NiTi2, a New Liquid Glass

Many endothermic liquid–liquid transitions, occurring at a temperature Tn+ above the melting temperature Tm, are related to previous exothermic transitions, occurring at a temperature Tx after glass formation below Tg, with or without attached crystallization and predicted by the nonclassical homogenous nucleation equation. A new thermodynamic phase composed of broken bonds (configurons), driven by percolation thresholds, varying from ~0.145 to Δε, is formed at Tx, with a constant enthalpy up to Tn+. The liquid fraction Δε is a liquid glass up to Tn+. The solid phase contains glass and crystals. Molecular dynamics simulations are used to induce, in NiTi2, a reversible first-order transition by varying the temperature between 300 and 1000 K under a pressure of 1000 GPa. Cooling to 300 K, without applied pressure, shows the liquid glass presence with Δε = 0.22335 as memory effect and Tn+ = 2120 K for Tm = 1257 K.


Introduction
The classical homogenous nucleation (CHN) equation [1][2][3][4] cannot be applied above the melting temperature T m because the melt is viewed as being homogenous, excluding the presence of intrinsic nuclei such as superclusters and superatoms, inducing crystallization after undercooling and liquid-liquid transitions above T m [5].The liquid-liquid transitions cannot be related to a first-order transition occurring below T m .The undercooling of liquid elements gradually increases with overheating and reaches an undercooling plateau, which is attributed to heterogenous nucleation from oxide impurities, surface cavities, and growth kinetics dependent with the size instead of homogenous nucleation [6].Scientists were quasi-unanimous at the end of 20th century, adopting CHN, believing that the glassy state results from liquid freezing instead of homogenous nucleation of phase transition.
The undercooling plateau is defined by the weakest reduced temperature θ = (T − T m )/T m where the crystallization occurs for a volume v and a nucleation time t depending on the cooling rate.The product θ 2 (1 + θ) for vt = 10 −8 ± 1 m 3 s in the CHN equation is a linear function of α 3 S m where α is proportional to the surface energy of the growth nuclei and S m is the entropy of melting of 38 elements.In fact, this law, deduced from the CHN equation, excludes the heterogenous nucleation and promotes the homogenous nucleation and the presence of intrinsic nuclei in melts [7].
The nonclassical homogenous nucleation (NCHN) equation includes the possible presence of homogeneous nuclei, introducing in the classical equation their contribution to the Gibbs free energy [7].The homogenous nucleation temperatures (T n− ) are defined by equations of two liquid states, Liquid 1 and Liquid 2 [5,[8][9][10][11].The formation of glassy phases occurs by percolation of bonds at T g during the first cooling [12][13][14].After quenching the melt below T g , the enthalpy of formation of all bonds can relax.There are two homogeneous nucleation temperatures in Liquid 2. The highest is (T n− = T g ); the lowest is the temperature where this enthalpy excess begins to be recovered using a slow heating rate.
The enthalpy excesses and the recovery temperatures are found in several glasses [15][16][17][18][19][20][21][22].The glass phase is stabilized after relaxation below the percolation threshold of broken bonds named configurons, occurring during the first heating at T g .The specific heat undergoes a second-order phase transition at T g with critical exponents during heating which are absent during the first cooling [23][24][25][26].The formation of the glass phase after relaxation and broken bonds during heating is governed by the presence of a new phase, initially called Phase 3, discovered in supercooled water [27,28], having an enthalpy coefficient ∆ε lg equal to the difference between those of Liquids 1 and 2 and defining the configuron enthalpy (∆ε lg H m ) [29].
The second class of homogenous nucleation reduced temperatures θ n+ above θ = 0 respects the relation with the configuron enthalpy coefficient (∆ε lg H m = θ n+ H m ) and (H m ) the melting heat of surviving entities and crystals.The configuron phase (Phase 3) undergoes a first-order transition characterized by an endothermic or exothermic latent heat equal to |θ n + H m |.Consequently, the relation (−∆ε lg = θ n+ ) is also respected because ∆ε lg can be positive or negative [11].The reduced temperature θ n+ above θ = 0 is equal to the Scher and Zallen percolation threshold of configurons [15,30].
Exothermic transitions are induced below T m during heating or cooling, depending on the heating and cooling rates.Undercooling with or without crystallization is observed during the first cooling depending on the cooling rate.The enthalpy coefficient ∆ε lg , equal to singular values of Phase 3, results from this first-order transition [31].Heating from the glass state can produce a first-order transition between T m and T g with or without crystallization.The fraction ∆ε lg is involved in the formation of a liquid glass up to T n+ and the crystallized fraction (1 − ∆ε lg ) melts at T m [32].The glass fraction stabilizes nanocrystallization of any material because T n+ is higher than T m .For example, volcanic rocks are composed of crystals of various sizes surrounded by glasses containing variable fractions of SiO 2 .
A liquid glass is created by a first-order transition of the configuron phase to stabilize its enthalpy to a negative singular value (−∆ε lg ) up to T n+ with a specific heat equal to zero.The upper limit of ∆ε lg is defined by T g /T m , the glass transition temperature of the liquid glass being equal to (T g = 2 T m − T g ) [33,34].
Our project in this paper is to induce a first-order transition under extreme pressures in NiTi 2 by molecular dynamics simulations and observe the stabilization of the liquid glass, varying the temperature under constant pressure to induce a memory effect [35].The choice of NiTi 2 is guided by molecular dynamics simulations showing that the glass transition occurs at (T g = 800 K) at high cooling rate, revealing structural changes via radial distribution functions at (T g ) [36].

Predictions of NCHN and Configuron Models 2.1. The Nonclassical Homogeneous Nucleation (NCHN) and the Classical Homogeneous Nucleation (CHN)
In the NCHN equation, the Gibbs free energy change for a nucleus formation in a melt is given by Equation (1) [7]: where R is the nucleus radius and following Turnbull [1], σ ls is its surface energy for ε ls = 0, given by Equation (2), θ is the reduced temperature (T − T m )/T m , H m is the enthalpy of melting at T m, and V m is the molar volume: The complementary enthalpy ε ls × H m /V m , introduced in the classical homogeneous nucleation (CHN) equation, authorizes the presence of growth nuclei above T m .The prediction of an exothermic or endothermic enthalpy change, at the same temperature T n+ above T m , viewed as due to the presence of antibonds or bonds, sets the NCHN equation [33].The CHN equation is obtained for ε ls = 0, and leads to a homogeneous liquid above T m in contradiction with the presence, in the liquid above T m , of first-order transitions, and single crystal formations by cooling overheated liquid droplets.
The nucleation rate (Jvt) in a melt of volume v, after a time t, and the thermally activated energy barrier ∆G* ls /k B T are given in Equations ( 3) and ( 4) [4]: where S m is the entropy of melting of crystals and condensed entities [4,7].The critical parameter ( ) is not infinite at the melting temperature T m when (ε ls ) is present.In this case, this event now occurs above T m for θ = ε ls.The nucleation rate is equal to 1, and Ln(Jvt) = 0 when Equation ( 5) is respected: The surface energy coefficient α ls in Equation ( 2) is determined from Equations ( 3)-( 5) and equal to Equation ( 6): (S m α ls 3 ) is effectively proportional to (1 + θ) θ 2 with (θ) being the reduced temperature of undercooling of 38 liquid elements, obtained after overheating the melt, using volumes v and time t with vt = 10 −8 ± 1 m 3 s, K v = 90, Ln (K v vt) = 71.9± 1 in agreement with the predictions of Equation ( 6), as shown in Figure 1 [4].
Materials 2023, 16, x FOR PEER REVIEW 3 of 21 [33].The CHN equation is obtained for εls = 0, and leads to a homogeneous liquid above Tm in contradiction with the presence, in the liquid above Tm, of first-order transitions, and single crystal formations by cooling overheated liquid droplets.The nucleation rate (Jvt) in a melt of volume v, after a time t, and the thermally activated energy barrier ΔG*ls/kBT are given in Equations ( 3) and ( 4) [4]: where Sm is the entropy of melting of crystals and condensed entities [4,7].The critical parameter ( ∆ * ) is not infinite at the melting temperature Tm when (εls) is present.In this case, this event now occurs above Tm for θ = εls.The nucleation rate is equal to 1, and Ln(Jvt) = 0 when Equation ( 5) is respected: The surface energy coefficient αls in Equation ( 2) is determined from Equations ( 3)-( 5) and equal to Equation ( 6): (Smαls 3 ) is effectively proportional to (1 + θ) θ 2 with (θ) being the reduced temperature of undercooling of 38 liquid elements, obtained after overheating the melt, using volumes v and time t with vt = 10 −8 ± 1 m 3 s, Kv = 90, Ln (Kvvt) = 71.9± 1 in agreement with the predictions of Equation ( 6), as shown in Figure 1 [4].This law is beautifully respected in the two nucleation models and is due to homogeneous nucleation phenomena.The phase diagram of liquid elements predicts these homogenous nucleation temperatures [34].A first-order transition occurs at the undercooling reduced temperature θ.The exothermic enthalpy is either totally attributed to crystallization with CHN without considering any homogeneous nucleation because the liquid is viewed as being homogenous above Tm, or only partially to the nucleation of a glassy fraction depending on the nucleation temperature of first-order transition with NCHN This law is beautifully respected in the two nucleation models and is due to homogeneous nucleation phenomena.The phase diagram of liquid elements predicts these homogenous nucleation temperatures [34].A first-order transition occurs at the undercooling reduced temperature θ.The exothermic enthalpy is either totally attributed to crystallization with CHN without considering any homogeneous nucleation because the liquid is viewed as being homogenous above T m , or only partially to the nucleation of a glassy fraction depending on the nucleation temperature of first-order transition with NCHN [29,32,37].Recent examples of these two approaches are devoted to the undercooling of Pt 57 Cu 23 P 20 alloy [31,38].

The Enthalpy Coefficients of Two Liquid States Depending on the Glass Transition Temperature at T g below T m
The NCHN model describes two liquid states characterized by enthalpy coefficients.The coefficient ε ls of the initial liquid state called Liquid 1 is a quadratic function of Equation θ in Equation (8) [7]: where θ 0m is the Vogel-Fulcher-Tamann (VFT) reduced temperature leading to ε ls = 0 for θ = θ 0m [8].(θ 2 0m ), minimizing the complementary surface energy, is given in Equation (9) [9]: with ε ls0 = θ g + 2.
The glass transition gives rise to Liquid 2 with a new coefficient ε gs , also depending on θ 2 in Equation (10) with the second VFT reduced temperature (θ 0g ) given in Equation (11): with ε gs0 = 1.5θ g + 2, as expected with Equation ( 7), θ g is the reduced glass transition temperature, and (ε ls0 ) and (ε gs0 ) are values for which the activation energies of Liquids 1 and 2 are equal at T g , leading to an activation energy equal to zero for the Liquid 3 characterized by ∆ε lg = (ε ls − ε gs ) [29].

Formation of New Phases through First-Order Transitions below T m
First-order transitions are observed below T m , depending on the heating rate.They lead to new glassy phases having a weaker enthalpy than that of the initial glass state, which is equal to zero.The enthalpy coefficient ∆ε lg is equal to −∆ε after the transition, and the first-order transition reduced temperature θ x is determined with Equation (12): where ε gs0 is the enthalpy coefficient of Liquid 2 at θ = 0 given in Equation (10).This first-order reduced temperature is called θ x , even if it is not accompanied by crystallization.
This equation is used after application of very high temperatures, and the undercooling reduced temperature is directly equal to θ x and ∆ε lg = −∆ε (see for example [31]).

The Glassy State of Phase 3 Up to the Melting Temperature T n+ above T m
The difference ∆ε lg in Equation ( 13) between the coefficients ε ls and ε gs determines the enthalpy coefficient of Phase 3 during heating and the configuron enthalpies when the quenched liquid has escaped crystallization: This phase explains the homogenous nucleation of phase transformations in supercooled water [28], and corresponds to the enthalpy coefficients of broken bonds (configurons) at the glass transition [29].
In the absence of first-order transition below T m during the first heating, Phase 3 undergoes an exothermic transition at a reduced temperature θ n+ = ∆ε lg , respecting Equation ( 7) above T m .This transition occurs at T n+ ~1.145 T m and corresponds to the temperature where the liquid viscosity ceases to respect the Arrhenius law during cooling from high temperature [29,58].Rapidly quenched glass formers are amorphous and are transformed into glass phases by relaxing enthalpy during the first heating.Two liquids give rise, at first, to an intermediate Phase 3 below T 3 < T g respecting the entropy constraints, and then, the enthalpy increases towards that of the glass phase up to T g .Phase 3 carries a medium-range order above T g , which can be superheated above the melting temperature up to T n+ .The configuron model is successfully applied to 54 glasses, explaining the transitions at T n+ by percolation and the existence of a glassy 'ordered' fraction equal to the critical threshold of Scher and Zallen (Φ c = 0.15 ± 0.01), producing T g [25,30].The temperature T n+ is the melting temperature of residual bonds involved in this glassy fraction [29].
The first-order transition at θ x induces either an enthalpy of Phase 3 being constant and equal to −∆ε H m between θ x and θ n+ at high rates of cooling without crystallization, or with a crystallized (or condensed) fraction (1 − ∆ε) melted at T m .For a fully crystallized volume, the enthalpy of crystals is assumed to be −H m , considering that the denser crystalline phases lead to ∆ε = 1 [64].In the second case, the melt above T m contains a glassy fraction ∆ε and a homogeneous liquid fraction equal to (1 − ∆ε).This mixed state leads to the formation of crystals, coexisting with a glass fraction up to T m with a glass state stable up to T n+ [11,26,[31][32][33].

The Formation of a Liquid Glass above T m
The formation of a liquid glass is expected at a temperature (T g = 2 T m − T g ) [32].A glass transition at θ g is accompanied at high temperatures by a second glass transition occurring at −θ g above T m .This phenomenon is a consequence of the θ 2 dependence of the enthalpy of glass phases.The corresponding VFT temperature squares are negative at high temperatures and the values of ε gs0 and ε ls0 become higher than 2.These glass transitions have been already observed three times without being recognized as liquid glasses by the experimentalists [65][66][67][68].The two transitions occur at T g = 715 K and T g = 1535 K in Cu 46 Zr 46 Al 8 with T m = 1125 K, during cooling of the melt, accompanied by an exothermic transition [32,65].In the second material, the composite (ZIF-62) (Al-rich) (50/50) has a solidus melting temperature at 650 K, T g = 591 K and T g = 709 K observed during heating and accompanied by an endothermic latent heat.A glass transition still exists in tin far above T m , and looks like a partial glassy fraction by [34,68].A liquid glass is observed in suspensions of ellipsoidal colloids [69].

Validation of NCHN with Molecular Dynamics Simulations
A phase diagram of ice in single-walled carbon nanotubes at atmospheric pressure is established by numerical simulations, and predicts ice melting points as a function of their diameter up to 1.7 nm [70].All of these melting points agree with the temperatures T n+ given by the NCHN equation [37].A first-order transition from liquid to homogenous glass, denoted L-glass, is predicted with the NCHN equation since 2016, in liquid elements, having a Lindemann constant close to 0.103, accompanied by a latent heat of 10.5% of the melting heat [10].The transition under pressure of 4 He is the first example of this phenomenon [71].Recent molecular dynamics simulations identify, in addition, a firstorder transition at T x from liquid (L) to a metastable heterogenous solid-like phase, denoted as G-glass, when a supercooled liquid evolves isothermally below its melting temperature T m at deep undercooling [59,61].The NCHN model describes the first-order transitions from liquid to L-glass and to G-glass in agreement with these simulations [62,72].The G-glass is a heterogenous phase consisting of regions fully embedded in a surrounding disordered medium.
Molecular dynamics simulations show full melting at T n+ = 1.119T m for Zr [60], 1.126 T m for Ag [61], 1.219 T m for Fe, and 1.354 T m for Cu [73].The NCHN model applied to liquid elements is based on the increase of the Lindemann coefficient with the heating rate.The glass transition at T g and the nucleation temperatures of G-phases at T x and their melting at T n+ are predicted.A universal law relating T n+ and T x shows that T x cannot be higher than 1.293 T m for T n+ = 1.47 T m .The enthalpies of G-phases have singular values, corresponding to the increase of percolation thresholds with T g and T x above the Scher and Zallen invariant at various heating and cooling rates [62].
As a conclusion of this chapter, the liquid glass states extend far above T m .Varying pressures and temperatures and formation conditions of liquid glasses could be determined.For that purpose, a study of glass states and liquid glass states of (NiTi 2 ) [36,74] under pressure by molecular dynamics simulations is developed in the next chapter.

Numerical Simulations
Molecular dynamics (MD) computer simulation using a open source code -software package for classical molecular dynamics Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS, https://www.lammps.org/#gsc.tab=0accessed on 29 October 2020) at periodic boundary conditions was used for modeling [75].The simulation was performed at a 1 fs time step using the embedded atom potential derived for Ti-Ni alloys [76].An atomic cell containing 128,000 atoms was heated to 2500 K to melt, and then cooled down to produce a glass.Melting was confirmed by the radial distribution function and stabilization of the density variation with time.A thermostat was used to control the temperature [77,78], while pressure was maintained by a barostat [79].
Deviations and irreproducibility have been studied using these types of coupling to determine possible deviations in dynamic properties for time constants as low as 0.01 ps.Reliable and thermostable dynamic properties can be derived for coupling time constants above 0.1 ps [80].The atomic volume is calculated here using coupling time constants much higher than 10 −13 s.The weakest time (10 −13 s) would correspond to a temperature variation of 1 K without a change of dV at /dT.The highest time used in these simulations is 3 × 10 −10 s.The atomic volume variation under a pressure of 1000 GPa is dV at /dp = 1.7881 × 10 −7 with p in bars at a constant temperature of 299.7 K.The typical pressure variation is ~10,000 bars, leading to ∆V at = 1.788 × 10 −3 Å 3 .Changing the temperature of 1 K has no influence on dV at /dp.Typical variations (∆T) at ambient pressure are ~5 K.The reproducibility of the simulation during several cycles of temperature and pressure shows that the limit of ~10 −13 s has no influence on the results because the times of coupling are much higher than 0.1 ps.The glass transition of NiTi 2 is equal to 800 K with a cooling rate of 10 12 −10 13 K/s and ~700 K [36] for an experimental value obtained on heating [74].NiTi 2 is chosen because its glass transition, at T g = 800 K, and at high cooling rate, was initially determined by molecular dynamics simulations [36].
The enthalpy coefficients calculated with Equations ( 7)-( 13), representing the enthalpy coefficient of Phase 3 in Figure 2, are given below in Equation ( 14), and are those of a fragile glass with T m = 1257 K [80] and T g = 800 K: influence on dVat/dp.Typical variations (ΔT) at ambient pressure are ~5 K.The reproducibility of the simulation during several cycles of temperature and pressure shows that the limit of ~10 −13 s has no influence on the results because the times of coupling are much higher than 0.1 ps.The glass transition of NiTi2 is equal to 800 K with a cooling rate of 10 12 −10 13 K/s and ~700 K [36] for an experimental value obtained on heating [74].NiTi2 is chosen because its glass transition, at Tg = 800 K, and at high cooling rate, was initially determined by molecular dynamics simulations [36].
The liquid glass transition would be T = 2T − T = 1714 K (θ = +0.36356)at the end of the first heating after the formation of broken bonds (configurons).
The liquid glass transition would be T g = 2T m − T g = 1714K θ g = +0.36356at the end of the first heating after the formation of broken bonds (configurons).
The enthalpy coefficient (∆ε lg ) of Phase 3 is −0.18178 for T g = 800 K [36] with cooling rates of 10 12 −10 13 K/s and −0.22335 for T g = 695.5K at a low heating rate [74].A transition is expected at T n+ = 1434 K for a liquid with T g = 800 K, being quenched from temperatures near T m and heated at 0.3 K/s [32].Many exothermic or endothermic transitions have already been observed at these temperatures [42,[44][45][46]48,[52][53][54].Recent study confirms the existence of these temperatures T n+ > T m in all liquids previously vitrified at low temperatures [29].All these transitions correspond to the temperature below which the Arrhenius law is no longer working, occurring at T n+ ∼ = 1.145T m [58].They are glass transitions of the fraction ( ∼ =0.1445) of unbroken bonds, equal to the percolation threshold predicted by the configuron model and the Scherr and Zallen invariant [25,26].
The enthalpy coefficients for T g = 695.5K and T m = 1257 K, used in Figure 3, are given below in Equation ( 15

The Atomic Volume Hysteresis above the Melting Temperature up to 2120 K
The NiTi2 atomic volume Vat in Angström 3 , represented by the red line in Figure 4, is obtained by cooling the melt from 3000 K to 300 K, and is equal to 15.2829 Å 3 at 300 K.  [34].Simulations of the volume are used to predict this coefficient of 1.5661 ± 1%.The transition at T n+ is expected at 1475 K, a temperature where θ n+ = ∆ε lg = 0.17306.

The Atomic Volume Hysteresis above the Melting Temperature Up to 2120 K
The NiTi 2 atomic volume V at in Angström 3 , represented by the red line in Figure 4, is obtained by cooling the melt from 3000 K to 300 K, and is equal to 15.2829 Å 3 at 300 K.The red line is obtained by decreasing the temperature from 3000 K down to 300 K under pressures weaker than 700 bars.The first glass transition occurs at 800 K during cooling.The melt is submitted to a glass transition at 2120 K during heating.The atomic volume along the blue line is very stable from 300 to 800 K, whatever the pressure of ±700 bars is, and results from a first-order transition, under a pressure of 1000 GPa at 1000 K, varying the temperature to 300 K, and finally decreasing the pressure to 1 bar.The glass phase totally disappears at 2120 K instead of 1714 K.
The atomic volume represented by the blue line starts at 300 K from Vat = 14.9331Å 3 obtained after applying a pressure of 1000 GPa at 1000 K, lowering the temperature down to 300 K, and finally decreasing the pressure to 1 bar.The hysteresis cycle between 300 and 3000 K is measured under weak pressures P, varying between −660 bar and +700 bar.The line thicknesses include the volume variations with pressure.The atomic volume follows the blue line during reheating up to 2120 K. Above this temperature, the blue and red lines merge, and the volume is reproducible with pressure increase or decrease.The glass transition temperature occurs at 2120 K without volume jump.
The temperature T′g = 1714 K is equal to (2 Tm − Tg) with Tm = 1257 K and Tg = 800 K.The temperature of 2120 K corresponds to θn+ = 0.68632, being equal to the sum of singular enthalpy coefficients (0.14085 + 0.18178 + 0.36369) of Phase 3 with Tg = 800 K, and determining the highest glass transition temperature of Phase 3 and the highest melting temperature of NiTi2, as already observed for several liquid elements [62].
The first-order transition of the glass phase expected at T′g = 1714 K is not observed by heating from low temperatures.Nevertheless, ΔVat, the difference between the red and blue lines, is maximum at this temperature and decreases to zero from 1714 K to 2120 K.In addition, it is constant from 300 K up to the first glass transition at Tg = 800 K where ΔVat begins to increase with the temperature.
The glass transition at 800 K is observed on the red curve by the slope change of Vat at this temperature while a new glass state is induced by a first-order transition, occurring at 1000 GPa by decreasing the temperature from 1000 to 300 K, followed by lowering the pressure down to 1 bar.This new glass state disappears during heating at 2120 K.The NiTi2 melt cooled from 3000 K to 800 K does not undergo liquid glass transformation.

The First-Order Transition under 1000 GPa
A liquid glass state is revealed in Figure 5 by a first-order transition as expected.An atomic volume change, equal to −0.00979 Å 3 under 1000 GPa, is induced by a reduction of temperature from 1000 K to 300 K and by the memory of this transition, as shown in Figure 4.The red line is obtained by decreasing the temperature from 3000 K down to 300 K under pressures weaker than 700 bars.The first glass transition occurs at 800 K during cooling.The melt is submitted to a glass transition at 2120 K during heating.The atomic volume along the blue line is very stable from 300 to 800 K, whatever the pressure of ±700 bars is, and results from a first-order transition, under a pressure of 1000 GPa at 1000 K, varying the temperature to 300 K, and finally decreasing the pressure to 1 bar.The glass phase totally disappears at 2120 K instead of 1714 K.
The atomic volume represented by the blue line starts at 300 K from V at = 14.9331Å 3 obtained after applying a pressure of 1000 GPa at 1000 K, lowering the temperature down to 300 K, and finally decreasing the pressure to 1 bar.The hysteresis cycle between 300 and 3000 K is measured under weak pressures P, varying between −660 bar and +700 bar.The line thicknesses include the volume variations with pressure.The atomic volume follows the blue line during reheating up to 2120 K. Above this temperature, the blue and red lines merge, and the volume is reproducible with pressure increase or decrease.The glass transition temperature occurs at 2120 K without volume jump.
The temperature T g = 1714 K is equal to (2 T m − T g ) with T m = 1257 K and T g = 800 K.The temperature of 2120 K corresponds to θ n+ = 0.68632, being equal to the sum of singular enthalpy coefficients (0.14085 + 0.18178 + 0.36369) of Phase 3 with T g = 800 K, and determining the highest glass transition temperature of Phase 3 and the highest melting temperature of NiTi 2 , as already observed for several liquid elements [62].
The first-order transition of the glass phase expected at T g = 1714 K is not observed by heating from low temperatures.Nevertheless, ∆V at , the difference between the red and blue lines, is maximum at this temperature and decreases to zero from 1714 K to 2120 K.In addition, it is constant from 300 K up to the first glass transition at T g = 800 K where ∆V at begins to increase with the temperature.
The glass transition at 800 K is observed on the red curve by the slope change of V at at this temperature while a new glass state is induced by a first-order transition, occurring at 1000 GPa by decreasing the temperature from 1000 to 300 K, followed by lowering the pressure down to 1 bar.This new glass state disappears during heating at 2120 K.The NiTi 2 melt cooled from 3000 K to 800 K does not undergo liquid glass transformation.

The First-Order Transition under 1000 GPa
A liquid glass state is revealed in Figure 5 by a first-order transition as expected.An atomic volume change, equal to −0.00979 Å 3 under 1000 GPa, is induced by a reduction of temperature from 1000 K to 300 K and by the memory of this transition, as shown in Figure 4.A memory effect has been numerically predicted in a three-dimensional model for structural glass when submitted to a temperature cycle [35].Here, a first-order transition is observed and builds a memory effect.
This transition is reversible, as shown in Figure 6.Applying a pressure of 1000 GPa at 300 K and increasing the temperature to 1000 K induces an increase of the atomic volume of 0.0098 Å 3 , as shown in Figure 6.This A memory effect has been numerically predicted in a three-dimensional model for structural glass when submitted to a temperature cycle [35].Here, a first-order transition is observed and builds a memory effect.
This transition is reversible, as shown in Figure 6.A memory effect has been numerically predicted in a three-dimensional model for structural glass when submitted to a temperature cycle [35].Here, a first-order transition is observed and builds a memory effect.
This transition is reversible, as shown in Figure 6.Applying a pressure of 1000 GPa at 300 K and increasing the temperature to 1000 K induces an increase of the atomic volume of 0.0098 Å 3 , as shown in Figure 6.This Increasing the pressure on the glass state at T = 300 K up to 1000 GPa and increasing the temperature from 300 K to 1000 K induces a first-order transition from the glass to the liquid state, and a volume change of +0.00980 Å 3 , equal to that observed in Figure 5.
Applying a pressure of 1000 GPa at 300 K and increasing the temperature to 1000 K induces an increase of the atomic volume of 0.0098 Å 3 , as shown in Figure 6.This transfor-mation reduces the difference (∆V at ) = −0.3498Å 3 at 300 K, as shown in Figure 4, to a value which is 35.69 times weaker.This factor (35.69) is related to the increase of the melting heat under pressure.

The Phase Diagram of Glass Phases in Fragile Liquids
The phase diagram in Figure 7 is devoted to Phase 3 of fragile liquids, having a melting temperature (T m ) and submitted to first-order transitions induced at T x /T m and T g /T m at various heating rates from their low temperature glass state.The ratio of T g /T m of fragile glasses is always higher than 0.5 (θ g > −0.5).This limit corresponds to Vogel-Fulcher-Tamann temperatures higher than T m /3.The reduced glass transition at θ g is varying from −0.5 to 2.581 and T g /T m from 0.5 to 3.581.For each value of θ g , there are first-order transitions of Phase 3, occurring at θ x and θ n+ , accompanied by enthalpy coefficient changes respectively equal to −∆ε and +∆ε.For example, T x /T m = 0.22335 = ∆ε leads to two glass transitions at T g /T m = 0.5487 and 3 (θ g = −0.77665and 2) at zero pressure and constant melting temperature.

The Melting Temperature Increases with Pressure
The melting entropy is 9.3 J/K/mole in many metals, and the melting heat is qua proportional to Tm [4].The pressure application increases the liquid entropy proporti ally to the induced melting temperature.Consequently, the melting heat is expected to proportional to Tm where T0 is the melting temperature at ambient pressure and P is the pressure in GPa.T equation for Tg = 695.5K works and results from the decrease of Tg from 800 K to 695.5 at ambient temperature and pressure.The glass transition temperature at the pressure For each value of T x /T m , higher than 0.8 at zero pressure, there are two glass transitions corresponding to T g and T g .For T x /T m > 0.13962, there is a unique glass transition T g /T m < 3.22076.A pressure has, for consequence, to reduce T x /T m and to increase T g /T m beyond 3.22076.This ratio tends to the upper limit equal to 3.581 when the pressure tends to infinite.
There are two types of first-order transitions at T x .In the first case, the Phase 3 enthalpy falls to −H m at T x , remains constant up to T m , and an enthalpy fraction ∆ε is recovered through a new first-order transition at T n+ far above T m .Consequently, the melting enthalpy, recovered at T m , is reduced and equal to (1 − ∆ε) H m .The temperature (T n+ ), corresponding to a reduced temperature equal to θ n+ = ∆ε, is the glass transition reduced temperature of the liquid fraction ∆ε, which is not crystallized or condensed at T x [31,34].This diagram is also used to predict the glass transitions under pressure, taking account of the pressure dependence of T m and H m .
In the second case, the transition at T x is not accompanied by a crystallized or a condensed fraction, and is only due to the formation of various Phases 3, having enthalpies equal to −∆ε H m with ∆ε also equal to singular values of percolation thresholds of config-urons in Equation ( 14): ∆ε sg0 (θ = 0), ∆ε sg0 /2 (θ = θ g ), ∆ε lg (θ = θ 0m ), ∆ε lg (θ n+ ), and zero.There is no crystalline phase.The volume difference ∆V at increases with temperature up to T g = 1714 K.The liquid fraction [∆ε = (T g − T m )/T m ] participates to the glass state up to T g .
The enthalpy changes, associated with the first-order transitions at T n+ = T g and T x , are, respectively, equal to ±∆ε H m .The melting heat H m increases with pressure, while (∆ε) decreases as 1/H m .The value of ∆ε is divided by 35.69 under a pressure of 1000 GPa because the melting heat is multiplied by 35.69.Consequently, (∆ε) = 0.0098 at 1000 GPa leads to T g /T m = (3.581− 0.0098/0.581)= 3.5641 near the upper limit of 3.581, as shown in Figure 7.

The Melting Temperature Increases with Pressure
The melting entropy is 9.3 J/K/mole in many metals, and the melting heat is quasiproportional to T m [4].The pressure application increases the liquid entropy proportionally to the induced melting temperature.Consequently, the melting heat is expected to be proportional to T m 2 and a linear function of the pressure (P), as shown for NiTi 2 in the following equations: assuming ∆ε = 0.22335 for T g = 695.5K and assuming ∆ε = 0.18178 for T g = 800 K where T 0 is the melting temperature at ambient pressure and P is the pressure in GPa.The equation for T g = 695.5K works and results from the decrease of T g from 800 K to 695.5 K at ambient temperature and pressure.The glass transition temperature at the pressure of 1000 GPa is T g = 800 K.Then, the most probable melting temperature is T m = 6779 K at this pressure, given by Equation (17) for T g = 800 K.At P = 0, T m is equal to T 0 = 1257 K [80].For P =1000 GPa, (T m /T 0 ) 2 = 35.69or 29.08, in agreement with the numerical simulations.(T m ) under P = 1000 GPa is 7509 K or 6779 K.The dependence of T m with P is given in Table 1, using the two values (0.22335) and (0.18178) of ∆ε.For P = 500 GPa, ∆ε and ∆V at are multiplied by 1.948 compared to ∆V at at 1000 GPa.The volume change, calculated at 1000 GPa, is 2% or 3.6% weaker than the simulated value (0.00979).The glass transition even increases at pressures much higher than 1000 GPa, with ∆V at tending to zero.
This linear law of variation with T m 2 with pressure has never been used.It is then important to apply it to other known metals.Copper and iron are chosen.There are many measurements of copper at low pressures [81][82][83][84][85].The initial slope varies between 36.4 and 47.7 K/GPa.These measurements lead to the two following laws, Equations ( 18) and ( 19) for copper, in good agreement with many experimental and theoretical results: which are represented in Figure 8, together with those of NiTi 2 and Fe.For Cu, our extrapolated values in 500 GPa are 7253 K and 8299 K, in agreement with T m = 7900 K [82].
From another determination in 100 GPa, T m = 3900 K [86], and our values are 3463 K and 3905 K.
Table 1.Four first columns on the left are devoted to the liquid having T g = 695.5 K, while the four last columns on the right to T g = 800 K. Values of T m , ∆ε, and ∆V at are given as a function of pressure in GPa respecting Equations ( 16) and (17).This linear law of variation with Tm 2 with pressure has never been used.It is then important to apply it to other known metals.Copper and iron are chosen.There are many measurements of copper at low pressures [81][82][83][84][85].The initial slope varies between 36.4 and 47.7 K/GPa.These measurements lead to the two following laws, Equations ( 18) and (19) for copper, in good agreement with many experimental and theoretical results:

P (GPa
( ) = 1 + 47.7/1000 × P (GPa), which are represented in Figure 8, together with those of NiTi2 and Fe.For Cu, our extrapolated values in 500 GPa are 7253 K and 8299 K, in agreement with Tm = 7900 K [82].From another determination in 100 GPa, Tm = 3900 K [86], and our values are 3463 K and 3905 K.For Fe, recent measurements up to 103 GPa are determined by X-ray absorption spectroscopy up to 103 GPa, and used to predict a melting temperature of 4850 ± 200 K at the inner core boundary (ICB) of the earth for P = 330 GPa [87].Another melting temperature of 5500 ± 220 K at the ICB has been extrapolated from a measurement under P = 290 GPa in a resistance-heated diamond-anvil cell [88].
An extrapolation of this experimental result yields a melting point of 5500 ± 220 K at the ICB, higher than the previous reported result.Two laws, Equations ( 20) and ( 21) for iron, are deduced from our model, respectively, based on the works of Aquilanti et al. [87] and Sinmyo et al. [88].
4.6.Increase of the Enthalpy Difference between Liquid and Glass with Temperature from 300 K to 1734 K The difference ∆V at is equal to 0.3498 Å 3 under pressure variations between ±700 Bars from 300 K to 700-800 K, as shown in Figures 4 and 9.
Our extrapolation, at 330 GPa from measurements below 103 GPa, leads to Tm = 4834 K, in agreement with the evaluation of 4850 ± 200 K [88].

Increase of the Enthalpy Difference between Liquid and Glass with Temperature from 300 K to 1734 K.
The difference ΔVat is equal to 0.3498 Å 3 under pressure variations between ±700 Bars from 300 K to 700-800 K, as shown in Figures 4 and 9.The glass state, induced by the first-order transition, exists up to T′g =1714 K for Tg = 800 K with ΔVat, increasing from 0.3498 to 0.6808 Å 3 , as shown in Figure 10.The glass state, induced by the first-order transition, exists up to T g =1714 K for T g = 800 K with ∆V at , increasing from 0.3498 to 0.6808 Å 3 , as shown in Figure 10.The value of ∆V at at T g = 1714 K would correspond to a first-order transition under 1000 GPa induced by a temperature increase from 300 to 1724 K with ∆V at = 0.6808/36.73= 0.0185 ± 0.0016.This continuous increase of ∆V at up to T g indicates that a high fraction volume of the sample belongs to a glass phase.
The enthalpy coefficient of Phase 3 with ∆V at = 0.6808 corresponds in fact to ∆ε = 0.6808/1.5661= 0.4387, a value a little weaker than θ g = (1818.5− 1257)/1257 = 0.4467, and much higher than θ g = (1714 − 1257)/1257 = 0.36356.The melt has a glass transition temperature much closer at T g = 1818.5K (T g = 695.5 K) than T g = 1714 K (T g = 800 K).The coefficient ∆ε tends to decrease toward 0.36356, corresponding to T g = 1714 K and T g = 800 K, as shown in Figure 11.The value of ΔVat at T′g = 1714 K would correspond to a first-order transition under 1000 GPa induced by a temperature increase from 300 to 1724 K with ΔVat = 0.6808/36.73= 0.0185 ± 0.0016.This continuous increase of ΔVat up to T′g indicates that a high fraction volume of the sample belongs to a glass phase.
The enthalpy coefficient of Phase 3 with ΔVat = 0.6808 corresponds in fact to Δε = 0.6808/1.5661= 0.4387, a value a little weaker than θ′g = (1818.5− 1257)/1257 = 0.4467, and much higher than θ′g = (1714 − 1257)/1257 = 0.36356.The melt has a glass transition temperature much closer at T′g = 1818.5K (Tg = 695.5 K) than T′g = 1714 K (Tg = 800 K).The coefficient Δε tends to decrease toward 0.36356, corresponding to T′g = 1714 K and Tg = 800 K, as shown in Figure 11.The atomic volume change at the melting temperature T m = 1257 K is equal to the volume variation (∆V = 1.5661Å 3 ) of the melt between T m and 2 T m [34].A new liquid glass state is stabilized at 300 K.The glass transition temperature T g becomes equal to 695.5 K instead of 800 K after a cooling rate at 10 12 −10 13 K/s.The Phase 3 enthalpy coefficient ∆ε lg = ε ls − ε gs is equal to 0.22335 at low temperatures, and corresponds to a volume change of 0.22335 × 1.5661 = 0.35, in agreement with that of Figure 8.
The ratio T g /T m in Figure 11 is expected to be equal to 1.4467 for T g = 1818.5K and T g = 695.5K and 1.36356 for T g = 1714 K and T g = 800 K. Its maximum value is 1.4387 at T g = 1714 K instead of 1.4467.Above T m = 1257 K, the dominant liquid glass phase with T g = 1818.5K is progressively replaced by the liquid glass with T g = 1714 K and T g = 800 K, because ∆V at begins to decrease above 1714 K.
These results show that ∆ε progressively varies between the two maximum values θ n+ = ∆ε = 0.4467 and 0.36356.The first-order transitions expected at 1714 K and 1818.5 K are not observed because the crystal melting in NiTi 2 extends up to 2120 K in Figure 3  There is a competition between two glass phases.The first one has a glass transition temperature equal to T n+ = 2120 K, which is equal to the temperature T n+ of NiTi 2 full melting, as shown in Figure 12, studying the crystal melting temperature dependence with high heating rates at 10 12 −10 13 K/s.The transition is sharp with 10 12 K/s, and continuous with 10 13 K/s.
are not observed because the crystal melting in NiTi2 extends up to 2120 K in Figure 3  There is a competition between two glass phases.The first one has a glass transition temperature equal to Tn+ = 2120 K, which is equal to the temperature Tn+ of NiTi2 full melting, as shown in Figure 12, studying the crystal melting temperature dependence with high heating rates at 10 12 −10 13 K/s.The transition is sharp with 10 12 K/s, and continuous with 10 13 K/s.This temperature Tn+, being the liquid glass transition temperature, corresponds to θn+ = 0.68632 equal to the sum of singular values (0.14085 + 0.18178 + 0.36369) of the enthalpy coefficient of Phase 3. The NiTi2 melting starts earlier at Tn+ = 1891 K and θn+ = 0.50454 = 0.14085 + 0.36369, and is prolongated toward θn+ = 0.68632.The second glass phase results from a first-order transition obeying to the nucleation law in Equation (7) with a weaker and weaker fraction Δε.The maximum difference Δε increases up to 0.4387 without attaining T′g/Tm = 0.4467, and declines toward zero at 2120 K.The first-order This temperature T n+ , being the liquid glass transition temperature, corresponds to θ n+ = 0.68632 equal to the sum of singular values (0.14085 + 0.18178 + 0.36369) of the enthalpy coefficient of Phase 3. The NiTi 2 melting starts earlier at T n+ = 1891 K and θ n+ = 0.50454 = 0.14085 + 0.36369, and is prolongated toward θ n+ = 0.68632.The second glass phase results from a first-order transition obeying to the nucleation law in Equation (7) with a weaker and weaker fraction ∆ε.The maximum difference ∆ε increases up to 0.4387 without attaining T g /T m = 0.4467, and declines toward zero at 2120 K.The first-order transition cannot take place at T g = 1714 K because the glass phase is prolongated by that of a liquid glass fraction equal to 0.686 with T g = T n+ = 2120 K.
The two liquid states found by numerical simulations are already observed in Co-B melts [39].The temperature-dependent liquid structures are studied in situ, measuring the magnetization.A magnetization anomaly in terms of the non-Curie-Weiss temperature dependence of magnetization was observed in the overheated state, demonstrating a temperature-induced liquid-liquid structure transition.This anomalous behavior was found to be a universal formula for the Co-B binary alloy system.The transition point at T n+ (called T 0 in the publication), above which there is a unique liquid state and below which two paramagnetic Curie temperatures (θ p (L I ), θ p (L II )) corresponding to two distinct kinds of liquids (i.e., high-magnetization liquid (HML) I and low-magnetization liquid (LML) II ), are measured.With the increased concentration of Co, T n+ , θ p (L I ) and θ p (L II ) shift to higher temperatures, and the Curie constants for the HML and LML decrease.Based on the location of T n+ , a guideline is drawn above the liquidus in the Co-B phase diagram.
As a conclusion of this chapter, the liquid glass states extend far above T m .Table 2, after ref. [26], illustrates the positioning of liquid glasses in terms of connectivity and ordering increase (e.g., on temperature decrease).The connectivity between atomic species can be diminished not only by an increase of temperature: the irradiation of glasses, which breaks the interatomic chemical bonds, leads to fluidization of glasses [89].
The right column of Table 2 is for solid-like matter, which occurs when the degree of connectivity between atomic or molecular constituents is high.Here, we have glasses with a topologically disordered distribution of atoms or molecular species, or crystals.At a medium degree of order, we have composite materials composed of both vitreous, crystalline with periodicity, and even quasi-crystalline (QC) phases with nonperiodic order [29,90].When the degree of connectivity is low, as shown in the second column of Table 2, we have fluid-like behavior of matter.It starts with melts containing icosahedral superclusters when there is low connectivity between species at a low degree of their ordering.If the degree of ordering becomes high, we have the case of liquid crystals, which is a distinct phase of matter observed between the crystalline (solid) and isotropic (liquid) states.The liquid glass state of matter was missing within classification of matter until recently, because it is a combination of both glassy and liquid fractions, which cannot be readily revealed, e.g., via X-ray diffraction.It is a flowing state of matter placed within a classification scheme at a low degree of connectivity characteristic to liquids, and has a medium degree of ordering because the fraction of glass within liquid adds some degree of ordering, being distinct from the molten state of matter at least by its symmetry signature, i.e., the Hausdorff dimensionality of bonds, which is D = 3 for glasses and D = 2.5 for liquids; see, e.g., Table 2 of Refs.[25,36,[91][92][93].

Conclusions
The formation of liquid glasses induced by first-order transitions at temperatures T x < T m , with glass transition temperatures at T n+ > T m has been proposed many times during the last years.Glass transitions have been recently observed above T m in the composite (ZIF-62) (Al-rich) (50/50), in tin, and in suspensions of ellipsoidal colloids.The existence of such liquid glasses can be extended by molecular dynamics simulations.NiTi 2 is chosen because its glass transition, at T g = 800 K (T m = 1257 K) and at high cooling rate, was initially determined by MD simulations, revealing structural changes at this glass transition via radial distribution functions.Applying very high pressures attaining 1000 GPa at 1000 K, and abruptly decreasing the temperature from the liquid state to 300 K or increasing the temperature from 300 K to 1000 K from the glass state, induces a weak reversible first-order transition of the atomic volume with ∆V at = 0.0098 Angström 3 .A new liquid glass state exists at 300 K up to 2120 K after reduction of the pressure to one bar, characterized by an atomic volume difference at 300 K, with the classical glass with T g = 800 K being 35.69 times higher than under a pressure of 1000 GPa.This decrease of ∆V at with pressure is associated with the strong increase of T m .[T m (1000 GPa)/T 0 ] 2 is equal to 35.69 because the entropy of melting linearly increases with pressure instead of being constant.The melting temperature T m of NiTi 2 would attain 6779 K for T 0 = 1257 K.We control this relation, calculating the melting temperatures of copper and iron as a function of pressure using known experimental values at various low pressures.Our extrapolated values agree with other extrapolations for copper and iron.The observed first-order transitions show that this liquid glass exists up to temperatures T n+ = 1.686T m , and that memory effects are present at ambient pressure by abruptly varying the temperature under extreme high pressures.Liquid glasses are a new class of materials which deeply disrupt the concept of homogenous liquids without memory of their solid and glass transformations below T m .
The continuous increase of the volume difference ∆V at between the liquid and the glass from 300 K up to T g = 1714 K shows that the volume fraction involved in this glass fraction attains ~44%.
This finding could lead to a reconsideration of glass formation in volcanic rocks.Up to now, the explanation is based on the high content of SiO 2 in amorphous rocks with T g ≤ 1473 K.A first-order transition at T g = 2 T m − T g ≤ 2519 K could exist during cooling or, in the absence of spontaneous first-order transition at T g ≤ 2519 K, and depending on the thermal history, a first-order transition could be induced during the simultaneous pressure and temperature decreases at different rates.

4 .
NiTi 2 , a New Liquid Glass 4.1.The Vitreous States of NiTi 2 with T g = 800 K and 695.5 K

4. 1 .
The Vitreous States of NiTi2 with Tg = 800 K and 695.5 K

Figure 2 .
Figure 2. Temperature dependence of NiTi2 configuron enthalpy for Tg = 800 K.The enthalpy coefficient (Δεlg) of Phase 3 given in Equation (13) is plotted as a function of the temperature in Kelvin for a fragile liquid with Tg = 800 K.The singular values (Δεlg) = −Δεgs0 = −0.18178 of Phase 3 and zero in the glass state are observed.The two others Δεgs0/2 = 0.09089 and Δεlg(θ0m) = 0.36369 are not obtained at room temperature.An exothermic first-order transition is expected at Tn+ = 1434 K with θn+ = Δεlg = 0.14085 at a low heating rate.

Figure 2 .
Figure 2. Temperature dependence of NiTi 2 configuron enthalpy for T g = 800 K.The enthalpy coefficient (∆ε lg ) of Phase 3 given in Equation (13) is plotted as a function of the temperature in Kelvin for a fragile liquid with T g = 800 K.The singular values (∆ε lg ) = −∆ε gs0 = −0.18178 of Phase 3 and zero in the glass state are observed.The two others ∆ε gs0 /2 = 0.09089 and ∆ε lg (θ 0m ) = 0.36369 are not obtained at room temperature.An exothermic first-order transition is expected at T n+ = 1434 K with θ n+ = ∆ε lg = 0.14085 at a low heating rate.

15 )Figure 3 .
Figure 3. Temperature dependence of NiTi2 configuron enthalpy for Tg = 695.5 K.The enthalpy coefficient (Δεlg) of Phase 3 given in Equation (15) is plotted as a function of the temperature in Kelvin for a fragile liquid with Tg = 695.5 K.The singular values (Δεlg) = −Δεgs0 = −0.22335 of Phase 3 and zero of the glass state are observed.The two others Δεgs0/2 = 0.11167 and Δεlg(θ0m) = 0.29442 are not obtained at room temperature.An exothermic first-order transition is expected at Tn+ = 1475 K with θn+ = Δεlg = 0.17306 at a low heating rate.A transition is expected at slow heating at Tn+ = 1475 K for a liquid being quenched from temperatures much closer to Tm.The enthalpy coefficient of Phase 3 is −0.22335.The volume change calculated by numerical simulations is equal to 0.22335 × 1.5661 = 0.3498 ± 2%.The coefficient 1.5661 ± 1% transforms the enthalpy coefficient −0.22335 in a volume change in Å 3 at the melting temperature Tm because the enthalpy of melting is equal to the enthalpy variation between Tm and 2 Tm [34].Simulations of the volume are used to predict this coefficient of 1.5661 ± 1%.The transition at Tn+ is expected at 1475 K, a temperature where θn+ = Δεlg = 0.17306.

Figure 3 .
Figure 3. Temperature dependence of NiTi 2 configuron enthalpy for T g = 695.5 K.The enthalpy coefficient (∆ε lg ) of Phase 3 given in Equation (15) is plotted as a function of the temperature in Kelvin for a fragile liquid with T g = 695.5 K.The singular values (∆ε lg ) = −∆ε gs0 = −0.22335 of Phase 3 and zero of the glass state are observed.The two others ∆ε gs0 /2 = 0.11167 and ∆ε lg (θ 0m ) = 0.29442 are not obtained at room temperature.An exothermic first-order transition is expected at T n+ = 1475 K with θ n+ = ∆ε lg = 0.17306 at a low heating rate.A transition is expected at slow heating at T n+ = 1475 K for a liquid being quenched from temperatures much closer to T m .The enthalpy coefficient of Phase 3 is −0.22335.The volume change calculated by numerical simulations is equal to 0.22335 × 1.5661 = 0.3498 ± 2%.The coefficient 1.5661 ± 1% transforms the enthalpy coefficient −0.22335 in a volume change in Å 3 at the melting temperature T m because the enthalpy of melting is equal to the enthalpy variation between T m and 2 T m [34].Simulations of the volume are used to predict this coefficient of 1.5661 ± 1%.The transition at T n+ is expected at 1475 K, a temperature where θ n+ = ∆ε lg = 0.17306.

Figure 4 .
Figure 4. NiTi2 atomic volume variation with temperature using cooling and heating rates of 10 13 K/s.Two values of the atomic volume are observed and fixed by the thermal history under pressure.The red line is obtained by decreasing the temperature from 3000 K down to 300 K under pressures weaker than 700 bars.The first glass transition occurs at 800 K during cooling.The melt is submitted to a glass transition at 2120 K during heating.The atomic volume along the blue line is very stable from 300 to 800 K, whatever the pressure of ±700 bars is, and results from a first-order transition, under a pressure of 1000 GPa at 1000 K, varying the temperature to 300 K, and finally decreasing the pressure to 1 bar.The glass phase totally disappears at 2120 K instead of 1714 K.

Figure 4 .
Figure 4. NiTi 2 atomic volume variation with temperature using cooling and heating rates of 10 13 K/s.Two values of the atomic volume are observed and fixed by the thermal history under pressure.The red line is obtained by decreasing the temperature from 3000 K down to 300 K under pressures weaker than 700 bars.The first glass transition occurs at 800 K during cooling.The melt is submitted to a glass transition at 2120 K during heating.The atomic volume along the blue line is very stable from 300 to 800 K, whatever the pressure of ±700 bars is, and results from a first-order transition, under a pressure of 1000 GPa at 1000 K, varying the temperature to 300 K, and finally decreasing the pressure to 1 bar.The glass phase totally disappears at 2120 K instead of 1714 K.

Figure 5 .
Figure 5. NiTi2 atomic volume variation at 1000 GPa, decreasing the temperature from 1000 to 300 K. Increasing the pressure at T = 1000 K up to 1000 GPa and lowering the temperature down to 300 K induces a first-order transition in the glass state accompanied by a volume change equal to ΔV = −0.00979Å 3 .

Figure 6 .
Figure 6.NiTi2 atomic volume variation at 1000 GPa, increasing the temperature from 300 to 1000 K. Increasing the pressure on the glass state at T = 300 K up to 1000 GPa and increasing the temperature from 300 K to 1000 K induces a first-order transition from the glass to the liquid state, and a volume change of +0.00980 Å 3 , equal to that observed in Figure5.

Figure 5 .
Figure 5. NiTi 2 atomic volume variation at 1000 GPa, decreasing the temperature from 1000 to 300 K. Increasing the pressure at T = 1000 K up to 1000 GPa and lowering the temperature down to 300 K induces a first-order transition in the glass state accompanied by a volume change equal to ∆V = −0.00979Å 3 .

Materials 2023 , 21 Figure 5 .
Figure 5. NiTi2 atomic volume variation at 1000 GPa, decreasing the temperature from 1000 to 300 K. Increasing the pressure at T = 1000 K up to 1000 GPa and lowering the temperature down to 300 K induces a first-order transition in the glass state accompanied by a volume change equal to ΔV = −0.00979Å 3 .

Figure 6 .
Figure 6.NiTi2 atomic volume variation at 1000 GPa, increasing the temperature from 300 to 1000 K. Increasing the pressure on the glass state at T = 300 K up to 1000 GPa and increasing the temperature from 300 K to 1000 K induces a first-order transition from the glass to the liquid state, and a volume change of +0.00980 Å 3 , equal to that observed in Figure5.

Figure 6 .
Figure 6.NiTi 2 atomic volume variation at 1000 GPa, increasing the temperature from 300 to 1000 K.Increasing the pressure on the glass state at T = 300 K up to 1000 GPa and increasing the temperature from 300 K to 1000 K induces a first-order transition from the glass to the liquid state, and a volume change of +0.00980 Å 3 , equal to that observed in Figure5.

Materials 2023 ,Figure 7 .
Figure 7. Phase diagram of fragile liquids: (Tx/Tm) versus (Tg/Tm).For each value of Tx/Tm, hig than 0.8 at zero pressure, there are two glass transitions corresponding to Tg and T′g.For Tx/T 0.13962, there is a unique glass transition T′g/Tm < 3.22076.A pressure has, for consequence, to red Tx/Tm and to increase Tg/Tm beyond 3.22076.This ratio tends to the upper limit equal to 3.581 wh the pressure tends to infinite.

Figure 7 .
Figure 7. Phase diagram of fragile liquids: (T x /T m ) versus (T g /T m ).For each value of T x /T m , higher than 0.8 at zero pressure, there are two glass transitions corresponding to T g and T g .For T x /T m > 0.13962, there is a unique glass transition T g /T m < 3.22076.A pressure has, for consequence, to reduce T x /T m and to increase T g /T m beyond 3.22076.This ratio tends to the upper limit equal to 3.581 when the pressure tends to infinite.

Figure 9 .
Figure9.NiTi2 atomic volume variation with pressure at 300 K. Two stable volumes at room temperature and under atmospheric pressure are obtained after the first-order transition induced at 1000 GPa.The volume difference ΔVat at 300 K between liquid, noted 1, and glass, noted 2, is equal to 0.3498 ± 0.004 Å 3 , and corresponds to that of configurons for Tg = 695.5 K.

Figure 9 .
Figure9.NiTi 2 atomic volume variation with pressure at 300 K. Two stable volumes at room temperature and under atmospheric pressure are obtained after the first-order transition induced at 1000 GPa.The volume difference ∆V at at 300 K between liquid, noted 1, and glass, noted 2, is equal to 0.3498 ± 0.004 Å 3 , and corresponds to that of configurons for T g = 695.5 K.

Materials 2023 , 21 Figure 10 .Figure 10 .
Figure 10.NiTi2 atomic volumes between 1500 and 1800 K.The atomic volume difference increases with temperature up to ΔVat = 0.6808, corresponding to Δε = 0.6808/1.5661= 0.4387, a value a little weaker than T′g/Tm = 1818.5/1257= 0.4467, and much higher than T′g/Tm = 1714/1257 = 0.36356 (q′g = 0.36356).The melt has a glass transition temperature much closer at T′g = 1818.5K (Tg = 695.5 K) than T′g = 1714 K (Tg = 800 K).The value of ΔVat at T′g = 1714 K would correspond to a first-order transition under 1000 GPa induced by a temperature increase from 300 to 1724 K with ΔVat = 0.6808/36.73= 0.0185 ± 0.0016.This continuous increase of ΔVat up to T′g indicates that a high fraction volume of the sample belongs to a glass phase.The enthalpy coefficient of Phase 3 with ΔVat = 0.6808 corresponds in fact to Δε = Figure 10.NiTi 2 atomic volumes between 1500 and 1800 K.The atomic volume difference increases with temperature up to ∆V at = 0.6808, corresponding to ∆ε = 0.6808/1.5661= 0.4387, a value a little weaker than T g /T m = 1818.5/1257= 0.4467, and much higher than T g /T m = 1714/1257 = 0.36356 (q g = 0.36356).The melt has a glass transition temperature much closer at T g = 1818.5K (T g = 695.5 K) than T g = 1714 K (T g = 800 K).

Figure 11 .
Figure 11.NiTi2 atomic volume difference between liquid and glass above Tm = 1257 K.The initial liquid glass state at room temperature is characterized by T′g = 1818.5K and Tg = 695.5 K. Its enthalpy coefficient is expected to be equal 0.4467.The maximum calculated value is 0.4387 ± 0.01 at 1714 K.It begins to decrease and attain 0.36356 at T = 1931 K and zero at 2120 K in Figure 3.The influence of the glass phase with T′g = 1818.5still exists up to 1931 K.The glass phase disappears without firstorder transition at 2120 K due to the prolongation of the glass state.

Figure 11 .
Figure 11.NiTi 2 atomic volume difference between liquid and glass above T m = 1257 K.The initial liquid glass state at room temperature is characterized by T g = 1818.5K and T g = 695.5 K. Its enthalpy coefficient is expected to be equal 0.4467.The maximum calculated value is 0.4387 ± 0.01 at 1714 K.It begins to decrease and attain 0.36356 at T = 1931 K and zero at 2120 K in Figure 3.The influence of the glass phase with T g = 1818.5still exists up to 1931 K.The glass phase disappears without first-order transition at 2120 K due to the prolongation of the glass state.
at 10 13 K/s.The glass fraction first increases with temperature from ∆ε = 0.22335 to 0.4387, and attains 0.36356 at 1931 K.
at 10 13 K/s.The glass fraction first increases with temperature from Δε = 0.22335 to 0.4387, and attains 0.36356 at 1931 K.

Table 2 .
Phases of materials as function of connectivity and ordering of atomic constituents.