coarse mesh. Then, each processor read the mesh data of its subdomain and built an isolated finite-element space. Afterwards, all the element degrees of freedom in each subdomain were indexed on the global domain. At the step of solving the partial differential equations, to build the sparse linear system of equations, Ax = f, the matrix A and the right hand side f were calculated first on a single element and then their values were added to the local matrix of a subdomain, and finally these matrices were assembled together on the whole domain among all employed processors. The linear equations were solved by a multigrid solver in HYPRE.²²⁾ After several time steps of calculation, the solid-liquid interface changed its position, so that mesh adaption was implemented on each processor to guarantee the computing accuracy. After mesh adaption, the number of degrees of freedom for calculating on a certain processor might be much larger or smaller than on other processors, so that the computing load had to be balanced on all the processors by decomposing the global domain again. The code was programmed based upon the AFEPack package²³⁾ with adaptive and parallel computing libraries.

Table 1. Physico-chemical parameters used in the numerical simulations of equiaxed dendritic solidification of Al–4 wt.% Cu alloy.

Solute partition coefficient, k	0.1417)
Liquidus temperature, Tl	647°C18)
Solute diffusion coefficient in solid, Ds	1.15 × 10–8 cm2/s19)
Solute diffusion coefficient in liquid, Dl	2.4 × 10–5 cm2/s19)
Gibbs-Thomson coefficient, Γ	2.36 × 10–5 cm°C19)
Liquidus slope, m	−3.5°C/wt.%
Cooling rate, R	0.5°C/min
Initial concentration, c0	4 wt.% Cu
Interface width parameter, ξ	5410)
Surface energy anisotropy strength, ε4	0.010620)

For single isolated free dendrite growth, to eliminate the influence of the domain boundaries on the solidification front, a large rectangular simulation domain 4000 × 7000 (4233 μm × 7408 μm) was chosen. At the beginning of simulation, a circular seed with a radius equal to 10.6 μm was placed at the center of the domain. For the growth of two interacting dendrites, two identical seeds with <100>-directions of preferred growth opposed to each other were attached at opposite sides of a rectangular domain 3000 × 6000 (3175 μm × 6350 μm). In the initial state, the value of the phase-field variable $\psi \text{(}\overrightarrow{x},0\text{)}=-tanh\text{(}\overrightarrow{x}/\sqrt{2}\text{)}$ along the normal to the interface and solute concentration was set equal to the alloy concentration c_l = c₀ in the liquid, identical to the initial uniform distribution of solute in experiment, and c_s = kc₀ in the solid, which corresponds to $U\text{(}\overrightarrow{x},0\text{)}=0$ in the whole system. No-flux Neumann boundary conditions were imposed for both the phase field and solute field. During simulations, the minimum dimensionless length of the adaptive mesh element was controlled with dx_min ≈ 0.25 by giving a very small adapting tolerance which determined the threshold for coarsening or refining a grid element. As an adaptive mesh was used in the calculations, the numbers of degrees of freedom for calculating at the beginning of the simulations was suitable for a small number of CPU cores, so that 24 CPU cores (Xeon(R), X5650, 2.67GHz) were employed for computing. With dendrite growth, the shape of the solid-liquid interface became more and more complicated and the degrees of freedom concomitantly increased to a larger amount, so that the simulations had to be distributed over a higher number of computational nodes, till 96 cores were used at the end of the simulations.

3. Results and Discussion

3.1. Characterization of the Simulated Dendritic-equiaxed Growth in Continuous Cooling-down of the Melt

The simulated growth morphology of the dendritic equiaxed grain and the two dendritic grains in interaction are shown in Fig. 2, in which iso-concentration lines are superimposed as well. For the isolated dendrite (Fig. 2(a)), the far-field liquid concentration at the upper boundaries remained equal to the initial concentration of the alloy, so that the dendrite arm with tip1 was growing freely, without feeling any interaction with domain boundary. For the growth of the two dendrites eventually in interaction (Fig. 2(b)), at the beginning of solidification the two grains were “far” away from each other, and the solute boundary layers ahead of the advancing dendrite tips were very thin and completely distinct. Therefore, the growth of each tip was free from the influence of the solute rejected by the other one. However, this free growth regime rapidly ceased, as soon as the solute fields ahead of each dendrite tip started to overlap due to the decrease of the distance between the two tips. Actually, after hundreds of seconds of solidification, the solute boundary layers along the normal direction of the tip interface were overlapping, yielding to a soft impingement without effective contact between grains.

Fig. 2.

Phase-field simulation of dendrite growth morphology in Al–4 wt.% Cu alloy solidifying from melt by continuous cooling-down at rate R = 0.5°C/min. Solute iso-concentrations in the liquid are superimposed, with 0.1 wt.% Cu interval between consecutive lines. (a) Single isolated equiaxed grain, the vertical dendrite arm, which grows freely, is labeled Tip1 for characterization; the solute concentration at the outmost line is 4.10 wt.% Cu. (b) Two equiaxed grains with dendrite arms growing face to face and becoming in interaction; the horizontal dendrite arm of the left grain is labeled Tip2 for characterization.

The main characteristic parameters of the dendritic equiaxed grain are the length of the primary arm, L_t, the tip velocity, V_i, and the tip radius, ρ_i, which evolved during the grain formation as a function of time, or equivalently applied undercooling ΔT as ΔT = Rt. The variation of the length of primary dendrite arm as a function of cooling time is plotted in Fig. 3 for both growth situations (with and without a neighbor grain ahead). During the initial stage (t < 440 s), the data points coincide and the arm lengths continuously increase with cooling-down of the melt, which indicates that the growth of both dendrite arms is identical and free during this period. After 440 seconds, the length of the isolated dendrite (Tip1) continues to increase while the other one (Tip2) is undergoing interaction so that its length reaches a plateau as a result of increasing solute poisoning by the opposite grain, which hinders its growth. These phase-field simulations are in agreement with the analysis performed by Bogno et al.¹³⁾ in the case of dendritic equiaxed growth of Al – 10 wt.% Cu alloys.

Fig. 3.

Variation of the primary-dendrite arm length with time in the phase-field simulations for free dendrite growth and growth with neighbor interaction.

The growth velocity is given by the slope of the curves giving the evolution of arm length with time in Fig. 3. As the applied thermal undercooling driving dendrite growth is proportional to time, the velocity evolution is plotted as a function of the thermal undercooling ΔT in Fig. 4. (like in Fig. 5 in Ref. 13). As already mentioned above for dendrite arm length, the variation of velocity for Tip1 and Tip2 is identical in early stage, as long as Tip2 stays in the accelerated growth regime. The two tips show different growth dynamics as soon as the solutal interaction between the two grains comes into play. The free dendrite, for which the solute concentration “far” away from its tip remains almost equal to the initial composition of the alloy, continues accelerated growth as a consequence of the continuous increase of undercooling ΔT. For the interacting dendrite, the Tip2 slows down as a result of the overlap of the solute boundary layers around the two dendrite tips. Indeed, solute pile-up in the region between the two tips leads to a decrease of the melt supercooling, even though the temperature is keeping decreasing with cooling down. This solutal impingement is at the origin of the decrease of the growth rate of Tip2 to almost zero at the end of simulation.

Fig. 4.

Evolution of the velocities of Tip1 and Tip2 with applied thermal undercooling for free dendrite growth and growth with neighbor interaction.

Fig. 5.

Variation of the tip radius as a function of the melt undercooling for the equiaxed dendrite arms Tip1 and Tip2.

Another important feature that characterizes the dendritic equiaxed growth morphology is the tip radius. The variation of the tip radius with the applied thermal undercooling is reported in Fig. 5 for Tip1 and Tip2. For both cases, the tip radius initially increases with the applied thermal undercooling, which is contrary to the behavior ρ_i²V_i = Ct as V_i is increasing (Fig. 4). Very likely, this should be linked to the fact that the dendrite is actually still a circular grain as the interfacial energy anisotropy has not yet effectively selected the preferred <100>-growth direction. About ΔT ≈ 0.744°C, the circular grain reaches its maximum radius (≈ 40 μm) when the circular solid-liquid interface rapidly bulges along the preferred growth direction, showing prominent anisotropy. Once more, the radius evolution does not show visible difference between the two different growth conditions before dendrite Tip2 starts to interact. Thereafter, dendrite Tip2 stops sharpening and coarsens as denoted by the increase of the tip radius, while the Tip1-radius continues decreasing, which is in good agreement with the predictions by theoretical model for free dendrite growth that the tip radius is a monotone decreasing function of undercooling.²⁴⁾ From the characterization of the simulated evolution dynamics in dendritic equiaxed growth, it can be concluded that the process of dendrite interaction follows a first stage of free growth, as long as the dendrite is “far” away from neighbors, and starts with the inception of solutal poisoning with the overlapping of the solute boundary layers in-between opposite grains. This agrees well with the experimental characterization of dendritic equiaxed growth interaction by means of in situ and real-time observation of Al–10 wt.% Cu alloy.¹³⁾

3.2. Comparison of the Phase-field Simulations with Data from in Situ Observation

During a real experiment, unlike dendrite growth in simulations, a great number of grains nucleate in the melt during the solidification process so that grain interaction inevitably exists after some time. When solidification is achieved on thin alloy samples, most equiaxed grains do not grow into a perfect shape with fourfold symmetry, as shown in Fig. 1, because of the reduced thickness necessary for in situ X-ray observation. Thus, direct comparison of the phase-field simulations with experimental measurements is rendered delicate. To circumvent this difficulty as much as possible, the dendrite arm length is firstly measured on equiaxed grains visibly growing with <100>-primary arms, such as the grain in a yellow circle in Fig. 1, and tip growth velocity is then obtained as in the numerical simulations. The measurements are shown in Fig. 6. The primary arm length defined here is following that in the Ref. 13),¹³⁾ and the error bar estimated to ±22 μm (i.e. ±3 pixels) for all measurements of dendrite arm length. As can be seen in the figure, the lengths of the four dendritic arms of the grain first increase before reaching a plateau, well before the end of solidification (Fig. 6(a)). Plotting the growth rates deduced from the evolution of the arm lengths shows a continuous decrease with time (Fig. 6(b)), which indicates that all dendrite arms had already started getting poisoned by the solute rejected from neighbor grains that results in the darker areas visible around the grains in radiography (Fig. 1). For this equiaxed grain in the radiographs, the accelerated growth stage is likely too short to be noticeable. Furthermore, the distance to neighbor grains is less than that in the phase-field simulation, so that quantitative comparison with simulation of the interaction dynamics would not be reasonable. Consequently, it is appropriate to select another grain, encircled by a yellow curve in Fig. 7(a), for comparison with numerical predictions. As can be seen in Fig. 1, the dendrite on the left can grow into an unobstructed space for a long time after its nucleation, which gives us the opportunity to analyze its initial stage of accelerated growth, free from interaction with surrounding grains.

Fig. 6.

Evolution with time of the dendrite arm lengths and tip velocities of the equiaxed grain encircled at the center of Fig. 1. (a) Dendrite arm length; (b) Tip velocity.

Fig. 7.

Selection of a primary dendrite in the experiment convenient for the comprehensive investigation of its growth dynamics, and comparison with phase-field simulation. (a) Morphology and position of the selected dendrite. In addition, the yellow arrow shows a dendrite whose trunk width decreases with arm lengthening (free growth) while the red arrow shows a dendrite whose trunk is widening towards the tip (neighbor interaction). (b) The tip velocity of the dendrite selected in (a) fitted to the velocity predicted by PP-model.^25,26) (c) Phase-field prediction for the dendrite tip undergoing interaction (Tip2).

The tip velocity of the selected grain is measured and plotted in Fig. 7(b), together with the predictions by Pelcé-Pomeau (PP) analytical model^25,26) in three dimensions, derived from Ivantsov solution. The growth rate of the simulated interacting dendrite is shown aside in Fig. 7(c). It should be noted that the measured growth velocity in experiment does not start from zero, due to the limited spatial resolution limitation of the X-ray radiography. Indeed, when the dendrite gets visible in the experiment, its diameter is already more than 100 μm. Since the melt was started to be cooled above the liquidus temperature, the thermal undercooling in the experiment was obtained following the procedure used in the Ref. 13)¹³⁾ based on the adjustment of free dendrite growth rate in experiment to the rigorous three-dimensional PP-model. In Fig. 7(b), the evolution of the tip velocity displays a trend analogous to that of the simulated dendritic growth under interaction (Fig. 7(c)), and the two growth regimes can in particular be distinguished. The accelerated growth regime, which follows PP-model, corresponds to a free dendrite. After the peak value, the tip velocity gradually decreases with the increase of the applied thermal undercooling due to solute poisoning of growth by surrounding grains. Nevertheless, quantitative comparison between simulation and experiment of the tip velocity during the free dendrite growth exhibits significant discrepancy. Namely, the dendrite growth velocity is underestimated in the numerical simulation for the cooling-down applied in experiment. This difference should be attributed to the fact that, as demonstrated in Ref. 13),¹³⁾ the experiment must be treated as three dimensional because the thickness of the sample is around 200 μm, which is much larger than the tip radius of dendrite, whereas the simulation is in two dimensions. According to the theoretical treatment of free dendritic growth in PP-model, the growth dynamic is typically two orders of magnitude slower in two dimensions.¹³⁾ The simulated tip growth velocity would thus be higher if the simulations were performed in three dimensions, probably in better quantitative agreement with experiment, as it has been evidenced by three-dimensional phase-field simulation of crystal growth of a pure substance from undercooled melt.^16,27) Furthermore, residual gravity-driven convection in the experiment might not be fully negligible, which would to some extent alter the solute field ahead of the dendrite in the melt and thus the tip growth velocity.

When comparing the simulated dendrite tip shape with experiment, comparison is not possible for the tip radius, due to insufficient contrast and spatial resolution in the radiographs for Al−4 wt.% Cu alloys. Anyway, qualitative agreements is found between phase-field simulation and experiment for dendrite arm length and tip velocity. In light of the evolution trends shown by the tip radii of the simulated free dendrite and interacting dendrite in Fig. 5, the tip radius of the free dendrite in experiment should decrease with time, and the tip becomes sharper and sharper with the cooling down of the melt. As the width of the dendrite trunk is known to scale with the tip radius, this is reflected in the shape of the dendrite trunk visible in the radiographs, as e.g. pointed out by the upper yellow arrow in Fig. 7(a). For interacting dendrite, the tip and the trunk conversely broaden significantly when the arm closely approaches another one as exemplified in the experiment by the lower red arrow.

4. Conclusions

First, the dynamics of dendrite growth from the melt during alloy solidification by continuous cooling-down observed in in situ experiments is qualitatively recovered in the phase-field simulations. As long as a dendrite is growing freely, far enough apart from neighbors, the growth velocity of its tip gradually increases with the increasing applied thermal undercooling as expected, in a way that mimics the accelerating growth regime of equiaxed crystals in the early stage of solidification in experiment. In contradistinction, for the case of two dendrites undergoing interaction, it is found that, after some time of free growth, the tip velocity reaches a maximum before gradually decreasing towards zero when face-to-face dendrite arms nearly touch each other. The slowing-down of dendrite growth due to solute poisoning in simulation exhibits the same evolution with time as that was monitored by synchrotron X-ray radiography. The evolution process of the tip shape in simulations agrees with experimental observation. Second, quantitative comparison between the present phase-field two-dimensional simulations with experiment of the free dendrite growth velocity reveals a large discrepancy, which may be primarily ascribed to the fact that the experiment is actually three-dimensional. In future work, the residual gravity-driven natural convection that exists in the experiment despite the reduced thickness of the alloy sample and possibly redistributes the solute around the dendritic envelope should be taken into account to assess its effects on dendrite tip dynamics. This holds also for the nucleation undercooling which may have some impact in the very early stage of dendrite growth. The ultimate objective will be to carry out quantitative three-dimensional phase-field simulation of multiple-grain growth in the limit of diffusion transport in the melt first, and then properly couple phase-field with hydrodynamics in liquid phase to realistically describe the intricate interaction of fluid flow with the solidification process, in order to quantitatively predict solidification microstructure and grain structure in practical situations encountered in materials casting, in particular when equiaxed growth is dominant.

Acknowledgements

This work is supported by the National Key Science & Technology Special Project, China (Grant No. 2011ZX04014- 052), the National Natural Science Foundation for Young Scientists of China (Grant No. 51001096) and the National Natural Science Foundation of China (Grant No. 51271184). The authors thank N. Mangelinck-Noël, Abdoul-Aziz Bogno and Xiu Hong Kang for discussions on Al–Cu experiments and phase-field simulations.

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