1 Introduction

Because amorphous materials have high strength mechanical properties and excellent corrosion resistance and chemical compatibility [1], and crystal materials have unique structural characteristics and excellent physical and chemical properties [2], both of them have been attracted much attention in the field of materials science. Meanwhile, the variation of the microstructure during crystallization and amorphization phase transformation will cause the change of physical properties [3]. Therefore, crystallization and amorphization phase transformation have been the subject of a considerable significance in both theoretical and experimental work.

It is found that there are many factors affecting crystallization and amorphization phase transformation. For example, the amorphous structures are formed at the cooling rates of 1 × 1013 and 1 × 1012 K/s in Cu10Ag90 alloy, and the crystallized structures are formed when the cooling rate is lower than 1 × 1011 K/s, and the degree of crystallization increases with decreasing cooling rate [4]. It has been performed to simulate ion-beam-induced epitaxial crystallization and interfacial amorphization at an amorphous/crystalline Si interface, and a divacancy and di-interstitial pair can induce amorphization and crystallization [5]. Ti50Al45Si5 multi-component powders which are synthesized by mechanical alloying and heat treatment [6, 7] can present an amorphous phase or crystallization phase due to the degree of mechanical alloying and micro-structural properties [8]. Taking pure copper as an example, crystal nucleation occurs due to the emergence of adequate crystal-like clusters, while glass formation in some critical cooing rate region is determined by atomic structural fluctuation [9].

In addition to above factors, composition on crystallization and amorphization in alloys are also important. For example, the minor addition of Nb can improve the glass-forming ability by shortening the start time of the amorphization reaction in Zr66.7-xCu33.3Nbx alloys, and the amorphous Zr66.7Cu33.3 phase gradually transforms into a metastable FCC-Zr2Cu phase [10]. The amorphous phase appears in Fe-50Ti, and the amorphous phase is formed directly and at much lower energies in Fe-33.3Ti system during mechanical alloying of Fe-xTi (x = 33.3 and 50 at.%) alloys [11]. The crystallization and amorphous transition are observed in different components of Fe–Ni alloys during the isothermal relaxation process [12]. Therefore, the alloy composition not only is important for crystallization and amorphization, but also has an impact on material properties. Currently, Ni-Co alloys, as a representative of transition-metal, are widely studied for their unique properties such as strong adsorption [13, 14]; however, there is no analysis in crystallization and amorphization on the atomic scale. In addition, due to the limitations of even the most advanced experimental techniques, there is still a lack of detailed understanding of the microstructure evolution. Therefore, a computational simulation has been performed for studying the crystallization and amorphization in Ni-Co alloy. In this paper, the crystallization and amorphization are studied in Ni73Co27 and Ni74Co26 alloys during the rapid solidification process through molecular dynamics (MD) simulation.

2 Simulation Details

In this paper, MD simulation is performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) [15]. The 10,000 atoms of NixCo(100−x) (x = 73, 74) alloys are randomly placed in individual cubic boxes. According to a periodic boundary condition, the time step for the system operation is set at 10−15 s. In the fixed NPT ensemble [16], the interaction potential between the atoms is indicated by an embedded-atom model (EAM) developed and optimized by G. P. Pun and Y. Mishin [17]. The relevant analog computation starts at 1800 K that is much higher than the experimental melting temperature (Tm ~ 1500 K) [18]. At first, a uniform initial velocity obey is a Gaussian distribution is assigned to all atoms in the system, and the system is isothermally relaxed for 1,000,000 steps at the initial temperature to enter the equilibrium state. Secondly, the system temperature is rapidly reduced to 300 K at the cooling rate of 1 × 1012 K/s, and they are recorded with an interval of 10 K to acquire the microcosmic information, including the spatial coordinates, atomic numbers, and atomic potential energy of all atoms in the system, which is used to explore microstructure evolution at different temperatures during the rapid cooling process. At last, the structural configuration and formation and evolutionary law of microstructures are deeply investigated by pair distribution function (g(r) or PDF) curve [19], the energy-temperature (E-T) curve, Largest Standard Cluster Analysis (LSCA) [20], three-dimensional (3D) visualization [21], and tracking method. And the crystallization and amorphization are studied between Ni73Co27 and Ni74Co26 alloys under the same solidification conditions on microstructure.

3 Results and Discussion

3.1 Pair Distribution Function

In materials science, the pair distribution function is extensively used to research the distinguishing features of the microstructure owing to its objectivity and reliability, which can refer to the spherical shell within a certain distance of an appointed atom and the probability of other atoms appearing in this range. Figure 1 shows the g(r) curves of ten NixCo(100-x) alloys at 300 K. The first peak of Ni73Co27 alloy is higher than that of Ni74Co26 alloy, which indicates that the order degree of the first neighbor atoms is stronger in Ni73Co27 alloy. There is a new peak between the first and second main peaks in Ni73Co27 alloy, which indicates that the crystal nucleus has been formed [22]. However, the second peak splits into two sub-peaks in Ni74Co26 alloy, where the intensity of the left sub-peak is slightly greater than that of the right sub-peak, indicating the formation of an amorphous structure [23]. When the Ni content (CNi) ≤ 73, there are many sharp peaks in the g(r) curves, and long-range ordered crystal structures are formed in the system. When CNi ≥ 74, the second peak of g(r) curve splits, which indicates that amorphous structures are formed. In order to study the formation process of crystallization and amorphization in Ni-Co alloys, two critical Ni73Co27 and Ni74Co26 alloys are selected as research objects.

Fig. 1
figure 1

The g(r) curves of NixCo(100-x) alloys at 300 K

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3.2 Evolution of Energy

The average potential energy (APE), as a simple and effective method, can objectively show the preliminary characteristic of phase transition by tracing the thermal history in material science. As shown in Fig. 2, it can be seen that the APEs of two alloys decrease with the decrease of temperature. And the APEs are almost coincident at a temperature within the range of 1800 to 1250 K, which indicates that the difference of APE is not obvious in areas of elevated temperatures. The APE of Ni73Co27 suddenly drops at 1250 K, which is thought to be the crystallization temperature [24]. And the second drop occurs at 1180 K, which undergoes a second-order phase transition. The APE curve suddenly dropped, which indicates that a crystalline system should be formed. But the APE of Ni74Co26 gently declines without sudden change, so hence amorphous structures should be formed. When the slope of E-T curve changes, the temperature T at intersection can be regarded as the glass transition temperature Tg [25], which is about 950 K. In addition, the APE of Ni74Co26 is slightly higher than that of Ni73Co27 at 300 K, which indicates that the final structure of crystalline system is more stable than that of the amorphous system.

Fig. 2
figure 2

Average potential energy of Ni73Co27 and Ni74Co26 alloys evolves with T

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3.3 Bond-Type Index Analysis

The bond-type index analysis can clearly reflect the geometric characteristics of short-range atom arrangement. It can clearly show the bonding between atoms and neighboring atoms (see the inset in Fig. 3). In general, using the bond index method to analyze the structure, S421, S422, S444, and S666 are related to crystalline structures, whereas S555, S544, and S433 [26] are associated with amorphous structures. As shown in Fig. 3(a), the number of S421 changed most obviously during the rapid solidification in Ni73Co27 alloy. Its proportion is 3.6% at 1800 K, and suddenly increases to 72.84% at 1250 K, and then, it increases to 83.72% for the second time at 1180 K. At T < 1180 K, the amount of S421 increases slowly to 96.2% at last. On the contrary, the number of other bond types begins to decrease suddenly at 1250 K, and decrease steadily after 1180 K. The total number of other bond types accounted for only 5.85% at 300 K. Finally, the main crystalline structures composed of S421 are formed in Ni73Co27 alloy. As shown in Fig. 3(b), the number of S555 changes most obviously during the rapid solidification in Ni74Co26 alloy. Its proportion is 23.57% at 1800 K, and increases to 34.57% at 950 K, and finally reaches to 33.17% at 300 K. In addition, the numbers of S544 and S433 maintain at about 15% during the cooling process. The total number of three amorphous bond types ranges from 58.7% in initial liquid state to 64.44% in final amorphous state.

Fig. 3
figure 3

Relative numbers of the main CNSs in Ni73Co27 and Ni74Co26 alloys

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3.4 The Largest Standard Cluster Analysis

Although bond-type index analysis can be successfully used to describe the bonding relationship between an atom and its nearest neighbors, it still cannot describe the different basic clusters formed by an atom with its nearest neighbors [27]. Therefore, the LSCA method based on CTIM [28] is selected in this study to analyze the microstructure of the system [20]. The biggest advantage of this analysis method is that it can automatically confirm the unique cutoff distance rc [29], which indicates that the local structure characteristics of each atom can be uniquely and completely identified and characterized, so that it can objectively describe the local structure formed by each atom and its neighbor atoms in the system. The face-centered cubic structure (FCC) and hexagonal close-packed structure (HCP) LSCs are respectively denoted by [12/S421] and [6/S421, 6/S422] (see the inset in Fig. 4(a)) in crystalline structure.

Fig. 4
figure 4

Relative number of basic clusters in Ni73Co27 and Ni74Co26 alloys

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At the same time, topologically close-packed (TCP) LSCs [30] are a group of LSCs that can be denoted as [n4/S444, n5/S555, n6/S666] with n5 > 0. Based on the Euler theorem, an equation set of {n4 + n5 + n6 = CN, 2n4 + n5 = 12} is satisfied for TCP LSCs. TCP structures are ubiquitous in liquid and metallic glass [31]. Therefore, the topological feature of a TCP LSC can be described by a TCP index of “Ln”, where “L” is Z, A, B, C, D, or E, indicating n4 = 0, 1, 2, 3, 4, 5; and n refers to CN. For example, typical TCP LSC of [12/S555] is Z12, i.e., icosahedron (ICO); a familiar TCP LSC of [1/S444, 10/S555, 2/S666] is A13 (see the inset in Fig. 4(b)) in amorphous structure.

As shown in Fig. 4(a), the number of FCC changes most obviously during the rapid solidification in Ni73Co27 alloy. The proportion of FCC is 0 at 1800 K, and suddenly increases to 58.98% at 1250 K, and then, it increases to 71.46% for the second time at 1180 K, corresponding to the second increase of energy. The amount of FCC increases slowly to 92.91% after 1180 K, and the system finally forms a standard FCC crystal structure. As shown in Fig. 4(b), there are different types of amorphous clusters in Ni74Co26 alloy, which are composed of amorphous bonds. Seven types of TCP structures are formed in Top10, and the number of amorphous clusters changes slightly. The amorphous clusters increase slowly at T > Tg, and the number of A13 changes most obviously. Its proportion 0.62% at 1800 K increases to 2.74% at 950 K. The growth of amorphous clusters is starting to flatten at T < Tg. The number floating of amorphous clusters is in consequence of unstable thermal fluctuation. In summary, only one crystal structure FCC influences structural characteristics during the crystallization process in Ni73Co27 alloy, but a variety of complex structures are formed during the amorphization process in Ni74Co26 alloy.

3.5 Topologically Close-Packed Analysis

TCP cluster phase with the lower average potential energy inhibits the growth and formation of crystals because of the interconnections between crystal and TCP clusters [4]. Therefore, TCP can reflect the degree amorphous of the system. As shown in Fig. 5, the number of TCP in two alloys is almost coincident in liquid state at T > Tc. At the beginning of Tc, the number of TCP in Ni73Co27 alloy suddenly decreases from 1555 to 36 at final, indicating that the degree of amorphization decreases and crystalline structures are formed. The number of TCP in Ni74Co26 alloy suddenly increases to 2051 at Tg, and then keeps a steady state. The evolution of TCP can demonstrate the beginning of crystallization and amorphization, which characterizes the local structure characteristics of amorphous structures.

Fig. 5
figure 5

The numbers of the TCP in Ni73Co27 and Ni74Co26 alloys

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3.6 Three-dimensional Visualization Analysis

It is necessary to delve into the formation of local structures at the atomic level. With the help of 3D visualization, the details of structural evolution can be intuitively presented. As shown in Fig. 6 (a), the central atom number of FCC, HCP, and BCC clusters in Ni73Co27 alloy suddenly increases from 200 at 1250 K to 3216 at 1220 K, and then suddenly increases from 6179 at 1180 K to 7426 at 1100 K for the second time. The atomic arrangement becomes more and more regular with a drop in temperature. The second-order phase transformation leads to an increase in the degree of long-range order. When the number of atoms is 9394 (FCC: 9294, HCP: 99, BCC: 1) at 300 K, the near-perfect crystal structure with FCC ratio of 92.94% is formed at final. According to our previous research [12], the HCP atoms on the planar grain boundaries between FCC grains in Ni73Co27 are easy to be converted to FCC atoms at 1180 K, which results in a solid–solid phase transition, and more stable state, even a single-phase FCC crystal. The central atoms of the main TCP clusters are shown in Fig. 6(b) in Ni74Co26 alloy, such as A13, B14, C13, ICO, A14, C14, and D14. Compared with the crystal structure, the distribution of atoms is irregular, and the number of TCP increases slowly. The number of TCP increases from 705 at 950 K to 781 at 900 K, and remains a steady amount at T < 900 K, indicating that the amorphous structures change a little after Tg. Above all, it can be concluded that atomic proportions of Ni-Co alloys have a significant influence on microstructure during solidification. Even if the Ni atoms of two Ni-Co alloys differ by only 1%, it can cause the system to be crystallization or amorphization, and finally form crystal and amorphous structures.

Fig. 6
figure 6

Atomic distribution of two alloys near the phase transition temperature

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3.7 The Evolution of Crystal and Amorphous Microstructures

From Fig. 4(a), it can be seen that FCC is the main crystal cluster type in the system; therefore, FCC is selected for further investigation in this study. FCC clusters in Ni73Co27 alloy are further investigated by the tracking method, as shown in Fig. 7. An FCC nano-cluster composed of 28 atoms (Fig. 7(a)) with 4 central atoms (Fig. 7 (d)) at 1270 K grows to be an FCC nano-cluster composed of 41 atoms (Fig. 7(b)) with 5 central atoms (Fig. 7 (e)) at 1260 K, and it becomes an FCC nano-cluster composed of 65 atoms (Fig. 7(c)) with 6 central atoms (Fig. 7(f)) at 1250 K. It reveals that the growing FCC nano-clusters are always distributed regularly, and the growth pathway of the nano-FCC clusters is shown in Fig. 7(g)–(i). The FCC with central atom 1893 (Fig. 7(g)) grows an FCC nano-clusters (Fig. 7(h)) by attaching 14 green atoms and 1 orange atom, and atoms 2038, 7555, 8106, 8980, 7059, and 6096 form a hexagonal structure with central atom 8092 through another hexagonal structure with central atom 1465, which form two new FCC clusters by intercross-sharing connection. Then, atoms 8092 and 9842 are bonded at 1250 K (Fig. 7(i)), and finally, one new FCC nano-cluster with 1893–1465-8092–9842 as the central atom is formed, in which atoms 8207, 8236, 9799, 7375, 6239, and 9002 form a hexagonal structure with central atom 9842. The symmetry of hexagon satisfies the basic symmetry characteristics of crystal structure, so it is determined that the crystal growth pathway is hexagonal nucleation in Ni73Co27 alloy.

Fig. 7
figure 7

The evolution of FCC at 1270–1250 K in Ni73Co27 alloy. (Blue and green-orange balls represent inherited and newly formed atoms, respectively)

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Crystal and amorphous structures are different in the growth pathway. At present, the growth process of icosahedron has been tracked in amorphous MgZn alloys by the molecular dynamics method [32]. It is found that one icosahedron can grow to be nano-cluster by attaching a 6-atom pentagonal pyramid. TCP clusters with more complex than icosahedron in Ni74Co26 are further investigated by tracking method from 1270 to 1250 K, as shown in Fig. 8. A TCP nano-cluster composed of 37 atoms (Fig. 8(a)) at 1270 K with 4 central atoms (Fig. 8(d)) grows to be a new TCP nano-cluster composed of 42 atoms (Fig. 8(b)) with 5 central atoms (Fig. 8(e)) at 1260 K, and it becomes a TCP nano-cluster composed of 53 atoms (Fig. 8(c)) with 6 central atoms (Fig. 8(f)) at 1250 K. It reveals that the growing TCP clusters are always distributed in no regular arrangement, and demonstrates the growth pathway of TCP clusters as shown in Fig. 8(g)–(i). Central atoms 365 and 9041 of TCP clusters (Fig. 8(g)) are bonded at 1270 K to form a new TCP (Fig. 8(h)) by intercross-sharing connection, and atoms 9992, 9502, 5702, 2372, and 5467 form a pentagonal structure relating to a fivefold symmetry S555 with central atom 9041. And the location of atoms at 1260 K is different from that at 1270 K due to the motion of atoms. Then, central atoms 9041 and 5702 of TCP clusters (Fig. 8(i)) are bonded at 1250 K, and finally, an TCP nano-cluster with 365–9041-5702 as the central atoms is formed. Atoms 100, 1965, 9502, 9041, and 6364 form a pentagonal structure with central atom 5702, in which atoms 9414 and 5702 form a fivefold symmetry S555. The fivefold symmetry satisfies the basic characteristics of amorphous structure, so it is determined that the amorphous growth pathway is pentagonal formation through the necessary bond type such as S555 [33].

Fig. 8
figure 8

The evolution of TCP at 1270–1250 K in Ni74Co26 alloy. (Blue and green balls represent inherited and newly formed atoms, respectively)

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4 Conclusions

The microstructure evolutions have been systematically studied during crystallization and amorphization processes in Ni73Co27 and Ni74Co26 alloys by employing classical MD simulation. The following conclusions are obtained.

  1. (1)

    The average potential energy of Ni73Co27 alloy suddenly drops at Tc, and the second drop occurs at 1180 K, which undergoes a second-order phase transition. The average potential energy of Ni74Co26 alloy declines without sudden change, so hence amorphous structures have been formed.

  2. (2)

    S421 is the most obvious bond type in Ni73Co27 alloy, which suddenly increases twice, and the system finally forms a standard FCC crystal structure. S555 is the most obvious bond type in Ni74Co26 alloy, and the number of TCP clusters keeps a steady state after Tg, which finally forming amorphous structure with a variety of complex structures.

  3. (3)

    By the tracking method, the sixfold symmetry satisfies the characteristics of crystal structure, so the hexagonal nucleation is the growth pathway in Ni73Co27 alloy. While the fivefold symmetry satisfies the characteristics of amorphous structure, so the pentagonal formation is the growth pathway in Ni74Co26 alloy.