Correlation between the Arrhenius crossover and the glass forming ability in metallic glasses

The distinctive characteristic of the metallic glass-forming system is that the variation in viscosity with temperature obeys Vogel-Fulcher-Tammann (VFT) relationship in the undercooled state and Arrhenius relationship in the high temperature region. A dimensionless index has thus been proposed based on the Arrhenius-VFT crossover and the classical nucleation rate and growth rate theory to evaluate the glass-forming ability (GFA). The indicator G(a) is expressed with the combination of T g, the glass transition temperature, T x, the onset crystallization temperature, T l, the liquidus temperature, T 0, the VFT temperature, and a a constant that could be determined according to the best correlation between G(a) and the critical cooling rate (R c). Compared with other GFA indexes, G(a) shows the best fit with R c, with the square of the correlation coefficient (R 2) being 0.9238 when a = 0.15 for the 23 various alloy systems concerned about. Our results indicate the crossover in the viscosity variation has key effect on GFA and one can use the index G(a) to predict R c and GFA for different alloys effectively.

Viscosity characterizes the relaxation time of the atoms or molecules in a liquid, and its magnitude plays a key role in the formation of glass phase. Different materials have different viscosity-temperature relationships, as Fig. 1 shows. For a strong liquid system which is a natural glass such as SiO 2 and GeO 2 , the relationship between the viscosity and temperature (η-T, with η the viscosity and T temperature) obeys the Arrhenius relationship (line a), and the viscosity increases strongly with the decrease of temperature. For a metallic system which cannot form glass, η-T also obeys the Arrhenius relationship (line b). However, the relaxation time of the atom in this kind of materials is short and the viscosity increases slowly with the decreasing temperature. For the metallic glass-forming systems, the investigations in recent years have revealed that η-T obeys the Vogel-Fulcher-Tammann (VFT)-type relationship 1 : f 0 0 0 as line c in Fig. 1 gives. In the above equation, η 0 is the pre-exponential constant, D f the fragility parameter, and T 0 is the VFT temperature. At temperature above the liquidus T l , the viscosity still behaves in an Arrhenius-like way. However, it increases sharply with undercooling and obeys the VFT relationship with deviations from the linear Arrhenius. As the temperature decreases and comes close to the glass-forming temperature, the viscosity obeys the Arrhenius relationship again 2 . This slope change, or Arrhenius-VFT (non-Arrhenius) crossover, has aroused much research interest in recent years [3][4][5][6] .
Since the increase of viscosity leads to the decrease of diffusivity, and the smaller diffusivity results in shorter characteristic solute diffusion length (D/V, D is diffusion coefficient and V is the growth velocity of crystals), it will cause the solute diffusion length to nano scale, which is a necessary condition to obtain glass in metallic system. This raises the question that how to consider the crossover from Arrhenius-VFT in viscosity in judging GFA in metallic systems.
The primary GFA indexes do not consider the contribution of the crossover, such as = T T T / rg g l and γ = + T T T /( ) x g l relatively small and not desirable. Later, the slope of the η-T curve was introduced into indexes. Oleg N. Senkov 9 proposed an indicator: g is the fragility index which determines whether a liquid is strong or fragile, m min is the minimum fragility index value approximating to 16 for metallic glass, and τ is the relaxation time or viscosity. This parameter combines the kinetic critical indication T rg and fragility index m from VFT relation. On one hand, with higher T rg , the nucleation frequency is restrained which stimulates the formation of pure glass phase. On the other hand, a smaller fragility index m denotes the characteristics of a stronger liquid. Long et al. 10 denotes the relaxation time that varies proportionally with viscosity at the nose of TTT (time-temperature-transition) curve, and the application of these two parameters shows that they have better correlation with GFA for various alloy systems than T rg and γ. The improvement of the indexes indicates that except for the kinetic process, temperature-related viscosity also contributes to the glassy phase transformation. Takeuchi et al. 11  is proposed as only a derivative from the new VFT plot for viscosity and an analog to T rg with its physical meaning probably to be explained from the aspect of VFT-type viscosity. Based on the discussions on the above parameters, it reveals that the temperature-dependent viscosity, especially the slope change at the Arrhenius-VFT crossover, has a significant influence on R c and GFA. A large slope at high temperatures and a small slope at low temperatures make the curve approach to the shape of the strong liquids and will be beneficial to form glass phase as can be seen in Fig. 1. In this study, we will consider this point and correlate GFA and the effect of crossover. The correlation is performed firstly by considering the relationship between R c and the nucleation rate and growth rate [12][13][14] , then the parameter is proposed by connecting the slope change in the viscosity curve with R c . The derivation process is displayed in the "Method" section. The decent correlation between the parameter and R c in various glass-forming systems proves the validity of starting from the classical theories. Finally, the new index is compared with some parameters proposed before and the result turns out that it has the best correlation (R 2 = 0.9238) with R c among them, validating our indicator is more reliable to characterize GFA.

Results and Discussions
Now, the index G(a) derived in the "Method" section is applied to different alloys to verify its validity. The data of 23 glass-forming alloys including bulk metallic glass (BMG), like vitreloy, etc. and marginal glass-forming systems (R c is more than 10 3 K/s) is collected, as given in Table 1, for their parameters used in equation (15) can be found in literatures.
The parameter a is determined according to the best fit between R c and G(a) and the relationship between various a and R 2 is then given in Fig. 2. Each a corresponds to a certain R 2 and the maximum of R 2 is 0.9238 at a = 0.15. When a = 0, i.e. only considering the influence of m Tl on GFA, R 2 = 0.9128. This means compared to the "critical temperature item", the influence of "viscosity item" on GFA is much more significant.
By using the value of a = 0.15 and a = 0 respectively, G(0.15) and G(0) are calculated as functions of R c and shown in Fig. 3, in which the variations in T rg , γ, F 1 , and ω are also presented for comparison. Data is from the 23 various alloy systems and R 2 is decided by statistical analysis. Among these GFA indicators, G(0.15) has the best correlation with R c , and their relationship can be expressed as: This equation could be used to predict R c for the 23 various metallic glass systems and the data of more systems are needed to further validate the equation. Now, the mechanisms for the better correlation of G(a) than other parameters will be discussed. Both critical temperatures and high viscosity are key factors to influence GFA. For G(a), it considers the contribution of the Arrhenius-VFT crossover in viscosity curve. Other parameters, however, consider little about it. When concentrating on the particular viscosity at a certain temperature, from equations (1) and (10), we have: VFT g 0 0 0 Therefore, the relationship between η l and T l could be obtained as: the same as about 10 −5 Pa s for many liquids. In this light, larger η l leads to bigger m Tl which is related firmly with better GFA. According to the schematic Fig. 1, GFA is proportional to the viscosity at the liquidus temperature. For showing this effect, the viscosities of four different La-based metallic glasses are calculated by using equation (3) − − which is also an indication for bigger m Tl and smaller m Tg . This inverse relationship between η l and R c indicates that our indicator is reliable. In this sense, temperature-dependent viscosity is crucial for determining GFA. When considering the classical nucleation rate and growth rate equations (5) and (6), we find that compared to nucleation rate I, growth rate U is much more dependent on the temperature-dependent viscosity because the value of its square brackets is in the range from 0 to 1. From this perspective, what contribute more to the glass formation are the sluggish diffusion (high viscosity) and the resulting low growth rate. The nucleation rate could be high but the nuclei could not grow because of the nano-scale diffusion length.

Conclusions
From the above analysis, it can be concluded that a new GFA indicator G(a) for BMG and marginal metallic glasses is proposed based on the Arrhenius-VFT crossover at T l as well as the classical nucleation rate and growth rate theory. This index is proved to have better correlation with R c and GFA for various alloy systems than other parameters proposed before. Furthermore, the calculated results also validate the dependability of using the classical theories mentioned before as the foundation of finding out a new indicator for GFA. Meanwhile, from the analysis result, for simplicity the attention could be paid on the "viscosity item" m Tl , which is the slope at the crossover temperature. This parameter reveals that the temperature dependent viscosity, especially the crossover at T l and the corresponding viscosity, are crucial for GFA. This could be guidance for developing new glass-forming systems. To be specific, researchers could measure the viscosity at T l , which is applicable because the temperature is relatively high. They could choose the systems with high viscosity at T l and try to synthesize bulk glass in it. In this sense, the parameter is a theoretical guidance for fabricating new glass-forming systems and could save lots of unnecessary efforts.

Methods
To correlate the viscosity with the nucleation and growth theory, the following equations for the homogeneous nucleation rate I and growth rate U are used [12][13][14] :  From the amorphous perspective, the crystalline phase has been suppressed until the glass-forming temperature reaches. Therefore, the fraction of the crystallized volume fraction f c is usually set to be less than 10 −6 . As a result, R c required for glass formation is determined as 19,20 :