Critical evaluation of glass forming ability criteria

The available glass forming ability criteria have been examined by classifying them into four basic categories depending on critical temperatures, thermodynamic quantities, topological and kinetic aspects of glass forming alloys. A large number of glass forming alloys of widely varying natures and origin have been analysed with their experimentally measured properties to assess their glass forming ability. A novel approach using kinetic viscosity of glass forming alloys obtained by the Vogel–Fulcher–Tamman equation and the critical cooling rate calculated from the TTT diagram is demonstrated as an excellent universal glass forming ability criterion. Moreover, thermodynamic and topological modelling results through computation of a novel PHSS parameter for various alloy compositions spanning different alloy systems have rendered qualitative guidelines on propensity for glass formation in multicomponent alloy systems. Besides, the importance of kinetic interpretation of PHSS range observed for glass forming alloys is also elaborated.


Introduction
Glass forming ability (GFA) is defined as the ease of vitrification for a material. In the case of cooling from liquid state, a higher GFA suggests a lower critical cooling rate and a higher section thickness for the glass. There has been extensive research to identify alloys with higher GFA with the help of various parameters comprising of different properties of a material.  Different GFA criteria can be categorised into four basic classes as shown in Table 1. Category A consists of criteria depending on critical temperatures of glass forming alloys, 1-21 category B shows the criteria depending on the thermodynamic parameters calculated based on Miedema's approach, [22][23][24][25][26][27][28] category C enlists the GFA criteria based on internal atomic structure and atomic arrangement of constituting elements in a glass forming alloy [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] and the category D suggests the criteria depending on the kinetic properties of the glassy materials. [45][46][47][48][49][50][51][52][53][54] Only those GFA criteria are listed where one criterion has been applied on various bulk metallic glass (BMG) forming systems and has been proven for its compatibility. If a criterion is proposed and proven only on the basis of one single alloy system with varying compositions or is proven on only a particular order of alloys (say binary alloys), then that GFA criterion has not been considered for this present comparison. Out of the entire list in Table 1, the following criteria that are very widely used are discussed in detail and examined for their compatibility in the present paper.
Category A: These are criteria based on the critical temperatures of a glassy material, i.e. the liquidus temperature T l , the temperature of onset of crystallisation during heating T x and the glass transition temperature T g . Several GFA parameters have been defined in this category with different combinations of these characteristic temperatures. T rg ( ¼ T g /T l ) by Turnbull, 3  Category B: These have parameters consisting of thermodynamic properties evaluated through mainly Miedema's model 22,25,26 like chemical enthalpy of mixing DH chem , elastic enthalpy DH elast , topological enthalpy DH topo , total enthalpy of amophisation DH amor , enthalpy for formation of intermetallic compounds DH inter and enthalpy for formation of solid solution DH ss . Das et al. 55,56 have shown that plot of DH amor versus DH ss can be effectively utilised to predict amorphous forming composition range (AFCR). Some of the criteria have been derived from these thermodynamic properties such as c * [(DH amor )/ (DH inter 2DH amor )] by Xia et al. 28 and c9 [(DH liquid-DH amor )/(DH inter ) 2 ] by Ji et al. 24 Category C: These have parameters based on atomic sizes/arrangements such as mismatch entropy DSs proposed by Mansoori et al., 29 configurational entropy Table 1  Zhang et al. 20 14 T rK T rK ¼ T K /T l , where, T K is the Kauzmann's temperature Cao et al. 16 15 where T 0 is the temperature corresponding to zero configurational entropy Senkov et al. 9 16 F 2 parameter Kozomidis et al. 9 17 Kozomidis et al. 9 18 F KA parameter Zhang et al. 19 24 Weienberg's parameter Weinberg et al. 14

25
Hruby's parameter Hruby et al. 21 26 Saad and Poulin's parameter where g 0 is the pre-exponential factor. D is the fragility parameter, which is a measure of how strong a liquid is to retain its amorphous atomic structure on solidification. The more is the value of D, the stronger is the liquid and better is GFA Angell 45 53 Modified fragility parameter D The VFT equation has been solved analytically with three sets of (g,T) values at T l , T x and T g for BMGs, and the D value obtained in that way reflects the GFA in much better way These parameters consist of mainly mobility of atoms in liquid phase and possibility of formation of a particular structural configuration, which can be known through fragility of a liquid alloy. Fragility is an inevitable part of viscosity of a glassy material. Moreover, nucleation and growth based factors like critical cooling rate R c , which depicts how slowly a liquid can be cooled to avoid crystallisation, can be a very good measure of GFA. For the criteria of this category, the critical cooling rate can be theoretically determined by the application of the knowledge of viscosity as a function of temperature according to Weinberg et al. 52 Earlier, Guo et al. 60 have examined the compatibility of various GFA criteria consisting of critical temperatures only on the basis of critical cooling rate. Further, Yang et al. 61 divided the available GFA parameters into four classes and examined them. However, any kind of universality was not found out with any criterion so far. The present paper carefully examines the utility of different criteria from all the above four categories while predicting GFA of a number of glassy alloys, oxides and organic glasses with an aim to find out universality of performance by GFA criteria. Recently, a new technique has been proposed by the current authors 47 for the evaluation of viscosity in the form of Vogel-Fulcher-Tamman (VFT) equation in the temperature range of T l to T g . In the present work, a modified fragility parameter of the VFT equation is considered to understand the GFA. In addition, with the help of modified viscosity of the liquid as a function of undercooling, refinement in theoretical critical cooling rate for the formation of glass has also been carried out. Moreover, the refined theoretical critical cooling rates and modified fragility parameters have also been utilised for checking the GFA in comparison to critical thickness achieved for a number of glasses. A comparative study has also been carried out to examine the suitability of current approach of defining GFA with already existing criteria.

Existing GFA criteria
Glass forming ability criteria based on critical temperatures of glassy alloys T rg : Turnbull 3 has first shown that the ease of formation of glass can relate to critical temperatures of the glass.
It has been observed that the higher the T g and the lower the T l of a glassy alloy, the better is its GFA. Since GFA / T g and GFA / 1/T l , a combination of these two produces GFA / T g /T l (T rg ). It has been shown that, if T rg is w0.6, the alloy can be considered to have higher GFA.
DT x : Inoue et al. 7 have shown that the tendency of devitrification increases with the decrease in T x , i.e. the higher the T x , the more is the chance of a glassy alloy to retain its amorphous nature. Thus, the range of stability of a glass is from T g to T x . Inoue proposed that the GFA can be correlated to the glass stability range, i.e. T x 2T g , which has been termed as DT x . A larger DT x indicates higher GFA.
a Parameter: Mondal and Murty 15 have proposed that, as the T x increases, the stability of a glass increases and ease of vitrification increases with decrease in T l . Thus, GFA / 1/T l and GFA / T x results in GFA / T x / T l , which has been termed as a parameter.
b Parameter: Mondal and Murty 15 have shown that, in b parameter, GFA / (T x /T g ) þ (T g /T l ) or GFA / 1 þ (T x /T l ) when T x ¼ T g . Yuan et al. 18 c Parameter: Since different glasses have varied T g , DT x ¼ T x 2T g has been normalised to T g as per Lu and Liu,13 i.e. (T x 2Tg)/T g ¼ [(T x /T g )21]. Thus, GFA is proportional to T x /T g . Moreover, GFA is also directly related to T x /T l resulting in GFA / (T x /T l )(T x /T g ). In another way, GFA is proportional to [(T g /T x )(T l /T x )] 21 . Averaging the term inside second bracket results in GFA / 1/ 2[(T g þ T l )/T x ] 21 and, finally, GFA / T x /(T g þ T l ).
d Parameter: Since GFA / T g /(T l 2T g ) and GFA / T x /T g , Chen et al. 2 proposed that GFA / [T g / (T l 2Tg)](T x /T g ), which results in GFA / T x /(T l 2T g ).
Q Parameter: According to Fan et al., 5 GFA is proportional to T rg , and they have found out that GFA is directly related to (DT x /T g ) a , where a has been estimated to be ¼ 0.143. Thus, GFA'T rg (DT x /T g ) 0.143 .
v Parameter: Ji and Pan 8 have proposed a new GFA criterion based on the free energy change for liquid-solid transformation according to Thomson and Spaepen, 62 c c Parameter: Guo et al. 6 have shown that GFA can be related toð3T x 2 2T g Þ=T l .

Parameters consisting of thermodynamic properties
Miedema has proposed ways to calculate enthalpies assuming the macroscopic model of atoms. 22,25,26 The enthalpies have been calculated assuming additive rules of enthalpies extending the Miedema's approach. The interactions of higher order than binary have been assumed to be absent. All forms of enthalpies have been calculated for binary subsystems and added in combination to form ternary, quaternary and quinary, e.g. for quinary alloy ABCDE Thus, enthalpy of chemical mixing where DH chem ij is the interaction parameter between i and j, and c i and c j are the atom fractions of elements i and j respectively. Here where x i is the mole fraction of element i, and DH sol i in j is the enthalpy change of solution where j is solvent and i is solute. The values of solution enthalpies are taken from Niessen et al. 63 where V i is the molar volume of element i The enthalpy of formation of amorphous phase is where l ¼ 5 for amorphous forming enthalpy because of the short range order (SRO) observed in the glassy structures. 64,65 Equation (5) can be used for the calculation of enthalpy of formation of solid solution DH SS chem by incorporating l ¼ 0 and enthalpy of formation of intermetallic compound DH inter chem by l ¼ 8. The solid solution has been assumed to have no order, and the intermetallic compound has medium to long range order. 59,64 Besides, the elastic enthalpy change has been calculated according to Eshelby 65 and Friedel 66 as where DH elastic ij is the elastic enthalpy parameter of a constituent binary system, which can be expressed as where DH elastic i in j is the elastic enthalpy of ith element in jth in the dilution limit, and it is defined as where K, G and W are bulk modulus, shear modulus and corrected molar volume 63 respectively. The topological enthalpy 22 is where x i and T m ,i are the mole fraction and melting temperature of ith element respectively. Another form of enthalpy for alloy formation is structural enthalpy. However, in the present work, it has been neglected. It is to be mentioned that Basu et al. 67 have also neglected this term. The total enthalpies of formation are calculated with the combination of the above fundamental enthalpies. The total enthalpy of amorphous phase formation Total enthalpy change for formation of solid solution is and total enthalpy of formation of intermetallic compound is In the present work, the enthalpy for liquid formation DH liq has been assumed to be equal to the enthalpy for chemical mixing only. Amorphous forming composition range: Das et al. 55,56 have proposed a new approach to evaluate AFCR effectively. The widely used approach based on Miedema's model considers DH amor ,0 and DH amor ,DH ss . However, it does not take care of intermetallic phase formation separated out from amorphous phase formation field. Thus, Das et al. 55,56 proposed two simultaneous conditions to be followed to precisely define only amorphous forming composition range. The two conditions are DH amor ,DH ss as well as DH amor ,DH mixture . c * Parameter: Xia et al. 28 have shown that two factors affect glass formation: (i) the driving force for the formation of amorphous phase, i.e. 2DH amor (ii) the resistance against formation of amorphous phase with respect to formation of intermetallic compounds, i.e. DH amor 2DH inter . The higher and lower the values of the former and latter respectively, the higher is the GFA. Thus, Xia et al. 28 have proposed a new GFA criteria based on the formation enthalpies of amorphous phase and intermetallic compound as c9 Parameter: Ji et al. 24 have suggested that the stability of liquid phase in competition with intermetallic phase can be expressed as GFA'DH liq /DH inter . On the other hand, the stability of amorphous phase in competition with intermetallic compound is (GFA') DH amor /DH inter . Thus, a new GFA parameter has been defined as Parameters with atomic structure/arrangements

Mismatch entropy
Mansoori et al. 29 have proposed an entropy term that arises when more than one element are mixed together from the mismatch in atomic sizes of the constituent elements. They have concluded the change of entropy to be where k B is Boltzman's constant, and f ¼ 1/(12j), where j is the packing fraction of the alloy. Here, the packing fraction has been considered to be 0.64, i.e. a dense random packing is assumed as suggested by Takeuchi and Inoue 68 c i and c j are the atom fractions, whereas d i and d j are the atomic diameters of element i and j respectively. Here where k ¼ 2 or 3. It means that k can take either of two integer values 2 or 3.

Configurational entropy
The configurational entropy has been estimated to be where R is the universal gas constant, and x i is the molar fraction of element i. Bhatt et al. 30,31,32,33 have shown that two parameters can be utilised to decide the amorphous forming ability of a particular alloy. Two compound parameters are formed from these two entropy terms, and they are as follows: (i) DH chem (DS s /k B ), which is the multiplied enthalpy of chemical mixing by mismatch entropy change normalised to Boltzman's constant (ii) (DS config /R), which is the configurational entropy change normalised to the universal gas constant. P HSS : Rao et al. 34 have developed another parameter that takes care of enthalpy of mixing, configurational entropy and mismatch entropy altogether. The parameter has been termed as P HSS , which has been defined as Lattice strain model: Ray and Akinc 59 have proposed a new parameter that involves thermodynamic enthalpy with local lattice strain energy evaluated through modified Miedema's model and modified Miracle's model 57,58 respectively. The criterion has been defined as follows DH ch;ss pq þ min n r¼1 ðDH el molar;r Þ where DH ch;am uv is the enthalpy of u-v amorphous binary, c k and T k m are composition and melting point of the kth element, DH ch;ss pq is the enthalpy of formation of p-q binary solid solution and min n r¼1 ðDH el molar;r Þ is the minimum possible lattice strain energy of the alloy.

Viscosity of glassy alloys and compounds
Viscosity of a glassy alloy or compound in the temperature range between T l and T g has been well described by several researchers. 45,[69][70][71] Among these, the VFT equation is widely used. The equation describes viscosity at any temperature where g 0 is a pre-exponential factor, D is the fragility parameter of the liquid and T 0 is the ideal glass transition temperature. Here, the parameter D reflects how strong a liquid is to retain its liquid structure when solidified. The larger the D, the stronger the liquid is and the better is the GFA. So far, the VFT parameters of viscosity equation are used to be evaluated through a number of experimental viscosity data at respective temperatures followed by fitting the data according to the VFT equation. A solution of VFT equation thus obtained could reflect GFA of several alloys through its D values. However, it could not decide GFA for a number of alloys as well. Recently, the present authors 47 have proposed a new route of estimation of viscosity of a glassy alloy or compound according to VFT equation. 47 Equation (22) contains three arbitrary constants (g 0 , D and T 0 ). Thus, three sets of viscosity-temperature data (viscosity data at three different temperatures) can analytically solve equation (22) for its three arbitrary constants. The authors have shown with reasoning that the three temperatures have to be chosen in an optimum way, and the temperatures are preferably three critical temperatures of a glassy alloy, i.e. T l , T x and T g . The values of fragility parameter, thus obtained for the alloys, have been compared with the critical casting thickness Z max values of the respective alloys. Moreover, the viscosity data at or very near to these three critical temperatures are not available for a number of alloys and compounds in the literature. However, for those alloys, temperature-viscosity data at highest and lowest temperatures of the given temperature ranges are chosen as two sets of data. The third set of data is taken at that temperature, which is nearest to the midpoint of the given temperature range. The D values have been compared to the Z max values of the alloys and compounds first taking all the materials irrespective of available experimental viscosity data in the entire temperature range or not. Afterwards, the D values, where the complete data are available, are examined with respect to Z max , separately. According to the work by the authors, 47 the solution with the three data points of equation (22) is far more accurate. Thus, available D values in the literature as well as refined D values obtained using the analytical solution approach of VFT equation are used for examining fitting with Z max .

Theoretical critical cooling rate Rc
Johnson-Mehl-Avrami 72,73 evaluated a relationship of fraction transformed and time elapsed in a phase transformation as where X is the fraction of the material transformed during a phase transformation process, I is the rate of nucleation, U is the rate of growth of the critical sized nuclei and t is the time elapsed. At the beginning stages of crystallisation (transformation), the value of X can be approximated to be Assuming homogeneous nucleation, the rate of nucleation at a temperature T where D n is the diffusivity of atoms, N v is the number of atoms per unit volume, a 0 is the average jump distance of atoms and k B is the Boltzman's constant and where s is the interfacial energy of liquid and crystal, and DG v is the free energy change per unit volume. The interfacial energy s has been calculated according to Mondal and Murty 74 where V m is the molar volume, and the DG m is the free energy for solidification, which can be estimated according to Turnbull, 75 at a particular temperature T to be where DH m and T m are the enthalpy and temperature of melting respectively. Growth rate Here, the diffusivity term as in nucleation rate equation (equation (25)) has been replaced by k B T/4pg 0 according to the Stokes-Einstein relationship. 76 Now, combining the equations from equations (24)-(29), time elapsed t for a critical fraction transformed can be approximated to be where x c is the critical fraction transformed to realisation of nucleation and growth, and it is taken to be 10 26 in the present work. The plot of temperature T versus log (t) can produce theoretical TTT diagram, and the critical cooling rate R c can be calculated using the following formula where t n is the nose time.

Evaluation of various GFA criteria
A number of binary to multicomponent alloys  have been tested for their GFA according to each GFA criterion discussed above, and compared with the maximum casting dimension Z max reported in the literatures.  Each GFA criterion has been fitted with the Z max , and the goodness of fit is examined for the compatibility of the particular criterion. Tables 2 and 3 summarise the considered alloys with their T l , T x , T g , Z max and calculated values of all the GFA criteria comprising of these critical temperatures. Correspondingly, Fig. 1 shows the fitting of each GFA criterion to Z max of the alloys. Best fitting has been observed in case of b parameter proposed by Yuan et al., 18 where R 2 is 0.5 (Fig. 1e). The rest all show poor correlation with the Z max of the glassy alloys with R 2 in the range of 0.   Tables 2 and 3 have been evaluated and shown in Tables 6 and 7. Each of the criteria has been fitted to Z max , and thereafter, the goodness of fit in terms of R 2 for each case has been evaluated. Figure 2 shows all the linear fits of each criterion of category B with respect to Z max . Unfortunately, all the criteria have failed to satisfactorily correlate with Z max . Basically, Miedema's approach has been developed for binary alloys, 22,26 and it has been demonstrated to predict AFCR in some ternary 109 and higher order systems. However, it has not been found to result in useful GFA criteria over a wide range of alloys based on enthalpy alone.
Glass forming ability criteria consisting of internal structural properties are summarised in Tables 6 and 7. Entropy change due to mismatch in atomic sizes DSs and the change in atomic configuration DS config and two other GFA criteria proposed by Bhatt et al. [30][31][32][33] and Ray and Akinc 59 have also been evaluated and tabulated in Tables 6 and 7. Accordingly, the results have been fitted with respect to Z max , and the fitting results are shown in Fig. 3. Configurational entropy DS config (Fig. 3a) and mismatch entropy DSs (Fig. 3b) have shown a fitting with R 2 ¼ 0.179 and 0.0054 respectively. However, the derived criterion show poorer fit with R 2 ¼ 0.0128 (Fig. 3c) and 0.0054 (Fig. 3d) respectively. Table 8 shows the solutions of VFT equation in the form of three VFT parameters (g 0 , D and T 0 ) obtained from the viscosity data taken at three different temperatures as discussed in the section on 'Viscosity of glassy alloys and compounds'. Viscosity data of a very few alloys are available in the literature; moreover, the viscosity data in the range of T l to T g are available for even much fewer alloys and compounds. In Table 8, those alloys, for which analytical solution of the equation (22) is possible, i.e. the viscosity data at or very near to T l , T x and T g are available, are shown with italic letters and numbers. For other alloys, the viscosity data are not available in the temperature range between T l and T g . For them, the VFT parameters have been obtained by taking two points at two extremities of the data range, and the third point has been taken at or nearest to the middle of the available range.
The theoretical critical cooling rate to avoid crystallisation R c has been calculated utilising modified VFT parameters via analytical solution approach as described before. 47 Figure 4a shows the fitting of D values obtained via analytical solution of VFT equation of all the glassy materials listed in Table 8 with their respective Z max values. In Fig. 4b, only those D values of the alloys have been fitted against Z max , where the full range viscosity data are available (italic lettered materials in Table 8). Figure 4c shows the fitting of inverse of calculated R c with Z max of respective alloys of Table 9. Figure 4d shows the fitting of inverse of R c calculated via VFT parameters obtained only through full range viscosity data (italic lettered materials in Table 8) with Z max . In Fig. 4a, the solution produces a reasonable fit with R 2 ¼ 0.675. Whereas in Fig. 4b, the D values for those italic lettered alloys result in an excellent fit with R 2 ¼ 0.952. The italic lettered alloys have better result with respect to D values because of the solution obtained through the necessary three data points according to equations (22).
The D values of Pd based alloys Pd 40 Cu 30 Ni 10 P 20 and Pd 40 Ni 40 P 20 show anomalous nature with respect to their Z max values in Table 8. In spite of having very high Z max , these two alloys show high fragility, i.e. lower D values. Moreover, the Pd 40 Cu 30 Ni 10 P 20 has much higher Z max with respect to Pd 40 Ni 40 P 20 (72 and 25 mm respectively; see Table 8), but the D values show an inverse relationship. This anomalous behaviours of Pd based alloys have been explained by Fan et al. 110 who argue that that only kinetic factors like viscosity are not sufficient to explain GFA of Pd based BMGs. Utilising these viscosities, the critical cooling rate has been calculated where viscosity plays a very important role (equation (30)). All the calculated critical cooling rates are shown in Table 9. Figure 4c shows a fit of the R 21 c , where it shows a reasonably good fit with R 2 ¼ 0.692. The alloys with italic letters have been chosen separately to examine the fitting of their critical cooling rate with Z max (Fig. 4d). The fit is outstandingly well with Glass forming ability criteria of category A are sometimes considered to signify glass stability rather than GFA. Glass stability is generally considered as the stability of the glass against devitrification. According to one of the temperature dependent GFA criterion, the wider the difference between T g and T x , the greater is the glass stability. Another way of depicting the glass stability is through T g itself. The higher the T g , the greater is the fragility parameter D as per VFT equation. A higher T g suggests that viscosity of the melt increases more rapidly to 10 12-13 Pa s (the viscosity for glass formation) as it is cooled from above its melting point. Increase in fragility parameter also suggests that the melt structure becomes more complex leading to greater hindrance to diffusion of atoms for crystallisation during cooling of glass forming melts or during heating of already formed glass. Moreover, if viscosity increases, the TTT diagram    would shift to the right, favouring the glass formation with lower cooling rate, thus improving the GFA. Thus, glass stability and GFA are to some extent interrelated. Therefore, all four major categories, as indicated in the section on 'Introduction', are somehow interrelated. However, it is not easy to correlate all the four major criteria and come up with a unified criterion. However, from the above discussion, it is very clear that glass formation and glass stability are not independent parameters; rather, they are closely related, which has been demonstrated by Nascimento et al. 111 The reason for poor fit (average R 2 *0.25) of GFA criteria consisting of critical temperatures can be explained in different ways. The poorest fit shown by DT x has its inherent cause, since it measures the difference between T x and T g . 7 It reflects the stability of a glass or amorphous structure up to T x from T g during heating. However, GFA requires an understanding of the liquid at T l . How easily the liquid structure can be frozen into glassy state, and it does not have direct relevance with the stability of a glassy structure during heating. The glass formation depends on several other factors of liquid properties at T l like mobility of atoms. T rg , though it measures a ratio of Tg and T l , 3 still fails to describe GFA well (Fig. 1a). This is due the fact that Tg is measured by heating the glass but not while cooling the liquid.
It is true that Z max is not intrinsic in nature, and it depends on the processing condition. The usual methods for deciding the Z max is by solidifying the liquid into different diameter rods or by wedge shaped ingots. It is true that there could be fine nanocrystals even if the glass is amorphous as indicated by X-ray diffraction (XRD). In our discussion, we have actually used the data presented by different authors. In all these cases, the XRD patterns show amorphous structure. Even low magnification bright or dark field TEM images have shown monolithic amorphous structure. However, there are cases where, even if the low magnification TEM bright or dark field images show no crystals, high resolution images do show nanocrystalline/SRO presence. For example, a paper by Mondal et al. 112 has clearly demonstrated these differences in structures of XRD amorphous alloys. Considering these facts, we have taken the Z max reported by different researchers and tried to find a correlation in the absence of more satisfactory parameter. However, the critical cooling rate, which has to be a very good replacement for Z max , is comprised of a time factor, too, which is not taken into account in this case. Thus, the same undercooling may result in a wide variation in GFA depending on the time required to avoid the minimum crystallisation. In line with that, all the criteria composed of a combination of critical temperatures of glassy alloys do not fit well with Z max because they invariably take care of either the undercooling or the glass stability or the both. Thus, the average R 2 remains *0.25. However, only one of the nine different criteria tested here, v, 8 shows a better fit as it is proportional to inverse of driving force for crystallisation, which, to some extent, measures the ability to suppress crystallisation. Still, the goodness of fit with this parameter remains not so good with R 2 ¼ 0.456. This indicates that only the thermodynamic part of the crystallisation cannot predict GFA. goodness of fit, R 2 of each of them for their compatibility . a enthalpy of chemical mixing, b enthalpy of mixing to f orm a n amorphous phase, c topological enthalpy, d elastic enthalpy, e total enthalpy of amorphous phase formation, f enthalpy of intermetallic compound formation, g enthalpy of solid solution formation, h γ' parameter, i γ * parameter The reason may lie in the fact that the most crucial part of crystallisation avoidance is the kinetics of the process, which is not taken care by the criteria of category A.
The GFA criteria in category B are measures of thermodynamic enthalpies of several transformations.
However, not a single criterion shows satisfactory correlation with Z max as is evident in Fig. 2. In the present work, the extended Miedema's model has been adopted as shown by Bhatt et al. [30][31][32][33] Takeuchi and Inoue 64 have explained the method of calculation of formation 3 Different GFA criteria depending on internal atomic structure have been fitted to Z max . a configurational entropy, b mismatch enthalpies of chemical mixing, elastic enthalpies and topological enthalpies assuming no interaction of elements of higher order than binary. Ray and Akinc 59 have explained the way of successful extension of Miedema's approach 22,26 for higher order alloys. In the present work, the same approach has been utilised, and any interaction of higher order than binary has been neglected. Bhatt et al., [30][31][32][33] Takeuchi and Inoue, 64 Basu et al. 67 and Murty et al. 109 have shown successful applications of extended Miedema model 22,26 for predicting AFCR for several alloys of ternary and even quaternary nature. However, all the criteria individually does not prove satisfactory to explain GFA if they are compared to Z max of the alloys from binary to quinary in nature. The ignorance of higher order elemental interaction factors above binary could be the sole reason for such discrepancy. Second, all the successful applications of this extended approach of Miedema's model 22,26 have been carried out in the mutual intersection area of the curves of ternary diagrams, where vertices of triangles represent different enthalpy related property or different elemental combinations plotted one against another. [30][31][32][33]64,109 It is to be noted that the variation of these criteria listed in Tables  4 and 5 with respect to Z max has not been shown satisfactory correlation. The regular solution model and the macroscopic model take a very critical form, when it is applied to ternary or higher order system. The surrounding atoms of a particular element become different, and the electronic interactions on the atom surfaces are not as simple as in case of binary. Each case would contain a probability factor of interaction of an atom of a particular element with another atom of a different element. The situation becomes more and more complicated when the interaction steps up to quaternary and quinary level. Thus, the calculations made with such simply dealt situations of atomic interactions (by summation of the enthalpies of binary interactions) cannot be utilised to assess GFA of alloys of higher order alloys. The complexity in the interactions must be dealt for ternary, quaternary and quinary alloys.
Glass forming ability criteria of category C have been related to the change of entropy for the mixing of elements of the alloy and the change of entropy arising from the mismatch of atomic sizes of different elements when mixed together. Figure 3 describes the goodness of fit of each criteria relating entropy change due to mixing of different elements. No criteria has proved itself to be compatible with Z max as depicted by Fig. 3. It is not possible for the measure of entropy change to predict about GFA or Z max in reality. According to the laws of thermodynamics, with the increase in number of elements, the configurational entropy increases. Tables 4  and 5 shows that there is an increase in average DS config and DSs as the alloys move from binary to ternary, quaternary and quinary. It is evident from Fig. 3a and b that DS (of both types) increases to a very high value. However, the Z max , does not increase in the same order. Several authors have shown the application of entropy change in forecasting Amorphous Forming Range (AFR) [30][31][32][33]64,109 with intersection of several properties taking them in different axes. How ever, individually, they were not good reflectors of GFA.
Glass forming ability criteria in category D take care of kinetic part of crystallisation, and they have been proven to be far better with respect to the previous ones within the few cases that were studied. Unfortunately in this case, the kinetic data are not available for a large number of alloys, and hence, these criteria are not statistically tested over large number of cases. Table 8 shows the values of VFT parameters obtained from solution of VFT equation (equation (22)) according to the procedure explained in the section on 'Viscosity of glassy alloys and compounds'. Angell 45 has calculated the viscosity behaviour of a number of alloys of widely varying origins using the VFT equation and correlated it with their glass forming nature in the form of strength of liquid phase in terms of the fragility parameter D in the VFT equation (equation (22)). The D value reflects the mobility of atoms in a liquid. The more is the D value, the less mobile are the atoms in the liquid, and the more viscous is the liquid. Therefore, the fragility is a measure of how weak a liquid is, i.e. it is equivalent to the inverse of inertia of liquid structure to retain itself under freezing. Thus, it tells the kinetics of crystallisation by itself. The effect is elaborated in Fig. 4a and b. D values obtained for all the alloys and compounds produce a fit with Z max with R 2 ¼ 0.675. In this case, the polymeric liquids like salol and glycerol also were considered with their D values, as their experimental viscosity data are available in the literature. However, the polymeric liquids are not considered for comparison of D values with the other materials. 47 Thus, it further deteriorates the goodness of fit. On the other hand, the alloys with accurate D values (italic lettered materials in Table 8, section on 'Viscosity of glassy alloys and compounds') fit much better with Z max of all the alloys giving rise to Another GFA criterion of this kind is theoretically calculated critical cooling rate R c . According to Weinberg et al., 52 the calculated R c is not exactly equal to the experimentally found out ones for the same materials. Rather, the theoretically calculated ones are always one order of magnitude larger than the experimentally obtained ones. The basic reason cited by them 52 is the value of Turnbull parameter, which has to be assumed properly. Decrease in the value of Turnbull parameter increases the temperature of maximum nucleation rate T Imax . The temperature of maximum growth rate is always higher than T Imax . Thus, the decrease in the value of Turnbull parameter increases the overlap area of nucleation and growth curves, resulting in increment in overall crystallisation rate and thereby increasing R c . In reality, the Turnbull parameter must be higher than it is taken in the calculation of nucleation rate. Here also, it has been observed that the calculated R c and experimentally found out R c are different ( Table 9). The inverse of R c calculated with the viscosity values of all the alloys has been fitted with Z max , and the fitting has been proven to be good with R 2 ¼ 0.692. On the other hand, the inverse of R c calculated with the viscosities of those alloys, which have been reported with italic letters in Table 8, have given an excellent fit with Z max with R 2 ¼ 0.971. Therefore, even though the absolute values of critical cooling rate may not be accurate in theoretically calculated ones, still the theoretical R c values are quite able to assess the GFA. It is true that the correct D values (Fig. 4b) and the correct R c values (Fig. 4d) shown for the alloys are very less in number because the unavailability of viscosity data required for the calculation (section on 'Viscosity of glassy alloys and compounds'). The materials are of the category from inorganic compound (SiO 2 ) to inorganic alloys, from inorganic alloys to ionic solids (K þ Ca 2þ NO 2 3 ) and from ionic solid to organic liquid (o-terphenyl). The Z max varies from 400 to 0.1 mm (Tables 8 and 9). The R 2 ¼ 0.952 and 0.971 ( Fig. 4b and d) proves that the procedure applies to all categories of alloys and compounds equally well.
In summary, it can be mentioned that if the glass formation is considered critically, it involves a certain amount of driving force for amorphisation; there has to be optimum atomic structure, and there always exists a time factor for amorphous formation that needs suppression of crystallisation. Thus, thermodynamic factor, structure factor and kinetic factor altogether simultaneously play important roles in amorphisation of any glass forming material. From categories A to C, the GFA parameters take care of either only one factor of thermodynamic and internal atomic structure or at maximum two of them together. However, in amorphous phase formation, kinetic factor plays the most important part. The fragility parameter D is related to the diffusivity of atoms for atomic rearrangements, which takes care of the kinetic factor as well as the atomic structural part. Therefore, the fragility parameter D is very consistent with Z max values of the respective materials. Moreover, the calculated critical cooling rate R c takes care of the rate of nucleation and growth and the diffusivity and, thus, the thermodynamic, kinetic and structural part of phase transformation. Thus, the most successful GFA criterion can be taken as a properly calculated theoretical R c .

Thermodynamics and topology based P HSS model for multicomponent alloys
The majority of BMG formers that have been reported till date are based on one or two dominating elements. However, in order to discover novel glass forming alloys with higher GFA, multicomponent alloys that belong to high configurational entropy regime offer potential unexplored domain. In order to determine GFA of these alloys, capturing the mixing enthalpy DH mix and topological parameter (d and DS s /k B ) of these alloys in a twodimensional map is currently the most popular and also the effective method to initiate exploration in vast compositional space. 113 Through this method, a qualitative estimate can be made as to whether a selected multicomponent alloy has optimum chemical and topological interactions for glass formation. However, as configurational entropy plays one of the dominant roles for multicomponent alloys, this is incorporated in the P HSS parameter.
P HSS parametric values have been computed for different glass forming compositions of several multicomponent alloy systems, in addition to various high entropy alloys (HEAs) and high entropy metallic glasses (HEMGs). It has been identified that many of the alloy systems group into different diffuse zones in a twodimensional map of mixing enthalpy and mismatch entropy. The computed P HSS values of alloys that were reported in the literature are appended at Tables S1, S2 and S3 in the supplementary material (www.maney online.com/doi/supp/10.1179/1743284715y.0000000104 ) Figure 5 represents plot of DH mix versus DS s /k B . P HSS parametric ranges of all computed alloys are depicted in Fig. 6. P HSS parameter versus critical diameter W of various metallic glasses (MGs) is plotted in Fig. 7. P HSS parameter versus DT x (width of supercooled liquid region) pertaining to several MGs is plotted in Fig. 8. The discussion per taining to the above mentioned figures is elaborated in subsequent paragraphs. Figure 5 depicting DH mix versus DS s /k B illustrates the clear distinction between the DH mix and DS s /k B requirements between solid solution forming HEAs and various BMGs. The solid solutions (HEAs) are formed at DH mix w215 kJ mol 21 and DS s /k B v0.1. It is in line with the Hume-Rothery criterion for the formation of solid solutions. In order to form solid solution phases in HEAs, DS s /k B should be as small as possible and DH mix should be close to zero, either slightly positive or not too much negative. This result is similar to that reported earlier in the literature. 113 It is evident that, excluding a few Cu based MGs and alkali, alkaline earth metal based BMGs, glass formation in all the transition and rare earth (RE) based glasses occurs at DH mix v223 kJ mol 21 (Fig. 5). In the case of DS s /k B , wide variation is noticeable among BMGs. Iron based BMGs form due to large DS s /k B in the range of 0.5-0.8 (Fig. 5). This is attributable to a large of number of alloying elements in addition to significant variation of atomic sizes of metalloids as compared to that of transition metal (TM) in the Fe based MGs. 114 Rare earth based MGs also group under the category of Fe based BMGs that contain large DS s /k B . The atomic size of RE elements being larger among all the elements is responsible for the observed mismatch entropy (alloys 263-273 in Table S2 of the supplementary data material Transition metal based MGs form under wide range of DH mix depending upon the alloying elements involved. However, the DS s /k B for their formation is significantly less as compared to that of Fe and RE based MGs (Table  S2). Excluding a few zirconium based BMGs, the entire TM based glass family lies within the DS s /k B of 0.3 (Fig. 5). Most of the TM glasses like Cu, Zr, Pd, Hf and Ni based BMGs cluster around the smaller window of 235vDH mix v225 kJ mol 21 and 0.19vDS s /k B v0.25 (Fig. 5). High entropy metallic glasses form in a narrow window within the heavily clustered TM glass forming region. From the DH mix versus DS s /k B (Fig. 5) plot, BMGs can be broadly categorised into two types: (i) transition metal based BMGs, forming within wide DH mix range ( 72 to ( 15 kJ mol 21 and narrow range of DS s /k B of 0.15-0.3 (ii) iron, Ca and Mg, RE, Co based MGs forming within narrow DH mix range of ( 30 to ( 10 kJ mol 21 and wide range DS s /k B of 0.2-0.8. Figure 5 demonstrates that DS s /k B plays the dominant role for glass formation and is the most important parameter separating solid solutions from BMGs. Regardless of the alloy system, a minimum DS s /k B of 0.1 exists below which BMG formation is not feasible according to the calculated results (Fig. 5). Such topological strain induced due to atomic size variation among the constituents of alloy influences the formation of SRO. 115 As the atomic size radius ratio of solute and solvent is different for different classes of alloy systems, topological clusters of coordination numbers from 8 (CN 8) to 20 (CN 20) that are incompatible with translational symmetry have been hypothesised to be candidate clusters of MG structure. 116 In the results of the present work, large DS s /k B range of 0.1-0.8 justify the huge variation in cluster topology for various alloys systems. The variation in DH mix along with DS s /k B is expected to generate many variants of the ideal MG clusters specified in the literature. 116 The TM based glasses are grouped around P HSS range of 211 to 25 kJ mol 21 (Fig. 6). To attain the glassy phase through equiatomic concentration of elements, the selection of elements should be such that the mismatch entropy and mixing enthalpy of the alloy should be considerably higher than that of solid solution formation criteria, thereby resulting in larger P HSS range for glasses (Fig. 6). Figures 7 and 8 depict the plots of P HSS against widely used GFA parameters such as critical diameter, W and DT x respectively of the experimentally reported BMGs. Both the graphs indicate similar trend in which GFA correlating parameters are maximum between P HSS range of 211 to 25 kJ mol 21 . Figures 7 and 8  provide very good correlation with GFA universally among all systems. The results (Figs. 7 and 8) prove that each class of alloy system has different GFA and thermal stability and does not enable a generic comparison. The GFA and thermal stability of a particular alloy system must be assessed independently. Moreover, P HSS parameter is thermodynamic in nature and cannot aid in visualisation of local SRO dynamics during glass formation. As most of the MG formers cluster within the P HSS range of 211 to 25 kJ mol 21 , the results in the present work can be a primary guideline for compositional optimisation in high configurational entropy domain of multicomponent alloy systems. However, the validity of the proposed P HSS range need to be experimentally verified for each of the individual multicomponent alloy systems to further establish this parameter as reliable glass forming locator.

Kinetic interpretation of P HSS range of glass forming alloys and conclusions
Metallic glass structure exhibits randomness at atomic scale due to their thermodynamic metastability and is also prone to variation with the processing technique. 117 The degree of metastability or deviation from thermal equilibrium is dependent on the degree of quenching from molten state. In the process of quenching, the manner in which both entropy and enthalpy of supercooled liquid change as function of temperature primarily influence GFA and the atomic scale structure of glass. It is widely known that the linear scaling relationship of viscosity g and inverse of temperature (1/T) on logarithmic scale is vague for MGs, due to the mysterious manner in which evolution of configurational entropy of supercooled liquid contributes to non-Arrhenius ascent in viscosity. Hence, viscosity evolution of supercooled liquid is governed by the dynamics of local SRO. As different local structural motifs for different alloy systems might be due to different ranges of P HSS parameter, precise quantitative understanding of these parameters is a promising pathway for qualitative guidance to BMG design. Such knowledge provides subsequent capability to tailor structural and functional applications of BMGs. Furthermore, establishing the accurate connection between thermodynamic and kinetic properties of BMGs based on SRO perspective has the ability to unlock many unanswered mysteries in this field. It has been observed that the available GFA criteria based on four major categories do not go well with the GFA of the glassy materials when maximum critical thickness Z max is compared. The reasons for not so good fit with Z max have critically been examined. Most of the cases, either structure or thermodynamic or kinetic parts are considered separately. Some cases where Miedema's model forms the backbone of a particular criterion have not considered multiple interactions. They can be good for a specific system; however, they fail to assess GFA when too many different systems are considered. The kinetic parameters have been proven to be very good for predicting GFA since it reflected structure as well as kinetic part by considering viscosity of the liquid. Moreover, critical cooling rate approach by considering refined VFT equation seems to work well since it incorporates structure of the liquid, thermodynamic driving force as well kinetic factors like diffusion of atoms.
The strength of the present finding is that the P HSS parameter is able to depict the distinct boundary between non-glass formers and BMGs. P HSS range of 211 to 25 kJ mol 21 can be used as a guide line to design bigger dimension BMGs. Glass formation in TM based glasses occurs over wide DH chem range of 272 to 215 kJ mol 21 and narrow range of DS s /k B of 0.15-0.3. Iron, Ca and Mg, RE, Co based glasses form with narrow DH chem range of 230 to 210 kJ mol 21 and wide range DS s /k B of 0.2-0.8. This implies glaring structural differences at atomic scale between these two groups of glasses. Mismatch entropy, a measure of the atomic polydispersity, is the dominating prerequisite for glass formation. Calcium and Mg based alloy systems form bulk glasses despite having DH chem as low as ( 7 kJ/mol. Copper based alloys can form glasses at DH chem as low as 212 kJ mol 21 . However, no alloy system forms glass below DS s /k B of 0.1.