# Multiple Glass Transitions in Bismuth and Tin beyond Melting Temperatures

Robert F Tournier

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Robert F Tournier. Multiple Glass Transitions in Bismuth and Tin beyond Melting Temperatures. 2022. hal-03870558

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# Multiple Glass Transitions in Bismuth and Tin beyond Melting Temperatures

Robert F. Tournier

**Abstract:** Liquid-liquid transitions were discovered above the melting temperature ($T_m$) in Bi and Sn up to $2T_m$ and 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 $T_g = T_{n+} = 2T_m$. 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 ($2T_m$) for $T_x/T_m < 0.7069$. A specific heat equal to zero is expected after cooling from $T \le 2T_m$ and could correspond to a fully ordered 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 $T_x/T_m < 0.7069$.

**Keywords:** Metals, glass transitions, melting enthalpy, Liquid-liquid transitions, density, first-order transitions, configurons, Phase 3, structural transitions.

## 1. Introduction

**Citation:** Lastname, F.; Lastname, F.; Lastname, F. Title. *Metals* **2022**, *12*, x. https://doi.org/10.3390/xxxxx

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Liquid-liquid phase transitions occur in glass-forming melts at temperatures $T_{n+}$ above $T_m$, 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 $T_{n+} = T_g$ [ (23) (24) (22)]. 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 ($2T_m$), 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 $T_g = T_{n+} = 2T_m$ in Bi and Sn as already envisaged for a component of $Cu_{46}Zr_{46}Al_8$ melt [ (23) (12)].

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 $10^{12}$ K/s [ (25) (26)]. A first order transition from the
liquid-phase (L) to a metastable, heterogeneous, ordered phase viewed as a glass phase called G-phase was observed. 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 Hm at various temperatures Tn+ = Tg in Ag [ (25)], in Zr [ (31)], in Cu and in Fe [ (32)]. The values of Tn+ were determined from singular values of G-phase frozen enthalpy, employing the nonclassical homogeneous nucleation (NCHN) model to predict T<sub>x</sub>, Tn+ and the enthalpy. Singular values of (f) leading to (Tn+) correspond to percolation thresholds of broken bonds (configurons) leading to glassy phases up to Tn+ [ (33) (34) (24) (29) (30)].

High undercooling rates of bulk liquid elements are known since many years [ (35)]. Consequently, we plan to confirm, in this publication, that the nucleation of G-phases and their glass transition temperatures above Tm are observable without employing heating and cooling rates of the order of 10<sup>10</sup> to 10<sup>13</sup> K/s to escape from crystallization. A phase diagram of glassy fractions, occurring in molten elements at T<sub>x</sub> < T<sub>m</sub>, is proposed, completing the diagram already established for T<sub>x</sub> > T<sub>m</sub> [ (30)] in agreement with previous molecular dynamics simulations [ (25) (26) (31) (32)]. The specific heat values at T<sub>g</sub> = Tn+ is predicted up to T<sub>g</sub> = 2 T<sub>m</sub> where a fully ordered glassy phase is expected. The density far above T<sub>m</sub> 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 enthalpies equal to ε<sub>ls</sub>Hm, ε<sub>gs</sub>Hm and Δε<sub>lg</sub>Hm [ (36)]. For liquid elements, that are easily crystallized, the liquid enthalpy coefficients obey the following [ (30), (27)]:

$$ \epsilon_{ls} = \epsilon_{ls0} \left(1 - \frac{\theta^2}{\theta_{0m}^2}\right) = \epsilon_{ls0} \left(1 - 2.25 \theta^2\right), \qquad (1) $$

$$ \epsilon_{gs} = \epsilon_{gs0} \left(1 - \frac{\theta^2}{\theta_{0g}^2}\right) = \epsilon_{gs0} \left(1 - \theta^2\right) \qquad (2) $$

$$ \Delta\epsilon_{lg}(\theta) = [\epsilon_{ls} - \epsilon_{gs}] = [\epsilon_{ls0} - \epsilon_{gs0} - \theta^2(2.25 \epsilon_{ls0} - \epsilon_{gs0})], \qquad (3) $$

where Hm is the melting enthalpy, εls and εgs are fractions of Hm, θ<sub>0m</sub> = -2/3 and θ<sub>0g</sub> = -1 are the reduced Vogel-Fulcher-Tammann (VFT) temperatures in liquid elements for which the minimum value of T<sub>g</sub> is fixed by the Lindemann coefficient (δls) of each element [ (27)]. Eq. (3) is transformed into Eq. (4) for the minimum value of ε<sub>ls0</sub> = ε<sub>gs0</sub>:

$$ \Delta\epsilon_{lg} = -1.25 \epsilon_{gs0} \theta_g^2, \qquad (4) $$

where Δε<sub>lg</sub>(θ<sub>g</sub>) is the latent heat coefficient, accompanying the glass transition during the first cooling, the formation of Phase 3 below the percolation threshold of bonds, and the partial breaking of bonds after reheating above T<sub>g</sub>, occurring without latent heat [ (30)] Figure 3. These relaxation effects due to the development of bonds below the percolation
threshold at T<sub>g</sub> are observed after quenching melt in amorphous state and heating at 20 K/min [ (37), (38) (39) (40) (41) (42)]. 83

Coefficient minima (ε<sub>lso</sub> = ε<sub>gso</sub>) were initially determined to be equal to 0.217 [ (43)]. 86
They were the average of many liquid element coefficients deduced from the undercool-
ing rate of each of them corresponding to the mean value 0.103 of their Lindemann coef-
ficient [9,18]. The number 2.25, initially equal to 2.5 in Eq. (1), had for consequence to fix
the VFT temperature to T<sub>m</sub>/3 [ (44)]. It appears later that (Δε<sub>lg</sub>) in Eq. (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, attributed to all glassy phases,
which are formed through a first-order transition [ (37)]. Phase 3 could be the congruent
bond lattice predicted for disordered oxide systems [ (48)], extended later to critical pack-
ing density formation applied to broken bonds (configurons) producing the glass transi-
tions at T<sub>g</sub> [ (33), (49)]. An ordered fraction of atoms equal to various percolation thresh-
olds exists up to T<sub>n+</sub>, in all glass-forming melts [ (34; 50; 23)]. Phase 3 results from the
formation of configuron phases [ (50)]. The NCHN model predicted the formation condi-
tions of glacial phases (Phase 3) in Ag-Cu and Ag liquids at various heating rates as pre-
viously described by molecular dynamics (MD) simulations, showing full melting at T<sub>n+</sub>=
1.119T<sub>m</sub> for Zr, 1.126T<sub>m</sub> for Ag, 1.219T<sub>m</sub> for Fe and 1.354T<sub>m</sub> for Cu [ (25) (26) (31) (32)]. These
phenomena are governed by singular values of the enthalpy Δε<sub>lg</sub> of glassy Phase 3, formed
at nucleation temperatures T<sub>nG</sub> [ (29), (30)]. 100

Two families of nucleation temperatures are given in Eqs. (5,6) where θn- is equal to two opposite values of θg because εgs is a function of θ² in Eq. (2): 104

$$
\theta_{n-} = \theta_g = \frac{\pm(\epsilon_{gs}-2)}{3}, \qquad (5)
$$

$$
\theta_{n+} = \Delta\epsilon \qquad (6)
$$

where Δε is equal to singular values of the enthalpy coefficient (-Δε<sub>lg</sub>) of Phase 3. 108

There are two methods to produce a glassy phase with an enthalpy coefficient equal to (-Δε). The first one predicts the undercooling temperature for each value of Δε. These nucleation temperatures (θₓ) are calculated using the (NCHN) model applied to Liquid 2: 109
110

$$
\epsilon_{gs}(\theta = 0) = (3\theta_x + 2 - \Delta\epsilon)/(1 - \frac{\theta_x^2}{\theta_g^2}) \quad (7)
$$

Here, θ₀g² = 1. For Δε = 0, a second order-like phase transition temperature takes place at T₀g during heating for the minimum value of εgs0. Values of θₓ for various values of Δε are deduced for the same value of εgs0. 113
114
115

Glass phases, formed at θₓ after undercooling, lead to glassy phases with θg = θn+ = Δε
in agreement with Eq. (6). They are accompanied by a crystallized fraction (1-Δε). A com-
posite crystal-glass is built below Tm with a melting enthalpy (1-Δε) Hm and a missing
enthalpy (Δε Hm) recovered at Tn+ [ (23)]. The glassy phase fraction is not destroyed at Tm
and depends on the sample thermal history and the last value of Δε obtained at Tₓ during
heating. 116
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120
121

The minimum glass transition at T<sub>g</sub> is masked by spontaneous crystallization of many liquid elements and determined with Eq. (7) using Δε = 0 and the minimum value of ε<sub>gso</sub> = ε<sub>lso</sub>, depending on the Lindemann coefficient δ<sub>ls</sub> of each element [ (27)]: 122
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$$
\epsilon_{gs0} = \epsilon_{ls0} = (1 + \delta_{ls})^2 - 1 \qquad (8)
$$

In Figure 1, a second method is used to calculate new values of θₓ during heating. Each value of θg varying from -0.5 to 2 corresponds to a value of Tg/Tm between 0.5 and 3. All ratios Tg/Tm, represented in Figure 1, could be those of phases resulting from a first-order transition at Tₓ, with Tₓ and Tg depending on heating rates. Applying Eq. (7), determines the value of εgs0 = εls0, (positive or negative [ (23)]), for each (θg) and Δε = 0. Each first-order transition at θₓ respects Δε = θn+ = θg > 0, in agreement with Eq. (6). Negative values 126
127
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129
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of θg (Tg/Tm < 1) are higher than θx because Tg/Tm is always higher than Tx/Tm. The values of (θx) obtained with Eq. (7) are weaker than those predicted in Figure 1 along the dashed curve.

<figure><img src="image_4.png" /><figcaption>**Figure 1:** Glassy phase diagram (Tg/Tm) resulting from first-order transitions at Tx/Tm during heating of liquid elements. Valuable for all glass transitions due to Lindemann coefficients. For Bi, Tx/Tm = 0.8068 reveals Tg/Tm = 1.8675 as observed from differential thermal analysis (DTA). For Sn, Tx/Tm = 0.71022 reveals Tg/Tm = 1.5523. For Tx/Tm > 0.7069, there are 5 glass transitions. For Tx/Tm < 0.7069, the number of glass transitions is equal to 3. (Tg/Tm) can be higher than 2. Transitions at Tg/Tm = 1.4291 and 1.4384 are initiated by other (Tx/Tm) values. Tg/Tm = 0.5, 1.5 and 3 for Tx/Tm = 0.22451. For Tx/Tm = 1, transitions were observed at Tg/Tm with latent heat much weaker than (Δε Hm) above Tm.</figcaption></figure>

The two lines between Tg/Tm = 0.5 and 1.5 are symmetrical with respect to 1. There are 5 values of Tg/Tm for Tx/Tm higher than 0.7069, and 3 values of Tg/Tm for Tx/Tm < 0.7069, depending on the singular values of Δε determined by each thermal history. For Tx/Tm = 0.7069, Tg/Tm = 1.475. Note that the NCHN model predicts nucleation temperatures at Tn+ > 2 Tm, 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, leading, in fact, to vitreous transitions at Tg/Tm = 1.4291, 1.4384, 1.8675 and 2 for Tx/Tm = 0.70811, 0.70767, 0.80681 and 1 respectively without recovery of the endothermic heat (Δε Hm) at these temperatures [ (1) (3) (4)]. The value of εgs0 for bismuth was equal to 0.1907 corresponding to δls = 0.0912 [4].

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

The Bi glass transition was not reproduced at Tg/Tm = 2 during cooling because all bonds were erased during heating far above Tg/Tm = 2 as observed with heating and cooling rates of 2 K/min [ (4)] (Figure 13b).

In contrast, the Sn transition at Tg/Tm = 2.168, during heating, disappeared during cooling and gave rise to a new transition at Tg/Tm = 1.832 characterizing a first-order transition at Tg/Tm = 2 observing that [(2.168+1.832)/2 = 2] with heating and cooling rates of 7.5 K/min [ (4)] (Figure 13e). The first-order transition at Tg/Tm = 1.832 (Tg = 652 °C) was reproduced during the second heating and cooling at 10 °C/min and could correspond to Δε = 0.832.

The resistivity [ (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 Tm, 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 $\Delta\varepsilon_{lg}$ ($\theta$) = -$\Delta\varepsilon$ at various temperatures $T_x$ would correspond to various percolation thresholds of configurons. The enthalpy coefficient ($\Delta\varepsilon_{lg} = -\Delta\varepsilon$) of Phase 3 is constant up to $T_{n+} = T_g$. The transition at $T_g$ leads to a new liquid state with $\Delta\varepsilon_{lg}$ linearly decreasing with temperature in agreement with Eq. (6). Consequently, the melt specific heat, being proportional to the derivative ($\Delta\varepsilon_{lg}/dT$), undergoes a jump equal to $\Delta\varepsilon H_m/T_m$ 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 $\Delta\varepsilon$ were determined: $\Delta\varepsilon_{lg} = 0$, $\Delta\varepsilon_{lg0} = 0.1907$, $-\Delta\varepsilon_{lg}$ ($\theta_g$) = 0.094065, $-\Delta\varepsilon_{lg}$ ($\theta_{0m} = -2/3$) = 0.10594, $-\Delta\varepsilon_{lg}$ ($\theta = -1$) = 0.238375 and $\Delta\varepsilon_{lg} = -1$. The experimental coefficients ($\Delta\varepsilon$) giving rise to $\theta_{n+} = 778/544.5 - 1 = 0.429$ 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 $\Delta\varepsilon = 0.8675$ [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 ($\Delta\varepsilon = 1$) along Line (4), nucleated at 395 K, without crystallization at $T_m$, disappears at $T_g = 2T_m = 1089$ K. We expect, by reversing heating to cooling at a reduced temperature slightly lower than $T_g$, that the enthalpy coefficient $\Delta\varepsilon_{lg}$ will fall to zero, without forgetting that the glassy fraction is equal to $\Delta\varepsilon$.

The enthalpy coefficient variation of Phase 3 along Line (5) in Figure 2, obeying to Eq. (6), is obtained after a transition at $T_g = T_{n+}$ followed by continuous heating through the various glass transitions.

<figure><img src="image_5.png" /><figcaption>**Figure 2:** Bismuth glassy phase fractions $f = \Delta\varepsilon$ expected after undercooling and reheating. Four liquid-liquid transitions at $T_{n+} = T_g$ after reheating were observed. Horizontal lines: 1- $\Delta\varepsilon_{lg} = 0$ of glassy fractions; 2- $\Delta\varepsilon = \theta_{n+} = \theta_g = 0.42908$ up to $T_g = 778$ K, 0.43838 up to $T_g = 783$ K, (not represented); 3- $\Delta\varepsilon = 0.8675$ up to $T_g = 1016$ K; 4- $\Delta\varepsilon = 1$ up to $T_g = 1089$ K without crystallization at $T_m$. Vertical lines: $T_x = 286.2$ K leading to $\Delta\varepsilon = 0.42908$ and a crystallized fraction (1-$\Delta\varepsilon = 0.57092$); $T_x =$</figcaption></figure>
370 K leading to $\Delta\varepsilon = 0.8675$ and a crystallized fraction ($1-\Delta\varepsilon = 0.1325$); $T_x = 395$ K leading to $\Delta\varepsilon = 1$; $T_m = 544.5$ K: melting of crystallized fractions; $T_{n+} = 778.1$ K; $T_{n+} = 1016$ K; $T_{n+} = T_g = 2T_m = 1089$ K. The glass transitions leading to $\Delta\varepsilon_{lg} = 0$ could be observed by reversing heating to cooling slightly below $T_g$. Line (5) represents Eq. (6). 205
206
207

## 4.2 Tin

The weakest glass transition temperature of tin is $T_g = 185.2$ K ($\theta_g = -0.63332$), applying Eq. (7) for $\Delta\varepsilon_{gs0} = 0.167$, ($\delta_{ls} = 0.08028$). The singular enthalpy coefficients of liquid tin are $\Delta\varepsilon_{lg} = 0$, $\Delta\varepsilon_{lg0} = 0.167$, $-\Delta\varepsilon_{lg} (\theta_g) = 0.08373$, $-\Delta\varepsilon_{lg} (\theta_{0m} = -2/3) = 0.09278$, $-\Delta\varepsilon_{lg} (\theta = -1) = 0.20875$ and $-\Delta\varepsilon_{lg} = 1$. A sum of all basic coefficients leads to $\Delta\varepsilon = 0.5523$. Here too, the combination of singular enthalpy coefficients determines the glass transition at $T_g/T_m$. 208
209
210
211
212
213

<figure><img src="image_6.png" /><figcaption>**Figure 3:** Tin glassy phase fraction $f = \Delta\varepsilon$ expected after undercooling and reheating. $T_x = 284$ K ($\theta_x = -0.43757$), leading to a liquid-liquid transition at $T_{n+} = T_g = 783.9$ K after reheating. ($\Delta\varepsilon = 0.5523$) corresponding to the enthalpy coefficient of the gray phase. The crystallized fraction ($1-\Delta\varepsilon$) of the white phase is (0.4477). The glassy phase with $T_g = 2 T_m = 1010$ K and $\Delta\varepsilon = 1$ is nucleated at $T_x = 366.7$ K ($\theta_x = -0.33336$) and heated without crystallization at $T_m$. 214
215
216
217
218</figcaption></figure>

In Figure 3, the transition occurring at $T_x = 284$ K ($\theta_x = -0.43757$) 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 $\varepsilon_{gs0} = 0.167$ and Eqs. (6,7). 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 $T_m \approx 284$ K of gray tin, is difficult to observe because of the concomitant formation of a glass phase at the same temperature [ (51)]. 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 $T_g \approx 780$ K, confirming the existence of a glassy fraction above $T_m$ [ (5)]. The glass phase corresponding to $f = \Delta\varepsilon = 1$ 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)]. 219
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230

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. 231
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234

## 5. Experimental densities of Bi and Sn during heating

Density varies linearly with increasing temperature, $T$ (K), above $T_m$ in liquid elements [ (3) (51) (52) (53) (54) (55)]: 235
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237

$$d = d_0 - aT. \qquad (9)$$

where $d_0$ 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 239
240
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 $\Delta\epsilon$ belonging to lower enthalpy phases determines the temperature $T_{n+} = T_g$, and gives rise, at very low heating rate, to relaxation of weak endothermic latent heat in the neighborhood of temperatures $T_{n+}$. These relaxation times also have for consequence, to disperse the liquid density measurements and are due to the development of bonds above $T_m$. Density, d, is expected to be reduced for bismuth and increased for tin by this enthalpy relaxation up to $T_g = 2T_m$. Then, Eq. (9) can be written as a function of $\theta = \Delta\epsilon$, applying Eqs. (4,6):

$$d = d_0 - aT_m (1 + \theta_{n+}) = d_0 - aT_m (1 \pm \Delta\epsilon_{lg}). \quad (10)$$

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 [56], is strongly dispersed as reproduced in Figure 4. Few singular values of $T_{n+}$ are indicated. The deepest one occurs at 862 K ($\Delta\epsilon = 2 \times 0.23838 + 0.10594 = 0.5827$) and the highest ones at 1017 K ($\Delta\epsilon = 0.8675$) and 1089 K ($\Delta\epsilon = 1$). 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 ($\Delta\epsilon$) associated with $T_{n+}$. The same dispersion is expected for density measurements below the line defined by Eq. (9).

<figure><img src="image_7.png" /><figcaption>Figure 4: Specific heat of bismuth above $T_m$. Reproduced from [56] with permission of Springer Ed. The line $C_p = 0.1464$ KJ/(Kg K) leading to the value of $C_p$ at $T_m = 505$ K is added. The specific heat is constant from $T_m$ to 2 $T_m$ in the absence of relaxed enthalpy. $T_g = 674$ K for $\Delta\epsilon = 0.23838 + 0.09401 = 0.3324$, $T_g = 778$ K for $\Delta\epsilon = (0.42908 - 0.1907 + 0.23838)$, $T_g = 882$ K for $\Delta\epsilon = (0.429 + 0.23838 + 0.09456 = 0.761)$, $T_g = 1017$ K for $\Delta\epsilon = 0.8675$, $T_g = 1089$ K for $\Delta\epsilon = 1$.</figcaption></figure>

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.
<figure><img src="image_8.png" /><figcaption>**Figure 5:** Bismuth density versus temperature, T (°C). Reproduced with authorization of EPL, EPI [ (3)]. The density variation between T<sub>m</sub> (K) and 2 T<sub>m</sub> (K) is equal to the density change at T<sub>m</sub> while the enthalpy coefficient varies from 0 to 1. Singular values of the enthalpy coefficient are added to the original figure. Line (1) is the density before enthalpy relaxation of glassy phases. A melting heat, 1.23853 H<sub>m</sub> includes the latent heat of the glassy fraction occurring for T<sub>g</sub>/T<sub>m</sub> = 1 in the phase diagram of Figure 1.</figcaption></figure>

The density change at T<sub>m</sub> corresponds to a melting heat of 1.23838 H<sub>m</sub> instead of H<sub>m</sub>, because it contains the enthalpy (0.23838 H<sub>m</sub>) given by Eq. (4), which corresponds to the initial formation of Phase 3. There is no visible transition in Figure 5 at T = 2T<sub>m</sub> as shown by the quasi-continuity of the density.

**5.2 Tin density**

The density of tin is represented Lines (1-3) in Figure 6: solid tin, Line (1) [ (54)], and liquid tin, Lines (2) [ (53)] and (3) [ (57)]. Lines (2,3) are chosen among measurements with only 50 Kg/m³ of error reviewed by Alchagirov and Chochaeva [ (53)]. The difference in density, 82 Kg/m³ at 2 T<sub>m</sub> 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<sub>m</sub> and beyond as predicted. The density change at T<sub>m</sub> and the melting heat depend on the thermal history and may include glassy fractions that are melted at very high temperatures beyond 3 T<sub>m</sub> as predicted by the glassy phase diagram.

<figure><img src="image_9.png" /></figure>

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**Figure 6:** Tin density (d) in Kg/m³, versus T (K). Line (1) [ (54)]; Line (2) [ (53)]; Line (3) [ (57)]. Tₓ = 284 K, the glass formation temperature after undercooling and the melting temperature of gray tin. 2 Tₓ = 1010 K.

## 6. The heat capacities during heating

The specific heat (Cₚ) is reduced by a contribution (δCₚ) above Tₓ during heating in the presence of a glassy fraction f = Δε after an enthalpy change (-Δε Hₘ) and a first-order transition at Tₓ:

$$ \delta C_p = T (\delta S/\delta\theta)_p (\delta\theta/\delta T)_p = -TS_m/T_m \quad (11) $$

where S = -Δε Hₘ/Tₘ = -θ Hₘ/Tₘ represents the entropy of the glassy phase fraction at Tₙ+ with Δε = 0, applying Eq. (6). There is no specific heat and density added in the absence of glassy fraction for Δε = 0 in all liquids. Eq. (11) is not applied in the absence of first-order transition at Tₓ.

Applying Eq. (11) leads to Cₚ = 0 at Tₓ = 2 Tₘ:

For Sn:

$$ C_p = 28.43 - 0.0563 \times (T - T_m), \quad (12) $$

with Hₘ = 7179 J/mole [ (58)] and Tₘ = 505 K

At Tₘ, Cₚ = 28.43 J/mole in agreement with Chen's measurements [ (59)] after adding 0.9 mJ/mole corresponding to the electronic specific heat contribution of tin [ (60)].

For Bi:

$$ C_p = 30.2 - 0.05546(T - T_m), \quad (13) $$

with Tₘ = 544.5 K. At Tₘ, Cₚ = 30.2 J/mole [ (61)] is used to determine Hₘ = 8613 J/mole. The measured melting enthalpy is 1.23838 Hₘ leading to 10662 J/mole inside an uncertainty of measurements varying from 10480 to 11300 J/mole [ (62)].

The constants 28.43 and 30.2 J/K/mole are values of Cₚ at Tₘ in the absence of glassy phases. These heat capacities are equal to zero at T = 2Tₘ and to 28.43 and 30.2 J/mole at temperatures higher than 2Tₘ as shown for Sn in Figure 7 and bismuth in Figure 8. When a fraction (f) of liquid is no longer in a glassy state above the temperature (T), (Cₚ) 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 Figures 7 and 8.

<figure><img src="image_10.png" /></figure>
**Figure 7:** Heat capacities of bismuth in J/K/mole. $C_p$ = 30.2 from T = 544.5 to 1089 K in the absence of glass phase. $C_p$ = 17.2 from T_m = 544.5 to T_g = 778.1 K in the presence of glassy fraction f = 42.908 %. $C_p$ = 4 from T_m = 544.5 to T_g = 1017 K in the presence of a glassy fraction f = 86.75 %. $C_p$ = 0 for f = 100%. Heating beyond T_g, leads to $C_p$ = 30.2 inside the transition width. Cooling from a temperature slightly below T_g leads to the glassy state.

<figure><img src="image_11.png" /></figure>

**Figure 8:** Heat capacities of tin in J/K/mole. $C_p$ = 28.43 from T = 505 to 1010 K in the absence of glassy phase. $C_p$ = 12.7 from T_m = 505 to T_g = 783.9 K in the presence of glassy fraction f = 55.23%. $C_p$ = 0 for f = 100%. Heating beyond T_g, leads to $C_p$ = 28.43 inside the transition width. Cooling from a temperature slightly below T_g leads to the glassy state.

## 7. Other experimental observations of glassy states above T_m

325

### 7.1 Tin

326

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 12.7 J/K/mole corresponding to a glassy fraction f ≈ 1.8/28.4 = 6.3 % instead of 55.23 %. 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 minutes. The peak of Δ$C_p$, observed during this slow heating, accompanied by critical phenomena, could be attributed to the thermodynamic transition of configurons [33), (63)]. The jump of Δ$C_p$ measured at 4 K/min was weaker because the nucleation time of new bonds was much lower.

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### 7.2 Bismuth

336

DTA at 0.1 °C/min reveals an endothermic latent heat of the order of 20 to 100 J/mole [33] 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 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.

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Configuron thermodynamic transition was revealed by structural changes at the glass transition via radial distribution functions [64), (65)]. 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

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348
dependence of the structure factor of molten bismuth, -S(q). The authors observed, in their neutron diffraction analysis [ (3)] , 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<sub>Shoulder</sub>(T). The derivative of N<sub>Shoulder</sub> 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% 359

DSC revealed that C<sub>p</sub> was equal to zero at 1070 °C with a heating rate of 20 °C /min [ (4)] Figure 15b. 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<sub>m</sub> = 1396 K. The specific heat increased from 1070 °C to 1123 °C. The transition width (2000 K) is expected to be 10 times wider than at 2 K/min. Consequently, the recovery of ΔC<sub>p</sub>, for a temperature increase of 53 K, is of the order of 53/2000 ≅ 2.6 % of its value at T<sub>m</sub>. We conclude that the glassy fraction (f = 100 %) was induced at a temperature T<sub>x</sub> < T<sub>m</sub>. The rapid increase of the heating rate had for consequence to induce the first order transition at T<sub>x</sub> < T<sub>m</sub>. This experiment showed for the first time that a glassy phase of 100 % can be obtained with a heating rate of 20 K/min.

7.4 InSn80wt% 370

An internal friction method was used to study the structural changes of InSn80wt% [ (66)]. This alloy has a melting temperature of about 190°C (463 K) and a glass transition temperature, expected at 2 T<sub>m</sub> = 926 K (653 °C). A minimum of internal friction occurs at 625 °C et un 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 given by [ (4)] is known as being due to the percolation threshold of configurons [ (33)].

7.5 PbSn61.9wt% 380

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)].

7. Another method to stabilize glassy states above (2 T<sub>m</sub>) 387

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 to various temperatures T<sub>x</sub> < T<sub>m</sub>, followed by successive reheating as shown in Figure 7. Each value of T<sub>x</sub>/T<sub>m</sub> < 0.7069 lead to a glass transition much higher than 2 T<sub>m</sub> as shown by the glassy phase diagram of Figure 1. These various reheating enriched the total glassy fraction as shown by resistivity reductions of 40 % after three heating of Cu-Sb76.5wt%, of 22 % in tin and 16.7% in bismuth after two heating. In addition, depending on heating rate, another glass transition occurs at T<sub>m</sub> < T<sub>n+</sub> < 2 T<sub>m</sub> in various alloys such as InSn80wt%, InBi32wt%
at R = 3°C/min, and CuSb76.5wt% at 5°C/min. They were due to the formation of very 396
weak glassy fractions (f), depending on heating and cooling rates. The residual resistivity 397
is equal to 100 µΩ.cm at Tm during the first heating. Consequently, the formation of high 398
glassy fractions with Tg >> 2 Tm could be attained after adding new thermal cycles, leading 399
to a maximum resistivity fall. Reheating cycles could stabilize the glassy phase above (Tn+ 400
= 2Tm). 401

402

<figure><img src="image_12.png" /><figcaption>**Figure 9:** Resistivity falls of Cu-Sb 76.5 wt% after thermal cycling. Reproduced from [4] (Figure 13f).</figcaption></figure>

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8. Conclusions

406

Liquid-liquid transitions are observed below and above T<sub>m</sub>.

407

First-order transitions, predicted by the NCHN model, occur at Tₓ < Tm, building glassy phase fractions (Δε) after supercooling and crystallized fractions (1-Δε) of melt. Melting of glassy fractions occurs at temperatures Tn+ > Tm at the percolation threshold of configurons (broken bonds). One of them induces a glass enthalpy equal to the melting enthalpy (Δε = 1), without crystallized fraction, up to a glass transition (Tg = 2Tm). All glassy fractions would have a heat capacity equal to zero, revealed by reversing heating to cooling below Tn+.

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Liquid-liquid transitions between the melting temperature (T<sub>m</sub>) and (2 T<sub>m</sub>) are ob-
served in the absence of first order transition at T<sub>x</sub> < T<sub>m</sub> and are reminiscent of glassy frac-
tions (f) formed at temperatures (T<sub>x</sub>) weaker than (T<sub>m</sub>) through first-order transitions.
These weak fractions correspond to singular values (Δε) of enthalpy coefficients (Δε<sub>lg</sub>) =
-Δε) of a new phase called "Phase 3", and to typical percolation thresholds of configurons
at various temperatures (T<sub>n+</sub> = T<sub>g</sub>). These fractions are slowly induced near (T<sub>n+</sub>) by relaxa-
tion in the absence of first order transition prior to melting.

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Several situations are encountered in bismuth and tin:

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1- After melting Bi and Sn at T = T<sub>m</sub>, 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<sub>g</sub> = θ<sub>n+</sub> = (T<sub>n+</sub> - T<sub>m</sub>)/T<sub>m</sub> = Δε. Liquid-liquid transitions were observed at 1.8675 T<sub>m</sub> for bismuth and 1.5523 T<sub>m</sub> for Sn. The glassy character of these transitions is confirmed by structural transitions that we attribute to melting of configurons. The 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<sub>m</sub> due to the thermodynamic character of a transition obeying to critical exponents associated with

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configuron percolation. There is no endothermic latent heat equal to (Δε Hm), only a weak endothermic heat equal to a weak fraction (f << Δε). 430
431

2- After melting Bi and Sn at T = Tm, transitions were also observed at T = 2 Tm by DTA and (or) resistivity that we consider as new glass transitions. The transition was reversible only for Sn. A weak endothermic heat was observed for Bi at 2 Tm, as expected for a weak value of (f). 432
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435

3- We show that the density variation between Tm and 2 Tm is equal to that from solid to liquid at Tm. The glassy state of Bi has a density equal to that of the liquid at Tm while, that of Sn is equal to that of the solid at Tm. 436
437
438

4- After melting Bi at T = Tm, 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 Tm corresponds to 1.23838 Hm instead of Hm including, in addition, the latent heat of glassy phase after the first cooling as predicted by the NCHN model. 439
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5- A glassy phase diagram is proposed for systems having their lowest transition determined by their Lindemann coefficients. Each first order transition at T<sub>x</sub> < T<sub>m</sub> leads to multiple glass transitions. The possible existence of weak glassy fractions (f) for T<sub>x</sub>/T<sub>m</sub> < 0.7069, with glass transition temperatures much higher than (2 T<sub>m</sub>) is envisaged (beyond (3 T<sub>m</sub>) for f < 22.45%). Resistivity measurements showed decreases in Bi and Sn from T<sub>m</sub> to 2 T<sub>m</sub> and beyond, after thermal cycling between the solid and undercooled liquid states. 443
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448

6- The glassy phase formations at T<sub>x</sub> are accompanied by latent heats, being recovered at T<sub>n+</sub> with (T<sub>m</sub> < T<sub>n+</sub> < 2T<sub>m</sub>) which decrease the liquid specific heat. Predictions of their contribution equal to (θ<sub>n+</sub> H<sub>m</sub>/T<sub>m</sub>) are proposed for T<sub>n+</sub> = T<sub>g</sub> ≤ 2T<sub>m</sub>. The heat capacity linearly decreases down to zero when Δε increases up to 1 (T<sub>n+</sub> = 2T<sub>m</sub>). A specific heat equal to zero, down to T<sub>m</sub>, could be induced by reversing heating to cooling from a temperature slightly weaker than T<sub>g</sub> = T<sub>n+</sub>. We show, for the first time, that the liquid specific heat is constant between T<sub>m</sub> and 2T<sub>m</sub> in the absence of glassy phase. 449
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7- The stability of glassy fractions for T<sub>x</sub>/T<sub>m</sub> < 0.7069 can be very high because their (T<sub>g</sub>) could be much higher than (2 T<sub>m</sub>) as proved by resistivity decreases observed up to 2 T<sub>m</sub> 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%. 456
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**Author Contributions:** only one author. 591

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**Funding:** This research received no external funding. 593

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**Institutional Review Board Statement:** Not applicable. 594

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**Informed Consent Statement:** Not applicable. 595

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