Multiple Glass Transitions in Bismuth and Tin beyond Melting Temperatures

Abstract

Liquid-liquid transitions were discovered above the melting temperature (T_m) in Bi and Sn up to 2 T_m, viewed as glass transitions at T_g = T_n+ > T_m of composites nucleated at T_x < T_m and fully melted at T_n+. A glassy fraction (f) disappeared at 784 K in Sn. (T_n+) increases with singular values of (f) depending on T_x with (f) attaining 100% at T_g = T_n+ = 2 T_m. The nonclassical model of homogeneous nucleation is used to predict T_x, T_n+ and the specific heat. The singular values of (f) leading to (T_n+) correspond to percolation thresholds of configurons in glassy phases. A phase diagram of glassy fractions occurring in molten elements is proposed. The same value of (T_x) can lead to multiple (T_g). Values of (T_g = T_n+) can be higher than (2 T_m) for T_x/T_m < 0.7069. A specific heat equal to zero is predicted after cooling from T ≤ 2 T_m and would correspond to a glassy phase. Weak glassy fractions are nucleated near (T_n+) after full melting at (T_m) without transition at (T_x). Resistivity decreases were observed after thermal cycling between solid and liquid states with weak and successive values of (f) due to T_x/T_m < 0.7069.

1. Introduction

Liquid-liquid phase transitions occur in glass-forming melts at temperatures T_n+ above Tm, the equilibrium thermodynamic melting transition of crystals [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. These transitions often result from the separation of two liquid states, occurring in all supercooled materials at a temperature T_x < T_m, including pure elements in which two liquid phases of the same composition coexist. A liquid fraction crystallizes at T_x and melts at T_m while the complementary glassy fraction melts at T_n+ = T_g [22,23,24]. The purpose of this publication is to relate all the liquid-liquid transitions, occurring above the melting temperature (T_m) in Bi and Sn up to (2 T_m), and beyond, to glass transition temperatures [1,3,4]. A weak glassy fraction disappeared at 784 K in Sn as recently observed with specific heat measurements [5]. Here, we recall that (T_n+) increases with singular values of (f) depending on T_x and we examine if (f) can attain 100% at T_g = T_n+ = 2 T_m in Bi and Sn as already envisaged for a component of Cu₄₆Zr₄₆Al₈ melt [12,23].

Molecular dynamics simulations were employed to study the thermodynamics and kinetics of the glass transition and crystallization in deeply undercooled liquid Ag and Ag-Cu at high cooling rates of the order of 10¹² K/s [25,26]. A first order transition from the liquid-phase (L) to a metastable, heterogeneous phase called G-phase was observed, leading to a glassy phase below the recovery temperature T_n+ of the enthalpy above T_m. The L–G transition occurred by nucleation of the G-phase from the L-phase. The lowest glass transition temperature of liquid-phase (L) depends on its Lindemann coefficient and is much weaker than the nucleation temperature of G-phase [27,28,29,30]. Increasing the heating rate increased the glass transition temperature of liquid (L). A first order transition from liquid (L) to G-glass was observed, when a supercooled liquid evolved isothermally below its melting temperature at deep undercooling [26]. Several simulations of G-phases in various elements showed full melting heat H_m at various temperatures T_n+ = T_g in Ag [25], in Zr [31], in Cu and in Fe [32]. The values of T_n+ were determined from singular values of G-phase frozen enthalpy, employing the nonclassical homogeneous nucleation (NCHN) model to predict T_x, T_n+ and the enthalpy. Singular values of (f) leading to (T_n+) correspond to percolation thresholds of broken bonds (configurons) leading to glassy phases during cooling below T_n+ [24,29,30,33,34].

The high undercooling rates of bulk liquid elements have been known for many years [35]. Consequently, we plan to confirm, in this publication, that the nucleation of G-phases and their glass transition temperatures above T_m are observable without employing heating and cooling rates of the order of 10¹⁰ to 10¹³ K/s to escape from crystallization. A phase diagram of glassy fractions, occurring in molten elements at T_x < T_m, is proposed, completing the diagram already established for T_x > T_m [30] in agreement with previous molecular dynamics simulations [25,26,31,32]. The specific heat values at T_g = T_n+ is predicted up to T_g = 2 T_m where a glassy phase transition is expected. The density far above T_m depends on the formation time of bonds increasing glassy phase fractions. Resistivity decreases are due to the increase in (f) in these liquids [4].

2. Diagram of Glassy Phases

Three liquid states are present in all melts with complementary enthalpies introduced in the classical Gibbs free energy change which are equal to ε_lsH_m, ε_gsH_m and Δε_lgH_m [36], with H_m being the melting enthalpy. The coefficient (ε_ls) is attached to the initial liquid state before adding new atomic bonds induced by cooling while (ε_gs) is obtained by including them. The coefficient Δε_lg, equal to the difference (ε_ls − ε_gs), is attributed to the formation of a new liquid state describing the contribution of new bonds. For liquid elements, that are easily crystallized, the liquid enthalpy coefficients obey to the following [27,30]:

$${\epsilon }_{\mathrm{ls}}={\mathsf{\epsilon }}_{\mathrm{ls}0}\left(1-\frac{{\theta }^2}{{\theta }_{0m}^2}\right)={\epsilon }_{ls0}\left(1-2.25 {\theta }^2\right),$$

(1)

$${\mathsf{\epsilon }}_{\mathrm{gs}}={\mathsf{\epsilon }}_{\mathrm{gs}0}\left(1-{\theta }^2/{\theta }_{0g}^2\right)={\epsilon }_{gs0}\left(1-{\theta }^2\right)$$

(2)

$$∆{\mathsf{\epsilon }}_{\lg }\left(\mathsf{\theta }\right)=\left[{\mathsf{\epsilon }}_{\mathrm{ls}}-{\mathsf{\epsilon }}_{\mathrm{gs}}\right]=-1.25{\mathsf{\epsilon }}_{\mathrm{gs}0}{\theta }^2,$$

(3)

where θ_0m = −2/3 and θ_0g = −1 are the reduced Vogel–Fulcher–Tammann (VFT) temperatures in liquid elements for which the minimum value of T_g is fixed by the Lindemann coefficient (δ_ls) of each element [27]. Equation (3) is transformed into Equation (4) for the minimum value of ε_ls0 = ε_gs0:

$$∆{\epsilon }_{lg}=-1.25 {\epsilon }_{gs0 }{\theta }_g^2,$$

(4)

where $∆{\epsilon }_{lg}$(θ_g) is the latent heat coefficient, accompanying the glass transition during the first cooling and the formation of Phase 3 below the percolation threshold of bonds. The partial breaking of bonds during reheating occurs without latent heat [2]. This type of enthalpy relaxation due to the development of bonds below the percolation threshold at T_g was observed after quenching glass-forming melts in amorphous state and heating them at 20 K/min [37,38,39,40,41,42]. After these relaxation phenomena, the glass transition occurs without latent heat.

Coefficient minima (ε_lso) and (ε_gso) were initially determined to be equal to 0.217 [43]. They were the average of many liquid element coefficients deduced from the undercooling rate of each of them corresponding to the mean value 0.103 of their Lindemann coefficient [27,28,29]. The number 2.25, initially equal to 2.5 in Equation (1), consequently, fixed the VFT temperature to T_m/3 [44]. It appeared later that (Δε_lg) in Equation (3) is the enthalpy coefficient of a true thermodynamic phase called “Phase 3” discovered for the first time in supercooled water [45,46,47]. It is now a generic name that we extend to all glassy phases, and to those resulting from a first-order transition [37]. Phase 3 could be the congruent bond lattice predicted for disordered oxide systems [48], extended later to critical packing density formation applied to broken bonds (configurons), producing the glass transitions at T_g<<< [33,49]. A fraction (Δε_lg) of atomic bonds equal to various percolation thresholds still exists up to T_n+, in all glass-forming melts [23,34,50]. Phase 3 results from the formation of configuron phases [50].

A configuron is an elementary configurational excitation in an amorphous material, formed by breaking of a chemical bond (or transformation of an atom from one to another atomic shell) and the associated strain-releasing local adjustment of centers of atomic vibration [48]. For metals, evidence on configuron formation was first demonstrated by Iwashita et al. [51]. The temperature dependence of configuron contents is provided by Gibbs statistics [52,53]. The melting of an amorphous solid differs from the melting of a crystal. Glasses, on heating continuously, change most of their properties to those of a liquid-like state in contrast to crystals, where such changes occur abruptly at a fixed temperature (the melting point) which is explained by the high mobility of broken bonds tending to condense on irregularities of crystals, such as surfaces and inclusions. Therefore, there are no discontinuities in the volume and entropy changes at T_g [33].

The NCHN model predicted the formation conditions of glacial phases (Phase 3) in Ag-Cu and Ag liquids at various heating rates as previously described by molecular dynamics (MD) simulations, showing full melting at T_n+ = 1.119 T_m for Zr, 1.126 T_m for Ag, 1.219 T_m for Fe and 1.354 T_m for Cu [25,26,31,32]. These phenomena are governed by singular values of the enthalpy Δε_lg of glassy Phase 3, formed at nucleation temperatures T_nG [29,30].

Two families of nucleation temperatures are provided in Equations (5) and (6) where θ_n- is equal to two opposite values of θ_g because ε_gs is a function of θ² in Equation (2):

$${\theta }_{n-}={\theta }_g=\frac{\pm ({\epsilon }_{gs}-2)}{3},$$

(5)

$${\theta }_{n+}=∆\epsilon$$

(6)

where Δε is equal to singular values of the enthalpy coefficient (−Δε_lg) of Phase 3.

There are two methods to produce a glassy phase from an enthalpy coefficient equal to (−Δε). The first predicts the undercooling temperature for each value of Δε. These nucleation temperatures (θ_x) are calculated using the (NCHN) model applied to Liquid 2:

$${\mathsf{\epsilon }}_{\mathrm{gs}}\left(\mathsf{\theta }=0\right)=\left(3{\mathsf{\theta }}_x+2-∆\mathsf{\epsilon }\right)/(1-\frac{{\theta }_x^2}{{\mathsf{\theta }}_{0\mathrm{g}}^2})$$

(7)

where θ_0g² = 1. For Δε = 0, a second order-like phase transition temperature takes place at T_g during heating for the minimum value of ε_gs0. Values of θ_x for various values of Δε are deduced for the same value of ε_gs0. Consequently, Equation (6) demonstrates that enthalpy above θ_x is constant and equal to −ΔεH_m, belongs to a glassy fraction with a zero specific heat. The glass transition at (T_n+) is not accompanied during the first heating by the latent heat recovery (+Δε H_m).

Glassy phases, formed at θ_x after undercooling, lead to glassy phase fractions which are volume fractions occupied by the glassy phases with θ_g = θ_n+ = Δε in agreement with Equation (6). It has been shown that the set theory provides clear evidence of structural differences between glasses and melts; both glassy and liquid structures near T_g are disordered, however they have different Hausdorff–Besicovitch dimensions [54]. The set theory, as a branch of mathematical logic that studies abstract sets, can be used to characterize the configuron phase formed in amorphous materials out of broken chemical bonds termed configurons. Glasses have a 3D geometry of bonds with point-type broken bonds with a 0D geometry and, because of that, they exhibit a solid-like behavior. The configuron phase behaves differently in glasses and melts, forming a condensed phase in melts and occurring as a gaseous phase in glasses. It therefore has different Hausdorff–Besicovitch dimensions in melts and glasses [55]. The stepwise change in the Hausdorff–Besicovitch dimension of the set of configurons is due to the formation of the macroscopic (condensed fractal cluster) configuron phase above the T_g and as a result has the appearance of a kink in the first sharp diffraction minimum of scattered X-ray or neutrons [56].

The formation of a glassy fraction is accompanied by a crystallized fraction (1 − Δε). A composite crystal–glass is built below T_m with a melting enthalpy (1 − Δε) H_m and a missing enthalpy (Δε H_m) recovered at T_n+ [23]. The glassy phase fraction is not destroyed at T_m and depends on the sample thermal history and the last value of Δε obtained at T_x during heating.

The minimum glass transition at T_g is masked by spontaneous crystallization of liquid element and determined with Equation (7) using Δε = 0 and the minimum value of ε_gso = ε_lso, depending on the Lindemann coefficient δ_ls of each element [27]:

$${\epsilon }_{gs0}={\epsilon }_{ls0}={\left(1+{\delta }_{ls}\right)}^2-1$$

(8)

In Figure 1, a second method is used to calculate new values of θ_x during heating. Each value of θ_g varying from −0.5 to 2 corresponds to a value of T_g/T_m between 0.5 and 3. All ratios T_g/T_m, represented in Figure 1, would be those of phases resulting from a first-order transition at T_x, with T_x and T_g depending on heating rates. Applying Equation (7) determines the value of ε_gs0 = ε_ls0, (positive or negative [23]), for each (θ_g) and Δε = 0. Each first-order transition at θ_x respects Δε = θ_n+ = θ_g > 0, in agreement with Equation (6). Negative values of θ_g (T_g/T_m < 1) are higher than θ_x because T_g/T_m is always higher than T_x/T_m. The values of (θ_x) obtained with Equation (7) are weaker than those predicted in Figure 1 along the dashed curve.

The two lines between T_g/T_m = 0.5 and 1.5 are symmetrical with respect to 1. There are five values of T_g/T_m for T_x/T_m higher than 0.7069, and three values of T_g/T_m for T_x/T_m < 0.7069, depending on the singular values of Δε determined by each thermal history. For T_x/T_m = 0.7069, T_g/T_m = 1.475. Note that the NCHN model predicts nucleation temperatures at T_n+ > 2 T_m, ignoring the structure of these new ordered phases. We assume that glassy phases are formed.

3. Diagram Applications

Liquid-liquid transitions in bismuth were observed along lines (1,2,4,5) in Figure 1, by several authors using differential scanning calorimetry (DSC) or differential thermal analysis (DTA) at various heating rates. In fact, this leads to vitreous transitions at T_g/T_m = 1.4291, 1.4384, 1.8675 and 2 for T_x/T_m = 0.70811, 0.70767, 0.80681 and 1, respectively, without recovery of the endothermic heat (Δε H_m) at these temperatures [1,3,4]. The value of ε_gs0 for bismuth was equal to 0.1907 corresponding to δ_ls = 0.0912 [23].

The vertical lines (3,5) in Figure 1 are those of tin and correspond to T_g/T_m = 1.5523 (T_g = 783.9 K) and 2 (T_g = 2 T_m). The horizontal line T_x/T_m = 0.71022 determines T_g/T_m = 1.5523. Two liquid-liquid transitions, occurring at T_g/T_m = 1.5523 and 2 are known up until now. Only T_g/T_m = 1.5523 is characterized as a glass transition [5].

The Bi glass transition was not reproduced at T_g/T_m = 2 during cooling because all bonds were erased during heating far above T_g/T_m = 2, as observed with the heating and cooling rates of 2 K/min [4]

In contrast, the Sn transition at T_g/T_m = 2.168, during heating, disappeared during cooling and gave rise to a new transition at T_g/T_m = 1.832, characterizing a first-order transition at T_g/T_m = 2, and observing that [(2.168 + 1.832)/2 = 2] with heating and cooling rates of 7.5 K/min [3]. The first-order transition at T_g/T_m = 1.832 (T_g = 652 °C) was reproduced during the second heating and cooling at 10 °C/min and could correspond to Δε = 0.832.

The resistivity measurements [4] after the second heating were much weaker than those obtained during the first heating of Sn and Bi. At the highest temperature T = 2.28 T_m, the ordered fractions did not disappear because T_g is expected to be on the order of 3 T_m. The resistivity decreases were equal to 22% for tin and 16.7% for Bi.

4. Singular Enthalpy Coefficients

We examine the case where the glassy phase is formed at T_x. Characteristic values of Δε_lg (θ) = −Δε at various temperatures T_x would correspond to various percolation thresholds of configurons. The enthalpy coefficient (Δε_lg = −Δε) of Phase 3 is constant up to T_n+ = T_g. The transition at T_g leads to a new liquid state with Δε_lg linearly decreasing with temperature in agreement with Equation (6). Consequently, the melt-specific heat, being proportional to the derivative (Δε_lg/dT), undergoes a jump equal to Δε H_m/T_m at T_g = T_n+.

4.1. Bismuth

The weakest glass transition temperature of bismuth was predicted at T_g = 202.46 K [23]. The singular enthalpy coefficients Δε were determined: Δε_lg = 0, Δε_lg0 = 0.1907, −Δε_lg (θ_g) = 0.094065, −Δε_lg (θ_0m = −2/3) = 0.10594, −Δε_lg (θ = −1) = 0.238375 and Δε_lg = −1. The experimental coefficients (Δε) giving rise to θ_n+ = 778/544.5 −1 = 0.429 and 784.6/544.5 −1 = 0.44096 were obtained with a heating rate of 5 K/min and were nearly equal to the theoretical values (0.42908 = 0.1907 + 0.23838) and (0.43838 = 0.094065 + 0.10594 + 0.23838) [1]. In Figure 2, these two transitions are nucleated at T_x = 286.2 and 288 K during undercooling. Only the transition at T = 286.2 K is represented, leading to the horizontal line (2) up to T_g = 778.1 K. The observed coefficient Δε = 0.8675 [3], nucleated at T = 370 K, was equal to the sum (0.42908 + 0.43838) up to T_g = 1016 K along Line (3) [23]. The coefficient (Δε = 1) along Line (4), nucleated at 395 K, without crystallization at T_m, disappears at T_g = 2 T_m = 1089 K. We expect, by reversing heating to cooling at a reduced temperature slightly lower than T_g, that the enthalpy coefficient Δε_lg = −Δε will fall to zero after an incubation time, starting from a glassy fraction equal to Δε.

The enthalpy coefficient variation of Phase 3 along Line (5) in Figure 2, obeying to Equation (6), is obtained after a transition at T_g = T_n+ followed by continuous heating through the various glass transitions. The latent heat (Δε H_m) would be recovered by reversing heating to cooling slightly below T_g.

4.2. Tin

The weakest glass transition temperature of tin is T_g = 185.2 K (θ_g = −0.63332), applying Equations (7) and (8) for Δε_gs0 = 0.167, (δ_ls = 0.08028). The singular enthalpy coefficients of liquid tin are Δε_lg = 0, Δε_lg0 = 0.167, −Δε_lg (θ_g) = 0.08373, −Δε_lg (θ_0m = −2/3) = 0.09278, −Δε_lg (θ = −1) = 0.20875 and −Δε_lg = 1. A sum of all basic coefficients leads to Δε = 0.5523. Here, too, the combination of singular enthalpy coefficients determines the glass transition at T_g/T_m.

In Figure 3, the transition occurring at T_x = 284 K (θ_x = −0.43757) gives rise to a glassy fraction f = 0.5523 and a glass transition at T_g/T_m = 1.5523 (T_g = 783.9 K) calculated with ε_gs0 = 0.167 and Equations (6) and (7). The temperature (284 K) separates two crystalline phases corresponding to gray and white tin. The temperature T_x = 284 K, corresponding to the melting temperature T_m ≅ 284 K of gray tin, is difficult to observe because of the concomitant formation of a glass phase at the same temperature [57]. Above 284 K, the liquid fraction of gray phase is a glass with T_g = 783.9 K, coexisting with a crystallized fraction (1 − 0.5523 = 0.4477) of white tin up to T_m. A glass transition temperature was observed in tin at T_g ≅ 780 K, confirming the existence of a glassy fraction above T_m [5]. The glassy phase corresponding to f = Δε = 1 would be nucleated at T_x = 366.7 K and heated without crystallization at T_m, up to T_g = 2 T_m = 1010 K. We will see that the glass transition at 1010 K observed by resistivity measurements is reversible [4].

As a conclusion of this chapter, the existence of singular glassy fractions (f) in liquid Bi and Sn is predicted. Liquid-liquid transitions are observed and occur at the predicted glass transition temperatures. The latent heat (Δε H_m) is absent during heating as expected for glass transitions and could be present by reversing heating to cooling slightly below T_g.

5. Experimental Densities of Bi and Sn during Heating

Density varies linearly with increasing temperature, T (K), above T_m in liquid elements [3,57,58,59,60,61]:

$$d=d_0-aT.$$

(9)

where d₀ is a density at 0 K. In general, density measurements were made after melting the crystalline phase in the absence of temperature T_x resulting from undercooling and reheating. Nevertheless, liquid-liquid transitions were observed with heating rates between 0.1 and 20 °C/min at temperatures (T_n+) [23]. Consequently, new atomic bonds are induced by relaxation of liquid state near T_n+ [22]. The singular coefficient Δε belonging to lower enthalpy phases determines the temperature T_n+ = T_g. Bond formation gives rise, at very low heating rate, to weak endothermic latent heat, dispersing the liquid density measurements above T_m. Density, d, is expected to be reduced for bismuth and increased for tin by this enthalpy relaxation up to T_g = 2 T_m. Then, Equation (9) can be written as a function of θ = Δε, applying Equations (4) and (6):

$$d=d_0-a{T}_{m }\left(1+{\theta }_{n+}\right)=d_0-a{T}_{m }\left(1\pm ∆{\epsilon }_{lg}\right).$$

(10)

Density measurements would lead to dispersed results because they would be dependent on the measurement time and on the amplitude of the relaxed enthalpy.

5.1. Bismuth Density

The specific heat of bismuth above T_m, measured point after point [62], is strongly dispersed as reproduced in Figure 4. Few singular values of T_n+ are indicated. The deepest one occurs at 862 K (Δε = 2 × 0.23838 + 0.10594 = 0.5827) and the highest ones at 1017 K (Δε = 0.8675) and 1089 K (Δε = 1). The specific heat (0.147 J/K/g) at T_m is recovered at 1180 K showing that the transition width at 1089 K is about 100 K in this case. These results show that the dispersion of measurements attains 12%, and that these weak glassy fractions induced by relaxation have glass transition temperatures equal to T_n+. Consequently, liquid-liquid transitions observed around each temperature T_n+ would correspond to fractions much weaker than the singular value (Δε) associated with T_n+ Dispersion is expected for density measurements below the line defined by Equation (9).

The density of liquid bismuth was measured by gamma attenuation from the melting point to 1000 °C in discrete steps of 5 °C and reproduced in Figure 5 [3]. Values of density were stabilized at singular coefficients, 0.42908, 0.8093 and 0.90594, each of them being a sum of basic coefficients corresponding to a glassy phase. A transition between 0.8093 and 0.90594, occurring at 1.8675 T_m (743.8 °C) was observed by DTA at 0.1 °C/min.

The density change at T_m corresponds to a melting heat of 1.23838 H_m instead of H_m, because it contains the enthalpy (0.23838 H_m) provided by Equation (4), which corresponds to the initial formation of Phase 3 for T_g/T_m = 1 as shown in Figure 1. There is no visible transition in Figure 5 at T = 2 T_m as shown by the quasi-continuity of the density.

5.2. Tin Density

The density of tin is represented by Lines (1–3) in Figure 6: solid tin, Line (1) [60], and liquid tin, Lines (2) [59] and (3) [63]. Lines (2,3) are chosen among measurements with only 50 kg/m³ of error reviewed by Alchagirov and Chochaeva [59]. The difference in density, 82 kg/m³ at 2 T_m is higher than the measurement error. Lines (2,3) could represent two liquid densities that are parallel and separated by approximately 75 kg/m³ or less inside the measurement error. This phenomenon, if confirmed, would be associated with the presence of a weak glassy fraction f << 0.20875 from 0 K to 2 T_m and beyond as predicted. The density change at T_m and the melting heat depend on the thermal history and may include glassy fractions that are melted at very high temperatures beyond 3 T_m as predicted by the glassy phase diagram.

6. The Heat Capacities during Heating

The specific heat (C_p) is reduced by a contribution (ΔC_p) above T_m during heating in the presence of a glassy fraction f = Δε after an enthalpy change (−Δε H_m) and a first-order transition at T_x:

$$\delta C_p=T{(\delta S/\delta \theta )}_p{\left(\delta \theta /\delta T\right)}_p=-T{S}_m/T_m$$

(11)

where $S=-∆\epsilon H_m/T_{m }=-\theta H_m/T_m$ represents the entropy of the glassy phase fraction at T_n+ with Δε = θ, applying Equation (6). There is no specific heat and density added in the absence of glassy fraction for Δε = 0 in all liquids. Equation (11) is not applied in the absence of first-order transition at T_x.

Applying Equation (11) leads to C_p = 0 at T_g = 2 T_m,

For Sn:

$$C_p=28.43-0.0563\times \left(T-T_{m }\right),$$

(12)

with H_m = 7179 J/mole [64] and T_m = 505 K.

At T_m, C_p = 28.43 J/mole in agreement with Chen’s measurements [65] after adding 0.9 mJ/mole corresponding to the electronic specific heat contribution of tin [66].

For Bi:

$$C_p=30.2-0.05546\left(T-T_m\right),$$

(13)

with T_m = 544.5 K. At T_m, C_p = 30.2 J/mole [67] is used to determine H_m = 8613 J/mole. The measured melting enthalpy is 1.23838 H_m leading to 10,662 J/mole inside an uncertainty of measurements varying from 10,480 to 11,300 J/mole [68].

The constants 28.43 and 30.2 J/K/mole are values of C_p at T_m in the absence of glassy phases. These heat capacities are equal to zero at T = 2 T_m and to 28.43 and 30.2 J/mole at temperatures higher than 2 T_m as shown for Sn and Bi in Figure 7 and Figure 8. When a fraction (f) of liquid is no longer in a glassy state above the temperature (T), (C_p) linearly decreases from 28.43 for Sn to zero and from 30.2 J/K/mole to zero for Bi. These specific heat variations, expected during heating, are plotted as a function of temperature for Sn and Bi in Figure 7 and Figure 8 without latent heat recovery as expected for glass transitions.

7. Other Experimental Observations of Glassy States above T_m

7.1. In Tin

A glass transition was detected for the first time around 783.9 K in agreement with our predictions [5]. This transition induced a specific heat jump of about 1.8 J/K/mole instead of 28.4 J/K/mole corresponding to a glassy fraction f ≅ 1.8/28.4 = 6.3%. This glassy fraction was induced in a “stepwise-scanning mode: the temperature during the thermal equilibration stage changed with time and gradually approached a constant value in about 60 min”. The peak of ΔC_p, observed during this slow heating, accompanied by critical phenomena, could be attributed to the thermodynamic transition of configurons [33,69]. The jump of ΔC_p measured at 4 K/min was weaker because the nucleation time of new bonds was much lower.

7.2. In Bismuth

DTA at 0.1 °C/min reveals an endothermic latent heat of the order of 20 to 100 J/mole [3] at T_g =1.8675 T_m. A transition width (ΔT) of 200 K was observed at 1089 K by resistivity measurements with 2 °C/min and a width (ΔT = 10 K) expected for R = 0.1 °C/min [4]. We attribute this liquid-liquid transition to a glassy fraction transition of the order of 6.6% with 20 J/mole and 33% with 100 J/mole] assuming a transition width of 10 K. A second endothermic heat was observed by DTA at T = 2 T_m = 1089 K corresponding to a second glass transition and to the enthalpy relaxed during 720 min between 1017 and 1089 K.

Configuron thermodynamic transition was revealed by structural changes at the glass transition via radial distribution functions [56,70]. The first sharp diffraction minimum in the pair distribution function was shown to contain information on structural changes in amorphous materials at the glass transition temperature (T_g). An additional feature of such configuron transition was determined by measuring the temperature dependence of the structure factor of molten bismuth, −S(q)” [3]. The authors observed, in their neutron diffraction analysis, an additional feature in the measurement that appeared around the melting temperature of bismuth. “Pair distribution function curves (g(r)) were calculated for each (S(q)) measurement”. “At and above melting, both the S(q) and g(r) curves were characterized by a shoulder located on the high q and r side of the first peak, respectively”. The temperature dependence of the coordination numbers lead to the number of atoms contributed by the shoulder, N_Shoulder(T). The derivative of N_Shoulder with respect to temperature, showed a discontinuity at the transition point at 1089 K, and was associated with this temperature-driven transformation and a structural change. This structural change is a signature of the thermodynamic transition of configurons [33].

7.3. BiSb20 wt%

DSC revealed that C_p was equal to zero at 1070 °C with a heating rate of 20 °C/min [4]. From our model, we deduce that the glassy fraction (f) was 100%. The liquidus temperature of this alloy is 425 °C (698 K). The glass transition, occurring at 1070 °C (1343 K), was weaker than 2 T_m = 1396 K. The specific heat increased from 1070 °C to 1123 °C without latent heat. The transition width (2000 K) is expected to be 10 times wider than at 2 K/min. Consequently, the recovery of ΔC_p, for a temperature increase of 53 K, is of the order of 53/2000 ≅ 2.6% of its value at T_m. We conclude that the glassy fraction (f = 100%) was induced at a temperature T_x < T_m. As a result, the rapid increase of the heating rate would have to induce the first order transition at T_x < T_m. This experiment showed for the first time that a glassy phase of 100% can be obtained with a heating rate of 20 K/min. The latent heat equal to H_m is not recovered as predicted in Figure 7.

7.4. InSn80 wt%

An internal friction method was used to study the structural changes of InSn80 wt% [71]. This alloy has a melting temperature of about 190°C (463 K) and a glass transition temperature expected at 2 T_m = 926 K (653 °C). A minimum of internal friction occurs at 625 °C and a maximum at 700 °C. Based on the results of a diffraction experiment around 700 °C, the liquid structures before and after the peak are very different [4]. “Before the change, there are residual covalent bonds of solid tin in the melt and during the transition, the residual bonds are broken and at the same time, new atomic bonds build up, with a relatively uniform melt forming”. This description provided by [4] is known to be due to the percolation threshold of configurons [33].

7.5. PbSn61.9 wt%

This eutectic composition has a melting temperature of 183 °C (456 K). The highest glass transition temperature is predicted at 912 K (639 °C). The internal friction has a maximum at 670 °C and a minimum at 560 °C with a heating rate of 2.5 °C/min and a maximum at 712 °C and a minimum at 600 °C with 6 °C/min. This liquid-liquid transition temperature increases with the heating rate as observed in all glasses [4].

8. Another Method to Stabilize Glassy States above (2 T_m)

A process was described by Zu F.Q. [4], ignoring at this time that resistivity decreases could be due to the formation of glassy fractions, added after several cooling cycles to various temperatures T_x < T_m, followed by successive reheating as shown in Figure 9. Each value of T_x/T_m < 0.7069 lead to a glass transition much higher than 2 T_m as shown by the glassy phase diagram of Figure 1. These various reheating cycles enriched the total glassy fraction as shown by resistivity reductions of 40% after three heating of Cu-Sb76.5 wt%, of 22% in tin and 16.7% in bismuth after two heating cycles. In addition, depending on heating rate, another glass transition occurs at T_m < T_n+ < 2 T_m in various alloys such as InSn80 wt%, InBi32 wt% at R = 3 °C/min and CuSb76.5 wt% at 5 °C/min. They were due to the formation of very weak glassy fractions (f), depending on heating and cooling rates. The residual resistivity is equal to 100 µΩ·cm at T_m during the first heating. Consequently, the formation of high glassy fractions with T_g >> 2 T_m could be attained after adding new thermal cycles, leading to a maximum resistivity fall. Reheating cycles could stabilize the glassy phase above (T_n+ = 2 T_m).

9. Conclusions

Liquid-liquid transitions are observed below and above T_m.

First-order transitions, predicted by the NCHN model, occur at T_x < T_m, building glassy phase fractions (f = Δε) and crystallized fractions (1 − Δε) after supercooling the melt. Melting of glassy fractions occurs at temperatures T_n+ > T_m at the percolation threshold of bonds. One of them induces a glass enthalpy equal to the melting enthalpy (Δε = 1) and a zero specific heat, without crystallized fraction, up to the glass transition (T_g = 2 T_m). All glassy fractions are expected to have enthalpies equal to zero, after reversing heating to cooling below T_n+ and a long time of incubation at a temperature close to T_g.

Liquid-liquid transitions between the melting temperature (T_m) and (2 T_m) are observed in the absence of first order transition at T_x < T_m and are reminiscent of glassy fractions (f) formed at temperatures (T_x) weaker than (T_m) through first-order transitions. These weak fractions correspond to singular values (Δε) of enthalpy coefficients (Δε_lg (θ) = −Δε) of a new phase called “Phase 3”, and to typical percolation thresholds of configurons at various temperatures (T_n+ = T_g). These fractions are slowly induced near (T_n+) by relaxation in the absence of first order transition prior to melting.

All these liquid-liquid transitions above T_m are glass transitions.

Several situations are encountered in bismuth and tin:

(1)

After melting Bi and Sn at T = T_m, weak fractions (f) were built by slow heating (0.1 °C/min for Bi and one hour between each measurement for Sn) and melted at θ_g = θ_n+ = (T_n+ − T_m)/T_m = Δε. Liquid-liquid transitions were observed at 1.8675 T_m for bismuth and 1.5523 T_m for Sn. The glassy character of these transitions is confirmed by structural transitions that we attribute to the melting of configurons. An observed glass transition in Sn was also characterized by a weak specific heat jump predicted by the NCHN model and a peak at 1.5523 T_m due to the thermodynamic character of a transition obeying critical exponents associated with configuron percolation. There is no endothermic latent heat (Δε H_m) during heating.

(2)

After melting Bi and Sn at T = T_m, transitions were also observed at T = 2 T_m by DTA and (or) resistivity that we consider as new glass transitions. The transition was reversible only for Sn. A weak endothermic heat instead of H_m was observed for Bi at 2 T_m. The glassy states of Bi and Sn could have a density equal to that of the liquid at T_m only after reversing heating to cooling and a long incubation time close to T_m.

(3)

After melting Bi at T = T_m, high-resolution measurements of the density showed the existence of singular values corresponding to those of the enthalpy of Phase 3. The melting heat at T_m corresponds to 1.23838 H_m instead of H_m including, in addition, the latent heat of glassy phase as predicted by the NCHN model.

(4)

A glassy phase diagram is proposed for systems having their lowest transition determined by their Lindemann coefficients. Each first order transition at T_x < T_m leads to multiple glass transitions. The possible existence of weak glassy fractions (f) for T_x/T_m < 0.7069, with glass transition temperatures much higher than (2 T_m) is envisaged (beyond (3 T_m) for f < 22.45%). Resistivity measurements showed decreases in Bi and Sn from T_m to 2 T_m and beyond, after thermal cycling between undercooled liquid and solid states.

(5)

The glassy phase formations at T_x are accompanied by latent heats, without being recovered at T_n+ with (T_m < T_n+ < 2 T_m) during heating. Predictions of their contribution equal to (θ_n+ H_m/T_m) are proposed for T_n+ = T_g ≤ 2 T_m. The heat capacity linearly decreases down to zero when Δε increases up to 1 (T_n+ = 2 T_m). An enthalpy equal to zero, down to T_m, would be induced by reversing heating to cooling from a temperature slightly weaker than T_g = T_n+. We show, for the first time, that the liquid-specific heat is constant between T_m and 2 T_m in the absence of glassy phase.

(6)

The stability of glassy fractions for T_x/T_m < 0.7069 can be very high because their (T_g) could be much higher than (2 T_m) as proved by resistivity decreases observed up to 2 T_m and beyond in Bi and Sn. The glassy fraction (f) is enhanced by successive thermal cycles between solid and liquid states. Each new glassy fraction could be added and could reinforce the total glassy fraction at very high temperatures up to f ≤ 100%.