Uncoupling Growth Mechanisms of Binary Eutectics during Rapid

Article pubs.acs.org/JPCC

Uncoupling Growth Mechanisms of Binary Eutectics during Rapid Solidification Can Guo,† Jincheng Wang,*,† Junjie Li,† Zhijun Wang,† Sai Tang,‡ and Yunhao Huang† †

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf 40237, Germany

‡

ABSTRACT: Eutectic solidification under rapid solidification conditions has enormous applications, as it can produce microstructure-refined and interface-stable combined composite structures with low cost. However, investigations reported that the coupled interfaces will be destroyed under the condition of large undercoolings. For further understanding of the mechanisms of uncoupling growth, we investigated the uncoupling growth process of binary eutectics during rapid solidification using atomistic simulations. We find that both the nucleation rate and the crystallization velocity for the first phase of eutectics are very high; however, nucleation of the second phase is very inactive: its nucleation rate is low and nucleation incubation time is very long. As the nucleation of the eutectic second phase is severely suppressed during rapid solidifications, the crystallization of the second phase lags far behind, therefore we speculate that the uncoupling growth of eutectics during rapid solidification is nucleation-induced. the combined effects of these mechanisms.10 Recent experimental investigations have shown that the grain orientation of different phases in anomalous eutectics has notable characters: one is well oriented, whereas the other is randomly oriented within an anomalous colony. The kinetic mechanism cannot explain this phenomenon. The dendrite remelting mechanism seems to deal this problem; however, recalescence has significant influence on the microstructure in the undercooled melts, which somewhat inhibits the accurate understanding of the formation mechanism of anomalous eutectic.10,13,14 Recently, through adding laser scans on the regions where large Ni dendritic crystals exist, Lin15 found that the orientations of the remelted Ni particles are the same, which is in contradiction with the remelting mechanism. In all, although these mechanisms can explain some experimental results, the exact formation mechanism of anomalous eutectics is still unclear. Li et al.16,17 found copious nucleation phenomenon taking place in the undercooled melts and suggested that anomalous eutectic microstructures could be nucleation controlled. Through investigating the formation of anomalous eutectics by laser remelting, Lin et al.15 speculated that it is the external conditions (such as unmelted powder particles at the bottom of a molten pool) that results in copious nucleation and free growth, which leads to the formation of anomalous eutectics.

1. INTRODUCTION Eutectic solidification is a typical three-phase reaction process in which a liquid that is cooled to the eutectic temperature results in two solid phases occurring at the same time, eutectic temperature

Liquid ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ αsolid + βsolid , thus eutectic structures cooling

are formed from two phases that coexisting in chemical equilibrium and arranging in periodicity.1 Because alpha and beta phases are solidified through coupling growth, the phase boundaries are often well combined and chemically stable. It is precisely because eutectic solidification can produce interfacestable combined multilayer or fiber composite structures with low cost that more and more advanced structural2−4 and functional materials5−8 are manufactured through eutectic reaction. Moreover, as a typical nonequilibrium self-assembly process, eutectic solidification concerns synergistic nucleation and growth of multiple phases. Therefore, eutectic solidification has long been an important issue in mathematics, physics, metallurgy, and materials science. Rapid solidification techniques, which can produce significant microstructural modifications including refining grain size, enhancing solid solubility, and reducing defects, have attracted much attention and been widely used in recent years. However, some investigations reported that the coupled interface of eutectics will be destroyed under large undercooling conditions.9−11 This kind of uncoupling eutectics is usually called anomalous eutectics. Some suggested forming mechanisms of anomalous eutectics are presented, including kinetics-induced uncoupling growth mechanism,12 remelting mechanism,13 or © XXXX American Chemical Society

Received: February 9, 2017 Revised: March 22, 2017 Published: March 28, 2017 A

DOI: 10.1021/acs.jpcc.7b01311 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. Temporal evolution of nucleation and subsequent solidification processes of the first phase with different initial compositions: (a) c = 0.6, (b) c = 0.65, (c) c = 0.7. Blue and green atoms represent phases with square and triangle symmetry, respectively. (d−f) Temporal evolution of atom numbers for different phases with different initial compositions, (d) c = 0.6, (e) c = 0.65, (f) c = 0.7.

efficiency22 than molecular dynamics (MD) and density functional theory (DFT). Herein, we will use the binary XPFC model23 to study the nucleation and the growth process of hypereutectic alloys. The objective of this work is to study the atomistic nucleation pathways and kinetic process of anomalous eutectics, so as to gain a further understanding of the uncoupling growth mechanism under rapid solidification conditions.

Unfortunately, they failed to clarify the mechanisms of how the nucleation behaviors influence the formation of anomalous eutectics due to the difficulty in relating the final microstructures with the original nucleation behaviors directly. As nucleation can significantly influence the solidification process, nucleation behavior and its effect on the uncoupling growth of anomalous eutectics should be further investigated. Eutectic nucleation and the subsequent microstructure evolution processes have been studied using various experimental methods, such as X-ray diffraction, TEM, and colloidal experiments, but the lack of in situ observations at atomistic scales makes it hard to clarify the kinetic pathway of eutectic nucleation. It is still unclear whether the nucleation of the second phase is by epitaxial growth or not, and how the nucleation influences the uncoupling growth of anomalous eutectics. Since directly observing the nucleation process in real atom systems with experimental methods is still very difficult or impossible for alloy systems, numerical methods have aroused widespread concerns. As an alternative numerical model for nucleation issues,18−21 the phase field crystal (PFC) model not only can describe phenomena happening on atomic length scales, but also has the advantage of higher computational

2. MODEL AND PARAMETERS The PFC model is a simplified classical density functional theory, in which the dimensionless free energy functional for binary alloys can be written as23 F=

∫ dr{ψ 2/2 − ηψ 3/6 + χψ 4/12 + (ψ + 1)Fmix(c) −ψ

∫ dr′Ceffψ ψ ′(|r − r′|)/2 + α2 |∇⃗c|2 }

(1)

where ψ is the dimensionless atom number density field, c is the concentration field, and η, χ, and α are fitting parameters relate to real material properties. The entropy of mixing Fmix(c) is B


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results.10,13,14 By comparing the morphologies of β-rich phases between the initial stage (104dt) and the final stage (25 × 104dt), one can find that the difference between the final microstructure and the original nuclei distribution pattern is quite distinct. The crystal orientation is completely random, and the grain size is small in the initial stage, but due to the effect of coarsening and phase transition, the crystal orientation is more orderly in the final stage. Moreover, from Figure 1, we find that only one α-square crystal can exist in each closed melt region finally, and the orientations for α-square are different in one anomalous eutectic colony because these grains are crystallized from individual nuclei. In experimental investigations, the orientations for the second phase (Ni3Sn) are the same for each anomalous eutectic colony, which is mainly because the discrete particles are often crystallized from one closed melt region27 (the discrete regions are often connected in the third dimension). From this point of view, our results are still consistent with experimental observations. To figure out the solidification processes clearly, we tracked the temporal evolution of the atom numbers of different phases. The results for c = 0.6, 0.65, and 0.7 are shown in Figure 1d−f, respectively. It is clear that, for all our simulations, the solidification sequence is β-triangle, β-square, and α-square, the solidification velocities (the slope of “atom number”-time curve) of the β-rich phase are much larger than that of αsquare, and the nucleation incubation time for β-rich phases is far less than that for α-square, which are consistent with the atomic configurations, as shown in Figure 1a−c. Further, we find that the solidification processes are composition-related: the lifetime and stability of the metastable β-triangle phase decrease rapidly with the increase of initial composition c, while both the phase transition velocity from β-triangle to β-square and the solidification velocity of β-square increase with the increase of c. Moreover, we find that the incubation time of nucleation for β-square decreases with c, while that for α-square increases rapidly with c. The results indicate that the nucleation of the two phases are out of sync, which further leads to the growth of two uncoupled eutectic phases. Therefore, we speculate that the uncoupled growth of anomalous eutectics is nucleation-induced. For further understanding the uncoupling growth mechanism, the nucleation process will be examined in the following parts. Classical nucleation theory (CNT) suggested that the structure and concentration or density of the newly formed nuclei should be the same as those of final crystals. However, this proposition faces challenges as mounting experimental and simulation investigations indicated that crystallizations often pass through some intermediate states.28−30 The above results indicate that the crystallization of β-square passed through metastable triangle phases, and the solidification kinetic pathway is liquid−triangle−square. This kind of multistep crystallization process has been widely investigated for onecomponent materials,26 which can be explained by the step-rule of Ostwald. Multistep nucleation can reduce the nucleation energy barrier26 and enlarge the nucleation rate,31 thus the nucleation incubation time32 is less, and the nuclei number is larger for β-rich phases. Figure 1a−c shows that the nucleation process can also be influenced by the initial composition. For c = 0.6, as shown in Figure 1a, β-triangle clusters appeared first, and then they transformed into β-triangle nuclei and grow up quickly. After that, β-squares nucleated at the grain boundary of β-triangles after some relaxation-time, and finally all β-triangles transformed into β-squares. When c is increased to 0.7, as

Fmix(c) = ω{c ln(c /c0) + (1 − c) ln((1 − c)/1 − c0)} (2)

where the coefficient ω is introduced to fit the entropic energy away from the reference composition c0. For binary alloys, Greenwood et al.23 introduced an effective correlation function as ψ Ceff = X1(c)C2αα + X 2(c)C2ββ

(3)

where X1(c) = 1 − 3c2 + 2c3 and X2(c) = 1 − 3(1 − c)2 + 2(1 − c)3 are interpolation functions, and α and β are the marks of ββ atom species. Cαα 2 and C2 are the correlation kernels for pure α and β respectively, which can be written as24 C2* * =

2

2

2

∑ e−σ /σ e−(k− k ) /2ξi2 i

i

i

(4)

In this work, two-mode correlation kernels for α and β are adopted. The modes (i) and the parameter selection in eq 4 are discussed in ref 25. The equations of motion for ψ and c are ∂ψ /∂t = ∇{M ψ ∇[ψ − ηψ 2/2 + χψ 3/3 + Fmix −

∫ Ceffψ ψ ′]} + ξψ ,

∂c /∂t = ∇{Mc∇[(ψ + 1)δFmix /δc − α∇2 c −ψ

∫ (δCeffψ /δc)ψ ′/2]} + ξc

(5)

where M is the dimensionless kinetic mobility parameter for density field or solute field (here, M = 1), and ξ is the stochastic noise term. In this work, noise with a wavelength shorter than the interatomic spacing is filtered.21 Equation 5 is solved by using a semi-implicit Fourier spectral method with the grid space dx = π/4 and time step dt = 1.5. Our simulations are performed in a square domain of Lx × Ly = 1024Δx × 1024Δx with periodic boundary conditions, and nucleation in undercooled melts is induced by stochastic noises. The simulation parameters (σ, c) in this work are chosen based on the binary alloy phase diagram in ref 23. In the following simulations, we set σ = 0, and c = 0.6−0.8. It should also be noted that, in this work, the phase with composition less than 0.5 is called the αrich phase, while that with composition larger than 0.5 is called the β-rich phase. Moreover, all the parameters in this model are dimensionless, therefore the simulation results do not correspond to any specific material, and the solidification mechanisms revealed by this model are universal.

3. RESULTS AND DISCUSSION Figure 1 shows the snapshots of atomic configurations of eutectic solidification with different initial compositions c = 0.6 (Figure 1a), 0.65 (Figure 1b), and 0.7 (Figure 1c) under relative large undercoolings. Five different phases are observed during the solidification process: liquid phase (marked in white in Figure 1a−c), α-square (red), α-triangle (yellow) phases, βtriangle (green) and β-square (blue) (herein, the triangle phase is metastable). The existence of the metastable phases can be explained by the step rule of Ostwald, which is discussed in reference 26 in detail. According to Figure 1a−c, we found that the metastable β-triangle is the primary phase to crystallize, but it transforms into stable β-square quickly. Moreover, the nucleation and growth of the eutectic second phase (α-square) lags far behind the first phase (β-square), and the eutectic growth is uncoupled. It is clear that the grain orientation of β phases is random, which is consistent with experiment C


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Figure 2. (a) Temporal evolution of atom number density field (ψ) for one arbitrary nucleus, as shown in Figure 1. (b) Evolution of concentration field around the selected nucleus in panel a.

Figure 3. Nucleation pathways of the second phase: (a) heterogeneous nucleation for c = 0.6, (b) homogeneous nucleation for c = 0.7. For panel a, from left to right, t = 33000dt, 47000dt, 50000dt and 55000dt, respectively. While for Figure 3b, t = 125000dt, 125500dt, 126000dt, 141000dt. Blue, green, yellow, and red atoms correspond to β-square, β-triangle, α-triangle, and α-square phases, respectively.

an important role in the coupling growth of eutectics. Therefore, it is necessary to investigate the nucleation process of the second phase in-depth. The atomistic nucleation pathways of α-squares with different initial compositions are shown in Figure 3, which are the local enlarged results in Figure 1. It is clearly shown that the nucleation of α-squares follows two different pathways: heterogeneous nucleation for c = 0.6 (Figure 3a), and homogeneous nucleation for c = 0.7 (Figure 3b). According to Figure 3a, the formation of the α-rich phase (for both square and triangle phases) is a typical island epitaxial growth. As the lattice constant of the new phase changes with time because the concentration of new nucleus varies with time, the epitaxial growth process is quite complicated. The changing of lattice constant complies with Vegard’s law.33 As a result, the lattice mismatch between α-squares and β-squares increases with time, which leads to the change of crystal orientation of the eutectic second phase and the separation of the two phases. Figure 3b shows one homogeneous nucleation process of the αsquare phase, where the new phase is formed inside the melts with random crystal orientations. Figure 3 shows that the nucleation pathway of the second phase is composition-related, which has rarely been reported in the solidification of eutectic alloys. Here, we also calculated the number fraction (Nhomo/(Nhomo+Nheter)) of nuclei obtained from homogeneous nucleation under different initial composition conditions, and the results are shown in Figure 4a. It is

shown in Figure 1c, precursors are mixtures of triangle and square atoms in the initial stage of nucleation (500dt), and square-like clusters transform into β-square nuclei directly. This is because, with the increase of c, the crystallization driving force of the stable β-square phase is increased, then nucleation passing through the triangle phase will be energetically unfavorable.26 Figure 2a shows the temporal evolution of atom density field (ψ) for one arbitrary nucleus in Figure 1; here each red peak represents one solid atom. Figure 2b shows the evolution of concentration field around the nucleus shown in Figure 2a. Comparing the results in Figure 2a,b, we find that the formation of local ordered structures is instantaneous, while the concentration of nucleus does not reach its equilibrium value (ceq = 0.88) at the same time. Only after a period of relaxation does the concentration reach its equilibrium value. This indicates that the time scale of solute diffusion is much larger than that of the rearrangement or ordering of local atoms. So it should be the fluctuation of local ordered structures that triggers the nucleation process. Russo29 also concluded that it is the fluctuations of short-range orders rather than density that triggers the nucleation process in water freezing. Therefore, for nucleation of multicomponent materials, both the structure transition and concentration evolution of the nuclei may pass through multiple stages. As binary eutectic nucleation concerns synergistic crystallization of two phases, the nucleation of the second phase plays D


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Figure 4. (a) Number fraction of the nuclei with homogeneous nucleation type under different initial compositions. (b) Temporal evolution of concentration field around the nuclei as shown in Figure 3a and Figure 3b, where the black curves are for heterogeneous nucleation, and the red curves are for homogeneous nucleation.

Figure 5. Gibbs-enthalpy changes as a result of the introduction of a nucleus for homogeneous nucleation (red) and heterogeneous nucleation (black) with different initial compositions: (a) c = 0.6, (b) c = 0.7.

interface is about 106−8m−1. Therefore, as the presence of concentration gradient, the nucleation barrier of the second phase could be largely increased, which then decreases the nucleation rate of the second phase. Further, the dimensionless Gibbs free energies corresponding to different nucleation pathways in Figure 3 are calculated semiquantitatively, and the results are shown in Figure 5. For c = 0.6 (Figure 5a), the energy barrier for heterogeneous nucleation is lower than that of homogeneous nucleation, thus heterogeneous nucleation is more energy-advantaged and with higher probability. While for c = 0.7 (Figure 5b), the concentration gradient is further increased, the energy barrier will increase monotonically with the size of nucleus for heterogeneous nucleation. Therefore, heterogeneous nucleation will be suppressed by the sharper concentration gradient and only homogeneous nucleation could be practical, which is consistent with the results shown in Figure 3b and Figure 4a. The nucleation pathway selection of the second phase is significantly affected by the magnitude of concentration gradient, because the energy barrier changes with concentration gradient. Ibrahim et al.35 also calculated the change of Gibbs-enthalpy by introducing a silicide nucleus in sharp, medium, and broad interfaces in experimental investigations. Their calculations about the effect of concentration gradient on Gibbs energy during nucleation are consistent with ours in Figure 5. It should also be noted that, although the ∇c is small in the initial stage of solidification according to Figure 1d, heterogeneous nucleation of the second phase is still very hard. This is because rapid growth of the first phase leads to the fast movement of the interface, which suppressed the

shown that, with the increase of c, the probability of homogeneous nucleation is increased. For c ≥ 0.7, only homogeneous nucleation is observed; for c ≤ 0.64, only heterogeneous nucleation is observed; while for 0.64 < c < 0.7, both nucleation types can be found. Figure 4b shows the temporal evolution process of concentration field in the nucleation region, where the black curves correspond to heterogeneous nucleation in Figure 3a, while the red ones correspond to homogeneous nucleation in Figure 3b. According to Figure 4b, with the formation of α-square crystals, the concentration of the new nucleus decreases continually, until its value reaches the equilibrium composition (ceq = 0.1). The lag of concentration field evolution relative to the symmetry transformation during the second phase nucleation is similar to that during the nucleation of the first phase. This indicates that, symmetry evolution can stimulate nucleation process, which may be generalized for alloy systems under unequilibrium solidification conditions. To reveal the pathway selection mechanism of α-square and the reason why the nucleation of α-square is so inactive and lagged far behind that of β-square, the energy barriers of nucleation for the eutectic second phase are calculated. The Gibbs energy for nucleation with a concentration gradient is34 ΔG(r) = ΔGCNT + γ(∇c)2r5, where ΔGCNT is the Gibbs energy defined in CNT. The second term is proportional to the fifth order of cluster size r and concentration gradient squared (γ is positive), which means that the gradient term will dominate the Gibbs energy at a sufficiently large ∇c. The solid lines (at t = t1) in Figure 4b show the distribution of concentration field before nucleation, where the magnitude of ∇c near the solid/liquid E



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nucleation of new phase.32 Aaronson36 provided a criterion for the nucleation on a moving interface, i.e., the interface moving velocity must be such that θν ≤ a, where θ is the incubation time of nucleation, v is the interface moving velocity, and a is the lattice parameter. In our simulations, the value of θv is about one magnitude larger than a in the initial stage of solidification, thus the nucleation of the second phase by epitaxial growth can hardly happen. In the presence of concentration gradient and moving interface, the nucleation of α-square is suppressed, and then the lagged nucleation of the second phase directly leads to the uncoupling growth of anomalous eutectics. From the above discussions, two physical parameters can affect the nucleation mode of the second phase, solid\liquid interface migration velocity and the magnitude of concentration gradient. Therefore, any factors that influence the two physical parameters could affect the width of the transition region from heterogeneous nucleation to homogeneous nucleation in Figure 4a. As to the reasons of why the nuclei number of α-square is much less than that of the β-rich phases, one is the existence of the concentration gradient, which increased the nucleation barrier and suppressed the nucleation behavior of the α-square, as discussed above. Another reason is that, although there is a range of nucleation sites (grain faces, edges, and corners) for heterogeneous nucleation, in general, nucleation tends to occur at the corner according to CNT.37 The number of the corner sites is limited, which can also lead to the reduction of nucleation rate. Moreover, once one α-square nucleates at one site, with the growth of this nucleus, the solute depletion around the nucleation site will reduce the local supercooling of the surrounding area, which can further restrained the nucleation on the other sites. Aizenberg38 reported similar nucleation behavior in the crystallization of calcite. Therefore, the effects of large nucleation barrier, less nucleation site, and the competition for solute together result in a smaller grain number of α-square.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Jincheng Wang: 0000-0003-3910-1020 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science foundation of China (Grant No. 51571165, 51371151), and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase). We also thank the Center for High Performance Computing of Northwestern Polytechnical University, China, for computer time and facilities.

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4. CONCLUSION In conclusion, the nucleation pathway and uncoupling growth mechanism of hypereutectics during rapid solidifications are investigated by atomistic simulations. We found that the nucleation for the first phase passes through multiple intermediate stages for large undercoolings: liquid-trianglesquare for 2D simulations. The nucleation pathway for the second phase is composition-related: homogeneous nucleation for compositions deviating far from the eutectic composition; epitaxial growth (heterogeneous nucleation) for compositions near the eutectic composition. As the intermediate phases can stimulate the nucleation process effectively, both the nucleation rate and the crystallization velocity are very high for the first phase. Compared with the first phase, however, the nucleation of the second phase is very inactive, and the nuclei number is much less. We found two reasons that lead to the inactivity of the second phase nucleation. One reason is that the large amount of nucleation and rapid solidification of the first phase lead to sharp concentration gradients and a fast moving solid \liquid interface, which suppressed the nucleation of the second phase. The other is that the competition of solute can also decrease the nucleation rate. From the analysis of the nucleation behaviors of different phases, we suggest that because the nucleation of the second phase is suppressed severely under large undercoolings, it leads to the uncoupling growth of anomalous eutectic. F


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Uncoupling Growth Mechanisms of Binary Eutectics during Rapid

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