Thermodynamic determination of fragility in La-based glass-forming liquid

Differences in the thermodynamic functions between the liquid and crystalline states of La-based bulk metallic glasses alloys were calculated with the specific heat capacity Cp and the fusion heat ΔHf, which we measured. Fragility indexes having different thermodynamic definitions were calculated from the temperature dependence of excess entropy ΔSliq-cry. It is ambiguous for La-based glass-forming liquid to evaluate fragility from the intercepts of ΔSliq-cry-temperature curves. We found that the thermodynamic fragility MΔS, denoted ΔS, decreases at the onset melting temperatures Tm rather than at glass transition temperatures Tg, and relates linearly with the kinetic fragility m. The correlations between thermodynamic fragility and kinetic fragility are discussed within the frameworks of the Adam-Gibbs relationship and potential energy landscape theory.

The concept of liquid fragility has been widely used to classify the dynamic behavior of glass-forming liquids since first being introduced by Angell in 1985 [1]. According to the concept, glass-forming liquids are classified between two extremes: strong glass formers showing Arrhenius behavior and fragile glass formers that deviate from Arrhenius behavior. However, the fragility parameter m can only be calculated for systems with stable undercooled liquid. Recently, a new concept, the fragility of superheated melts M, defined as the rate of variation of the viscosity of superheated liquids towards the liquidus temperature, was proposed by Bian et al. [2]. A negative correlation between M and glass-forming ability (GFA) has been found in Al-based [2][3][4] and rare-earth-based alloy systems [5,6]. Moreover, the correlation between m and M has also been investigated [3,7].
The two fundamental aspects of phase transition are thermodynamics and dynamics. The Kauzmann paradox suggests that the two are inherently related [8]. The Adam and Gibbs model [9], which is based on cooperatively relaxing regions in supercooled liquids, provided the first quantitative relation between the thermodynamic and kinetic properties of a liquid. Several attempts to correlate the fragility (the kinetic property) of supercooled liquids with thermodynamic behaviors have been made, and have mainly focused on changes in heat capacity [10] and entropy [11][12][13][14]. However, the thermodynamic fragility of metallic glasses has not been investigated systematically [11,[14][15][16][17].
Heat capacity is the most fundamental thermodynamic nature of matter. In this paper, the specific heat capacity of six La-based bulk metallic glasses (BMGs) is measured for the crystalline solid, the supercooled liquid, and the superheated liquid. The differences in thermodynamic functions between the liquid and the crystalline states are calculated. With these data, the thermodynamic fragilities defined previously [12,14] are calculated and new thermodynamic fragility indexes are proposed to investigate the interconnection of thermodynamics and kinetic fragility.  , and La 62 Al 14 -Cu 12 Ni 12 (La-6) were prepared by arc-melting with 99.9 wt%-99.999 wt% pure Al, Co, Cu, Ni and industrially pure La in a Ti-gettered argon atmosphere. To ensure the homogeneity of the samples, the ingots were remelted at least three times in addition to being electromagnetically whisked. BMG rods with a diameter of 2 mm and a length of 50 mm were prepared by copper-mould suction casting in an arc furnace. The amorphous structure (except in the case of La-1) of the transverse cross sections of the rods was identified by X-ray diffraction using a D/Max-rB diffractometer and Cu K radiation.
A Netzsch DSC404 differential scanning calorimeter (DSC) was used to determine the characteristic temperature and the specific heat capacity of the alloys. For heat-flux differential scanning calorimetry, the resultant signal is expressed as p DSC , where K is the sensitivity coefficient, DSC is the measurement result, C p is the specific heat capacity of unit mass, m is the sample mass and HR is the heating rate. With the same heating rate HR, the measured absolute heat flow minus the baseline (the heat flow of the pan itself) is directly proportional to the heat capacity (C p m), and it can be expressed at a certain temperature as sam pan p sam sam std pan p std std ( ) , ( ) where m is the mass in the unit of grams and C p is the specific heat capacity in the unit of Joules per gram per degree Kelvin. The subscripts sam, std and pan indicate the metal sample, the sapphire standard and the sample pan respectively.
At a heating rate of 20 K/min, by successively performing calorimetric measurements on a sample pan by itself, on a sapphire standard in the sample pan and on the metal sample in the sample pan, the absolute specific heat capacity of the metal sample is determined according to sam pan p std std p s a m std pan sam The mass of each sample is approximately 30 mg. The amorphous rods were heated nearly to their melting temperatures T m , determined beforehand using the DSC, for C p measurements and then cooled to room temperature freely to form the crystallized samples. The crystallized samples were heated to ~200 K above starting melting temperatures to obtain the specific heat capacity of both the crystallized solid and liquid metals. During heating, all samples were first heated to 373 K and held isothermally for 10 min and then heated at a constant rate of 20 K/min for C p measurements. Each experiment was carried out under a high-purity argon atmosphere.

Thermodynamics
Upon heating the amorphous alloys at a constant rate, the alloys go through glass transition, crystallization, and fusion. Such behavior of La 55 Al 25 Ni 5 Cu 15 (La-3) is shown in Figure  1 as an example. By integrating the area of the melting event (from solidus temperature T m to liquidus temperature T l ), the total heat of fusion H f is calculated. The entropy of fusion can then be estimated as S f = H f /T f . T f is the temperature at which the Gibbs free energy of the crystal is equal to that of the liquid. T f between T m and T l is likely to be very close to the melting start temperature T m . Thus, T m is adopted as T f in calculating S f . The characteristic temperatures and the heats and entropies of fusion of these Labased alloys are summarized in Table 1. T g of La-1, for which we did not obtain a fully amorphous rod, is cited from [18]. There are slight differences in characteristic temperatures between our work and [18][19][20][21], as is usually the case for different raw materials and laboratory facilities. The measured specific heat capacity is shown in Figure 2. The heat capacity of glass is similar to that of crystal at low temperatures. However, when heated above the glass transition temperature T g , the glass phase relaxes into the supercooled liquid state and its C p value assumes the higher heat capacity of a liquid [22]. Inoue et al. [21] and Lu et al. [20] reported that the specific heat capacity of La 55 Al 25 Ni 10 Cu 10 alloy is 42.0 and 39-41 J mol 1 K 1 respectively throughout the undercooled liquid region; our measurement results are similar. C p values for several La-based BMGs have so far been reported [21]; however, this is the first time that C p for   a) From Ref. [18].

Figure 2
DSC heat capacity measurements obtained from initially liquid (☆), crystalline (□) and glassy (△) samples, and the fitting lines of eq. (4) (solid lines) and eq. (5) (dashed lines). For La-1, the glassy data were not measured, but a value of 40 J mol 1 K 1 at 500 K, 10 K higher than T g , was assigned to allow the determination of the fitting constant for the liquid specific heat capacity (▲).
the both supercooled liquid and the melts has been measured to ensure fitting of the following eq. (5). The heat capacity of a crystal well above the Debye temperature can be described by [23] cry 2 and the heat capacity of a liquid can be described by where the gas constant R=8.3145 J mol 1 K 1 , and a, b, c and d are fitting constants. The fitting curves and T m and T l are shown in Figure 2. If an equilibrium liquid continuously cools to a supercooled liquid without solidification, which is required in glass formation, C p does not abruptly change and the C p curve throughout the equilibrium and supercooled range coincides with eq. (5) [21,[24][25][26][27]. Table 2 summarizes the fitting constants for both eqs. (4) and (5).
The specific heat capacity of La-1 (La 55 Al 25 Ni 20 ) in the undercooled liquid that initially forms upon heating BMGs could not be measured since a fully amorphous rod was not obtained. Of the other five alloys in this work, the heat capacity is highest at a temperature about 10 K higher than T g , and it has an average value of about 40 J mol 1 K 1 . The glass transition temperature T g of La 55 Al 25 Ni 20 is reported to be 490 K [18]. Thus, we assume a data point of 40 J mol 1 K 1 at 500 K for the data fitting of the liquid specific heat capacity.
Since the heat capacity of liquid and crystalline states and the enthalpy of fusion H f have been experimentally determined, the enthalpy and entropy differences between the liquid and crystalline states of the alloy can be calculated as functions of temperature: where liq-cry liq cry p p p . C C C    As mentioned above, T f is substituted with T m for the calculation. Figure 3 shows the calculated differences in the enthalpy and entropy between the liquid and crystalline states of La-1 as an example. On the plot, we mark T g , T m and the calculated Kauzmann temperature T k . At T k , the entropy of the liquid is equal to the entropy of the crystal. The values of T k are also listed in Table 1.

Fragility
Besides consideration of the viscosity, thermal scanning is thought to be a reliable way to obtain the fragility of the supercooled liquid [28]. Previous studies have demonstrated that the kinetic heating rate  dependence of the glass transition temperature T g describes the fragility as well as the temperature dependence of viscosity [29]. Values of m for La-1, La-2 and La-3, determined using the method employed in [6], are cited from [18], while values for the other three alloys were obtained employing the same method used in the DSC experiments at different heating rates; values are presented in Table 3. Ito et al. [12] used a Kauzmann plot to define the thermodynamic fragility of glass-forming liquids, and found that thermodynamic fragility and kinetic fragility have the same order for the different liquids. By this definition, a liquid for which there is a more rapid drop in S/S f with decreasing T/T f is more thermodynamically fragile. Two thermodynamic fragility parameters were assumed [12]. One is F 1/2,cal =T 1/2 /T m , with T 1/2 being the temperature at which S/S f =1/2. The other is F 0.8 , which is the fraction of fusion entropy lost (1-S/S f ) by T/T m =0.8 when cooling. Figure 4(a) shows the excess entropy S normalized by S f versus T normalized by T m . It is seen that F 1/2,cal and F 0.8 coincide; thus, we only calculate F 0.8 , as listed in Table 3.
In Figure 4(a), however, we find that F 0.8 does not show the rapidity of the S decrease thoroughly owing to the appearance of crossing points. S for La-4, with F 0.8 (0.227) close to that for La-2 (0.237) and that for La-5 (0.232), has a rate of decrease in the scaled excess entropy similar to that of La-1 (0.199) and that of La-3 (0.182). In Ref. [12], there is no crossing point between any two scaled excess entropy curves, and the larger F 0.8 results from a more rapid drop of S/S f with diminishing T/T m and thus larger T K /T m . Generally, this is an ambiguous definition. Breakdowns of the coherence between this kind of thermodynamic fragility and kinetic fragility have been observed for molecular glassforming liquids [13].
Similar to Ref. [12], Martinez and Angell [14] plotted excess entropy S scaled by S g (excess entropy at T g ) versus T g /T, which is similar to the Angell plot for kinetic fragility [1]. Following the idea of the mode of kinetic fragility F 1/2 =2T g /T 1/2 -1 [30], a thermodynamic fragility parameter F 3/4 =2T g /T 3/4 -1 is proposed, with T 3/4 the temperature at which S g /S=3/4 [14]. In Figure 4(b), we plot S g /S versus T g /T and mark the S g /S=3/4 line. To distinguish from F 0.8 defined in Ref. [12], we write the thermodynamic fragility parameter defined by S g scaled entropy in Ref. [14] as G 3/4 , and the values are listed in Table 3.
Thermodynamic fragilities F 0.8 and G 3/4 and kinetic fragility m are compared in Figure 5(a). Both F 0.8 and G 3/4 are positively correlated with the kinetic fragility for the most  part, although the correlation with G 3/4 appears better. Since both lie central to the fragility range from T g to T m [14,30], the difference between the trends of F 0.8 and G 3/4 stems from the different values of the scaled excess entropy S, at T g and T m . Both T g and T m vary for different metallic glasses, and variation in one characteristic temperature is thus ignored in the two definitions. S f /(T m T k ) [11], the average entropy-temperature dependence from T m to T k , has been found to positively correlate with kinetic fragility in metallic glass-forming liquids.
For the index of S f /(T m T k ), if the variation in entropy for different glasses is eliminated by scaling the index with S f , as in the method employed to deduce thermodynamic fragilities in Refs. [12] and [14], then 1/(T m -T k ) or the reduced isentropic temperature T k /T m seems to coincide with the original intention of thermodynamic fragility definition of the entropy slope [11,12,14]. As in the definition of kinetic fragility m, we suppose that thermodynamic fragility m S can be represented by the drop in excess entropy approaching T g , which can be written as Another concept of fragility, the fragility of superheated melts M [2], is defined as the rate of change in viscosity at the liquidus temperature. M is an index of the kinetic structure sensitivity to the temperature change approaching the phase transition of solidification [2], while m corresponds to the glass phase transition. Fusion and solidification are opposing phase transitions. Thus, thermodynamic fragility M S , reflecting the change rate of excess entropy approaching solidification/fusion, can be assumed according to Figure  4(a) as Values of m S and M S are listed in Table 3. We plot the relations between the two indexes and m in Figure 5(b).
It is interesting that we find nearly linear proportional correlations between M S and m. The correlation between M S (i.e., the excess entropy slope at T m instead of the excess entropy slope below T m ) and m may provide more detailed information about the correlation between the thermodynamics and kinetics of a liquid, which is discussed in the next section.

Discussion
Fragility was originally so named to infer that fragile liquids experience rapid breakdown of structure with decreasing temperature approaching T g [1]. In contrast to being "fragile", a liquid with slow structure change is deemed "strong". The Adam-Gibbs theory [9] predicts a dependence of a transition probability property on the configurational entropy S c as where S c is the configurational entropy, and A and C are constants without temperature dependence. The relation is based on the assumption of cooperatively rearranging regions. When configurational entropy S c increases, the cooperative rearrangements of the structure can take place independently in smaller and smaller regions of the liquid. It has often been assumed that S c can be approximated by the difference in entropy between liquid and crystal [12][13][14][23][24][25][26][27]30]. Thus, the gradient of structural changes can be quantified in terms of S liq-cry , and this is testified by the coincidence between F 0.8 and G 3/4 with m in the La-based glass-formation liquid.
Besides m (i.e., the slope of an Angell plot), the intercept of the Angell plot can be employed to quantify fragility [12,23,31]. The intercept is recommended as being a more reliable metric of the deviation from Arrhenius behavior, because it is central to the fragility range from T m to T g . Following this idea, the intercept of scaled excess entropy was chosen to represent thermodynamic fragility [12,14] as F 0.8 and G 3/4 , which we have calculated. However, the difference between the scaled S plot and Angell plot, which may introduce randomness in the thermodynamic fragility, was neglected. In the Angell plot, the decreasing scaled viscosity or relaxation time curves split from an intersection point at T g , and join up again as the temperature approaches infinity (i.e., as the scaled temperature approaches zero). Figure 4(a) and (b) shows that the scaled excess entropy curves do not cross again. Thus, for an Angell plot, the variations in the intercept and slope coincide, while for the excess entropy plot, the correlation is to a certain extent random. We calculated the slope of scaled excess entropy at both ends of the fragility range and the average slope as thermodynamic fragility indexes. The thermodynamic fragility M S has high unexpected coherence with the kinetic fragility m, relative to that for m S or the average slope.
The excess entropy that we have used to explain the coincidence of F 0.8 and G 3/4 with m is not the quantity that should appear in the Adam-Gibbs equation, even though it has been used in most experimental tests and the Adam-Gibbs equation usually tests well using the easily accessible excess entropy. The excess entropy should be divided into configurational entropy S c and vibrational entropy S vib , as in Figure 6 [3]. The potential energy landscape (PEL) is an ideal concept for combining dynamics and thermodynamics, and is advantageously used to analyze and explain the phenomenology of the supercooled liquid state [12,31,32]. As shown in Figure 6, in the case of constant basin shape Figure 6 Changes in entropy of a system due to the change in the energy landscape ("basin shape") during cooling.
(strong liquid), the drive to higher enthalpy comes only from the configurational entropy; for variable basin shape (fragile liquid), the increased rate of excitation of vibrational entropy increases the rate of generation of S c entropy, as well as that of the excess entropy [12].
By studying the PEL of glass-forming alloys, Hu et al. [33] found that the absolute value of chemical mixing enthalpy |H mix | is negatively related to the height of energy barriers separating the minima . The height of energy barriers between minima on the PEL determines when a system freezes, and a larger  indicates a more difficult change in the structure of the system. We calculate |H mix | on the basis of the extended regular solution model [34] and plot the M S -|H mix | relationship in Figure 7. A positive correlation is found between M S and |H mix |, and liquid with smaller M S thus has larger . Therefore, the alloy with a slower change in thermodynamics at the start of the fragility range below T m (a smaller M S ) has a more difficult change in structure approaching T g .
The coherence of thermodynamic fragility M S and kinetic fragility m and  suggests the possibility of a thermodynamic prediction of the structure stability. During rapid quenching, which generates BMGs, the drop in entropy at T m (i.e., the beginning of the fragility range) plays a crucial role in determining the structural stability of glassforming liquid in the fragility range.

Conclusion
Absolute values of the specific heat capacity C p of six Labased BMGs were measured with respect to sapphire standards for the crystal, supercooled liquid, and melts. The differences in entropy (S liq-cry ) and enthalpy (H liq-cry ) between liquid and crystal as functions of temperature were calculated with C p and the fusion heat H f . Thermodynamic fragility having different definitions was calculated and discussed for La-based glass-forming liquid on the basis of the S liq-cry -temperature dependence. In La-based glassforming liquid, using the interception of scaled excess entropy S curves as fragility indexes may result in randomness and ambiguity. We put forward thermodynamic fragilities m S and M S by evaluating the slope of the temperature dependence of the excess entropy at T g and T m respectively. The correlation between thermodynamic and kinetic fragilities was discussed on the basis of Adam-Gibbs and potential energy landscape theories. The thermodynamic fragility of the melt M S , rather than that of the supercooled liquid m ∆S , relates linearly with kinetic fragility m and the height of energy barriers separating the minima .