Evolution of glass forming ability indicator by genetic programming
Graphical abstract
The genetic programming structure for evolution of GFA expression in terms of Characteristics temperature (Tg, Tx and Tl) of bulk metallic glasses. Crossover of two parent computer programs, forming two offsprings. Parent:1 represents the expression (Tx/Tg) ∗ (Tx/Tg) ∗ (1/(Tg–Tl)), and Parent 2 the expression (Tx/Tg) + (1 − (Tx/Tl)), while their offsprings Child 1 and Child 2 respectively represent the expressions (Tx/Tg) ∗ (Tx/Tg) ∗ (1/(1 − (Tx/Tl))) and (Tx/Tg) + (Tg–Tl). The operators are represented by square nodes, containing one element from the set of functions F, while the terminals (constants or variables) by circular nodes containing one element from the set T.
Introduction
The critical diameter (Dmax), i.e., the maximum possible thickness of the metallic glass that can be produced from an supercooled liquid, has emerged as a practical consideration to compare the relative potential of glass formation in bulk metallic glasses (BMGs). The value of Dmax is in inversely proportion to the critical cooling rate (Rc) which is conceptually regarded as a good quantification of its glass forming ability. The experimental determination of Dmax is relatively easier than Rc. However, multiple experiments are still unavoidable to determine Dmax for each alloy composition which is eventually dependent on the fabrication method [1], [2].
The experimental determination of characteristic transformation temperatures of the metallic glasses is relatively easy using differential scanning calorimetry (DSC) or differential thermal analyser (DTA), as compared to other physical and thermal property measurement such as viscosity of a melt, in the course of estimation of glass forming ability [2], [3]. This fact has led to the development of expressions representing glass forming ability in terms of characteristic temperatures. A single experimental synthesis of an alloy followed by characteristic temperature measurements using DSC/DTA is sufficient for estimating the maximum possible diameter of glassy structure of the alloy. Nevertheless, such expressions in terms of characteristic temperatures should be well correlated with Dmax and be consistent with the physical metallurgy principals. Therefore, developing GFA expression in terms of characteristics temperatures, having high correlation with critical diameters of BMG alloys, can significantly reduce the experimental effort to determine glass forming ability of a composition with reasonable accuracy.
Inoue proposed to measure GFA on the basis of stability of the metallic glass against crystallization while heating, quantified as ΔTx (= Tx–Tg) [4]. It implies that a good glass former should have either high onset temperature of crystallization Tx or low Tg. It may be noted that the first GFA expression proposed by Turnbull (Trg = Tg/Tl) suggests that Tg should be high for good GFA, which is essentially contradicted by the Inoue’s proposition. The correlation coefficients with the experimentally measured Dmax (and log Rc) values were in better agreement with ΔTx compared to Trg, which favoured ΔTx to be accepted as a better GFA indicator.
Subsequently, several other indicators have been proposed on the basis of similar empirical considerations of glass formation in terms of Tg, Tx and Tl like γ, γm, δ, new-β, ω, etc. [2], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Despite of substantial efforts devoted in the development of reliable GFA expressions for past few decades, the frequently used expressions like γ and ω parameters have been recently reported to fail to represent the glass forming ability of the BMGs [15], [22], [23], [24]. Attempts have been made to develop expressions like Φ [10] and θ [25] by fitting in suitable indices to pre-fixed expression in terms of the characteristics temperatures to satisfy the requirement of high correlation with critical diameter, Dmax. However, with synthesis of new compositions, these expressions also have shown poor correlations with Dmax, since the fitting indices were dependent on the training data.
It comes by no surprise that while majority of the propositions regarding GFA expressions are based on thermodynamics and kinetics based considerations. Recently, Chattopadhyay et al. have classified the reported glass forming ability expressions into four categories on the basis of characteristic transformation temperatures, thermodynamic parameters, topology based parameters and phase transformation kinetics based parameters [26]. The authors have shown that kinetic viscosity based approach to estimate glass forming ability correlates well for few compositions (45 in number), with the critical cooling rate calculated from the TTT diagram, unlike other three proposed classes. However, this criterion is not statistically tested over large number of alloys since the kinetic data have not been reported. Similar approach has been considered by Johnson [27] and Kozmidis-Petrović [28] where the availability of kinetic data limits its practical application. Therefore, GFA expression in terms of characteristics temperatures with high correlation with Dmax remains a pragmatic solution to estimate glass forming ability of a composition.
There has been at least two attempts [10], [25] in developing the GFA expressions using statistical considerations. Multivariate statistical analysis has been attempted for feature extraction and classification of bulk metallic glasses by Tripathi et al. [29]. While statistical regression may be seen as the embryonic form for dealing with huge amount of data, giving meaning to it, and helping scientists to arrive at theories accurately. The advent and subsequent advances of the computing power and associated storage capacity of computers have laid open a new avenue for researchers to explore i.e., data analytics or informatics. Materials informatics, being a fast developing and highly successful paradigm for dealing with quantitative structure-attribute relationships [30], [31], [32], [33]takes advantage of this huge search potential in finding appropriate structure that would best fit into a given physical system, and provide plausible physical explanation to the obtained expression.
In view of the above, the present study attempts to evolve an expression using the transformation temperatures having high correlation with critical diameter Dmax by employing symbolic regression. It is imperative to mention here that symbolic regression is a very versatile technique capable of forming any arbitrary expression by combining a set of operators and given variables into a flexible form with no a priori assumptions regarding its structure. A set of 410 BMG data, compiled by Long et al. [12], has been undertaken to evolve the developed expression(s), referred as training data. Additional 184 BMG data have been compiled after publication of Long et al. [12] and the coefficient of correlations have been calculated to evaluate the performance of evolved expression(s).
In context of compositional design, evolutionary computational approaches have already been explored in the field of BMG composition design problem. One of the pioneer work on evolutionary computation applied to BMG composition design problem was done by Dulikravich et al. [34], where the composition evolution has been formulated in a multi-objective optimization sense considering simultaneously maximizing Tg, Tl and Tg/Tl and minimizing density of the designed alloys. In this work, the evolutionary tool of genetic programming has been used to evolve subsequent generations of flexible expressions representing GFA (constructed from the symbols of addition, subtraction, multiplication and division, operated on the characteristic transformation temperatures Tg, Tx and Tl) which gradually improve over the generations to produce the best possible expressions of a GFA indicator.
Section snippets
Methodology
Since the efforts made so far in developing a GFA parameter is constructed by the terms Tg, Tx and Tl using the operators addition (+), subtraction (−), multiplication (×) and division (/) following thermodynamic and kinetic interpretation of the operators {+, −, ×, /} in the expression and the constituent factors. Here, it is proposed to search for an expression that can be assembled using Tg, Tx and Tl and the operators {+, −, ×, /} which eventually describe the experimentally observed Dmax
Comparison of the correlations with Dmax
Table 1 compares six expressions evolved using genetic programming along with 15 reported GFA expressions. A GFA expression should ideally:
- (i)
Show high coefficient of correlation with Dmax of BMGs (training data, testing data, and all the reported data).
- (ii)
Log Rc being more fundamental measures of the GFA as compared with Dmax, the evolved expressions should show high correlation with log Rc for reported 66 BMG data [13].
- (iii)
The critical cooling rate, Rc, discretely separates the possibility of glass
Conclusion
To summarize, the parameter ; has been evolved with the help of genetic programming, using the characteristics temperatures, to estimate the glass forming ability of metallic glasses. The Gp yields a correlation coefficient of 0.67 for 594 bulk metallic glasses in contrast to its nearest best reported parameter new-β with a value of 0.57. The Gp, qualifies to be the improved expression for GFA not only in terms of the correlation value but also due to the fact that it
Acknowledgement
The authors are thankful to Prof. B.S. Murty, IIT Madras for stimulating discussions on the subject.
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