Crystal nucleation and growth processes in Cu-rich glass-forming Cu–Zr alloys

The origin of high nucleation and growth rates of crystals¹ in undercooled liquid pure metals, glasses,^2,3 and some compounds⁴ has been the subject of intensive studies for decades.^5,6 Despite that, a comprehensive theory of the glass formation process has not yet been established and remains an intriguing problem for materials science.⁷ Several simulation methods, including phase-field and Monte Carlo modeling, have been applied.^8,9 From the atomic viewpoint, molecular-dynamics (MD) simulations with various classical potentials have been used for studying the glass-transition process,^10,11 including evaluation of glass-forming ability.¹² It was also used to study crystallization of different substances,^13–15 including metals,^16,17 metallic thin films,¹⁸ and alloys.^19,20 In particular, crystallization of face-centered cubic (FCC) metals^21,22 and pure Cu was studied by MD simulations before.^23,24 It is also known that the addition of atoms, especially those immiscible in FCC-Cu, reduces the crystal nucleation and growth rates owing to solute partitioning, finally leading to formation of metallic glasses^25,26 and bulk metallic glasses,^27,28 even in binary Cu–Zr alloys.^29,30 As it was shown in a recent study,³¹ nucleation of monoatomic metallic crystals occurs in the regions with low degree of fivefold symmetry while translational and orientational ordering in nuclei takes place simultaneously.

Cu–Zr liquidus and metallic glasses are probably the most widely studied binary glass-forming substances by MD. The formation of Cu–Zr alloy system glasses^32,33 and their crystallization behavior^34,35 have been successfully modeled in the past. It was also suggested that high liquid–crystal interface energy is responsible for the high glass-forming ability of these alloys.^36,37 Glass-formation³⁸ and crystallization³⁹ of pure Zr was also studied.

It was also assumed that these melts show significant icosahedral short-range order,^40,41 especially around Cu atoms,⁴² but this fact was questioned in another work.⁴³ Entropy-driven docosahedral local structures were also suggested to be dominant in monatomic metallic liquids and glasses.⁴⁴ The evolution of clustered structures in Zr–Cu glass-forming alloys upon vitrification was studied in detail by MD simulation.⁴⁵ It is also found that the degree of short-range ordering in liquid Cu–Zr alloys is smaller than that in other bulk metallic glass formers.⁴⁶ Owing to a relatively weak Cu–Zr atomic interaction, the structure of Zr–Cu binary metallic glasses is found to be nearly an ideal disordered solid solution of Cu and Zr atoms without strong chemical ordering.⁴⁷ The formation of some ideally packed atomic clusters was also observed in these alloys.⁴⁸ The electronic structure of binary Cu–Zr metallic glasses was also studied in detail using x-ray absorption fine structure spectroscopy supported by MD simulation.⁴⁹ In addition, density functional theory calculations revealed that d-electrons of Zr dominate the states close to the Fermi level and they are responsible for the atomic shell to shell interactions.⁵⁰ 

Cu-rich compositions do not have high glass-forming ability but form high-strength crystalline alloys. Drawn wires of Cu-rich Cu–Zr binary alloys, such as Cu₉₅Zr₅ one, exhibit a high strength of 2200 MPa and good electrical conductivity.^51,52 These high values of strength are obtained owing to the fibrous hypoeutectic structure consisting of equilibrium pure FCC-Cu and Cu₉Zr₂ phases.⁵³ Compared to nearly equiatomic Cu–Zr alloys that have a high glass-forming ability (GFA) and can hardly be crystallized by homogeneous nucleation of crystals in MD simulation, Cu-rich alloys have a much lower GFA. Thus, the characteristic crystal nucleation times in these alloys are reasonably short enough to be studied by MD simulations. In addition, according to the Cu–Zr phase diagram, FCC Cu is a primarily forming equilibrium phase in Cu–Zr alloys up to 8.6 at. % of Cu. It can also presumably be formed in the alloys containing a higher amount of Zr under non-equilibrium solidification at a high cooling rate.

Thus, glass-transition and crystallization phenomena so far have mostly been modeled in nearly equiatomic Cu–Zr alloys, while less results are available for the solute/solvent-rich alloys. As mentioned earlier, in the present work, we study the glass-transition and its competing process: crystallization including homogeneous crystal nucleation and crystal growth kinetics. In order to study the crystal growth rate in a wide compositional range, an overcritical size nucleus of FCC Cu was introduced in the atomic liquid cell cooled down from 2500 K to 773 and 900 K.

Molecular dynamics simulation was performed using a software package for classical molecular dynamics (LAMMPS)⁵⁴ with a graphic processing unit acceleration⁵⁵ with an embedded atom method potential for Cu–Zr systems developed by Mendelev et al.⁵⁶ Simulations were performed under periodic boundary conditions with a time step of 1 fs. The temperature and pressure were controlled using Nose–Hoover style equations of motions as introduced by Shinoda et al.^57,58 A software package “OVITO”⁵⁹ was used to visualize and analyze the simulation results. Adaptive common neighbor analysis⁶⁰ was used to analyze the obtained atomic structure and identify crystalline regions.

The glass-transition process was studied, starting from a cubic box containing FCC atomic structures of Cu_100−xZr_x (x = 0–10 at. %) consisting of 100 000 atoms. Throughout this paper, atomic percentages are used. The 100 000-atom initial structures were obtained by starting by a perfect 108 000-atom FCC structure and removing 8000 atoms. During the simulation, the system was heated and melted at 2500 K for 80 ps, then cooled down to 300 K at different cooling rates.

FCC atomic arrangements of Cu_100−xZr_x (x = 0–4 at. %) consisting of 100 000 atoms contained in a cubic periodic cell were heated up and melted at 2500 K for 80 ps, then cooled down at 10¹² K/s to 800 K, and kept at this temperature until crystallization is completed. The procedure was repeated for 50 times for Cu₉₉Zr₁ and Cu₉₈Zr₂ as well as for 20 times for Cu₉₇Zr₃ and Cu₉₆Zr₄ alloys for statistical analysis. The beginning of crystallization was detected by a sudden cell volume decrease of 2 nm³. As the incubation time quickly increases with the Zr fraction, these calculations were not performed beyond x = 4 at. %. Instead, a crystal nucleus was inserted, as explained below.

FCC atomic arrangements of Cu_100−xZr_x (x = 0–10 at. %) consisting of 100 000 atoms (or 1 000 000 for Cu₉₁Zr₉ and Cu₉₇Zr₃ to visualize stacking faults) contained in a cubic periodic cell were heated up and melted at 2500 K for 80 ps and then cooled down at a constant cooling rate of 10¹² K/s to 773 and 900 K. The initial structures were obtained by removing 8000 (or 24000) atoms from an original cell consisting of 108 000 (or 1024000) atoms. Then, a spherical nucleus of FCC Cu of 1.5 nm radius was introduced in the atomic box consisting of a matrix supercooled liquid phase. The whole system is kept at either 773 or 900 K. With a cooling rate of 5 · 10¹³ K/s, depending on the estimation method, glassy Cu was formed earlier at about 720–780 K of the glass-transition range.⁶¹ Thus, 900 K clearly corresponds to the supercooled liquid, while 773 K is within the glass-transition region.

The details of the procedure are given as follows:

1. Heating from 300 to 2,473 K (773 K case)/2500 K (900 K case) in 20 ps.

2. Melting at 2473 K (773 K case)/2500 K (900 K case) for 80 ps.

3. Cooling down to 773 or 900 K with a cooling rate of 10¹² K/s.

4. Relaxation for 20 ps at 773 or 900 K.

5. Introduction of an FCC nucleus and annealing for some time, as shown in Table I.

The diffusion coefficient was measured for 100 000 atom configuration in the following way:

1. Heating from 300 to 2500 K (900 K case) or 2473 K (773 K case) in 20 ps.

2. Relaxation at 2500 or 2473 K for 80 ps.

3. Cooling down to 773 or 900 K at the cooling rate of 10¹² K/s.

4. Relaxation for 20 ps at 773 K or 900 K.

5. Annealing for 5 000 000 steps (5 ns).

6. Measuring mean squared displacements (MSDs) in the last 1 ns of annealing.

The self-diffusion coefficient (D) was estimated by calculating the slope of the MSD with respect to time using the expression⁶² 

where r_i are the atomic coordinates of the ith atom as a function of time (t).

Figure 1 shows (a) the density and potential energy (E) with respect to temperature, (b) a pair distribution function [PDF(R), also called g(R) in the literature] at 300 K, and (c) the PDF_min/PDF_max ratio^63,64 of the Cu₉₀Zr₁₀ alloy as a function of temperature (T). Two changes of the slope for E and PDF_min/PDF_max profiles can be observed, one at low temperature that corresponds to the glass-transition temperature (⁠$Tg*$⁠, where the symbol * indicates that the temperature corresponds to a relatively high cooling rate) and one at high-temperature (about 1500 K, yet below the equilibrium liquidus temperature of 1936 K). The change of slope in E at high temperature been described recently.⁶⁵ $Tg*$ can be determined using the computer program R⁶⁶ with the Akaike criterion⁶⁷ for segmented regression⁶⁸ as the intersection point of two fitting lines from the energy, density, or the PDF_min/PDF_max ratio curve. Figure 2(a) shows the evolution of $Tg*$ with respect to the Zr content, derived from the PDF_min/PDF_max ratio. From the energy plot in Fig. 1(a), $Tg*$ is estimated at 745 K, while a value of 740 K is found from the plot in Fig. 1(c). This value is close to that (700 K) obtained at a cooling rate of 5 · 10¹³ K/s from the slope dV/dT for a Cu₄₆Zr₅₄ bulk metallic glass former.^32,69

The glass transition temperature calculated from the PDF_min/PDF_max ratio shows little dependence on the atomic fraction of Zr and can be considered as nearly constant up to 10 at. %, as illustrated in Fig. 2(a). The value for pure Cu was obtained at the cooling rate of 10¹³ K/s because it partly crystallizes at 10¹³ K/s. $Tg*$ of pure Cu taken from Ref. 61 calculated in the same way is also shown for comparison. The critical cooling rate is shown as a function of Zr content in Fig. 2(b). Its logarithm shows a nearly linear dependence with respect to the Zr content. For the alloys containing more than 5 at. % of Zr, such simulation becomes too time consuming, and thus, the values were estimated by extrapolation. For the Cu₉₀Zr₁₀ alloy, the critical cooling rate is estimated to be about 10⁸ K/s. This is in line with the fact that no metallic glasses were reported to form by melt spinning for this composition.

It is known that T_g obtained at a high heating rate is higher than that obtained at a lower one.^70,71 Similarly, T_g* simulated at a high cooling rate for the Cu–Zr alloys is significantly higher than that obtained at a low cooling rate of 1 K/s at which 1000/T_g is estimated at 1.99 and T_g = 502 K, as shown in Fig. 2(c). The fitting is done using an Arrhenius-type equation,

as used in Ref. 72, where β is the cooling rate. The slope of the fitting line in Fig. 2(c) (natural logarithm) of β after inversion of the X and Y axes is −0.021. This value is close to those obtained for real glasses.⁷² As can be verified from the experimental Cu–Zr phase diagram,⁷³ the liquidus temperature (T_l) of the Cu₉₀Zr₁₀ alloy is about 1273 K. Then, the approximate T_rg* = T_g/T_l is 0.39, which is quite low. This finding is also in line with the fact that no metallic glasses were reported in this alloy even by melt spinning, though Cu–Zr thin films with an amorphous structure were prepared at Cu content up to 88 at. %.⁷⁴ 

Homogeneous nucleation was tested for Cu_100−xZr_x alloys (x = 0–4 at. %). Each atomic configuration consists of 100 000 atoms and was annealed at 800 K. 800 K corresponds to nearly 0.6T_l of pure Cu (T_l = 1358 K) at which the nose of the time–temperature-transformation diagram was found⁶¹ and the incubation period is short. The procedure was repeated for 50 times for Cu₉₉Zr₁ and Cu₉₈Zr₂ alloys as well as for 30 times for Cu₉₇Zr₃ and 20 times for Cu₉₆Zr₄. Figure 3(a) illustrates the volume change in 50 independent runs of Cu₉₉Zr₁ alloys. For each configuration, the incubation time was calculated, and a frequency distribution plot is shown in Fig. 3(b) and used to perform a statistical analysis. For the Cu₉₉Zr₁ alloy, several normality tests (Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors, Anderson–Darling, D’Agostino’s K-Squared, and Chen–Shapiro) indicated that the incubation time follows a normal statistical distribution at the alpha level of 0.05. For Cu₉₆Zr₄ alloys, all of the normality tests also confirmed normal distribution, while for Cu₉₈Zr₂ and Cu₉₇Zr₃ alloys, some tests did not confirm normal distribution.

As can be seen in Fig. 3(c), the incubation period (nucleation time) increases by 1.5 orders of magnitude from 1 to 4 at. % Zr. It correlates well with a decrease in the critical cooling rate for the same 1.5 orders of magnitude within this concentration range [Fig. 2(b)]. Such inverse correlation is expected as crystal nucleation is a process that competes with glass-formation. It should also be noted that one of the 20 Cu₉₆Zr₄ samples did not crystallize even after 50 ns. This configuration has been ignored in Fig. 3(c). As the incubation time quickly increases with the atomic fraction of Zr, the crystallization of Cu_100−xZr_x alloys (x > 4 at. %) is studied in Sec. III C by inserting an FCC nucleus to bypass the incubation stage.

We fitted the isothermal transformation curves at 800 K for the volume fraction of atoms that have transformed into crystal phases (V_f) as a function of time (t) [Fig. 4(a)] with the Johnson–Mehl–Avrami–Kolmogorov (JMAK) Eq. (3),

where I is the nucleation rate and U is the growth rate. Pair distribution functions [shown in Fig. 4(b)] indicate a clear intensification of the peaks related to FCC Cu.

Taking into account the value of U estimated from Table I, the resulted nucleation rates (I) obtained from the fits with Eq. (3) in the 0.1–0.5 range of volume fraction (V_f) are 2–4 · 10³⁴ m⁻³ s⁻¹. These values approximately correspond to those obtained by counting the number of nuclei formed in the cells with time. For example, seven overcritical nuclei (N) were counted to form within 0.4 ns of a part of crystallization time in the 11 × 11 × 11 nm³ Cu₉₈Zr₂ cell. The resulting nucleation rate is calculated using the formula

where V is the volume and is found to be 1.3 · 10³⁴ m⁻³ s⁻¹. It is close to the values obtained by Eq. (3) but somewhat higher than the values found for pure Cu.^23,61

The Avrami exponent (n) values of 2.8, 3.2, 3.1, 3.2, and 2.9 are found for Zr contents of 0, 1, 2, 3, and 4 at. %, respectively, for V_f in the 0.1–0.5 range. One can suggest that the initial nucleation stages are influenced by the incomplete relaxation of the liquid/glassy phase [Fig. 4(c)], which leads to reduced n values. When calculations are performed using a fraction transformed of 0.3–0.6 n, exponent values are 3.1, 4.0, 4.0, 4.5, and 3.8, respectively, indicating nucleation and three-dimensional interface-controlled growth of nuclei by the polymorphous mechanism.

For pure Cu, here simulated with the following potential,⁷⁵ a lower value of n = 3.1 was obtained and may indicate the growth of a pre-existing nuclei. In order to verify that size effects do not affect the fitting, the crystallization of a larger model (2 048 000 Cu atoms, ∼30 × 30 × 30 nm³ cell size) was also analyzed. In this case, about 15 nuclei were formed (leading to a number density of 5.9 · 10²³ nm⁻³), but the Avrami exponent n value remained about 2.8–3.1, thus confirming that the smaller system (100 000 atoms) should be large enough for this study.

As the time required for spontaneous nucleation rapidly increases with the Zr atomic fraction, a spherical nucleus of FCC Cu of 1.5 nm radius was introduced in the melt to bypass the nucleation time. We observed that the spherical nucleus starts to grow below the liquidus temperature and maintains its nearly spherical morphology. According to thermodynamic calculations, the critical nucleus radius for Cu in the studied temperature range is about 0.4–0.6 nm.^23,61 Although solute partitioning between the liquid and crystalline phases takes place in the studied Cu–Zr alloys (Table II), indicating diffusional redistribution of the constituent elements, the growth rate (U) of the FCC Cu region is nearly constant [though actually slightly increases with time Fig. 5(a)] before growing crystal impingement takes place. The nature of an increase in the growth rate after about 2.5 nm crystal radius observed in both pure Cu and alloys is beyond the scope of the present study. A linear fit in the 1.5–4 nm of radius was used to calculate the growth rate.

The growth rate values at different temperatures are shown in Table II. For pure Cu, the growth rate is the fastest and a maximum U value of 20.1 m/s (respectively, 30.2 m/s) was obtained at 773 K (respectively, 900 K). These values are close to the results reported in an earlier work in a similar temperature range.⁶⁴ They are also close to those obtained for pure Zr under levitation solidification.⁴ However, they are smaller than those found for planar (100) interface growth of pure FCC metals but closer to (111) one.²¹ It indicates that in the case of non-dendritic three-dimensional growth, the growth rate is limited by the slowest crystallographic growth direction. The dendrite-growth velocity was found to depend on solute trapping. The dendrite tip velocity in a Ni₉₉Zr₁ alloy was measured to be 1–25 m/s depending of undercooling.⁷⁶ The maximum value of 25 m/s obtained for large undercooling is close to the values calculated for the Cu₉₉Zr₁ alloy in Table II.

As can be seen from Fig. 5(b), the growth rate of FCC Cu decreases with an increase in Zr content at both temperatures. At the same time, FCC Cu crystals grow faster than diffusion can pull Zr away from the crystallization front. Thus, the Cu solid solution becomes supersaturated in Zr, which is practically insoluble in FCC Cu. In addition, the difference in growth rate at 773 and 900 K becomes almost an order of magnitude higher with addition of Zr up to 10 at. % (see Table II).

A linear fit of the nucleus radius as a function of time shown in Fig. 5(a) indicates that in FCC Cu, U drops with the Zr content (C_Zr) following an exponential decay [Fig. 5(b)]:

where A and B are constants and A is negative. After careful investigation using two fitting lines with segmented fitting,⁶⁷ it was found that there is a critical concentration of 3 at. % Zr at which the slope dU/dC_Zr changes [Fig. 5(b)]. This change of slope suggests a more significant influence of Zr solute partitioning on crystal growth at high Zr content. It is also related to the metastable solid solubility limit of Zr in FCC Cu of about 4.4 at. %, calculated from the time evolution of the Zr content inside the FCC Cu region for an overall Zr content ranging from 5 to 10 at. % Zr [as shown in Fig. 5(c)].

The snapshots of a series of concentration spectra in growing Cu particles indicate that in the alloys with low Zr content, the crystal growth follows a polymorphic interface-controlled growth mechanism, while at high Zr content, diffusional redistribution of the alloying element Zr clearly takes place in the FCC Cu solid solution (Fig. 6). Only in two alloys, Cu₉₁Zr₉ and Cu₉₀Zr₁₀ Zr concentration in the liquid phase is lowered near the interface at 900 K [Fig. 6(d)].

Growing FCC crystals have a nearly spherical but somewhat irregular shape (Fig. 7). The estimated main branches of the dendrites are in the ⟨111⟩ direction, which is different from an ordinary ⟨100⟩ growing direction for FCC metals. Unsurprisingly, growing FCC Cu particles have HCP type stacking faults at low Zr content [Fig. 7(a)]. However, the number of faults decreases drastically when the Zr fraction increases [Fig. 7(b)], which rather indicates an increase in stacking fault formation energy with Zr addition.⁷⁷ 

Notwithstanding on polymorphic type (congruent) crystallization, the dendritic growth of an intermetallic compound (likely cubic CuZr one) in the Cu₅₀Zr₅₀ alloy was found to be diffusion controlled,⁷⁸ and the crystal growth rate is as small as ∼10⁻⁹ m/s near the glass transition temperature (∼673 K).⁷⁹ 

Table III shows the diffusion coefficient (D) of Cu and Zr atoms in the Cu₉₅Zr₅ alloy, estimated using mean squared displacements plotted in Fig. 8. The results shown in Fig. 8 indicate that even at 773 K system relaxation is achieved in 1 nm. As the diffusion length L = 2√(D · t) (where t is time), then in 1 ns of computational time, the maximum particle size to be reached is only 0.3 nm corresponding to 0.3 m/s of the suggested growth rate for Zr, which is still smaller than ∼ 1 m/s of the actual growth rate in Table II. The rate of 0.4 m/s is obtained for Cu. 0.8 and 1.1 m/s growth rates were calculated from D for Zr and Cu at 900 K. The growth mechanism of FCC Cu crystals found in this work is collision limited and explains why some Zr atoms are trapped inside the crystals. Nevertheless, solute partitioning still takes place within growing Zr crystals afterward, as shown in Fig. 6.

Although $Tg*$ is nearly constant in Cu–Zr alloys up to 10 at. % Zr, their critical cooling rate increases by three orders in magnitude with an increase in Zr content from 0 to 5 at. % Zr in line with the reduction of the liquidus temperature by Zr.⁷³ It is estimated that the critical cooling rate drops to 10⁷–10⁸ K/s for the Cu₉₁Zr₉ alloy, provided that a competing Cu₅₁Zr₁₄ crystalline phase does not nucleate.⁷³ Unsurprisingly, the incubation period (nucleation time) correlates well with the inverse of the critical cooling rate (value changes of the same order of magnitude), since crystal nucleation is a process competing with glass-formation.

According to Ref. 80, the reduced glass transition temperature T_rg = T_g/T_l (where T_l is the liquidus temperature) for pure metals is 0.3, whereas for glass-forming alloys, T_rg exceeds 0.5. This study, in agreement with other recent studies,⁸¹ shows that even pure metals have relatively high T_rg values, while their low glass-forming ability is connected with high fragility^82,83 (which is the deviation of temperature dependence of their viscosity from the Arrhenius equation) of their liquids.⁸⁴ 

The growth rate of FCC Cu crystals in Fig. 5(a) is nearly constant but slightly increases with time after about 2.5–3 nm radius. The growth rate also significantly decreases with an increase in Zr content in the alloy, as the FCC Cu solid solution becomes supersaturated in Zr. There is a critical concentration of 3 at. % Zr at which the slope in dlog(U)/dC_Zr changes (at both temperatures, 773 and 900 K). Such a change in the growth rate is in line with the reduction of the number of stacking faults in crystalline Cu (Fig. 7). The analysis of the Zr concentration distribution showed that at less than 6 at. % of Zr concentration, a concentration gradient quickly appears in the FCC particle within the pre-inserted FCC crystal. However, it cannot proceed further during crystal growth, which takes place in a polymorphic way without solute partitioning at the interface (Fig. 6). On the other hand, only at Zr content more than 6 at. %, a concentration gradient is formed not at the liquid/crystal interface but within the crystals [see Figs. 6(b) and 6(c)]. This indicates that Cu crystals are growing too fast by interface-controlled growth leaving Zr to diffuse within crystalline phase. Growth is associated with the transport of “excess” zirconium within the entire growing crystal. However, the gradient of Zr content is not at the interface but within FCC Cu crystals at Zr content below 9 at. % [Fig. 6(b)], but it moves closer to the interface at 9 at. % Zr [Fig. 6(c)].

The metastable solid solubility limit of Zr in FCC Cu is about 4.4 at. % [see Fig. 5(b)]. Growing FCC crystals with nearly spherical morphology yet have some branches along the ⟨111⟩ crystallographic directions. The solute partitioning ratio (R = (C_Zr^l-C_Zr^c)/C_Zr^l) increases with Zr content higher than 8 at. %, which is, however, not reflected in the slope of dU/dC_Zr.

The JMAK method is found to be applicable for modeled crystallization of Cu–Zr alloys at large undercooling when several nuclei are formed. Using Eq. (3), fitting the nucleation rate (I) at 800 K was calculated to be 2–4 · 10³⁴ m⁻³ s⁻¹. The Avrami exponent values obtained for the volume fraction of crystallized region of 0.3–0.6 of 4.0, 4.0, 4.5, and 3.8 for Zr content of 1, 2, 3, and 4 at. %, respectively, indicate nucleation and three-dimensional interface-controlled growth of nuclei. This is consistent with the nearly constant growth rate found in Fig. 5(a) and the absence of the concentration gradient at the interface of alloys below 9 at. % of Zr. The initial nucleation stages are influenced by relaxation of the liquid/glassy phase, leading to gradual variation of volume and energy of the alloy with time, which lead to reduced n values.

The glass-formation of Cu–Zr alloys (Zr fraction from 1 to 10 at. %) was studied by MD simulation in line with the nucleation and growth of a FCC Cu crystalline phase. We found that the critical cooling rate increases by two orders of magnitude from 0 to 5 at. % Zr and is extrapolated to drop to 10⁷–10⁸ K/s for the Cu₉₀Zr₁₀ alloy.

Notwithstanding the negligible equilibrium solid solubility of Zr in FCC Cu, the growing Cu crystals are found to dissolve some amount of Zr with an overall metastable solubility limit of 4.4 at. % Zr. The growing FCC crystals have a nearly spherical morphology. In alloys containing up to 6 at. % Zr, a supersaturated FCC Cu solid solution grows polymorphically, while in alloys with higher Zr content, a Zr concentration gradient is observed in the FCC Cu region. At 9 at. % of Zr and more, the concentration gradient starts directly from the interface at 773 K and from the liquid phase at 900 K. Despite diffusional redistribution of the alloy components at high Zr content, the crystal growth rate before crystal impingement is nearly constant in all alloys (though somewhat increases with time), while it decreases exponentially with Zr content. There is a critical concentration of 3 at. % Zr at which the slope in dlog(U)/dC_Zr is changed, likely related to reduction of the number of stacking faults in crystalline Cu.

The authors sincerely thank Professor Hiroshi Suito for his valuable help with the software for statistical analysis of the experimental data. This study was supported by the JSPS Grant-in-Aid for Transformative Research Areas (A) “Hyper-Ordered Structures Science” (Grant No. 20H05883) and the JSPS Grant-in-Aid for Scientific Research (Grant No. 18H01143).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.