The formation of amorphous alloys is a complex process and is an intuitive manifestation of the comprehensive effect of dynamics, thermodynamics, structure, and other factors in the process of solidification transformation, which can characterize the difficulties experienced in the transition from the liquid to the glassy state in alloy systems. In decades of research, the ability to form amorphous alloys has been constantly discussed, and new insights have supplemented scientists’ knowledge and understanding of this process.

Amorphous alloys can be prepared by improving the cooling rate. It is generally believed that the stronger the metal’s ability to form an amorphous alloy, the larger the size of the amorphous sample that forms under the same cooling conditions. Based on this, Lin and Johnson (1995) have proposed the critical cooling rate (R _c) to characterize the forming ability of an amorphous alloy, as given by Eq. (1) R c = d T / d t = 10 / D 2 , (1) where T is temperature, t is time, and D is critical size. Johnson et al. (2011) have explained that nucleation is controlled by different mechanisms in the shallow supercooled liquid region from the melting temperature (T _m) to the “nose tip” temperature and in the deep supercooled liquid region from the “nose tip” temperature to the glass transition temperature (T _g). Figure 1 shows that, in the shallow supercooled liquid region, crystallization is controlled by nucleation. In this temperature range, the growth rate of the crystal nucleus is very large and, as long as the crystal nucleus appears, it is easier to form crystals. However, it is relatively easy to nucleate in the deep supercooled liquid region, but the growth of the crystal nucleus is more difficult, which is achieved by the long-range diffusion of atoms. Crystallization is inhibited by controlling the diffusion process of atoms to form the amorphous alloy.

Turnbull (1969) has put forward the famous criterion of amorphous forming ability according to classical nucleation theory; that is, the reduced temperature T _r is used to measure the amorphous forming ability of a liquid, as given by Eq. (2): T rg = T g / T m . (2) As shown in Figure 2, when T rg > 2 / 3 , there is no heterogeneous nucleation during solidification, the viscosity of the liquid metal is very high, and the rate of homogeneous nucleation is very low so that it is easier to form an amorphous state. When T rg = 1 / 2 , the amorphous alloy must be obtained at a higher cooling rate.

In addition to using the reduced glass temperature T _rg to characterize the interval between the melting temperature and the glass transition temperature, some researchers have also used the supercooled liquid region ΔT _x ( Δ T x = T x − T g , where T _x is the initial crystallization temperature) to indirectly reflect the glass-forming ability (Inoue et al., 1990). Generally speaking, the larger the width of ΔT _x in the supercooled liquid region, the higher the thermal stability and the stronger the crystallization resistance.

Based on the above research and the “confusion principle” put forward by Greer, the conclusion is that the more the alloy components, the more difficult the long-range diffusion of atoms and the higher the probability of random close packing (Greer, 1993).

The development and progress of science is a dialectical process. In 2004, Cantor et al. (2004) and others developed more than 20 kinds of alloy systems based on the principle of chaos. Amorphous alloys could not be obtained in any other way, which were called multicomponent alloys at the time. Yeh et al. (2004) have later put forward the concept of high-entropy alloys; then, in 2008, Zhang et al. (2008) have summarized the law of phase formation of multicomponent alloy systems. Thus, although the classical amorphous formation criterion has been a very important guiding factor, it also has some limitations. However, with the continuous development of the criterion of amorphous forming ability, from Greer’s “confusion principle” to “Inoue’s three principles,” the thermodynamic parameter of entropy plays an important role in amorphous alloy formation.

There has been a considerable amount of discussion about the role of entropy in amorphous alloy formation. Classical nucleation theory and crystal growth theory both emphasize the effect of the liquid–solid Gibbs free energy difference ΔG _l–s and the thermodynamic interfacial energy. The kinetic melt viscosity η (or the relaxation time τ and the diffusion coefficient) is an important parameter for determining the formation of amorphous alloys. In classical nucleation theory, when the temperature is lower than the melting point T _m, the melt enters a supercooled state, and the critical radius R _c at which stable nuclei can form in the supercooled liquid is given by thefollowing Eq. (3): R c = 2 σ Δ G l − s . (3) The critical nuclear energy E _c needed to form this crystal nucleus is given by Eq. (4): E c = 16 π σ 2 3 Δ G l − s 2 . (4) The free energy difference ΔG _l–s between the solid and liquid phases is the driving force for crystal formation. Thus, to obtain an amorphous alloy, it is necessary to reduce the free energy difference between the solid and liquid phases so that the shape of nuclear energy increases, which means that it is not easy to form crystals, which tend to form amorphously. The equation for the simplified free energy difference was given by Johnson (1986), as Eq. (5): Δ G l − s = Δ S f Δ T 2 T T m   +   T , (5) where ΔS _f represents the melting entropy, that is, the entropy difference between the solid and liquid phases near the melting point. It can be seen from Eq. 5 that when ΔS _f decreases, ΔG _l–s also decreases, which is beneficial to the formation of amorphous alloys. When the number of components in the alloy system increases, compared with a traditional alloy, ΔS _f near the melting point of the solid–liquid phase also decreases. Therefore, from the point of view of thermodynamics, the increase in entropy is beneficial to the formation of amorphous alloys.

As solidification is the process of atom nucleation and growth through long-range diffusion, the formation of amorphous crystals is also affected by kinetic factors. For a multicomponent alloy system, when R > R _c, the melt enters a supercooled state. As the temperature decreases, the viscosity of the melt becomes ever larger, causing the diffusion of atoms to be hindered, deviating from the equilibrium state of the system, and forming an amorphous state. Accordingly, the relationship between liquid configuration entropy and viscosity coefficient, established by Adam and Gibbs (1965), is given by Eq. (6): η = η 0 e x p ( A T S con ) , (6) where A is a constant and η0 is the reference dynamic viscosity coefficient under one atmosphere. It can be seen that with an increase in the number of components, the configuration entropy will increase, and the decrease in the viscosity coefficient η means that the fluidity of the melt increases, which is conducive to the long-range diffusion of atoms and is unfavorable for the formation of amorphous alloys. Therefore, from a dynamic point of view, the increase in entropy is not conducive to the formation of amorphous alloys.

In terms of structure, one of the “Inoue three principles” states that there must be a sufficiently large atomic size difference σ between the atoms of the constituent elements. An atomic size difference of greater than 12% is the most conducive. The compact and random stacking structure can improve the amorphous forming ability effectively (Egami and Waseda, 1984). The free volume model is used to analyze the influence of atomic size differences. In this model, the viscosity coefficient η is expressed as follows (Spaepen, 1977): 1 / η = A e x p ( − K / V f ) , (7) where A and K are constants and V _f is the free volume. The relationship between the free volume and the self-diffusion coefficient of the liquid can be approximately expressed by the Stokes–Einstein equation, as given by Eq. (8): D = ( k B T / 6 π r 0 ) / η , (8) where k _B is the Boltzmann constant and r ₀ is the molecular diameter. When the atomic size is uneven, a closely stacked structure will be formed, the free volume will decrease, the solid–liquid interface of the alloy can be increased, the viscosity of the melt will increase, and the long-range diffusion of atoms will be hindered. Unlike a crystal, the short-range local atomic arrangement of an amorphous alloy is composed mainly of icosahedral clusters, such as in the La-Al-TM (TM = transition metal) alloy, which has been shown by high-resolution transmission electron microscopy (XRD) and neutron scattering (Chen and Ma, 2011).

In contrast, the mismatch entropy S _σ (one component of the configuration entropy) is used to discuss the effect of S _σ on the formation of amorphous alloys. The standardized mismatch entropy S σ / k B is usually used to describe the degree of mismatch between atoms in the system and to reflect the elastic properties of the system. Previous calculations by Takeuchi et al. (2013) have shown that the standardized mismatch entropy S σ / k B is related to the atomic size difference, as given by the following Eq. (9) S σ / k B ∼ ( δ / 22 ) 2 . (9) When the atomic size difference is large, the interaction between atoms is strong, and it is not easy to form a stable structure. In addition, when nonmetallic elements such as Si are added to the alloy system, the bonding mode of the latter becomes more complex, and the metals with a strong ability to form amorphous alloys with Si or Sn can do so more easily. Therefore, the mismatch entropy is related to the atomic size difference δ. The larger the atomic size difference, the greater the mismatch entropy. From the structural conditions of amorphous formation, the increase in the misalignment entropy helps increase the amorphous formation capacity. However, it is worth noting that the influence of structure is also different in different alloy systems. Generally speaking, an increase in entropy is beneficial to the formation of amorphous alloys at the structural level. It can be seen from Figure 3 that the formation region of the amorphous phase is for δ > 9, −49 ≤ ΔHmix ≤ −5.5 kJ/mol, 7 ≤ ΔSmix ≤ 16 J/(K·mol) (Guo et al., 2013); that is to say, when the alloy composition parameters involved fall within this range, the multicomponent alloy formed will be amorphous, for which Li et al. (2017) and others have confirmed the accuracy of their predictions.

In researching and discussing amorphous formation ability, the effects of the various factors involved are found to be very complex. In terms of the effects of the above thermodynamic and kinetic parameters on amorphous alloy formation, there is no unified parameter for measuring, estimating, and predicting amorphous alloy formation, where the inclusion of too many parameters leads to confusion and a lack of understanding.

Recently, Wang et al. (2020) have discussed the synergistic and competitive relationship between enthalpy and entropy in the process of amorphous alloy formation. They have found a certain relationship between the melting entropy and melting point T _m, the critical cooling rate R _c, the classical amorphous formation parameter T g / T m , and the kinetic melt viscosity. Entropy is considered to be more suitable as a thermodynamic representative parameter of the amorphous formation energy. Unlike the relationship between the entropy of melting and its parameters in kinetics, the enthalpy or entropy caused by different structures in thermodynamics has different numerical values. Therefore, these thermodynamic parameters of different materials and different systems need a structural parameter to normalize the entropy of melting, aiding discussion of the differences. The structural parameter is the Beads number (Tu et al., 2016), reflecting the degree of freedom of molecular rotation or orientation and can be used as a normalized parameter to avoid this problem. The feasibility of using this parameter has been confirmed from studies of drug molecules and intermetallic compounds, but whether it is the most crucial parameter remains to be seen. Thus far, the above discussion has been about the parameters of amorphous forming ability, but its effectiveness and widespread use need to be proved for a greater number of different alloy systems.

The electronic shells structure of rare earth elements can usually be represented by 4f ⁿ5d¹6s², and n is from 0 to 14. With the increase of atomic number, the 4f electron shell is gradually filled, and the neighboring rare earth elements only differ by one electron in the 4f atomic orbital. Therefore, the physical and chemical properties of the elements are similar. From the perspective of the electronic shells structure of rare earth elements, they have a high electronic valence, large atomic radius, strong polarization, and therefore strong chemical activity, which can form oxides, sulfides, chlorides, fluorides, hydroxides, and carbonates and many other compounds. Compared with ordinary metals and alloys, it has some unique effects, among which adding a proper amount of rare earth elements into the amorphous alloy can effectively improve the glass-formation ability. As early as 1989, Ning Yuantao et al. systematically studied the influence of La, Ce, Pr, and other eight rare earth elements on the thermal stability of PdSi_16.5 amorphous alloy (Ning Yuantao et al., 1989). The results showed that rare earth elements, especially heavy rare earth elements, could significantly improve the thermodynamic parameters of the amorphous alloy, including characteristic transition temperature T _θ [T _g (glass transition temperature), T _x (initial crystallization temperature), T _p (peak crystallization temperature), and T _rg (reduced glass temperature)]. As mentioned above, T _g (glass transition temperature), T _x (initial crystallization temperature), and T _rg (reduced glass temperature) are classical parameters to describe the formation ability of amorphous alloys, and the improvement of these parameters contributes to the glass-formation ability (GFA). The crystallization activation energy is the increase in the critical nucleation energy of crystallization, as described in Eq. 4, which is conducive to the formation of amorphous alloys.

Adding rare earth elements to amorphous alloys is an effective method to increase the size of amorphous alloys and improve the GFA. Yong et al. (2000) investigated the changes of adding different contents of rare earth metal yttrium in Zr₅₅Al₁₅Ni₁₀Cu₂₀ alloys. The results were observed by XRD, DSC, and DTA measurements. The effect of yttrium addition on the alloy phase change was observed. As shown in Figure 4, with the addition of 1–4% yttrium, the crystalline peak diffraction becomes gradually lesser and weaker. When the amount of yttrium reaches 4%, almost no crystalline diffraction peaks were observed. However, when the yttrium content continued to increase to 6%, part of the crystal phase was precipitated. Therefore, the proper addition of yttrium could inhibit the formation of the crystal phase and improve the GFA. At the same time, the same trend was also confirmed by DTA and DSC measurements; that is, the appropriate addition of yttrium could reduce the melting temperature T _m of the alloy. As shown in Figure 5 and Figure 6, with the heating rate of 0.67 K/s, the DTA test results showed an obvious glass transformation process when the amount of yttrium was 2 and 4%, and the subcooled liquid region ( Δ T = T x − T g ) was widened. In order to explore the influence of rare earth element yttrium on the thermodynamic parameters, the samples were prepared under the conditions of raw material purity and low vacuum. With the heating rate of 0.33 K/s, DSC test results showed that the supercooled liquid phase ΔT reaches the maximum value when the addition amount reaches 2% and the glass transition temperature T _rg is reduced to the maximum value when the addition amount is 4%. The larger ΔT and Trg are, the easier it is to form an amorphous state and the lower the critical cooling rate will be. The results of the thermodynamic analysis also prove that the appropriate addition of yttrium can improve the GFA under low raw material purity and low vacuum conditions. In 2006, similar results were obtained by adding yttrium to Cu-Zr-Al amorphous alloy, as shown in Figure 7, as the yttrium content increases from 0.5 to 2.5 at.%, XRD patterns of the resultant alloys generally show a diffusive amorphous hump with only a few small crystallization peaks, and especially, there are no obvious crystallization peaks observed for the sample with 2% yttrium (Zhang et al. 2006). Moreover, the fusion entropy was calculated after adding yttrium as the added yttrium in the Cu-Zr-Al alloy system is larger than 2 at.%, ΔSf decreases. As a result, the total free energy barrier will decrease, thus facilitating the formation of the crystalline phases of Cu₁₀Zr₇ and CuZr.

The influence of adding a proper amount of rare earth elements to the amorphous alloy on the amorphous formation ability can be summarized as follows:

1) The atomic radius of rare earth elements is larger than that of ordinary metal elements, and the atomic size difference δ is larger after adding rare earth elements, making the entropy ΔS larger.

In the previous discussion, the forming ability of amorphous alloys was analyzed from the point of view of thermodynamics and kinetics. Entropy and enthalpy alone cannot control the formation of the amorphous phase. In the Gibbs thermodynamic equation, the formation of amorphous alloys is only expressed as the interaction of entropy and enthalpy. However, in order to describe which parameter of entropy and enthalpy contributes more to the formation of amorphous alloy, it is necessary to express the proportional relationship between entropy and enthalpy. Taking into account the factors of entropy of mixing, enthalpy of mixing, and melting temperature, Ω parameter is proposed, as expressed by the following Equ. (9)   Ω = T m Δ S mix Δ H mix , (10) where ΔS _mix is the entropy of mixing, ΔH _mix is the enthalpy of mixing, and T _m is the melting point of the mixture. When Ω > 1, the effect of TΔS _mix is larger than that of ΔH _mix, which becomes the dominant factor. According to Eq. 4, the free energy difference between the solid and liquid phases decreases and the alternative nuclear energy of the crystal nucleus increases so that it is more favorable to the formation of an amorphous phase. In contrast, if Ω < 1, ΔH _mix is the main factor, the difference in free energy between the solid and liquid phases is larger, and the driving force for the formation of the crystal phase is greater, which is not conducive to the formation of an amorphous phase. Synthesizing the role of the atomic size difference δ in the formation of the amorphous phase and drawing a two-dimensional diagram of Ω–δ, as shown in Figure 8, the amorphous phase falls in the range of parameter values Ω < 1, δ ≥ 5%. The Ω parameter can predict whether the amorphous phase can be formed before the formation of the alloy in order to adjust the composition and improve the accuracy and efficiency of the process.

The capacity for amorphous alloy formation belongs to the internal factors of the process, just as the cooling rate is the external factor affecting amorphous formation. Currently, of the more mature technologies available for the preparation of amorphous alloys, arc melting–copper mold suction casting is the most commonly used method. As shown in Figure 9, the main problem with this technique is that during the suction casting of the copper mold, the upper and lower temperature gradients are different, resulting in different cooling rates. Crystallization occurs when the melt close to the upper suction gate is cooled, resulting in the amorphous alloy sample not being a completely amorphous structure, and the size and quality of the amorphous sample are therefore limited. Based on this research, how to eliminate the cooling rate gradient has become the key to solving this problem; hence, the method of three-dimensional balanced growth is proposed. The core of this method is the balanced growth of amorphous alloys. The term “balanced growth” comes from the classical economic theory of balanced growth (Nath and Zong, 1963; Xia and Wang, 2005). This is designed to achieve mutual coordination and common growth of various industries and departments through large-scale investment in various sectors of the national economy at the same time. As representatives of the theory of balanced growth, Rosenstein-Rodin and Knox have believed that to promote the balanced development of the national economy, the government must invest economic and policy support according to the needs of the development of various regions, industries, and fields to ensure the development of backward areas, industries, and fields. Taking an opposing view of balanced growth, the economist Herman put forward the counterbalancing theory of “unbalanced growth” and thought that government support would lead to inertia in economic development and hence put forward the concept of an economic “development pole.” Some of the areas with rapid economic development promote the economic development of more backward areas. The policy of regional economic development implemented in China draws lessons from this theory. The current regional economic development, urban agglomeration, and the two Silk Road development belts promote the economic development of the eastern and western regions and the coastal inland areas to achieve the development of the point belt and the line to finally realize the highest ideal of common prosperity. Considering this economic theory, there are similarities and differences to be observed between the amorphous process of copper mold suction casting and the development of the economy. During suction casting, part of the melt’s temperature that first enters the mold is quite different from that of the mold, which can be cooled rapidly in a short time, and the cooling rate is greater than the critical cooling rate. The amorphous state is formed and, with a gradual increase in the melt, the temperature in the mold and the inner environment of the mold increases, which affects the cooling rate of the melt. (From the axial point of view, the size of the amorphous formation is uneven; from the radial point of view, at present, because the diameter of the existing bulk amorphous alloy is small, there is little difference in the cooling rate between the core and the surface.) Such a state of unbalanced growth is similar to the “development pole” of economic theory, with rapid development in some regions and a more prominent economic level. In the regions with slow development, the economy lags. Economic development and the growth of amorphous alloys both hope to finally achieve the goal of balanced growth. This concept is applied to solve the phenomenon of nonequilibrium growth in the preparation of amorphous alloys. As with the investment in primitive and regional fields in economics, there is a cooling rate gradient in the process of suction casting of amorphous alloys, and their formation in different parts of the process is different. By reforming the original copper mold, the part with a slow cooling rate can increase it and realize balanced growth of the amorphous alloy.

A schematic diagram of the copper mold is shown in Figure 10. On the basis of the original copper mold, the cylindrical copper mold has a uniform series of holes punched in the side, and the thermocouple is connected symmetrically on both sides. The thermocouple is a common device for measuring and controlling the temperature. The temperature signal can be converted into a voltage and the apparatus does not need an external power supply. Having high sensitivity, it is often used to measure the temperature of gases or liquids in the furnace and the pipe and the surface temperature of the solid. Using a thermocouple, the temperature changes during the suction casting of the copper mold can be monitored in real time, thus reflecting the rate of cooling. The position of the slower cooling rate is judged according to the signal transmitted by the thermocouple, and then the input cooling medium or phase change materials, such as gallium-indium alloy, absorb heat during phase transformation and achieve a cooling effect on the melt in the mold, which increases the cooling rate. The cooling rate gradient is approximately zero (there are cooling rate gradients in both axial and radial directions, as well as the axial cooling rate gradient, shown in Figure11), thus eliminating the cooling rate gradient to achieve the effect of balanced growth.

The core aim of the three-dimensional balanced growth theory is to eliminate the cooling rate gradient, but it is not limited to this objective. First, the design of alloy composition is based on the design concept of “entropy regulation:” 1) δ > 12%; 2) the enthalpy of mixing ΔH _mix should be sufficiently negative; 3) the entropy of mixing ΔS _mix should be within a certain range. Theoretically, the three different parameters described above take different values when measured from three different angles. Second, the cooling rate of each point reaches relative equilibrium and increases in the three spatial dimensions. Finally, the amorphous alloy is obtained by 3D printing, ultrasonic manufacturing, thermoplastic processing, and other preparation techniques, which is also balance in three-dimensional space. In general, there are three dimensions in terms of composition design, elimination of cooling rate gradient, and preparation technology.

In the latest research on the size of bulk amorphous alloys, the size of La amorphous alloy was 100 mm (Li et al., 2021); that is, multilayer components have succeeded in being stacked by thermoplastic processing. In this process, a cooling system was added to the thermoplastic equipment, the temperature was added to the supercooled liquid region, and pressure was formed when the alloy melt became viscous. The function of the cooling system is to quickly cool the viscous melt thus formed below the glass transition temperature in order to form an amorphous alloy. Because the size of the component is very small, when the sample is cooled in the cooling system, the cooling rate gradient can be ignored, which can ensure the integrity of the amorphous state of the alloy. An increase in the three-dimensional size of the amorphous alloy can be obtained by this new process.